Research ArticleStability and Convergence Analysis of DirectAdaptive Inverse Control

Muhammad Shafiq1 Muhammad A Shafiq2 and Hassan A Yousef1

1ECE SQU Muscat Oman2ECE GIT Atlanta GA USA

Correspondence should be addressed to Muhammad Shafiq shafiqsaeedayahoocom

Received 14 June 2017 Accepted 10 October 2017 Published 14 November 2017

Academic Editor Danilo Comminiello

Copyright copy 2017 Muhammad Shafiq et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In adaptive inverse control (AIC) adaptive inverse of the plant is used as a feed-forward controllerMajority ofAIC schemes estimatecontroller parameters using the indirect method Direct adaptive inverse control (DAIC) alleviates the adhocism in adaptive loopIn this paper we discuss the stability and convergence of DAIC algorithm The computer simulation results are presented todemonstrate the performance of the DAIC Laboratory scale experimental results are included in the paper to study the efficiencyof DAIC for physical plants

1 Introduction

Adaptive inverse control (AIC) is a well-established adaptivetracking methodology [1ndash4] Robust tracking and computa-tionally less expensive characteristics of AIC have attractedthe interest of many researchers for several decades [1ndash15]AIC schemes are applicable to stable or stabilized plants[2] AIC has been applied successfully in several practicalapplications such as real-time blood pressure control shocktesting control of the kiln real-time control of temperatureof a heating process real-time speed control of a brush DCmotor nonlinear ship maneuvering echo cancelation andnoise cancelation [1 15ndash20] Recently AIC is used to controlthe position of piezoelectric inchworm actuator [21] andthe acceleration of six-degree-of-freedom electrohydraulicshaking table [22]

Discrete time plants for which one or more zeros lieoutside the unit circle are called non-minimum phase plantsSimilarly continuous time plants in which one or more zeroslie on the right hand side of the S-plane are known as non-minimum phase plants [23] Discretization of continuoustime plants most often gives non-minimum phase discreteplants [24 25] Numerous techniques have been developedfor the control of non-minimum phase plants AIC based on

linear and nonlinear filtering AIC of linear and nonlinearsystems using dynamic neural networks Normalized LeastMeans Square (NLMS) based adaptive controller nonlinearadaptive inverse control systems based on filtered-120598 LeastMean Square (LMS) Algorithm internal model controlstructure using adaptive inverse control strategy k-step delaycontroller for robust tracking and causal inversion solutionare few of them [5ndash14 17 26 27] Majority of controlschemes for non-minimum phase plants are indirect [5ndash8 11] In some AIC schemes inverse is designed based onidentified plant [9 10] Most of the AIC schemes estimateright inverse 119876119877(119902minus1) and then it is used as left inverse119876119871(119902minus1) by considering left and right inverse are equal Butthey are not equal because practical plants most often havesome kind of nonlinearities Therefore directly estimatedleft inverse in principle accomplishes better tracking thanindirect algorithms Since the plant and its inverse are incascade they collectively form a unity gain transfer functionSimilarly the left inverse precedes plant Right and left inverseare shown in Figures 1(a) and 1(b) respectively

A direct adaptive inverse technique based on NLMS forcontrol of discrete time linear plants to alleviate the adhocismin adaptive loop is proposed in [1] However the stabilityand convergence analysis is missing A neural network based

HindawiComplexityVolume 2017 Article ID 7834358 12 pageshttpsdoiorg10115520177834358

2 Complexity

P(qminus1) QR(qminus1)

(a) Right inverse

P(qminus1)QL(qminus1)

(b) Left inverse

Figure 1

DAIC was then proposed in [28] Reference [29] proposeda prefilter inversion system on the similar lines of DAICThis DAIC structure was used for the adaptive control ofelectrohydraulic servo systems The modified DIAC schemewas proposed for the prediction of the compression strengthof concrete using neural network based on kernel ridgeregression [30] Another extension of the DAIC is proposedin [31] Based on DAIC proposed in [1] a closed loopdirect adaptive inverse control scheme is introduced in [32]that improves tracking error convergence and disturbancerejection properties of DAIC These extensions and usesof DAIC motivate us to provide stability and convergenceanalysis of DAIC along with some applications to experi-mental systems In this paper parameter estimation for plantmodel and adaptive inverse controller stability analysis anderror convergence for DAIC is discussed thoroughly Furthersimulation results in presence of disturbance are given inthe paper Laboratory scale experiments are presented toelaborate the performance of DAIC on physical plants DAICcan be used for tracking of stable or stabilized minimumor non-minimum phase discrete time linear plants Littlemodification can also establish model reference adaptivetracking as well

The rest of the paper is organized as follows Section 2presents problem statement Section 3 discusses existingindirect adaptive inverse control (IAIC) schemes Design ofDAIC scheme is given in Section 4 Parameter estimationalgorithms and stability analysis for DAIC are given inSection 5 Simulation results are presented in Section 6Experimental results are described in Section 7 Conclusionsare drawn in Section 8

2 Problem Statement

Let us consider 119875(119902minus1) as a discrete time stable or stabilizedlinear plant which is given by

119875 (119902minus1) = 119902minus119889119861 (119902minus1)119860 (119902minus1)

119860 (119902minus1) = 1 + 1198861119902minus1 + 1198862119902minus2 + sdot sdot sdot + 119886119899119902minus119899119861 (119902minus1) = 1198870 + 1198871119902minus1 + 1198872119902minus2 + sdot sdot sdot + 119887119898119902minus119898

(1)

where 119902minus1 is a back shift operator defined as 119902minus1119910(119896) =119910(119896 minus 1) 119896 is a positive integer that represents discrete timeinstant 119889 is a positive integer that represents delay of theplant 119899 and 119898 are positive integers and 119899 ge 119898 119860(119902minus1) and

119861(119902minus1) are relatively coprime polynomials We also assumethat the plant may be non-minimum phase that is inverseof plant is unstable Let 119903(119896) 119910119889(119896) and 119910(119896) be the referenceinput desired output and plant output respectively Furtherit is assumed that parameters of the plant are unknown orslowly time varying compared to the adaptation algorithmThe objective is to design a controller such that 119910(119896) tracks119910119889(119896) that is

lim119896rarrinfin

(119890ref (119896))2 = lim119896rarrinfin

(119910119889 (119896) minus 119910 (119896))2 997888rarr 0 (2)

where 119910119889(119896) = 119903(119896 minus 119871) 119871 is a positive integer that representsknown delay and 119890ref (119896) is error at instant 119896

3 Overview of Existing IAIC Schemes

Control scheme for linear Single Input Single Output (SISO)plants that uses IAIC proposed in [4] is shown in Figure 2

Right inverse 119876119877(119902minus1) is estimated using inverse modelidentification 119876119877(119902minus1) is then copied into feed-forward pathof plant that is 119876119877copy(119902minus1) 119871 is considered zero in Figure 2for controlling minimum phase plants [4] 119890119903(119896) is used toadapt theweights of adaptive filter where 119890119903(119896) = 119910119894(119896)minus119906(119896minus119871) 119910119894(119896) is output of 119876119877(119902minus1) When 119890119903(119896) rarr 0 then 119890ref (119896)will approach zero as well [2] In this case 119890ref (119896) is given by

119890ref (119896) = [119902minus119871 minus 119876119877copy (119902minus1) 119875 (119902minus1)] 119903 (119896) (3)

Due to commutability of linear filters

119876119877copy (119902minus1) 119875 (119902minus1) cong 119875 (119902minus1)119876119877 (119902minus1) (4)

AIC based on linear and nonlinear adaptive filteringdiscussed in [5] is shown in Figure 3 119872(119902minus1) is filter withdesired response For structure in Figure 3 (3) can berewritten as

119890ref (119896) = [119872(119902minus1) minus 119876119877copy (119902minus1) 119875 (119902minus1)] 119903 (119896) (5)

Indirect adaptive tracking schemes discussed above provegood for stable or stabilized plant IAIC schemes estimate119876119877(119902minus1) and then it is copied in feed-forward path as leftinverse 119876119871(119902minus1) There are situations in which 119876119877copy(119902minus1)may not be equal to 119876119871(119902minus1) because of nonlinearities in theplant So the use of 119876119877copy(119902minus1) instead of 119876119871(119902minus1) in suchsituations will not accomplish tracking [2]

Complexity 3

r(k)

qminusL

qminusL

u(k)

yd(k) +

minus

+

minus

y(k)

yi(k)

er(k)

u(k minus L)

sum

sumeＬ (k)

P(qminus1)QR＝ＩＪＳ(qminus1) QR(q

minus1)

Figure 2 Indirect control scheme for non-minimum phase plants [4]

r(k) u(k)

yd(k) +

minus

+

minus

y(k)

yi(k)

er1(k)sum

sumeＬ (k)

M(qminus1)

M(qminus1)

P(qminus1)QR＝ＩＪＳ(qminus1) QR(q

minus1)

Figure 3 Indirect AIC structure for linear SISO plants [5]

4 Design of DAIC

DAIC structure for controlling stable or stabilized mini-mumnon-minimum phase linear SISO plants [1] is shownin Figure 3 In this structure approximate inverse system119876119871(119902minus1) is directly estimated Control input to plant 119906(119896) issynthesized by

119906 (119896) = 119876119871 (119902minus1) 119903 (119896) (6)

The online estimation of 119876119871(119902minus1) is accomplished usingthree steps given below

(1) Adaptive plant model (119902minus1) is obtained using NLMSadaptive filter

(2) The mismatch error 119890ref (119896) between desired response119910119889(119896) and plant output 119910(119896) is propagated throughplant model (119902minus1)

(3) Output obtained from the second step 119890119891(119896) is used toadapt the weights of controller which is also anNLMSadaptive filter

In this algorithm the parameters of the controller119876119871(119902minus1) are estimated directly This means 119876119877copy(119902minus1) isnot used Plant is preceded by the controller There is nodirect feedback from the plant output Control scheme is notstrictly feed-forward because controller weights are updatedsuch that it contains information about the plant output andreference input As shown in Figure 4 we identify the plantas a moving average system (ie the plant is approximatedby an adaptive Finite Impulse Response (FIR) filter) Thenfor estimation of the adaptive inverse controller parameters119890119891(119896) is used as an error signal where

119890119891 (119896) = (119902minus1) 119890ref (119896) (7)

DAIC ismuch simpler as compared tomethods presentedin [9 10] We use NLMS algorithm to estimate the plant andthe adaptive inverse controller parameters whereas Jacobianmatrices of network are calculated using dual subroutine andBack PropagationThroughModel (BPTM) algorithm is usedto adapt plantmodel and constrained controller in [9 10]The

4 Complexity

r(k) u(k)

qminusLyd(k)

+

minus

+

minus

eＬ (k)

eＧＩ＞(k)

y(k)

y(k)

ef(k)

sum

sum

P(qminus1)

P(qminus1)

P(qminus1)

QL(qminus1)

Figure 4 Direct adaptive inverse control scheme

details of parameter estimation and stability analysis of theproposed DAIC are given in Section 5

Mean square error (MSE) between desired output andplant output for non-minimum phase plants can be madesmall by incorporating the delay 119902minus119871 Since 119876119871(119902minus1) is usedas feed-forward controller for 119875(119902minus1) this gives

