Energy Considerations for Tracking in DC to DC Power ...downloads.hindawi.com/journals/mpe/2019/1098243.pdf · ResearchArticle Energy Considerations for Tracking in DC to DC Power

Research ArticleEnergy Considerations for Tracking in DC toDC Power Converters

G Obregoacuten-Pulido 1 G Sol-s-Perales 1 J A Meda-Campantildea 2

G A Vega-Goacutemez1 and E Ruiz-Velaacutezquez 1

1Electronic Department University of Guadalajara Av Revolucion 1500 Guadalajara Jalisco Mexico2ESIME Zacatenco del Instituto Politecnico Nacional Ciudad de Mexico Mexico

Correspondence should be addressed to G Solıs-Perales gualbertosoliscuceiudgmx

Received 1 September 2018 Revised 6 November 2018 Accepted 16 December 2018 Published 6 February 2019

Academic Editor Jean Jacques Loiseau

Copyright copy 2019 G Obregon-Pulido 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

A design of an adaptive controller applied to the boost and buck-boost converters to deal with the problem of tracking a sinusoidalvoltage is proposed The main contribution is to provide conditions on the design procedure in order to obtain a reduction in theDC voltage (offset of the sinusoidal signal) of the reference signal in this way the AC energy is maximized A nonlinear stablesystem is designed in order to produce the necessary inputs to exactly track the inductor reference current which is a necessarycondition to achieve the tracking behavior of the voltage reference signal 119891(119905) = 119860 + 119861 sin(119908119905) A numerical example is providedto corroborate the result

1 Introduction

With the advances in the power semiconductor technologythe use and control of switched converters and systems areexponentially increasing due to their potential applications[1ndash4] The switched converters have been widely analysedthese converters can increase or decrease the magnitudeof the DC voltage or invert its polarity in the load Theboost and buck-boost converters are highly applied in powersystems such as power factor correction [5 6] in wind powerturbines [7 8] In general for renewable energy systems [9]moreover a double loop control of buck-boost converter hasbeen reported where the control loop is designed for widerange of resistance and reference voltage however the outputreference voltage is constant [10]

In recent years some works which deal with the track-ing problem in power converters have been reported forinstance in [11 12] the tracking problem of a sinusoidalreference signal in DC-DC converters is studied and thereported algorithm is locally stable and is based on slidingmode control and the Galerkin method which can beviewed as a generalization of the harmonic balance methodThe sliding mode control has been used to compensate

disturbances on the output voltage inDC-DC converters [13]In [14] an adaptive control topology is used in order to obtainsinusoidal tracking for the inductor current such proceduredesign does not guarantee a small DC energy Authors in [15]present an algebraic approach based on online identificationof uncertain parameters which are used to solve the trackingproblem in an arrangement of power converters Howeverthe DC voltage level is not considered as a parameter designIn [16] a sliding mode scheme was reported to perform aDC to AC conversion and the parameters of the outputvoltage are chosen such that certain conditions are satisfiedThere is a common issue in the previous reported resultsthat is the DC voltage level is a parameter of the outputvoltage signal Nevertheless this voltage represents energyconsumption since on this voltage the sinusoidal signal ismounted therefore it is desirable to reduce this value in orderto obtain the same sinusoidal signal but with a reduced DCvoltage level and thus reduce the energy consumption

Taking into account these published results we improvethe procedure reported in [17] in order to reduce the DC levelin the reference signal which represents a reduction in theconsumption of DC energy A basic problem in the controllerdesign consists in finding the zero dynamics of the system

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1098243 9 pageshttpsdoiorg10115520191098243

2 Mathematical Problems in Engineering

such dynamics is required to be stable in order for the systemoutput to track the given reference In order to overcome thisdifficulty the indirect control method is proposed where analternative system output is considered and for such a systemthe internal dynamics is stable However other problemsarise for instance the determination of the reference forthe new system output and the control synthesis which arecommonly studied within the field of power converters Oneof them is the current mode regulation which is justified in[11] which provides the manner to find an approach to theunstable periodic solution required for the tracking problemThis solution is obtained using the harmonic balancemethod[12] On the other hand a boost topology is used in [11]in order to obtain a system that provides an output voltagegreater than the input Nevertheless it has been shown thatto control the boost and buck-boost converters the indirectcontrol of the output voltage is required [17]

