ADAPTIVE CONTROL OF A PRESSURE TANK SYSTEMWITH SATURATION NONLINEARITIES

Z. Ma � , A. Jutan � and V. Bajic �

� Canadian Space Agency,6767 route de l'Aeroport St. Hubert,

Quebec J3Y 8Y9 Canadae-mail:zi_ma@hotmail.com

� Department of Chemical & Biochemical Engineering,The University of Western Ontario,London, Ontario, N6A 5B8, Canada

e_mail:ajutan@julian.uwo.ca

� Computational Immunology GroupKRDL 21 Hemg Mui Keng Terrace

Singapore 119613e-mail:bajicv@krdl.org.sg

Abstract

The paper presents the design of a new nonlinear real-time adaptive control algorithmwhich is suitable the systems with saturation nonlinearity at the input. The linear part ofthe system can be non-minimum phase and/or unstable. The control algorithm is of theself-tuning type. The algorithm utilizes an auxiliary control parameter that can be chosenon-line to keep the controller output in the linear zone of the saturation nonlinearity.Stability of the algorithm is discussed. Several simulation experiments are performed todemonstrate effectiveness of the algorithm. Finally, the algorithm is applied to the realtime pressure tank process control system successfully.

Keywords: Adaptive control, saturation, nonlinear plant, pressure tank.

1.Introduction

Due to a rapid increase in the power of computing technology, adaptive controlalgorithms have become ever more important for practical implementation. Research inadaptive control for linear systems has produced important developments in the lasttwenty years (see [1][2][3] and [4]). However, in practice, systems are often not linearand often have actuators at the input, which impose a natural saturation limit. A typicalexample of an actuator in the process industry can be a valve such as in the pressure tankpressure control system used here. The static input-output characteristic of the actuatorcan be approximated by a saturation type nonlinearity. The presence of such anonlinearity may lead to windup problems or loss of control, if the control signal enters

the saturation domain of the actuator. These problems are common in many types of realtime controllers, thus it is of an interest to develop an adaptive control scheme that canhandle this situation. Significant progress has been made in this direction using adaptivecontrol systems (see [5] and [7]). A minimum variance adaptive control of discrete-timesystem with a saturating actuator was presented in [7]. These authores developed anadaptive algorithm for their system using Goodwin's approach [6]. Unfortunately, theirresults cannot be applied to systems whose linear part is unstable and/or of nonminimumphase. In this paper we resolved this problem and develop a new adaptive controlalgorithm for time-discrete systems that have a saturation type nonlinearity. Ouralgorithm is based on the generalized minimum variance adaptive control scheme and isapplicable to systems whose linear part can be nonminimum phase and/or unstable. Thestability characteristics of the algorithm are discussed. The algorithm performance isillustrated by simulation experiments. Finally, it is applied to a real time pressure tankcontrol system. The successful real time control results show that the control algorithmcan maintain the control signal in the linear control zone, so that the adaptive controlalgorithm can be directly applied without the need for the any `ad hoc' work around thatare often used in engineering practice.

2. Controller design

We consider a linear plant with a nonlinear saturation component at its input. The modelof such a system

,...2,1,0

);(

,;****

***

�

��

��

�

�

k

uuusignBuAy

uuBuAy

skksdk

skkdk

(1)

where ky* and ku* represent the values of the variables *y and *u at the moment krespectively; d is a positive integer representing the time delay in the system; A and B arepolynomials of the backward shift operator 1�z . These polynomials are given by

b

b

a

a

nn

nn zbzbbBzazaA ����

������ ,...,;,...,1 110

11 (2)

To simplify notation we will write P for a polynomial )( 1�zP , except when shorternotation is inadequate. In (1), 0�su is the saturation value of the control output

determined by the saturation nonlinearity at the plant input. When sk uu �* , the control

signal will be )(*ksk usignuu � . In order to ensure that the control output sk uu �

* , anauxiliary control mechanism characterized by the adjustable parameter � is introducedinto the system. The complete control system is depicted in Fig. 1.