119876119871 (119902minus1) 119875 (119902minus1) cong 119902minus119871 (8)

The parameter 119871 is generally kept small for minimumphase and large for non-minimum phase plants In simula-tions we have observed that choosing 119871 cong (]+119889+119898)2 givesgood tracking in non-minimumphase systems where ] is theorder of 119876119871(119902minus1)

Using 119876119877(119902minus1) for 119876119871(119902minus1) in IAIC introduces at leastone step delay in the controller parameters DAIC dwindlesthe adhocism of adaptive loop by directly incorporatingan adaptive controller 119876119871(119902minus1) in feed-forward loop Sinceplant model is identified first DAIC is less sensitive to plantuncertainties and variations Further mild nonlinearities atthe output of plant may be learnt by 119876119877(119902minus1) in IAIC causingdeviation from desired signal Using 119876119877copy(119902minus1) as leftinverse may not then accomplish tracking as commutabilityis lost DAIC rectifies this deficiency In DAIC

lim119896rarrinfin

(119890ref (119896))2 997888rarr 0 (9)

provided

lim119896rarrinfin

(119890mod (119896))2 997888rarr 0 (10)

where 119890mod(119896) = 119910(119896) minus 119910(119896) and 119910(119896) is output of estimatedplant (119902minus1) given by

119910 (119896) = 120579 (119896) 120595119879 (119896) (11)

where 120579(119896) is a parameter vector for (119902minus1) defined as 120579(119896) =[1205730 1205731 120573119872] and 120595(119896) is regression vector defined as120595(119896) = [119906(119896) 119906(119896 minus 1) 119906(119896 minus119872)]

5 Development of Estimation Algorithm forSISO Systems

In this section estimation algorithms for linear SISO systemsare developed Parameter estimation is developed based onNLMS algorithm

51 Parameter Updating for Plant Model The parametersof the plant model (119902minus1) are obtained by minimizing theperformance index 120601 defined by

120601 = 121198902mod (119896) (12)

Parameters of the plant model should be updated in thedirection of negative gradient as

120579 (119896 + 1) = 120579 (119896) minus 1205831 120597120601120597120579 (13)

where 1205831 is the learning rateNLMS is self-normalized version of LMS Convergence

of NLMS is faster than LMS [33] Due to normalization ofinput NLMS is less sensitive to colored input signal and hasmore stable behavior than LMS [34] The parameter updateequation for the plant model based on NLMS is given below

120579 (119896 + 1)

=

120579 (119896) if 120595 (119896) 120595119879 (119896) = 0120579 (119896) + 1205831119890mod (119896) 120595 (119896)

120595 (119896) 120595119879 (119896) if 120595 (119896) 120595119879 (119896) = 0(14)

Complexity 5

Stability Analysis and Error Convergence For 120595(119896)120595119879(119896) = 0(14) gives

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) (15)

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) [119910 (119896) minus 120579 (119896) 120595119879 (119896)]

120595 (119896) 120595119879 (119896) (16)

120579 (119896) = 119902minus112058311 minus (1 minus 1205831) 119902minus1

120595 (119896) 119910 (119896)120595 (119896) 120595119879 (119896) (17)

[1 minus (1 minus 1205831)119902minus1] will be a Schur polynomial if 0 lt1205831 lt 1 This means that (17) is stable that is 120579(119896)will be bounded if 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) is boundedThe term 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) will remain bounded if120595(119896)120595119879(119896) = 0 Now convergence of error will be proved Let

119890mod (119896) = 119910 (119896) minus 119910 (119896) (18)

119890mod (119896) = 119910 (119896) minus 120579 (119896) 120595119879 (119896) (19)

119890mod (119896 + 1) = 119910 (119896) minus 120579 (119896 + 1) 120595119879 (119896) (20)

Subtracting (19) from (20)

119890mod (119896 + 1) minus 119890mod (119896) = minusΔ120579 (119896) 120595119879 (119896) (21)

where Δ120579(119896) = 120579(119896 + 1) minus 120579(119896) Substituting value of Δ120579(119896)from (15) in (20) we get

119890mod (119896 + 1) minus 119890mod (119896) = minus1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) 120595119879 (119896) (22)

119890mod (119896 + 1) = (1 minus 1205831) 119890mod (119896) (23)

= (1 minus 1205831)2 119890mod (119896 minus 1) (24)

= (1 minus 1205831)3 119890mod (119896 minus 2) (25)

= (26)

119890mod (119896 + 1) = (1 minus 1205831)119896+1 119890mod (0) (27)

Taking limits on both sides of (27)

lim119896rarrinfin

119890mod (119896 + 1) = lim119896rarrinfin

(1 minus 1205831)119896+1 119890mod (0) = 0 (28)

Convergence of 119890mod(119896 + 1) will be satisfied if 0 lt 1205831 lt 1

52 Parameter Updating for Controller Parameters Param-eters of controller are obtained by minimizing the perfor-mance index 119869 given by

119869 = 121198902ref (119896)

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) 119890ref (119896) = 119903 (119896 minus 119871) minus 119910 (119896)

(29)

Weights of the adaptive controller should be updated in thedirection of negative gradient as

120596 (119896 + 1) = 120596 (119896) minus 1205832 120597119869120597120596 (30)

where 1205832 is learning rate and 120596(119896) is the parameter vector forcontroller 119876119871(119902minus1) defined as 120596(119896) = [1205720 1205721 120572119873] Nowfinding partial derivative

120597119869120597120596 = 120597

120597120596 (121198902ref (119896)) (31)

120597119869120597120596 = minus119890ref (119896) 120597

120597120596 (119875 (119902minus1) 119906 (119896)) (32)

120597119869120597120596 = minus119890ref (119896) 119875 (119902minus1) 120593 (119896) (33)

where 120593(119896) is regression vector defined as 120593(119896) = [119903(119896) 119903(119896 minus1) 119903(119896 minus 119873)] Now final parameter update equation forcontroller can be obtained by substituting (33) in (30)

120596 (119896 + 1) = 120596 (119896) + 1205832119875 (119902minus1) 119890ref (119896) 120593 (119896) (34)

We can replace 119875(119902minus1) in (34) by its adaptive model (119902minus1)because it is shown in Section 51 that as 119896 rarr infin then 119890mod(119896+1) rarr 0 Therefore (34) becomes

120596 (119896 + 1) = 120596 (119896) + 1205832119890119891 (119896) 120593 (119896) (35)

where

119890119891 (119896) = (119902minus1) 119890ref (119896) (36)

Since NLMS is used weight updating for controller isgiven by

120596 (119896 + 1)

=

120596 (119896) if 120593 (119896) 120593119879 (119896) = 0120596 (119896) + 1205832119890119891 (119896) 120593 (119896)

120593 (119896) 120593119879 (119896) if 120593 (119896) 120593119879 (119896) = 0(37)

Stability Analysis and Error Convergence for Controller Param-eters Now sufficient conditions on 1205832 are obtained to ensurestability of DAIC 119890119891(119896) can be written as

119890119891 (119896) = 120579 (119896) 119864119879ref (119896) (38)

119890119891 (119896)= 120579 (119896) [119890ref (119896) 119890ref (119896 minus 1) 119890ref (119896 minus119872)]119879

(39)

119890119891 (119896)= minus1205730119910 (119896) minus 1205731119910 (119896 minus 1) minus sdot sdot sdot minus 120573119872119910 (119896 minus119872)

+ 1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1) + sdot sdot sdot+ 120573119872119903 (119896 minus 119871 minus119872)

(40)

6 Complexity

Substituting (40) in (37)

120596 (119896 + 1) = 120596 (119896) minus 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119910 (119896)

+ 1205731119910 (119896 minus 1) + sdot sdot sdot + 120573119872119910 (119896 minus119872)] + 120588 (119896) (41)

where

120588 (119896) = 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1)

+ sdot sdot sdot + 120573119872119903 (119896 minus 119871 minus119872)] (42)

Now (41) can be written as

120596 (119896 + 1) = 120596 (119896) + 120588 (119896) minus 1205832sdot 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119875 (119902minus1) 120596 (119896) 120593119879 (119896)

+ 1205731119875 (119902minus1) 120596 (119896 minus 1) 120593119879 (119896) + sdot sdot sdot+ 120573119872119875 (119902minus1) 120596 (119896 minus119872)120593119879 (119896)]

(43)

Grouping and rearranging the terms of (43) we get

120596 (119896 + 1)= 120588 (119896) + (1 minus 12058321205730119875 (119902minus1)) 120596 (119896)

minus 1205832119875 (119902minus1) [1205731120596 (119896 minus 1) + sdot sdot sdot + 120573119872120596 (119896 minus119872)] (44)

Using (44) the controller parameter vector can be written as

120596 (119896) = 120588 (119896 minus 1)119863 (119902minus1) (45)

where

119863(119902minus1) = 119878 (119902minus1) + 1198781 (119902minus1) 119878 (119902minus1) = 1 minus (1 minus 12058321205730119875 (119902minus1)) 119902minus11198781 (119902minus1) = 1205832119875 (119902minus1) [1205731119902minus2 + sdot sdot sdot + 120573119872119902minus119872minus1]

(46)

119878(119902minus1) will be Schur polynomial if100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 lt 1

minus1 lt 1 minus 12058321205730119875 (119902minus1) lt 10 lt 12058321205730119875 (119902minus1) lt 2

(47)

To avoid overcorrection range is given by

0 lt 12058321205730119875 (119902minus1) lt 1 (48)

Further 119863(119902minus1) is Schur and will remain stable if it isshown that

1205832 10038161003816100381610038161003816119875 (119902minus1)10038161003816100381610038161003816119872

sum119894=1

10038161003816100381610038161205731198941003816100381610038161003816 lt 100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 (49)

Using triangle difference inequality [35] and simplifyingwe get

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 sum119872119894=0 10038161003816100381610038161205731198941003816100381610038161003816

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 |120579 (119896)|

(50)

Controller output will remain bounded if 1205832 is chosensuch that 0 lt 1205832 lt 1|119875(119902minus1)||120579(119896)| |119875(119902minus1)| and |120579(119896)| can befound online and incorporated as learning rate butwe choosesmall learning rate for controller in order to avoid instabilityTo be more conservative if 1205832 gt 1 then we use 0 lt 1205832 lt 1Now convergence of error will be proved Let

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) (51)

119890ref (119896) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896) 120593119879 (119896) (52)

119890ref (119896 + 1) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896 + 1) 120593119879 (119896) (53)

Subtracting (52) from (53) we obtain

119890ref (119896 + 1) minus 119890ref (119896)= minus119875 (119902minus1) (120596 (119896 + 1) minus 120596 (119896)) 120593119879 (119896)

(54)

= minus119875 (119902minus1) Δ120596 (119896) 120593119879 (119896) (55)

Using (35) and (55) the following relationship is obtained

119890ref (119896 + 1)= (1 minus 1205832119875 (119902minus1) (119902minus1)) 119890ref (119896) 120593 (119896) 120593119879 (119896)

(56)

Equation (56) will be asymptotically stable that islim119896rarrinfin119890ref (119896 + 1) = 0 if 1205832 is chosen such that 0 lt 1205832 lt1|119875(119902minus1)||120579(119896)| NLMS adaptive filters are inherently stableand are used for plant estimating the parameters of the plantand controller Output of controller remains bounded as longas 1205832 is kept small Since a bounded input is applied to theplant and stable adaptive filters are used in conjunction withthe stabilized plant the controller output remains bounded