In this note we propose an adaptive method applied toa boost and buck-boost topologies to achieve tracking ofa sinusoidal voltage where the reference signal is given byf (119905) = 119860 + 119861 sin(119908119905) The procedure consists in designingan adaptive controller Besides it is required to propose astable nonlinear system in order to obtain the control inputsrequired to practically track the reference signal The maincontribution consists in finding the conditions under whichthe controller is implementable and with the capability ofreducing the necessary DC energy imposed by the conditionsgiven in [11 14] In this sense the aim of the proposal is toreduce the use of theDC component in the tracking signals asestablished in [11 14] this represents a reduction in the energyrequired to generate or to track a sinusoidal signal besidesthe error in the tracking signals is practically zero and the DCcomponent is almost the minimum required which is a greatadvantage in the tracking of sinusoidal signals using DC-DCconverters

The paper is organized as follows Section 2 presents themodel of the converters and some considerations on electricalconditions In Section 3 an analysis is performed in orderto obtain conditions under which the design loses less DCenergy than that given in [6 9] Section 4 presents the devel-opment of the adaptive scheme to track practically the voltagereference In Section 5 the proposed result is corroboratedvia numerical simulations Finally the contribution is closedwith some conclusions on the tracking voltage problem inDC-DC power converters

2 Model Description

The mathematical equations which describe the boost andbuck-boost converters are given by

119871119889119868119871119889120591 = (119878119908 minus 1) (119881119862 + 119896119881119862119862) + 119881119862119862119862119889119881119862119889120591 = (1 minus 119878119908) 119868119871 minus 119881119862119877

(1)

where if k=0 we have the boost converter (Figure 1(a))otherwise if k=1 we have the buck-boost converter (Fig-ure 1(b)) VC IL and VCC are the capacitor voltage inductor

current and input voltage respectively The switching signalSw is the control signal which takes values from the set S =0 1 120591 is the nonscaled time The next change of variable isproposed in order to simplify the analysis [12]

119909 = 119868119871119881119862119862radic119871119862

119910 = 119881119862119881119862119862 119905 = 120591

2radic119871119862119886 = 1119877radic 119871119862119906 = 1 minus 119878119908

(2)

Then the dynamical model takes the form bellows

= 1 minus (119896 + 119910) 119906119910 = minus119886119910 + 119909119906 (3)

Therefore the problem consists in finding a control signalinput such that the solution 119910(119905) 997888rarr 119891(119905) gt 0 where thereference 119891(119905) is a positive signal

Multiplying (3) by 119909 and (119896+119910) the following differentialequation is obtained

(119896 + 119910) ( 119910 + 119886119910) = 119909 (1 minus ) (4)

When 119910(119905) reaches the steady state 119910(119905) = 119891(119905) given bya periodic orbit the differential equation (119896 + 119891)( 119891 + 119886119891) =119909(1minus) has a unique solution given by an unstable limit cycleoslash(119905) when 119892(119905) = (119896 + 119891)( 119891 + 119886119891) gt 0 [11] Thus to obtainconvergence of the solution 119909 997888rarr oslash1(119905) an approximation ofoslash1(119905) gt 0 is proposed and is considered as the 1st harmonicof the Fourier series of oslash(119905) From this it is clear that theseconverters can track a sinusoidal signal

As it was mentioned previously the boost and buck-boost converters require indirect control of the output voltage[11] in order to avoid instabilities in the internal dynamicsConsidering the signal oslash1(119905) as the reference for x(t) themaintracking problem is defined to control the system in orderto obtain a zero error tracking between the reference andthe output using a smooth control From (4) and considering119909(119905) = oslash1(119905) we obtain

119910 = minus119886119910 + oslash1 (1 minus oslash1)119896 + 119910 (5)

and now with the change 119911119910 = (119896 + 119910)minus1 it can be seen thatsystem (5) has a stable limit cycle 120574(119905) as the dynamics of 119911119910(119905)is

119910 = 119886119911119910 minus 1198861198961199111199102 minus 1199111199103oslash1 (1 minus oslash1) (6)

where we can see that if oslash1(119905) gt 0 and (1 minus oslash1) gt 0 thenfor any initial condition 119911119910(0) gt 0 the solution 119911119910 of system