Kh KhCONTROLLER

ESTIMATOR

PLANT

rk +

_

uk u*k

_ +um

y k

Fig. 1 Control system structure

In Fig.1, mu is upper bound of the control signal, mu may be chosen as sm uu � , and�/1�hk . The estimator determines the values of the parameters of the linear part of the

plant, as well as the value of the adjustable parameter � .

To ensure that the system operates in the linear zone of the actuator, the parameter � willbe selected so that

sk uu ��/ (2) If we define

dkdkkk yyuu ��

��** /;/ �� (3)

then (1) implies (for 0�� ) that

�� // kdk BuAy ��

(4)

The adaptive control system in Fig.1 will be designed based on (1)-(4). This modelincludes the auxiliary control parameter � . If � is obtained to satisfy (2) this will make

ssk uu �* and eliminate the saturation component.

We use the following generalized error e in the design process

kkdkdk QuSrPye ** / ��� ��

�

and define the performance cost function of the control system by

� �2**2 / kkdkdk QuSrPyeJ ���� �� � (5)

Here P, Q and S are, in general, polynomials introduced to improve the system dynamicsand steady-state characteristics of the overall control system. By selecting P and Q on-

line we can make the closed-loop system stable, while selection of S will control systembehavior when ��k

We define an auxiliary output dkdk Py ��

��* , which we obtain by a predictor of the form

kkkdk EBuGy **/)(

*��� � (6)

where G and E are polynomials of 1�z with orders 1�� dne and 1�� ag nn ,respectively, i.e.

g

g

e

e

nn

nn zgzggGzezeeE �

���

������� ,...,;,..., 110

110

and which satisfy the following Diophantine equation

GzEAP d��� (7)

Now we can state the following

Theorem1: The control law minimizing (5) is described by

kkkdk QuSr */)(

* / ��� � � (8)

and it ensures that kk ry � as ��k

The closed-loop system equations are given by

�/)( *kdk BSryQABP �� � (9)

�/)( *kk ASruQABP �� (10)

Remark 1: The reason for having two equations that characterize the system in theclosed loop is that (9) determines the dynamics of system from the 'reference input - plantoutput' point of view, while (10) determines its behavior from the viewpoint of thereference input - plant input' relation. Since both have the same characteristicpolynomial BP+QA, then if this polynomial is made stable, we will achieve stableprocess output, as well plant input. In this manner stable behavior of both the plantoutput and the plant input is ensured.

Proof: First we will prove that the closed-loop system obeys (9) and (10). From thepredictor equation (6) and the control law (8), system input ku* can be written as

QEBGySru kk

k�

�

�

** /� (11)

Substituting this ku* in (4) and using Diophantine equation (7), the closed-loop equation(9) can be obtained as follows

�

�

�

�

�

/)(

/])([

/)(

/)(

/

*

*

**

**

***

kdk

kdkd

dkd

kdk

kkdk

kkkdk

BSryQABP

BSryQAGzEAB

GyBzBSrAyQEB

BGyBSrAyQEBQEBGySr

BBuAy

��

���

���

���

�

�

��

�

�

�

�

�

�

�

�

Also, from (11) one gets

GuEBQSr

y kkk

** )(/ ��

�

�

(12)

Substituting this ky* in (3) and (4), and using Diophantine equation (7), the closed-loopequation (10) can be obtained in the following way

kk

kkd

k

kdkk

kd

k

kdk

uAQBPASr

AQuuGzEABASr

BuzG

uQBESrA

BuzAyBuAy

*

**

**

**

**

)(/

)(/

)(/

��

���

�

��

�

�

�

�

�

�

�

�

�

Now we need to prove that the control (8) minimizes (5). In equations (9) and (10) of theclosed-loop system, orders of weight polynomials P and Q can be determined based onthe orders of polynomials B and A as 1�� ap nn and 1�� bq nn , respectively, so that

;,...,;,..., 110

110

q

q

p

p

nn

nn zqzqqQzpzppP ����

�������

Eqn. (11) implieskkk GySruQEB ** /)( ��� �

Since kd

dk yzy ** �

� � , then from (9) one obtains

QABPBSrz

y kd

k�

�

�

�/* (13)