6 Simulation Results

Computer simulations of DAIC and IAIC schemes arepresented to show effectiveness of DAIC Two linear non-minimumphase systems are chosen onewithout disturbanceand other with disturbance

61 Example 1 A disturbance free discrete time non-minimum phase linear plant is chosen having

119910 (119896) = 119902minus1 1 + 12119902minus11 + 05119902minus1 + 01119902minus2 119906 (119896) (57)

This is a stable non-minimum phase plant having zero atminus12000 and poles at minus02500 plusmn 01936i In this example we

Complexity 7

Desired output Indirect AIC DAIC

02 04 06 08 10Time

0

1

2

3

4

5

6A

mpl

itude

Figure 5 Tracking desired output first 1 sec

Desired output Indirect AIC DAIC

Am

plitu

de

minus05

05

15

minus1

1

0

5 10 15 20 25 300Time

Figure 6 Tracking desired output amplitude minus12sim15

choose 1205831 = 01 and 1205832 = 001 Similarly learning rate forIAIC is chosen as 001 Orders of 119876119877(119902minus1) and 119876119871(119902minus1) arechosen as 10 Sampling time is chosen as 0001 sec Simulationresults are depicted in Figures 5ndash12 Zoomed preview fordesired output tracking is shown in Figures 5 and 6 Plantoutput in DAIC has less overshoot and converges to desiredoutput quickly compared to IAIC Tracking error is shownin Figures 7 and 8 Tracking error has less amplitude andconverges to zero faster in DAIC compared to IAIC MSE forIAIC and DAIC is shown in Figures 9 and 10 MSE is less forDAIC compared to IAIC Control input is shown in Figure 11

Indirect AIC DAIC

minus5

minus4

minus3

minus2

minus1

0

1

Am

plitu

de

02 04 06 08 10Time

Figure 7 Tracking error first 1 sec

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

minus05

0

05

1

Am

plitu

de

Figure 8 Tracking error amplitude minus1sim1

Control input for DAIC converges quickly compared to IAICModel identification error 119890mod(119896) in DAIC converges to zerovery quickly and is shown in Figure 12

62 Example 2 A disturbance 119899(119896) is added to discrete timenon-minimum phase linear plant Now plant output can bewritten as

119910 (119896) = 119902minus1

sdot 1 minus 3119902minus1 + 35119902minus21 + 005119902minus2 + 005119902minus3 + 002119902minus4 (119906 (119896) + 119899 (119896))

(58)

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 3

r(k)

qminusL

qminusL

u(k)

yd(k) +

minus

+

minus

y(k)

yi(k)

er(k)

u(k minus L)

sum

sumeＬ (k)

P(qminus1)QR＝ＩＪＳ(qminus1) QR(q

minus1)

Figure 2 Indirect control scheme for non-minimum phase plants [4]

r(k) u(k)

yd(k) +

minus

+

minus

y(k)

yi(k)

er1(k)sum

sumeＬ (k)

M(qminus1)

M(qminus1)

P(qminus1)QR＝ＩＪＳ(qminus1) QR(q

minus1)

Figure 3 Indirect AIC structure for linear SISO plants [5]

4 Design of DAIC

DAIC structure for controlling stable or stabilized mini-mumnon-minimum phase linear SISO plants [1] is shownin Figure 3 In this structure approximate inverse system119876119871(119902minus1) is directly estimated Control input to plant 119906(119896) issynthesized by

119906 (119896) = 119876119871 (119902minus1) 119903 (119896) (6)

The online estimation of 119876119871(119902minus1) is accomplished usingthree steps given below

(1) Adaptive plant model (119902minus1) is obtained using NLMSadaptive filter

(2) The mismatch error 119890ref (119896) between desired response119910119889(119896) and plant output 119910(119896) is propagated throughplant model (119902minus1)

(3) Output obtained from the second step 119890119891(119896) is used toadapt the weights of controller which is also anNLMSadaptive filter

In this algorithm the parameters of the controller119876119871(119902minus1) are estimated directly This means 119876119877copy(119902minus1) isnot used Plant is preceded by the controller There is nodirect feedback from the plant output Control scheme is notstrictly feed-forward because controller weights are updatedsuch that it contains information about the plant output andreference input As shown in Figure 4 we identify the plantas a moving average system (ie the plant is approximatedby an adaptive Finite Impulse Response (FIR) filter) Thenfor estimation of the adaptive inverse controller parameters119890119891(119896) is used as an error signal where

119890119891 (119896) = (119902minus1) 119890ref (119896) (7)

DAIC ismuch simpler as compared tomethods presentedin [9 10] We use NLMS algorithm to estimate the plant andthe adaptive inverse controller parameters whereas Jacobianmatrices of network are calculated using dual subroutine andBack PropagationThroughModel (BPTM) algorithm is usedto adapt plantmodel and constrained controller in [9 10]The

4 Complexity

r(k) u(k)

qminusLyd(k)

+

minus

+

minus

eＬ (k)

eＧＩ＞(k)

y(k)

y(k)

ef(k)

sum

sum

P(qminus1)

P(qminus1)

P(qminus1)

QL(qminus1)

Figure 4 Direct adaptive inverse control scheme

details of parameter estimation and stability analysis of theproposed DAIC are given in Section 5

Mean square error (MSE) between desired output andplant output for non-minimum phase plants can be madesmall by incorporating the delay 119902minus119871 Since 119876119871(119902minus1) is usedas feed-forward controller for 119875(119902minus1) this gives

119876119871 (119902minus1) 119875 (119902minus1) cong 119902minus119871 (8)

The parameter 119871 is generally kept small for minimumphase and large for non-minimum phase plants In simula-tions we have observed that choosing 119871 cong (]+119889+119898)2 givesgood tracking in non-minimumphase systems where ] is theorder of 119876119871(119902minus1)

Using 119876119877(119902minus1) for 119876119871(119902minus1) in IAIC introduces at leastone step delay in the controller parameters DAIC dwindlesthe adhocism of adaptive loop by directly incorporatingan adaptive controller 119876119871(119902minus1) in feed-forward loop Sinceplant model is identified first DAIC is less sensitive to plantuncertainties and variations Further mild nonlinearities atthe output of plant may be learnt by 119876119877(119902minus1) in IAIC causingdeviation from desired signal Using 119876119877copy(119902minus1) as leftinverse may not then accomplish tracking as commutabilityis lost DAIC rectifies this deficiency In DAIC

lim119896rarrinfin

(119890ref (119896))2 997888rarr 0 (9)

provided

lim119896rarrinfin

(119890mod (119896))2 997888rarr 0 (10)

where 119890mod(119896) = 119910(119896) minus 119910(119896) and 119910(119896) is output of estimatedplant (119902minus1) given by

119910 (119896) = 120579 (119896) 120595119879 (119896) (11)

where 120579(119896) is a parameter vector for (119902minus1) defined as 120579(119896) =[1205730 1205731 120573119872] and 120595(119896) is regression vector defined as120595(119896) = [119906(119896) 119906(119896 minus 1) 119906(119896 minus119872)]

5 Development of Estimation Algorithm forSISO Systems

In this section estimation algorithms for linear SISO systemsare developed Parameter estimation is developed based onNLMS algorithm

51 Parameter Updating for Plant Model The parametersof the plant model (119902minus1) are obtained by minimizing theperformance index 120601 defined by

120601 = 121198902mod (119896) (12)

Parameters of the plant model should be updated in thedirection of negative gradient as

120579 (119896 + 1) = 120579 (119896) minus 1205831 120597120601120597120579 (13)

where 1205831 is the learning rateNLMS is self-normalized version of LMS Convergence

of NLMS is faster than LMS [33] Due to normalization ofinput NLMS is less sensitive to colored input signal and hasmore stable behavior than LMS [34] The parameter updateequation for the plant model based on NLMS is given below

120579 (119896 + 1)

=

120579 (119896) if 120595 (119896) 120595119879 (119896) = 0120579 (119896) + 1205831119890mod (119896) 120595 (119896)

120595 (119896) 120595119879 (119896) if 120595 (119896) 120595119879 (119896) = 0(14)

Complexity 5

Stability Analysis and Error Convergence For 120595(119896)120595119879(119896) = 0(14) gives

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) (15)

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) [119910 (119896) minus 120579 (119896) 120595119879 (119896)]

120595 (119896) 120595119879 (119896) (16)

120579 (119896) = 119902minus112058311 minus (1 minus 1205831) 119902minus1

120595 (119896) 119910 (119896)120595 (119896) 120595119879 (119896) (17)

[1 minus (1 minus 1205831)119902minus1] will be a Schur polynomial if 0 lt1205831 lt 1 This means that (17) is stable that is 120579(119896)will be bounded if 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) is boundedThe term 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) will remain bounded if120595(119896)120595119879(119896) = 0 Now convergence of error will be proved Let

119890mod (119896) = 119910 (119896) minus 119910 (119896) (18)

119890mod (119896) = 119910 (119896) minus 120579 (119896) 120595119879 (119896) (19)

119890mod (119896 + 1) = 119910 (119896) minus 120579 (119896 + 1) 120595119879 (119896) (20)

Subtracting (19) from (20)

119890mod (119896 + 1) minus 119890mod (119896) = minusΔ120579 (119896) 120595119879 (119896) (21)

where Δ120579(119896) = 120579(119896 + 1) minus 120579(119896) Substituting value of Δ120579(119896)from (15) in (20) we get

119890mod (119896 + 1) minus 119890mod (119896) = minus1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) 120595119879 (119896) (22)

119890mod (119896 + 1) = (1 minus 1205831) 119890mod (119896) (23)

= (1 minus 1205831)2 119890mod (119896 minus 1) (24)

= (1 minus 1205831)3 119890mod (119896 minus 2) (25)

= (26)

119890mod (119896 + 1) = (1 minus 1205831)119896+1 119890mod (0) (27)

Taking limits on both sides of (27)

lim119896rarrinfin

119890mod (119896 + 1) = lim119896rarrinfin

(1 minus 1205831)119896+1 119890mod (0) = 0 (28)

Convergence of 119890mod(119896 + 1) will be satisfied if 0 lt 1205831 lt 1

52 Parameter Updating for Controller Parameters Param-eters of controller are obtained by minimizing the perfor-mance index 119869 given by

119869 = 121198902ref (119896)

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) 119890ref (119896) = 119903 (119896 minus 119871) minus 119910 (119896)

(29)

Weights of the adaptive controller should be updated in thedirection of negative gradient as

120596 (119896 + 1) = 120596 (119896) minus 1205832 120597119869120597120596 (30)

where 1205832 is learning rate and 120596(119896) is the parameter vector forcontroller 119876119871(119902minus1) defined as 120596(119896) = [1205720 1205721 120572119873] Nowfinding partial derivative

120597119869120597120596 = 120597

120597120596 (121198902ref (119896)) (31)

120597119869120597120596 = minus119890ref (119896) 120597

120597120596 (119875 (119902minus1) 119906 (119896)) (32)