To obtain a stable system, we select a stable polynomial T in advance, so that thepolynomials P and Q are determined from the following Diophantine equation

TQABP �� (14)

From (13) it follows that if the following closed-loop transfer function is equal to one, i.e.if

1)()(

1

1

�

�

�

�

�

�

QABPBSz

zRzY d

where Y and R are the z-transforms of y and r, we get �/*kk ry � and kk ry � for all

k=0,1,2,... Therefore, we need to select S to satisfy the following relation

dBzTS�

� (15)

However, in general, (15) cannot be satisfied for all allowed values of z but it can besatisfied for z=1. Note that the case z=1 corresponds to the behavior of system when

��k . Thus S is selected as

)1()1(

BTS � (16)

Hence, for ��k , the control mechanism given by (8) implies that kk ry � is achieved.This also makes J=0 it is minimum value.

From (10), we have

QABPASr

u kk

�

�

�/* (17)

Q.E.D.

Remark 2: For practical purposes, in order to keep the input ku* within suitable limits,

we select an allowed (prespecified) upper bound for ku* as mu where

sm uu ��0

Then the parameter � can be determined on-line at the moment k by

,0;/ �� kmkk uu ��

where k� is the value of � at the moment k. By the above rule, � is chosen sufficiently

large so that mmk

k

k

kk u

uuuu

u ���

/*

�

. Therefore, when ��k (i.e. z=1) one has from

(14), (16) and (17) that

./)1()1(*

kkk rBAu ��

Here, since

mk uu �*

then k� can be chosen as

m

kk uB

rA)1()1(

�� (18)

where 0)1( �B is assumed.

3. Adaptive control algorithm

The system (1) may be written in vector notation as

Tkky ��� (19)

where],...,,...,[ *

1**

1*

ba nkknkkk uuyy ���� ���� (20)

],...,,...,[ 01 ba nn bbaa�� (21)

The following least squares (RLS) algorithm can be used to identify the parameters �

kkkk L ��� ���1

ˆˆ (22)

where]ˆ,...,ˆˆ,...,ˆ[ˆ

01 knkknk babbaa�� (23)

Tkkk PL �� (24)

kk

Tkkk

kkkT

kkk P

PPPP �

�/][

1

111

���

����

�

��

�

(25)

)1( 010 ���� ����kk (26)

)1(ˆ*���� ky kkk �� (27)

where k� is a variable forgetting factor [6].

Based on the parameter estimation k�̂ , the adaptive control algorithm is defined by thefollowing steps

Step 1. According to the saturation value su of the system, determine the upper bound ofcontroller output mu .

Step 2. Select a stable polynomial T to be used in connection with (14).

Step 3. Establish the initial value 0� of the auxiliary control parameter � and initialvectors values of the parameters 0� and 0� .

Step 4. Using RLS algorithm from (22) to (27), calculate the updated system parameters

knkkknkk babbBaaA ˆ,...,ˆˆ,ˆ,...,ˆˆ

01 ��

Step 5. Based on Diophantine equation (14), kP and kQ are calculated from

TAQPB kkkk �� ˆˆ (28)

Step 6. Using kA and kP , the values of kE and kG are calculated based on Diophantineequation (7)

kd

kkk GzAEP �

�� ˆ (29)

Step 7. Based on (16), the value kS is calculated by

)1()1(

BTSk � (30)

Step 8. The control parameter � is determined by (18).

Step 9. Based on (11), the controller output is calculated by

000

***

ˆ/

kkk

kkkkkkk

qbeuDyGrS

u�

��

�

�

(31)

where

000ˆˆ

kkkkkkk qbeQEBD ���� (32)

The above steps, except Steps 1, 2 and 3, are repeated at each step k.

4.Stability analysis of the algorithm

The system stability analyses is based on the following assumptions

Assumption A: The system structure is known, i.e. the order an and bn are known.Assumption B: Polynomial A and B are coprime.

The well known, the recursive least squares algorithm has the following properties,

�����

Mkk�̂lim (33)

��������

���

lN

kkkk

0;ˆˆlim1

2

1�� (34)

0ˆˆlim ���

��lkkk

�� (35)

The following lemma concerning input-output dynamics is developed.