120597119869120597120596 = minus119890ref (119896) 119875 (119902minus1) 120593 (119896) (33)

where 120593(119896) is regression vector defined as 120593(119896) = [119903(119896) 119903(119896 minus1) 119903(119896 minus 119873)] Now final parameter update equation forcontroller can be obtained by substituting (33) in (30)

120596 (119896 + 1) = 120596 (119896) + 1205832119875 (119902minus1) 119890ref (119896) 120593 (119896) (34)

We can replace 119875(119902minus1) in (34) by its adaptive model (119902minus1)because it is shown in Section 51 that as 119896 rarr infin then 119890mod(119896+1) rarr 0 Therefore (34) becomes

120596 (119896 + 1) = 120596 (119896) + 1205832119890119891 (119896) 120593 (119896) (35)

where

119890119891 (119896) = (119902minus1) 119890ref (119896) (36)

Since NLMS is used weight updating for controller isgiven by

120596 (119896 + 1)

=

120596 (119896) if 120593 (119896) 120593119879 (119896) = 0120596 (119896) + 1205832119890119891 (119896) 120593 (119896)

120593 (119896) 120593119879 (119896) if 120593 (119896) 120593119879 (119896) = 0(37)

Stability Analysis and Error Convergence for Controller Param-eters Now sufficient conditions on 1205832 are obtained to ensurestability of DAIC 119890119891(119896) can be written as

119890119891 (119896) = 120579 (119896) 119864119879ref (119896) (38)

119890119891 (119896)= 120579 (119896) [119890ref (119896) 119890ref (119896 minus 1) 119890ref (119896 minus119872)]119879

(39)

119890119891 (119896)= minus1205730119910 (119896) minus 1205731119910 (119896 minus 1) minus sdot sdot sdot minus 120573119872119910 (119896 minus119872)

+ 1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1) + sdot sdot sdot+ 120573119872119903 (119896 minus 119871 minus119872)

(40)

6 Complexity

Substituting (40) in (37)

120596 (119896 + 1) = 120596 (119896) minus 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119910 (119896)

+ 1205731119910 (119896 minus 1) + sdot sdot sdot + 120573119872119910 (119896 minus119872)] + 120588 (119896) (41)

where

120588 (119896) = 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1)

+ sdot sdot sdot + 120573119872119903 (119896 minus 119871 minus119872)] (42)

Now (41) can be written as

120596 (119896 + 1) = 120596 (119896) + 120588 (119896) minus 1205832sdot 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119875 (119902minus1) 120596 (119896) 120593119879 (119896)

+ 1205731119875 (119902minus1) 120596 (119896 minus 1) 120593119879 (119896) + sdot sdot sdot+ 120573119872119875 (119902minus1) 120596 (119896 minus119872)120593119879 (119896)]

(43)

Grouping and rearranging the terms of (43) we get

120596 (119896 + 1)= 120588 (119896) + (1 minus 12058321205730119875 (119902minus1)) 120596 (119896)

minus 1205832119875 (119902minus1) [1205731120596 (119896 minus 1) + sdot sdot sdot + 120573119872120596 (119896 minus119872)] (44)

Using (44) the controller parameter vector can be written as

120596 (119896) = 120588 (119896 minus 1)119863 (119902minus1) (45)

where

119863(119902minus1) = 119878 (119902minus1) + 1198781 (119902minus1) 119878 (119902minus1) = 1 minus (1 minus 12058321205730119875 (119902minus1)) 119902minus11198781 (119902minus1) = 1205832119875 (119902minus1) [1205731119902minus2 + sdot sdot sdot + 120573119872119902minus119872minus1]

(46)

119878(119902minus1) will be Schur polynomial if100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 lt 1

minus1 lt 1 minus 12058321205730119875 (119902minus1) lt 10 lt 12058321205730119875 (119902minus1) lt 2

(47)

To avoid overcorrection range is given by

0 lt 12058321205730119875 (119902minus1) lt 1 (48)

Further 119863(119902minus1) is Schur and will remain stable if it isshown that

1205832 10038161003816100381610038161003816119875 (119902minus1)10038161003816100381610038161003816119872

sum119894=1

10038161003816100381610038161205731198941003816100381610038161003816 lt 100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 (49)

Using triangle difference inequality [35] and simplifyingwe get

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 sum119872119894=0 10038161003816100381610038161205731198941003816100381610038161003816

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 |120579 (119896)|

(50)

Controller output will remain bounded if 1205832 is chosensuch that 0 lt 1205832 lt 1|119875(119902minus1)||120579(119896)| |119875(119902minus1)| and |120579(119896)| can befound online and incorporated as learning rate butwe choosesmall learning rate for controller in order to avoid instabilityTo be more conservative if 1205832 gt 1 then we use 0 lt 1205832 lt 1Now convergence of error will be proved Let

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) (51)

119890ref (119896) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896) 120593119879 (119896) (52)

119890ref (119896 + 1) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896 + 1) 120593119879 (119896) (53)

Subtracting (52) from (53) we obtain

119890ref (119896 + 1) minus 119890ref (119896)= minus119875 (119902minus1) (120596 (119896 + 1) minus 120596 (119896)) 120593119879 (119896)

(54)

= minus119875 (119902minus1) Δ120596 (119896) 120593119879 (119896) (55)

Using (35) and (55) the following relationship is obtained

119890ref (119896 + 1)= (1 minus 1205832119875 (119902minus1) (119902minus1)) 119890ref (119896) 120593 (119896) 120593119879 (119896)

(56)

Equation (56) will be asymptotically stable that islim119896rarrinfin119890ref (119896 + 1) = 0 if 1205832 is chosen such that 0 lt 1205832 lt1|119875(119902minus1)||120579(119896)| NLMS adaptive filters are inherently stableand are used for plant estimating the parameters of the plantand controller Output of controller remains bounded as longas 1205832 is kept small Since a bounded input is applied to theplant and stable adaptive filters are used in conjunction withthe stabilized plant the controller output remains bounded

6 Simulation Results

Computer simulations of DAIC and IAIC schemes arepresented to show effectiveness of DAIC Two linear non-minimumphase systems are chosen onewithout disturbanceand other with disturbance

61 Example 1 A disturbance free discrete time non-minimum phase linear plant is chosen having

119910 (119896) = 119902minus1 1 + 12119902minus11 + 05119902minus1 + 01119902minus2 119906 (119896) (57)

This is a stable non-minimum phase plant having zero atminus12000 and poles at minus02500 plusmn 01936i In this example we

Complexity 7

Desired output Indirect AIC DAIC

02 04 06 08 10Time

0

1

2

3

4

5

6A

mpl

itude

Figure 5 Tracking desired output first 1 sec

Desired output Indirect AIC DAIC

Am

plitu

de

minus05

05

15

minus1

1

0

5 10 15 20 25 300Time

Figure 6 Tracking desired output amplitude minus12sim15

choose 1205831 = 01 and 1205832 = 001 Similarly learning rate forIAIC is chosen as 001 Orders of 119876119877(119902minus1) and 119876119871(119902minus1) arechosen as 10 Sampling time is chosen as 0001 sec Simulationresults are depicted in Figures 5ndash12 Zoomed preview fordesired output tracking is shown in Figures 5 and 6 Plantoutput in DAIC has less overshoot and converges to desiredoutput quickly compared to IAIC Tracking error is shownin Figures 7 and 8 Tracking error has less amplitude andconverges to zero faster in DAIC compared to IAIC MSE forIAIC and DAIC is shown in Figures 9 and 10 MSE is less forDAIC compared to IAIC Control input is shown in Figure 11

Indirect AIC DAIC

minus5

minus4

minus3

minus2

minus1

0

1

Am

plitu

de

02 04 06 08 10Time

Figure 7 Tracking error first 1 sec

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

minus05

0

05

1

Am

plitu

de

Figure 8 Tracking error amplitude minus1sim1

Control input for DAIC converges quickly compared to IAICModel identification error 119890mod(119896) in DAIC converges to zerovery quickly and is shown in Figure 12

62 Example 2 A disturbance 119899(119896) is added to discrete timenon-minimum phase linear plant Now plant output can bewritten as

119910 (119896) = 119902minus1

sdot 1 minus 3119902minus1 + 35119902minus21 + 005119902minus2 + 005119902minus3 + 002119902minus4 (119906 (119896) + 119899 (119896))

(58)

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Complexity

r(k) u(k)

qminusLyd(k)

+

minus

+

minus

eＬ (k)

eＧＩ＞(k)

y(k)

y(k)

ef(k)

sum

sum

P(qminus1)

P(qminus1)

P(qminus1)

QL(qminus1)

Figure 4 Direct adaptive inverse control scheme

details of parameter estimation and stability analysis of theproposed DAIC are given in Section 5

Mean square error (MSE) between desired output andplant output for non-minimum phase plants can be madesmall by incorporating the delay 119902minus119871 Since 119876119871(119902minus1) is usedas feed-forward controller for 119875(119902minus1) this gives

119876119871 (119902minus1) 119875 (119902minus1) cong 119902minus119871 (8)

The parameter 119871 is generally kept small for minimumphase and large for non-minimum phase plants In simula-tions we have observed that choosing 119871 cong (]+119889+119898)2 givesgood tracking in non-minimumphase systems where ] is theorder of 119876119871(119902minus1)

Using 119876119877(119902minus1) for 119876119871(119902minus1) in IAIC introduces at leastone step delay in the controller parameters DAIC dwindlesthe adhocism of adaptive loop by directly incorporatingan adaptive controller 119876119871(119902minus1) in feed-forward loop Sinceplant model is identified first DAIC is less sensitive to plantuncertainties and variations Further mild nonlinearities atthe output of plant may be learnt by 119876119877(119902minus1) in IAIC causingdeviation from desired signal Using 119876119877copy(119902minus1) as leftinverse may not then accomplish tracking as commutabilityis lost DAIC rectifies this deficiency In DAIC

lim119896rarrinfin

(119890ref (119896))2 997888rarr 0 (9)

provided

lim119896rarrinfin

(119890mod (119896))2 997888rarr 0 (10)

where 119890mod(119896) = 119910(119896) minus 119910(119896) and 119910(119896) is output of estimatedplant (119902minus1) given by

119910 (119896) = 120579 (119896) 120595119879 (119896) (11)

where 120579(119896) is a parameter vector for (119902minus1) defined as 120579(119896) =[1205730 1205731 120573119872] and 120595(119896) is regression vector defined as120595(119896) = [119906(119896) 119906(119896 minus 1) 119906(119896 minus119872)]

5 Development of Estimation Algorithm forSISO Systems

In this section estimation algorithms for linear SISO systemsare developed Parameter estimation is developed based onNLMS algorithm

51 Parameter Updating for Plant Model The parametersof the plant model (119902minus1) are obtained by minimizing theperformance index 120601 defined by

120601 = 121198902mod (119896) (12)

Parameters of the plant model should be updated in thedirection of negative gradient as

120579 (119896 + 1) = 120579 (119896) minus 1205831 120597120601120597120579 (13)

where 1205831 is the learning rateNLMS is self-normalized version of LMS Convergence

of NLMS is faster than LMS [33] Due to normalization ofinput NLMS is less sensitive to colored input signal and hasmore stable behavior than LMS [34] The parameter updateequation for the plant model based on NLMS is given below

120579 (119896 + 1)

=

120579 (119896) if 120595 (119896) 120595119879 (119896) = 0120579 (119896) + 1205831119890mod (119896) 120595 (119896)

120595 (119896) 120595119879 (119896) if 120595 (119896) 120595119879 (119896) = 0(14)

Complexity 5

Stability Analysis and Error Convergence For 120595(119896)120595119879(119896) = 0(14) gives

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) (15)

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) [119910 (119896) minus 120579 (119896) 120595119879 (119896)]

120595 (119896) 120595119879 (119896) (16)

120579 (119896) = 119902minus112058311 minus (1 minus 1205831) 119902minus1

120595 (119896) 119910 (119896)120595 (119896) 120595119879 (119896) (17)

[1 minus (1 minus 1205831)119902minus1] will be a Schur polynomial if 0 lt1205831 lt 1 This means that (17) is stable that is 120579(119896)will be bounded if 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) is boundedThe term 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) will remain bounded if120595(119896)120595119879(119896) = 0 Now convergence of error will be proved Let

119890mod (119896) = 119910 (119896) minus 119910 (119896) (18)

119890mod (119896) = 119910 (119896) minus 120579 (119896) 120595119879 (119896) (19)

119890mod (119896 + 1) = 119910 (119896) minus 120579 (119896 + 1) 120595119879 (119896) (20)

Subtracting (19) from (20)

119890mod (119896 + 1) minus 119890mod (119896) = minusΔ120579 (119896) 120595119879 (119896) (21)

where Δ120579(119896) = 120579(119896 + 1) minus 120579(119896) Substituting value of Δ120579(119896)from (15) in (20) we get

119890mod (119896 + 1) minus 119890mod (119896) = minus1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) 120595119879 (119896) (22)

119890mod (119896 + 1) = (1 minus 1205831) 119890mod (119896) (23)

= (1 minus 1205831)2 119890mod (119896 minus 1) (24)

= (1 minus 1205831)3 119890mod (119896 minus 2) (25)

= (26)

119890mod (119896 + 1) = (1 minus 1205831)119896+1 119890mod (0) (27)

Taking limits on both sides of (27)

lim119896rarrinfin

119890mod (119896 + 1) = lim119896rarrinfin

(1 minus 1205831)119896+1 119890mod (0) = 0 (28)

Convergence of 119890mod(119896 + 1) will be satisfied if 0 lt 1205831 lt 1

52 Parameter Updating for Controller Parameters Param-eters of controller are obtained by minimizing the perfor-mance index 119869 given by

119869 = 121198902ref (119896)

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) 119890ref (119896) = 119903 (119896 minus 119871) minus 119910 (119896)

(29)

Weights of the adaptive controller should be updated in thedirection of negative gradient as

120596 (119896 + 1) = 120596 (119896) minus 1205832 120597119869120597120596 (30)

where 1205832 is learning rate and 120596(119896) is the parameter vector forcontroller 119876119871(119902minus1) defined as 120596(119896) = [1205720 1205721 120572119873] Nowfinding partial derivative

120597119869120597120596 = 120597

120597120596 (121198902ref (119896)) (31)

120597119869120597120596 = minus119890ref (119896) 120597

120597120596 (119875 (119902minus1) 119906 (119896)) (32)

120597119869120597120596 = minus119890ref (119896) 119875 (119902minus1) 120593 (119896) (33)

where 120593(119896) is regression vector defined as 120593(119896) = [119903(119896) 119903(119896 minus1) 119903(119896 minus 119873)] Now final parameter update equation forcontroller can be obtained by substituting (33) in (30)

120596 (119896 + 1) = 120596 (119896) + 1205832119875 (119902minus1) 119890ref (119896) 120593 (119896) (34)

We can replace 119875(119902minus1) in (34) by its adaptive model (119902minus1)because it is shown in Section 51 that as 119896 rarr infin then 119890mod(119896+1) rarr 0 Therefore (34) becomes

120596 (119896 + 1) = 120596 (119896) + 1205832119890119891 (119896) 120593 (119896) (35)

where

119890119891 (119896) = (119902minus1) 119890ref (119896) (36)

Since NLMS is used weight updating for controller isgiven by

120596 (119896 + 1)

=

120596 (119896) if 120593 (119896) 120593119879 (119896) = 0120596 (119896) + 1205832119890119891 (119896) 120593 (119896)

120593 (119896) 120593119879 (119896) if 120593 (119896) 120593119879 (119896) = 0(37)

Stability Analysis and Error Convergence for Controller Param-eters Now sufficient conditions on 1205832 are obtained to ensurestability of DAIC 119890119891(119896) can be written as

119890119891 (119896) = 120579 (119896) 119864119879ref (119896) (38)

119890119891 (119896)= 120579 (119896) [119890ref (119896) 119890ref (119896 minus 1) 119890ref (119896 minus119872)]119879

(39)

119890119891 (119896)= minus1205730119910 (119896) minus 1205731119910 (119896 minus 1) minus sdot sdot sdot minus 120573119872119910 (119896 minus119872)

+ 1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1) + sdot sdot sdot+ 120573119872119903 (119896 minus 119871 minus119872)

(40)

6 Complexity

Substituting (40) in (37)

120596 (119896 + 1) = 120596 (119896) minus 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119910 (119896)

+ 1205731119910 (119896 minus 1) + sdot sdot sdot + 120573119872119910 (119896 minus119872)] + 120588 (119896) (41)

where

120588 (119896) = 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1)

+ sdot sdot sdot + 120573119872119903 (119896 minus 119871 minus119872)] (42)

Now (41) can be written as

120596 (119896 + 1) = 120596 (119896) + 120588 (119896) minus 1205832sdot 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119875 (119902minus1) 120596 (119896) 120593119879 (119896)

+ 1205731119875 (119902minus1) 120596 (119896 minus 1) 120593119879 (119896) + sdot sdot sdot+ 120573119872119875 (119902minus1) 120596 (119896 minus119872)120593119879 (119896)]

(43)

Grouping and rearranging the terms of (43) we get

120596 (119896 + 1)= 120588 (119896) + (1 minus 12058321205730119875 (119902minus1)) 120596 (119896)

minus 1205832119875 (119902minus1) [1205731120596 (119896 minus 1) + sdot sdot sdot + 120573119872120596 (119896 minus119872)] (44)

Using (44) the controller parameter vector can be written as

120596 (119896) = 120588 (119896 minus 1)119863 (119902minus1) (45)

where

119863(119902minus1) = 119878 (119902minus1) + 1198781 (119902minus1) 119878 (119902minus1) = 1 minus (1 minus 12058321205730119875 (119902minus1)) 119902minus11198781 (119902minus1) = 1205832119875 (119902minus1) [1205731119902minus2 + sdot sdot sdot + 120573119872119902minus119872minus1]

(46)

119878(119902minus1) will be Schur polynomial if100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 lt 1

minus1 lt 1 minus 12058321205730119875 (119902minus1) lt 10 lt 12058321205730119875 (119902minus1) lt 2

(47)

To avoid overcorrection range is given by

0 lt 12058321205730119875 (119902minus1) lt 1 (48)

Further 119863(119902minus1) is Schur and will remain stable if it isshown that

1205832 10038161003816100381610038161003816119875 (119902minus1)10038161003816100381610038161003816119872

sum119894=1

10038161003816100381610038161205731198941003816100381610038161003816 lt 100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 (49)

Using triangle difference inequality [35] and simplifyingwe get

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 sum119872119894=0 10038161003816100381610038161205731198941003816100381610038161003816

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 |120579 (119896)|

(50)

Controller output will remain bounded if 1205832 is chosensuch that 0 lt 1205832 lt 1|119875(119902minus1)||120579(119896)| |119875(119902minus1)| and |120579(119896)| can befound online and incorporated as learning rate butwe choosesmall learning rate for controller in order to avoid instabilityTo be more conservative if 1205832 gt 1 then we use 0 lt 1205832 lt 1Now convergence of error will be proved Let

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) (51)

119890ref (119896) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896) 120593119879 (119896) (52)

119890ref (119896 + 1) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896 + 1) 120593119879 (119896) (53)

Subtracting (52) from (53) we obtain

119890ref (119896 + 1) minus 119890ref (119896)= minus119875 (119902minus1) (120596 (119896 + 1) minus 120596 (119896)) 120593119879 (119896)

(54)

= minus119875 (119902minus1) Δ120596 (119896) 120593119879 (119896) (55)

Using (35) and (55) the following relationship is obtained

119890ref (119896 + 1)= (1 minus 1205832119875 (119902minus1) (119902minus1)) 119890ref (119896) 120593 (119896) 120593119879 (119896)

(56)

Equation (56) will be asymptotically stable that islim119896rarrinfin119890ref (119896 + 1) = 0 if 1205832 is chosen such that 0 lt 1205832 lt1|119875(119902minus1)||120579(119896)| NLMS adaptive filters are inherently stableand are used for plant estimating the parameters of the plantand controller Output of controller remains bounded as longas 1205832 is kept small Since a bounded input is applied to theplant and stable adaptive filters are used in conjunction withthe stabilized plant the controller output remains bounded

6 Simulation Results

Computer simulations of DAIC and IAIC schemes arepresented to show effectiveness of DAIC Two linear non-minimumphase systems are chosen onewithout disturbanceand other with disturbance

61 Example 1 A disturbance free discrete time non-minimum phase linear plant is chosen having

119910 (119896) = 119902minus1 1 + 12119902minus11 + 05119902minus1 + 01119902minus2 119906 (119896) (57)

This is a stable non-minimum phase plant having zero atminus12000 and poles at minus02500 plusmn 01936i In this example we

Complexity 7

Desired output Indirect AIC DAIC

02 04 06 08 10Time

0

1

2

3

4

5

6A

mpl

itude

Figure 5 Tracking desired output first 1 sec

Desired output Indirect AIC DAIC

Am

plitu

de

minus05

05

15

minus1

1

0

5 10 15 20 25 300Time

Figure 6 Tracking desired output amplitude minus12sim15

choose 1205831 = 01 and 1205832 = 001 Similarly learning rate forIAIC is chosen as 001 Orders of 119876119877(119902minus1) and 119876119871(119902minus1) arechosen as 10 Sampling time is chosen as 0001 sec Simulationresults are depicted in Figures 5ndash12 Zoomed preview fordesired output tracking is shown in Figures 5 and 6 Plantoutput in DAIC has less overshoot and converges to desiredoutput quickly compared to IAIC Tracking error is shownin Figures 7 and 8 Tracking error has less amplitude andconverges to zero faster in DAIC compared to IAIC MSE forIAIC and DAIC is shown in Figures 9 and 10 MSE is less forDAIC compared to IAIC Control input is shown in Figure 11

Indirect AIC DAIC

minus5

minus4

minus3

minus2

minus1

0

1

Am

plitu

de

02 04 06 08 10Time

Figure 7 Tracking error first 1 sec

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

minus05

0

05

1

Am

plitu

de

Figure 8 Tracking error amplitude minus1sim1

Control input for DAIC converges quickly compared to IAICModel identification error 119890mod(119896) in DAIC converges to zerovery quickly and is shown in Figure 12

62 Example 2 A disturbance 119899(119896) is added to discrete timenon-minimum phase linear plant Now plant output can bewritten as

119910 (119896) = 119902minus1

sdot 1 minus 3119902minus1 + 35119902minus21 + 005119902minus2 + 005119902minus3 + 002119902minus4 (119906 (119896) + 119899 (119896))