Lemma 1: When the control algorithm is applied to the plant (1), the input-outputdynamics of the system is given by

��

���

���

���

�

��

��

���

���

���

���

�

������

������

���

���

�

��

���

�

���

���

���

�

�

�

�

�

�

k

dk

tkklklkl

lkklklkl

k

dk

klkklkkklklkklkkkl

klkklkkklklkklkkkt

kkkkktl

kkkkkl

k

dk

kkkk

kkkk

uy

BBEAAAEABBEBAAEB

uy

QAQABEBEAPAPAAEAEAQBQBBEBEBPBPBAEAEB

rSASASASBSBSB

uy

QAPBQAPB

*

*

*

*

*

*

)(.)(.)(.)(.

).()..().()..().()..().()..(

/).().(

00

�

(36)

where kA and lA denote the estimate values of parameter A at the k and k+l (l is afinite constant) moments respectively.

),(),,( 11 ��

�� zlkAzkAA lk (37)

kllk ABzlkBzkABA .),(),(. 11���

�� (38)

kkkk ABzkBzkABA ���� ),(),( 11 (39)

Proof: From (4), we have

klkdkklkldkl uBByAAuByA **** )()( ����� �� (40)

eqn. (40) can also be written

klkdkklkldkl uBByAAuByA **** )()(; �������� ��

(41)

Multiplying (41) by kE , one has

���� kklkdklk EuBEyAE ** .. (42)

from engs. (29), (38) and (39), the following equation can be obtained

�������� �� kdklkkkkkklkkkkkkdkk EyAEAEuBEBEuBEyGyP ***** ).().( (43)

Using eqns. (6), (8) and (43), we have

�������� �� kdklkkkkkklkkkkkdkk EyAEAEuBEBEuQrSyP **** ).().(/� (44)

Multiplying eqn. (44) by lB , gives us

������

��

�

�

kldklkkklkkklkl

kklkkldkkl

EByAEAEBuBEBEB

uQBrSByPB

.)..()..(

./..**

**�

(45)

using the system open-loop equation (4) and eqns. (38), (39), we obtain

�����

����

�����

�

�

klkklkklkkkl

dkklkklkkkt

kkkkkldkkkkk

EBuQBQBBEBEB

yPBPBAEAEB

rSBSBSByQAPB

.)].()..([

)].()..([

/]).[()(

*

*

*�

(46)

Multiplying eqn. (44) by lA , follows can be obtained

������

��

�

�

kldklkkklkkklkl

kklkkldkkl

EAyAEAEAuBEBEA

uQArSAyPA

.)..()..(

./..**

**�

(47)

using the system open-loop equation (4) and eqns. (38), (39), we obtain

�����

����

�����

�

klkklkklkkkl

dkklkklkkkl

kkkkkklkkkkk

EAuQAQABEBEA

yPAPAAEAEA

rSASASAuQAPB

.)].()..([

)].()..([

/]).[()(

*

*

*�

(48)

at this point, Lemma 1 has been proven.

The stability and convergence of the control algorithm are given by the followingtheorem

Theorem 2: The control algorithm has the following properties

��������

kkkkuy lim;lim (49)

0)(lim 2��

��

dkek

(50)

Proof: From the properties of least squares algorithm (34) and (35), it is shown that theparentheses terms in (36) tend to zero as ��k . According to the rule of determining �

0)1(ˆ

)1(ˆ��

mk

kk

uBAr

�

and from the boundedness of kr the property of the least squares algorithm (33), it canbe seen that

����

�

klim (51)

using eqn. (28), the boundedness of control signal ku * and ky* cab be obtained . Fromthe boundedness of � , the boundedness of system output ky and ku cab be obtained.From the definition of generalized error dke

� (see eqn. (5))

kkkkkdkkdk uQrSyPe ** / ��� ��

� (52)

using eqn. (44), we have

2**2 ).().( ������ �� kdklkkkkkklkdk EyAEAEuBEBEe (53)

From the convergence of the parameter estimation term (34) and (35), it can be shownthat eqn. (50) is satisfied.