(58)

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 5

Stability Analysis and Error Convergence For 120595(119896)120595119879(119896) = 0(14) gives

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) (15)

120579 (119896 + 1) = 120579 (119896) + 1205831120595 (119896) [119910 (119896) minus 120579 (119896) 120595119879 (119896)]

120595 (119896) 120595119879 (119896) (16)

120579 (119896) = 119902minus112058311 minus (1 minus 1205831) 119902minus1

120595 (119896) 119910 (119896)120595 (119896) 120595119879 (119896) (17)

[1 minus (1 minus 1205831)119902minus1] will be a Schur polynomial if 0 lt1205831 lt 1 This means that (17) is stable that is 120579(119896)will be bounded if 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) is boundedThe term 1205831(120595(119896)119910(119896)120595(119896)120595119879(119896)) will remain bounded if120595(119896)120595119879(119896) = 0 Now convergence of error will be proved Let

119890mod (119896) = 119910 (119896) minus 119910 (119896) (18)

119890mod (119896) = 119910 (119896) minus 120579 (119896) 120595119879 (119896) (19)

119890mod (119896 + 1) = 119910 (119896) minus 120579 (119896 + 1) 120595119879 (119896) (20)

Subtracting (19) from (20)

119890mod (119896 + 1) minus 119890mod (119896) = minusΔ120579 (119896) 120595119879 (119896) (21)

where Δ120579(119896) = 120579(119896 + 1) minus 120579(119896) Substituting value of Δ120579(119896)from (15) in (20) we get

119890mod (119896 + 1) minus 119890mod (119896) = minus1205831120595 (119896) 119890mod (119896)120595 (119896) 120595119879 (119896) 120595119879 (119896) (22)

119890mod (119896 + 1) = (1 minus 1205831) 119890mod (119896) (23)

= (1 minus 1205831)2 119890mod (119896 minus 1) (24)

= (1 minus 1205831)3 119890mod (119896 minus 2) (25)

= (26)

119890mod (119896 + 1) = (1 minus 1205831)119896+1 119890mod (0) (27)

Taking limits on both sides of (27)

lim119896rarrinfin

119890mod (119896 + 1) = lim119896rarrinfin

(1 minus 1205831)119896+1 119890mod (0) = 0 (28)

Convergence of 119890mod(119896 + 1) will be satisfied if 0 lt 1205831 lt 1

52 Parameter Updating for Controller Parameters Param-eters of controller are obtained by minimizing the perfor-mance index 119869 given by

119869 = 121198902ref (119896)

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) 119890ref (119896) = 119903 (119896 minus 119871) minus 119910 (119896)

(29)

Weights of the adaptive controller should be updated in thedirection of negative gradient as

120596 (119896 + 1) = 120596 (119896) minus 1205832 120597119869120597120596 (30)

where 1205832 is learning rate and 120596(119896) is the parameter vector forcontroller 119876119871(119902minus1) defined as 120596(119896) = [1205720 1205721 120572119873] Nowfinding partial derivative

120597119869120597120596 = 120597

120597120596 (121198902ref (119896)) (31)

120597119869120597120596 = minus119890ref (119896) 120597

120597120596 (119875 (119902minus1) 119906 (119896)) (32)

120597119869120597120596 = minus119890ref (119896) 119875 (119902minus1) 120593 (119896) (33)

where 120593(119896) is regression vector defined as 120593(119896) = [119903(119896) 119903(119896 minus1) 119903(119896 minus 119873)] Now final parameter update equation forcontroller can be obtained by substituting (33) in (30)

120596 (119896 + 1) = 120596 (119896) + 1205832119875 (119902minus1) 119890ref (119896) 120593 (119896) (34)

We can replace 119875(119902minus1) in (34) by its adaptive model (119902minus1)because it is shown in Section 51 that as 119896 rarr infin then 119890mod(119896+1) rarr 0 Therefore (34) becomes

120596 (119896 + 1) = 120596 (119896) + 1205832119890119891 (119896) 120593 (119896) (35)

where

119890119891 (119896) = (119902minus1) 119890ref (119896) (36)

Since NLMS is used weight updating for controller isgiven by

120596 (119896 + 1)

=

120596 (119896) if 120593 (119896) 120593119879 (119896) = 0120596 (119896) + 1205832119890119891 (119896) 120593 (119896)

120593 (119896) 120593119879 (119896) if 120593 (119896) 120593119879 (119896) = 0(37)

Stability Analysis and Error Convergence for Controller Param-eters Now sufficient conditions on 1205832 are obtained to ensurestability of DAIC 119890119891(119896) can be written as

119890119891 (119896) = 120579 (119896) 119864119879ref (119896) (38)

119890119891 (119896)= 120579 (119896) [119890ref (119896) 119890ref (119896 minus 1) 119890ref (119896 minus119872)]119879

(39)

119890119891 (119896)= minus1205730119910 (119896) minus 1205731119910 (119896 minus 1) minus sdot sdot sdot minus 120573119872119910 (119896 minus119872)

+ 1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1) + sdot sdot sdot+ 120573119872119903 (119896 minus 119871 minus119872)

(40)

6 Complexity

Substituting (40) in (37)

120596 (119896 + 1) = 120596 (119896) minus 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119910 (119896)

+ 1205731119910 (119896 minus 1) + sdot sdot sdot + 120573119872119910 (119896 minus119872)] + 120588 (119896) (41)

where

120588 (119896) = 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1)

+ sdot sdot sdot + 120573119872119903 (119896 minus 119871 minus119872)] (42)

Now (41) can be written as

120596 (119896 + 1) = 120596 (119896) + 120588 (119896) minus 1205832sdot 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119875 (119902minus1) 120596 (119896) 120593119879 (119896)

+ 1205731119875 (119902minus1) 120596 (119896 minus 1) 120593119879 (119896) + sdot sdot sdot+ 120573119872119875 (119902minus1) 120596 (119896 minus119872)120593119879 (119896)]

(43)

Grouping and rearranging the terms of (43) we get

120596 (119896 + 1)= 120588 (119896) + (1 minus 12058321205730119875 (119902minus1)) 120596 (119896)

minus 1205832119875 (119902minus1) [1205731120596 (119896 minus 1) + sdot sdot sdot + 120573119872120596 (119896 minus119872)] (44)

Using (44) the controller parameter vector can be written as

120596 (119896) = 120588 (119896 minus 1)119863 (119902minus1) (45)

where

119863(119902minus1) = 119878 (119902minus1) + 1198781 (119902minus1) 119878 (119902minus1) = 1 minus (1 minus 12058321205730119875 (119902minus1)) 119902minus11198781 (119902minus1) = 1205832119875 (119902minus1) [1205731119902minus2 + sdot sdot sdot + 120573119872119902minus119872minus1]

(46)

119878(119902minus1) will be Schur polynomial if100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 lt 1

minus1 lt 1 minus 12058321205730119875 (119902minus1) lt 10 lt 12058321205730119875 (119902minus1) lt 2

(47)

To avoid overcorrection range is given by

0 lt 12058321205730119875 (119902minus1) lt 1 (48)

Further 119863(119902minus1) is Schur and will remain stable if it isshown that

1205832 10038161003816100381610038161003816119875 (119902minus1)10038161003816100381610038161003816119872

sum119894=1

10038161003816100381610038161205731198941003816100381610038161003816 lt 100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 (49)

Using triangle difference inequality [35] and simplifyingwe get

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 sum119872119894=0 10038161003816100381610038161205731198941003816100381610038161003816

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 |120579 (119896)|

(50)

Controller output will remain bounded if 1205832 is chosensuch that 0 lt 1205832 lt 1|119875(119902minus1)||120579(119896)| |119875(119902minus1)| and |120579(119896)| can befound online and incorporated as learning rate butwe choosesmall learning rate for controller in order to avoid instabilityTo be more conservative if 1205832 gt 1 then we use 0 lt 1205832 lt 1Now convergence of error will be proved Let

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) (51)

119890ref (119896) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896) 120593119879 (119896) (52)

119890ref (119896 + 1) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896 + 1) 120593119879 (119896) (53)

Subtracting (52) from (53) we obtain

119890ref (119896 + 1) minus 119890ref (119896)= minus119875 (119902minus1) (120596 (119896 + 1) minus 120596 (119896)) 120593119879 (119896)

(54)

= minus119875 (119902minus1) Δ120596 (119896) 120593119879 (119896) (55)

Using (35) and (55) the following relationship is obtained

119890ref (119896 + 1)= (1 minus 1205832119875 (119902minus1) (119902minus1)) 119890ref (119896) 120593 (119896) 120593119879 (119896)

(56)

Equation (56) will be asymptotically stable that islim119896rarrinfin119890ref (119896 + 1) = 0 if 1205832 is chosen such that 0 lt 1205832 lt1|119875(119902minus1)||120579(119896)| NLMS adaptive filters are inherently stableand are used for plant estimating the parameters of the plantand controller Output of controller remains bounded as longas 1205832 is kept small Since a bounded input is applied to theplant and stable adaptive filters are used in conjunction withthe stabilized plant the controller output remains bounded

6 Simulation Results

Computer simulations of DAIC and IAIC schemes arepresented to show effectiveness of DAIC Two linear non-minimumphase systems are chosen onewithout disturbanceand other with disturbance

61 Example 1 A disturbance free discrete time non-minimum phase linear plant is chosen having

119910 (119896) = 119902minus1 1 + 12119902minus11 + 05119902minus1 + 01119902minus2 119906 (119896) (57)

This is a stable non-minimum phase plant having zero atminus12000 and poles at minus02500 plusmn 01936i In this example we

Complexity 7

Desired output Indirect AIC DAIC

02 04 06 08 10Time

0

1

2

3

4

5

6A

mpl

itude

Figure 5 Tracking desired output first 1 sec

Desired output Indirect AIC DAIC

Am

plitu

de

minus05

05

15

minus1

1

0

5 10 15 20 25 300Time

Figure 6 Tracking desired output amplitude minus12sim15

choose 1205831 = 01 and 1205832 = 001 Similarly learning rate forIAIC is chosen as 001 Orders of 119876119877(119902minus1) and 119876119871(119902minus1) arechosen as 10 Sampling time is chosen as 0001 sec Simulationresults are depicted in Figures 5ndash12 Zoomed preview fordesired output tracking is shown in Figures 5 and 6 Plantoutput in DAIC has less overshoot and converges to desiredoutput quickly compared to IAIC Tracking error is shownin Figures 7 and 8 Tracking error has less amplitude andconverges to zero faster in DAIC compared to IAIC MSE forIAIC and DAIC is shown in Figures 9 and 10 MSE is less forDAIC compared to IAIC Control input is shown in Figure 11

Indirect AIC DAIC

minus5

minus4

minus3

minus2

minus1

0

1

Am

plitu

de

02 04 06 08 10Time

Figure 7 Tracking error first 1 sec

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

minus05

0

05

1

Am

plitu

de

Figure 8 Tracking error amplitude minus1sim1

Control input for DAIC converges quickly compared to IAICModel identification error 119890mod(119896) in DAIC converges to zerovery quickly and is shown in Figure 12

62 Example 2 A disturbance 119899(119896) is added to discrete timenon-minimum phase linear plant Now plant output can bewritten as

119910 (119896) = 119902minus1

sdot 1 minus 3119902minus1 + 35119902minus21 + 005119902minus2 + 005119902minus3 + 002119902minus4 (119906 (119896) + 119899 (119896))

(58)

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Complexity

Substituting (40) in (37)

120596 (119896 + 1) = 120596 (119896) minus 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119910 (119896)

+ 1205731119910 (119896 minus 1) + sdot sdot sdot + 120573119872119910 (119896 minus119872)] + 120588 (119896) (41)

where

120588 (119896) = 1205832 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119903 (119896 minus 119871) + 1205731119903 (119896 minus 119871 minus 1)

+ sdot sdot sdot + 120573119872119903 (119896 minus 119871 minus119872)] (42)

Now (41) can be written as

120596 (119896 + 1) = 120596 (119896) + 120588 (119896) minus 1205832sdot 120593 (119896)120593 (119896) 120593119879 (119896) [1205730119875 (119902minus1) 120596 (119896) 120593119879 (119896)

+ 1205731119875 (119902minus1) 120596 (119896 minus 1) 120593119879 (119896) + sdot sdot sdot+ 120573119872119875 (119902minus1) 120596 (119896 minus119872)120593119879 (119896)]

(43)

Grouping and rearranging the terms of (43) we get

120596 (119896 + 1)= 120588 (119896) + (1 minus 12058321205730119875 (119902minus1)) 120596 (119896)

minus 1205832119875 (119902minus1) [1205731120596 (119896 minus 1) + sdot sdot sdot + 120573119872120596 (119896 minus119872)] (44)

Using (44) the controller parameter vector can be written as

120596 (119896) = 120588 (119896 minus 1)119863 (119902minus1) (45)

where

119863(119902minus1) = 119878 (119902minus1) + 1198781 (119902minus1) 119878 (119902minus1) = 1 minus (1 minus 12058321205730119875 (119902minus1)) 119902minus11198781 (119902minus1) = 1205832119875 (119902minus1) [1205731119902minus2 + sdot sdot sdot + 120573119872119902minus119872minus1]

(46)

119878(119902minus1) will be Schur polynomial if100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 lt 1

minus1 lt 1 minus 12058321205730119875 (119902minus1) lt 10 lt 12058321205730119875 (119902minus1) lt 2

(47)

To avoid overcorrection range is given by

0 lt 12058321205730119875 (119902minus1) lt 1 (48)

Further 119863(119902minus1) is Schur and will remain stable if it isshown that

1205832 10038161003816100381610038161003816119875 (119902minus1)10038161003816100381610038161003816119872

sum119894=1

10038161003816100381610038161205731198941003816100381610038161003816 lt 100381610038161003816100381610038161 minus 12058321205730119875 (119902minus1)10038161003816100381610038161003816 (49)

Using triangle difference inequality [35] and simplifyingwe get

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 sum119872119894=0 10038161003816100381610038161205731198941003816100381610038161003816

1205832 lt 11003816100381610038161003816119875 (119902minus1)1003816100381610038161003816 |120579 (119896)|

(50)

Controller output will remain bounded if 1205832 is chosensuch that 0 lt 1205832 lt 1|119875(119902minus1)||120579(119896)| |119875(119902minus1)| and |120579(119896)| can befound online and incorporated as learning rate butwe choosesmall learning rate for controller in order to avoid instabilityTo be more conservative if 1205832 gt 1 then we use 0 lt 1205832 lt 1Now convergence of error will be proved Let

119890ref (119896) = 119910119889 (119896) minus 119910 (119896) (51)

119890ref (119896) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896) 120593119879 (119896) (52)

119890ref (119896 + 1) = 119903 (119896 minus 119871) minus 119875 (119902minus1) 120596 (119896 + 1) 120593119879 (119896) (53)

Subtracting (52) from (53) we obtain

119890ref (119896 + 1) minus 119890ref (119896)= minus119875 (119902minus1) (120596 (119896 + 1) minus 120596 (119896)) 120593119879 (119896)

(54)

= minus119875 (119902minus1) Δ120596 (119896) 120593119879 (119896) (55)

Using (35) and (55) the following relationship is obtained

119890ref (119896 + 1)= (1 minus 1205832119875 (119902minus1) (119902minus1)) 119890ref (119896) 120593 (119896) 120593119879 (119896)

(56)

Equation (56) will be asymptotically stable that islim119896rarrinfin119890ref (119896 + 1) = 0 if 1205832 is chosen such that 0 lt 1205832 lt1|119875(119902minus1)||120579(119896)| NLMS adaptive filters are inherently stableand are used for plant estimating the parameters of the plantand controller Output of controller remains bounded as longas 1205832 is kept small Since a bounded input is applied to theplant and stable adaptive filters are used in conjunction withthe stabilized plant the controller output remains bounded

6 Simulation Results

Computer simulations of DAIC and IAIC schemes arepresented to show effectiveness of DAIC Two linear non-minimumphase systems are chosen onewithout disturbanceand other with disturbance

61 Example 1 A disturbance free discrete time non-minimum phase linear plant is chosen having

119910 (119896) = 119902minus1 1 + 12119902minus11 + 05119902minus1 + 01119902minus2 119906 (119896) (57)

This is a stable non-minimum phase plant having zero atminus12000 and poles at minus02500 plusmn 01936i In this example we

Complexity 7

Desired output Indirect AIC DAIC

02 04 06 08 10Time

0

1

2

3

4

5

6A

mpl

itude

Figure 5 Tracking desired output first 1 sec

Desired output Indirect AIC DAIC

Am

plitu

de

minus05

05

15

minus1

1

0

5 10 15 20 25 300Time

Figure 6 Tracking desired output amplitude minus12sim15

choose 1205831 = 01 and 1205832 = 001 Similarly learning rate forIAIC is chosen as 001 Orders of 119876119877(119902minus1) and 119876119871(119902minus1) arechosen as 10 Sampling time is chosen as 0001 sec Simulationresults are depicted in Figures 5ndash12 Zoomed preview fordesired output tracking is shown in Figures 5 and 6 Plantoutput in DAIC has less overshoot and converges to desiredoutput quickly compared to IAIC Tracking error is shownin Figures 7 and 8 Tracking error has less amplitude andconverges to zero faster in DAIC compared to IAIC MSE forIAIC and DAIC is shown in Figures 9 and 10 MSE is less forDAIC compared to IAIC Control input is shown in Figure 11

Indirect AIC DAIC

minus5

minus4

minus3

minus2

minus1

0

1

Am

plitu

de

02 04 06 08 10Time

Figure 7 Tracking error first 1 sec

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

minus05

0

05

1

Am

plitu

de

Figure 8 Tracking error amplitude minus1sim1

Control input for DAIC converges quickly compared to IAICModel identification error 119890mod(119896) in DAIC converges to zerovery quickly and is shown in Figure 12

62 Example 2 A disturbance 119899(119896) is added to discrete timenon-minimum phase linear plant Now plant output can bewritten as

119910 (119896) = 119902minus1

sdot 1 minus 3119902minus1 + 35119902minus21 + 005119902minus2 + 005119902minus3 + 002119902minus4 (119906 (119896) + 119899 (119896))

(58)

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 7

Desired output Indirect AIC DAIC

02 04 06 08 10Time

0

1

2

3

4

5

6A

mpl

itude

Figure 5 Tracking desired output first 1 sec

Desired output Indirect AIC DAIC

Am

plitu

de

minus05

05

15

minus1

1

0

5 10 15 20 25 300Time

Figure 6 Tracking desired output amplitude minus12sim15

choose 1205831 = 01 and 1205832 = 001 Similarly learning rate forIAIC is chosen as 001 Orders of 119876119877(119902minus1) and 119876119871(119902minus1) arechosen as 10 Sampling time is chosen as 0001 sec Simulationresults are depicted in Figures 5ndash12 Zoomed preview fordesired output tracking is shown in Figures 5 and 6 Plantoutput in DAIC has less overshoot and converges to desiredoutput quickly compared to IAIC Tracking error is shownin Figures 7 and 8 Tracking error has less amplitude andconverges to zero faster in DAIC compared to IAIC MSE forIAIC and DAIC is shown in Figures 9 and 10 MSE is less forDAIC compared to IAIC Control input is shown in Figure 11

Indirect AIC DAIC

minus5

minus4

minus3

minus2

minus1

0

1

Am

plitu

de

02 04 06 08 10Time

Figure 7 Tracking error first 1 sec

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

minus05

0

05

1

Am

plitu

de

Figure 8 Tracking error amplitude minus1sim1

Control input for DAIC converges quickly compared to IAICModel identification error 119890mod(119896) in DAIC converges to zerovery quickly and is shown in Figure 12

62 Example 2 A disturbance 119899(119896) is added to discrete timenon-minimum phase linear plant Now plant output can bewritten as

119910 (119896) = 119902minus1

sdot 1 minus 3119902minus1 + 35119902minus21 + 005119902minus2 + 005119902minus3 + 002119902minus4 (119906 (119896) + 119899 (119896))

(58)

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Complexity

Indirect AIC DAIC

05 1 15 20Time

0

2

4

6

8

10

12

14

Am

plitu

de

Figure 9 Mean square error first 2 sec

Indirect AIC DAIC

5 10 15 20 25 30Time

0

01

02

03

04

05

Am

plitu

de

Figure 10 Mean square error amplitude 0sim05

This is a stable non-minimum phase plant having zerosat 15000 plusmn 11180i and poles at 02500 plusmn 03708i and minus02500plusmn 01936i Here 119899(119896) is disturbance added to the plant and isshown in Figure 13 In this example we choose 1205831 = 001and 1205832 = 001 Similarly learning rate for IAIC is chosen001 Sampling time is chosen 0001 sec Simulation results areshown in Figures 14ndash16

Control input is depicted in Figure 14 Control input forDAIC is synthesized such that plant tracks desired outputeven in the presence of disturbance Desired output trackingfor DAIC is shown in Figure 15 Plant output not onlyconverges to desired output but good disturbance rejection is

Indirect AIC DAIC

5 10 15 20 25 300Time

minus1

0

1

2

3

4

Am

plitu

de

Figure 11 Control input

minus2

minus15

minus1

minus05

0

05A

mpl

itude

5 10 15 20 25 300Time

Figure 12 Model identification error in DAIC

minus1

minus05

0

05

1

Am

plitu

de

5 10 15 20 25 300Time

Figure 13 Disturbance

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 9

5 10 15 20 25 300Time

minus2

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

Figure 14 Control input in DAIC

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

Am

plitu

de

5 10 15 20 25 300Time

Figure 15 Desired output tracking in DAIC

also achieved in DAIC Plant output in IAIC does not followdesired output and is depicted in Figure 16

7 Experimental Results

The proposed scheme is implemented on laboratory scaletemperature control of a heating process speed and positiontracking of direct currentmotorThe temperature control of aprocess is a non-minimum phase system while the speed andposition control of a DC motor is a minimum phase systemTo accomplish the adaptive tracking the proposed DAICdoes not require a prior information of the system phaseIn these experiments a standard IBM PC-type Pentium IVis used for the computation in real time Data acquisition isaccomplished byNational Instrument cardNI-6024E and the