5. Simulation results

Simulation 1: The following unstable and nonminimum phase discrete model isconsidered

211 5.02.01.1���

��� kkkk uuyy

It is assumed that if the system control input 10*�ku , the system will enter the

saturation zone (i.e. the saturation value is 10�su ), the upper bound mu is chosen as 7.The initial data vector for the parameter identification algorithm is chosen as the zerovector, the system parameter initial values are chosen as a set random numbers in therang 10 � , the ideal system closed-loop characteristic equation T is chosen as 12.01 �

� zand the initial value of the auxiliary parameter chosen as 10 �� initially. The referenceinput signal is a square wave with amplitude 10� up to 200 time steps, then it is asinusoidal wave. Fig. 2-Fig.5 shows that the control result is very good. From Fig.3, withno auxiliary parameter � , the control signal ku is much larger than saturation value.After the auxiliary parameter is used, the control signal is limited to within the linearzone.

Simulation 2: In order to demonstrate the algorithm effectiveness for the plants withlarge delay, the simulation model is chosen as the following high order discrete modelwith delay 12�d

1312765321 01.02.0023.0125.0015.004.013.01.0��������

��������� kkkkkkkkk uuyyyyyyy

The saturation value of the control signal is chosen as , 15�su to ensure that the controlsignal is in linear zone, the upper bound mu is chosen as 10. Similar to Simulation 1, theinitial data vector for the system parameter identification algorithm is chosen as the zerovector, the system parameter initial values are chosen as random numbers in range

10 � , the system closed-loop characteristic equation T is chosen as 11.01 �

� z , and theinitial auxiliary parameter value is chosen as 10 �� . The reference input signal is asquare wave with amplitude 10� , Fig.6-Fig.9 shows the results. Again very good controlis achieved even for this high order large delay system with saturation nonlinearities.

0 100 200 300 400 500 600−15

−10

−5

0

5

10

15

20

time steps

syst

em in

put a

nd o

utpu

t sig

nals

No.1

No.2 No.1: reference input rNo.2: system output y

0 100 200 300 400 500 600−50

0

50

100

time steps

cont

rol s

igna

ls

No.1

No.2

No.1: control signal u*

No.2: control signal u

Fig.2 System tracking trajectories Fig. 3 Control signals (simulation 1) (simulation 1)

0 100 200 300 400 500 6003.2

3.4

3.6

3.8

4

4.2

4.4

time steps

auxi

liary

par

amet

er π

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

time steps

para

met

er e

stim

atio

n

No.1

No.2

No.3

No.1:parameter a1

No.2:parameter b0

No.3:parameter b1

Fig.4 Auxiliary parameter � Fig.5 Identified parameter curves (simulation 1) (simulation 1)

0 20 40 60 80 100 120 140 160 180 200−15

−10

−5

0

5

10

15

20

time steps

syst

em in

put a

nd o

utpu

t sig

nals

No.1

No.2

No.1: reference input rNo.2: system output y

0 20 40 60 80 100 120 140 160 180 200−50

0

50

100

time steps

cont

rol s

igna

ls

No.1

No.2

No.1: control signal u*No.2: control signal u

Fig.6 System tracking trajectories Fig. 7 Control signals (simulation2) (simulation 2)

0 100 200 300 400 500 6004.5

5

5.5

6

6.5

7

time steps

auxi

liary

par

amet

er π

0 20 40 60 80 100 120 140 160 180 200−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time steps

No.1

No.2

No.3No.4

No.5No.6

No.7

No.8

No.1:parameter a1No.2:parameter a2No.3:parameter a3No.4:parameter a4No.5:parameter a5No.6:parameter a6No.7:parameter b0No.8:parameter b1

Fig.8 Auxiliary parameter � Fig.9 Identified parameter curves (simulation 2) (simulation 2)