Desired outputDAIC

minus15

minus1

minus05

0

05

1

15

2

Am

plitu

de

5 10 15 20 25 300Time

Figure 16 Desired output tracking in IAIC

controller is implemented in SIMULINK real-time windowstarget environment The computations are performed infloating-point format and the sampling interval is selected as0001 sec

71 Temperature Control of Heating Process In this experi-ment we use process trainer PT326 manufactured by Feed-back Ltd UK This process is composed of a blower aheating grid tube and temperature sensor (bead thermistor)A variable power supply provides power to the heaterThis power can be controlled by initiating an appropriatecontrolling signal from the computer The process can beconsidered as a second-order time delay systemThis is a non-minimum phase system Input of the process is power andoutput is the temperature of air at somedesired location in theprocess tube This is a time delay system In this experiment10 parameters are selected for the plant estimation and 30 forinverse of the plant The proposed DAIC does not provide aprocedure for the selection of optimal number of the plantand the controller parameters However the control inputplant output and the tracking error remain bounded forany selected number of these parameters Experimental andsimulations studies show that large number of parametersachieves better tracking at the cost of computational burdenFigure 17 shows that output (temperature) of the processconverges to the desired temperature quickly Control inputto the plant is smooth and bounded Control signal isdepicted in Figure 18

72 Speed and Position Control of DC Motor In this experi-ment we use modular servo system (MSS) manufactured byFeedback Ltd UK All the modules used in this experimentare parts of MSS MT150F is a module containing DC motorand tacho-generator PS150E is the power supply and SA-150D is a servo amplifier Control input to the servo amplifier

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Complexity

0 20 40 60 80 1001

2

3

4

5

6

7

8

9

Time (seconds)Time (seconds)

(Vol

ts)

Desired temperaturePlant temperatureEstimated model temperature

Figure 17 Temperature control of heating process

0 20 40 60 80 1000

05

1

15

2

25

3

35

4

Time (seconds)

Cont

rol i

nput

(vol

ts)

Figure 18 Control input to heating process

is through a preamplifier PA150C In the speed control exper-iment the speed signal is measured by the tacho-generatorThis generatormeasures the speed inplusmn10 volts correspondingto plusmn1800 rpm Figure 19 show that themotor speed convergesto the desired speed There are some large spikes in theestimated speed but the actual speed of the motor is quitesmooth It can be observed that the speed control in thevicinity of the dead zone is also accomplished Speed controlof such systems at low speed is known be a difficult problemThe control input is shown in Figure 20 which is boundedand smooth Module OP150K is used to measure the positionof the motor shaft This module measures the shaft positionin the range 0ndash10 volts corresponding to 0ndash360 degreesFigure 21 shows that the motor shaft follows the desiredposition smoothly and quickly Figure 22 shows that controlinput is smooth and bounded

0 20 40 60 80 100minus4

minus3

minus2

minus1

0

1

2

3

4

Time (seconds)

Tach

omet

er o

utpu

t (vo

lts)

Desired speedMotor speedEstimated speed

Figure 19 Speed control of DC motor

0 20 40 60 80 100minus15

minus1

minus05

0

05

1

15

Cont

rol i

nput

(vol

ts)

Time (seconds)

Figure 20 Control input to speed control system

8 Conclusion

Stability and convergence of DAIC are discussed in detailSimulation results show that DAIC performs better thanIAIC in terms of mean square tracking error and disturbancerejectionThe stability of the closed loop is discussed in detailThe convergence of the error to zero and the boundedness ofthe controller parameters are proved However an algorithmto determine the optimal number of the estimated plantand the controller parameters is needed Laboratory scaleexperiments show that DAIC accomplishes tracking of theplant output to the desired smooth trajectory The synthe-sized control input in simulations and experiments remainssmooth and bounded

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 11

Time (seconds)Time (seconds)0 20 40 60 80 100

minus1

0

1

2

3

4

5

6

7

8

Pote

ntio

met

er o

utpu

t (vo

lts)

Desired positionMotor shaft positionEstimated position

Figure 21 Position control of Dc motor

Time (seconds)Time (seconds)0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

Cont

rol i

nput

(vol

ts)

Figure 22 Control input to position control system

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] M Shafiq and M A Shafiq ldquoDirect adaptive inverse controlrdquoIEICE Electronics Express vol 6 no 5 pp 223ndash229 2009

[2] B Widrow and E Walach Prentice-Hall Englewood CliffsNJ USA 1st edition 1995 Adaptive inverse control a signalprocessing approach

[3] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Longman publishing Co Inc Boston MA USA 2ndedition 1994

[4] Y D Landau Adaptive Control The Model Reference ApproachMarcel Dekker Inc New mdashYork NY USA 1st edition 1979

[5] B Widrow and M Bilello ldquoAdaptive Inverse Controlrdquo in Pro-ceedings of the International symposium on Intelligent Controlpp 1ndash6 IEEE Chicago IL USA 1993

[6] B Widrow and G L Plett ldquoAdaptive inverse control based onlinear and nonlinear adaptive filteringrdquo in Proceedings of theInternational workshop on neural networks for IdentificationControl Robotics and Signal Image processing pp 30ndash38 IEEEVenice Italy 1996

[7] BWidrow andG L Plett ldquoNonlinear adaptive inverse controlrdquoin Proceedings of the Decision and Control pp 1032ndash1037 IEEESan Diego CA USA

[8] N A Hizal ldquoImproved adaptive model controlrdquo in ARImdashAnInternational Journal for Physical and Engineering Sciences vol51 pp 181ndash190 3 edition 1999

[9] G L Plett ldquoAdaptive inverse control of linear and nonlinearsystems using dynamic neural networksrdquo IEEE Transactions onNeural Networks and Learning Systems vol 14 no 2 pp 360ndash376 2003

[10] G L Plett ldquoAdaptive inverse control of unmodeled stable SISOand MIMO linear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 16 no 4 pp 243ndash272 2002

[11] M Shafiq F M AL-Sunni and S O Farooq ldquoAdaptive controlof nonlinear hammerstein model using NLMS filterrdquo in Pro-ceedings of the International Conference on Electronics Circuitsand Systems vol 2 pp 439ndash442 IEEE Sharjah UAE 2003

[12] R Salman ldquoNeural networks of adaptive inverse control sys-temsrdquoAppliedMathematics and Computation vol 163 no 2 pp931ndash939 2005

[13] L Ming Y Cheng and S Yu ldquoAn improved nonlinear adaptiveinverse control systems based on filtered-120598 LMS algorithmrdquo inProceedings of the Chinese control conference pp 101ndash105 IEEEKunming China 2008

[14] B D O Anderson and A Dehghani ldquoChallenges of adaptivecontrol-past permanent and futurerdquoAnnual Reviews inControlvol 32 no 2 pp 123ndash135 2008

[15] A M Karshenas M W Dunnigan and B W WilliamsldquoAdaptive inverse control algorithm for shock testingrdquo IEEProceedings Control Theory and Applications vol 147 no 3 pp267ndash276 2000

[16] F M Dias and AM Mota ldquoDirect inverse control of a kilnrdquo inProceedings of the Direct inverse control of a kiln pp 336ndash3412000

[17] M Shafiq ldquoInternal model control structure using adaptiveinverse control strategyrdquo ISA Transactions vol 44 no 3 pp353ndash362 2005

[18] G Du X Zhan W Zhang and S Zhong ldquoImproved filtered-120576adaptive inverse control and its application on nonlinear shipmaneuveringrdquo Journal of Systems Engineering and Electronicsvol 17 no 4 pp 788ndash792 2006

[19] X Wang T Shen and W Wang ldquoAn approach for echocancellation system based on improvedrdquo in Proceedings ofthe International Conference on Wireless Communications Net-working andMobile Computing pp 2853ndash2856 IEEE ShanghaiChina 2007

[20] B-S Ryu J-K Lee J Kim and C-W Lee ldquoThe performance ofan adaptive noise canceller with DSP processorrdquo in Proceedingsof the 40th Southeastern Symposium on System Theory pp 42ndash45 IEEE New Orleans LA USA March 2008

[21] S Shubao S Siyang C Nan and X Minglong ldquoStructure andcontrol strategy for a piezoelectric inchworm actuator equippedwith MEMS ridgesrdquo Sensors and Actuators A Physical vol 264no 9 pp 40ndash50 2017

[22] S Gang L Xiang Z Zhencai T Yu Z Weidong and LShanzeng ldquoAcceleration tracking control combining adaptive

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Complexity

control and off-line compensators for six-degree-of-freedomelectro-hydraulic shaking tablesrdquo ISA Transactions vol 70 no9 pp 322ndash337 2017

[23] K J Astrom and B Wittenmark Computer controlled systemstheory and design Prentice-Hall 3rd edition 1996

[24] K J Astrom P Hagander and J Sternby ldquoZeros of sampledsystemsrdquo in Proceedings of the IEEE Conference on Decision andControl including the Symposium on Adaptive Processes vol 19pp 1077ndash1081 IEEE Albuquerque NM USA 1980

[25] M Ishitobi ldquoProperties of zeros of a discrete-time system withfractional order holdrdquo in Proceedings of the IEEE Conference onDecision and Control vol 4 pp 4339ndash4344 IEEE Kobe Japan1996

[26] E W Bai and S Dasgupta ldquoA minimal k-step delay controllerfor robust tracking of non-minimum phase systemsrdquo in Pro-ceedings of the IEEE Conference on Decision and Control vol1 pp 12ndash17 IEEE Lake Buena Vista FL USA 1994

[27] X Wang and D Chen ldquoCausal inversion of non-minimumphase systemsrdquo in Proceedings of the IEEE Conference onDecision and Control pp 73ndash78 IEEE Orlando FL USA 2001

[28] J Yao X Wang S Hu and W Fu ldquoAdaline neural network-based adaptive inverse control for an electro-hydraulic servosystemrdquo Journal of Vibration and Control vol 17 no 13 pp2007ndash2014 2011

[29] M Ahmed N Lachhab and F Svaricek ldquoNon-model basedadaptive control of electro-hydraulic servo systems using pre-filter inversionrdquo in Proceedings of the 9th International Multi-Conference on Systems Signals and Devices (SSD) IEEE Chem-nitz Germany March 2012

[30] M A Shafiq ldquoPredicting the compressive strength of concreteusing neural network and kernel ridge regressionrdquo in Proceed-ings of the Future Technologies Conference (FTC) pp 821ndash826IEEE San Francisco CA USA December 2016

[31] M A Shafiq ldquoDirect adaptive inverse control of nonlinearplants using neural networksrdquo in Proceedings of the Future Tech-nologies Conference (FTC) pp 827ndash830 IEEE San FranciscoCA USA December 2016

[32] M Shafiq M A Shafiq and N Ahmed ldquoClosed loop directadaptive inverse control for linear plantsrdquo The Scientific WorldJournal vol 2014 Article ID 658497 2014

[33] D T M Slock ldquoOn the Convergence Behavior of the LMS andthe Normalized LMS Algorithmsrdquo IEEE Transactions on SignalProcessing vol 41 no 9 pp 2811ndash2825 1993

[34] N J Bershad ldquoAnalysis of the normalized LMS algorithm withgaussian inputsrdquo IEEE Transactions on Signal Processing vol 34no 4 pp 793ndash806 1986

[35] O Smith Julius W3K Publishing 2nd edition 2007

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of