6. The Pressure Tank System

The real time system considered here is a Pressure Tank (Fig.10) through which air flowsfrom a regulated source. Control valves are installed on both inlet and exit flow. Thepressure in the vessel and the flow rate in exit flow are measured and transmitted to acomputer. Data collection and system control are achieved by use of a microcomputerwith an input-output (I/O) interface board. The controlled variables are pressure and flow

MUFFLERVALVE1

ORIFICEPLATE

AIR TUBES TOPRESSURE DIFFFERENTIAL

TRANSMITERPRESSURE TRANSMITER

WIRING TO A/D BOARDAND POWER SUPPLY TWO

LOAD DISTURBANCE

VALVE 2

PRESSURE TANK

Fig. 10 Pressure Tank System

The real time control is also implemented by Matlab/Simulink. When implementing realtime control, there are certain practical considerations, for example the controller outputcalculation is converted via a D/A board to produce an output signal in the range of 0-10volts. If the control calculation results in a conversion outside this range, the output is

clipped. Similarly for signal inputs, the range is converted by a A/D board which limitsthe range to 4-20 mA. Some `ad hoc' engineering methods such as deviation control areneeded to take into account the clipped values as opposed to the calculated values,otherwise, instability may occur. In our algorithm, the saturation nonlinearity has beenspecifically considered. Therefore the signal in and out of the process shouldautomatically be within the linear zone and clipping is longer required.

7. Experimental Runs

A number of experimental runs were conducted on the pressure tank to test out theefficiency of the control algorithm under real time conditions.

Run I. The controlled variable is flow rate. Since the saturation value of the system is1��su , as determined by the real time software, the upper bound 6.0�mu is chosen.

Based on our knowledge, this process is a high order system with unknown time delay sothat a self-tuning control algorithm based on a fourth order model with delay d=1 is used.The initial values of system parameters vector and data vector are given as a randomvector and zeros vector respectively. The value of auxiliary variable 100 �� is chosenfor startup. A stable pole placement equation is chosen as

11 15.01)( ��

�� zzT

In this experiment, the sample period is 1 second. The set-point of the flow rate inmin/3ft is varied as a square wave.

Fig.11-Fig.13 shows the closed-loop system response for the run. From Fig.11-Fig.13, agood control response with minor offset is obtained. This offset is due to the processnonlinearity which varies with setpoint level, but is minor in this range. Fig.12 shows theoutput of the controller, which for the most of part, maintains an absolute value of lessthan 0.6 (i.e. within linear control zone). Fig. 13 shows the trajectory of auxiliaryparameter k� with rapid convergence rate.

From Fig.12, we notice two points S1 and S2 where the control action overshoots it'sboundary magnitude of 0.6. This overshoot rapidly re-enters the linear zone and theoutput trajectory remains unaffected by this brief excursion. One reason for thisoccasional overshoot can be seen by examining the Diophantine equation (7). Here wesee that the controller parameters )( 1�zGt depend on the adaptive model parameters

)(ˆ 1�zAt and )( 1�zPt as well as the estimated time delay d. Generally the model timedelay is unknown and may even appear to vary due to process nonlinearities. Here afixed, estimated time dalay is used. This appears to be sufficient for all but brief instancesduring which the process output remains unaffected.

A second reason is that the rate of change of the adaptive auxiliary parameter � isdependant on the length of time over which the constraint is violated. Thus � will not

respond to brief departures and is better suited to more serious extended periods ofsaturation.

0 50 100 150 200 250 300 350 40025

26

27

28

29

30

31

32

time steps

Flo

w r

ate

and

set p

oint

( f

t 3 /m

in)

No.1

No.2

No.1: set point

No.2: flow rate

0 50 100 150 200 250 300 350 400−3

−2

−1

0

1

2

3

4

5

6

S1

S2

time steps

cont

rol s

igna

l

0 50 100 150 200 250 300 350 40020

25

30

35

40

45

50

55

60

65

time steps

auxiliary parameter π

Fig.11 Flow rate tracking Fig. 12 Control signal Fig.13 Auxiliary parameter Trajectories (run 1) (run 1) curves (run 1)

Run 2. The controlled variable is now pressure, the linear model is based on a fourthorder model with time delay d=1. The initial values of the system parameters vector anddata vector are given as a random vector and zero vector respectively. An upper bound of

6.0�mu is also chosen. The initial value of auxiliary 100 �� is chosen. The stable poleplacement equation is chosen as

11 1.01)( ��

�� zzT

In this experiment, the sample period is 0.2 second (smaller than the flow sample periodsince the pressure dynamics are faster). The set-point of the flow rate in psi is varied as asquare wave.

Fig.14-Fig.16 shows the closed-loop system response for the run. From Fig.14-Fig.16, agood control response with minor offset is obtained. Fig.15 shows the most of outputsignal value of the controller is within linear control zone. The control signal overshootappear in points P1, P2, D1 and D2. The overshoot in point P1 and P2 are effected by thechange of reference input signals, and the overshoot in point D1 and D2 are caused byunmeasurable disturbances. Since the choice of system time delay is a good enoughapproximation, the control signals return to the linear control zone quickly so that thesystem output trajectory is not affected. Fig. 16 shows the trajectory of the auxiliaryparameter k� .

0 50 100 150 200 250 300 350 4008.5

9

9.5

10

10.5

11

11.5

0 50 100 150 200 250 300 350 4008.5

9

9.5

10

10.5

11

11.5

time steps

Flo

w r

ate

and

set p

oint

No.1

No.2

No.1: set point;No.2: pressure (psi)

0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

P1

P2

D1D2

time steps

cont

rol s

igna

l

0 50 100 150 200 250 300 350 400199

199.2

199.4

199.6

199.8

200

200.2

200.4

200.6

200.8

201

time steps

auxiliary parameter π

Fig.14 Flow rate tracking Fig. 15 Control signal Fig.16 Auxiliary parameter Trajectories (run 2) (run 2) curves (run 2)

8. Conclusion

This paper presents an algorithm for dealing with saturation signals in adaptive controlsystems. When the system signals are not saturated, the control system runs in normaladaptive mode. If a signal becomes saturated, a scaling parameter � is used to bring thesignal back into the linear (unsaturated) zone. � is itself calculated adaptively, and tendsto a constant when the process parameters converge to a static state. This auxiliaryscaling parameter is chosen initially based on estimates of the plant parameters and themaximum value of the reference signal. An upper bound is also chosen for the controlsignal, which is less than the saturation limit. A suitable dead time must also be selected.The algorithm then operates to maintain the control signal within the linear (unsaturated)boundaries even in the initial stage when only rough process parameter values areavailable.

The new algorithm can be applied to unstable and/or nonminimum phase plants,including those of high order and large time delay. The system was tested via simulationand finally implemented on a real time pressure tank in SISO mode. The real timeexperiments demonstrate the success of the algorithm (except for a few transitory points)and is able to maintain the process operation within the linear control boundaries.

References

[1]. U. Borisson, Self-tuning regulator for a class of multivariable systems, Automatica,Vol. 15, pp. 209-215, 1979.

[2].G. C. Goodwin, P.J. Ramadge, and P.E. Caines, Discrete-time multivariable system,IEEE Trans. Automat. Contr. Vol. AC-25, pp. 449-456, 1980.

[3]. R.P. Singh and K. S. Narendra, Prior information in the design of multivariableadaptive controllers, IEEE Trans. Automat. Contr., Vol. AC-25, pp. 1241-1251, 1980.

[4].D. W. Clarke and P.J. Gawthrop, Self-tuning controller, IEE Proc. Pt. D., Vol. 122,pp. 929-935, 1975.

[5]Y. A. Daniel and G. F. Franklin, On the stability of adaptive pole-placementcontrollers with a saturating actuator, \QTR{it}{IEEE Trans. on Autom. Contr}., Vol. 35,No. 3, pp. 303-306, 1990.

[6]G. C. Goodwin and K. S. Sin, Adaptive filtering prediction and control (Informationand System Science Series), Englewood Cliffs, NJ: Prentice-Hall, 1984.

[7] D. Y. Abramovitch, R. L. Kosut, and G.F. Franklin, Adaptive control with saturatinginputs, Proc. 25th Conf. Decision Contr., 1986.