Mathematical Problems in Engineering 3

LD

C R

+

minussw６

Ｃ

ＣＣ２

(a)

sw

６

L

D

C R

+

minus

ＣＣＣ２

(b)

Figure 1 (a) Boost converter circuit (b) Buck-boost converter circuit

(6) tends to the limit cycle given by 120578(119905) = (119896 + 120574(119905))minus1 Nowconsidering the steady state continuous control input 1199060(119905)and 119906(119905) as the continuous control input calculated as themean value of 119906 1199060(119905) = (1 minus oslash1)120578(119905) Thus taking (3) andimplementing (6) we can write

= 1 minus (119896 + 119910) 119906119910 = minus119886119910 + 119909119906 = 119886119911 minus 1198861198961199112 minus 1199113oslash1 (1 minus oslash1)119906 = (1 minus oslash1) 119911

(7)

and we have that 119906(119905) 997888rarr 1199060(119905) 119909(119905) 997888rarr oslash1(119905) 119910(119905) 997888rarr120574(119905) and the switched input 119906 is replaced by its equivalentin continuous time 119906(119905) In order to show this assertionconsider the error signals 119890119910 = 119910 minus 120574 119890119909 = 119909 minus oslash1 and notethat 119911 997888rarr (119896 + 120574)minus1 since by design we consider oslash1 gt 0 and(1 minus oslash1) gt 0 then we write the dynamics for the error systemon the steady state of 119911 as

119890119910 = minus119886119890119910 + 119890119909 (1 minus oslash1)119896 + 120574119890119909 = minus(1 minus oslash1)119896 + 120574 119890119910

(8)

where 120574 = minus119886120574 + oslash1(1 minus oslash1)(119896 + 120574) Taking the Lyapunovfunction 119881(119890) = 11989021199092 + 11989021199102 and the derivative given as(119890) = minus1198861198902119910 then by LaSalle theorem 119890119910 997888rarr 0 and from thefirst equation 119890119909 997888rarr 0 Solving (6) the input for 119909 997888rarr oslash1 isfoundNevertheless the parameter 119886 is uncertain but it can beestimated by means of an adaptive scheme Now if the inputfunction oslash1 is an affine function of the parameter 119886 thus it ispossible to determine a globally stable controller

Remark 1 The reference for the current oslash1 could be anyfunction of time and if oslash1 gt 0 and (1 minus oslash1) gt 0 thenimplementing (6) we have that 119909 997888rarr oslash1

Let the reference voltage119910(119905) given as119891(119905) = 119860+119861sin(119908119905)and as it was described in [12] 120574(119905) is the first harmonic of119891(119905)

if oslash1 is properly chosen Using the harmonic balance methodthe function oslash1 must satisfy (see [12])

oslash1 = 1198861198600 + 1198611 sin (119908119905) + 1198621 cos (119908119905) (9)

1198611 = 119872 sin (120579) = 119886119861 [(2119860 + 119896) minus 11990821198600 (119896 + 119860)]1198862119860201199082 + 11198621 = 119872 cos (120579) = 119908119861 [11988621198600 (2119860 + 119896) + (119896 + 119860)]1198862119860201199082 + 11198600 = 119860 (119860 + 119896) + 11986122

(10)

and these give the parameters to find oslash1 which depends onthe values of the constants 119860 and 119861 Now consider that if119872and 120579 are independent of 119886 then oslash1 is an affine function of 119886

In the following we give some conditions for the ampli-tude of oslash1 to find an implementable controller such that itsatisfies 0 le 1199060 le 1 and oslash1 is an affine function of theparameter 119886

In order to show that it is possible to choose oslash1 as an affinefunction of 119886 consider the solution of oslash1 in (9) and considera fixed frequency given as 119908 = 2120587119891119903 2radic119871119862 where 119891119903 is the realfrequency therefore the frequency 119908 is determined by thecircuit parameters 119871 and 119862 It is clear that if the referencesignal is modified then parameters designs 119871 and 119862must bereadjusted

From B1 and C1 in (9) we have that the value

1199082 = 2119860 + 1198961198600 (119896 + 119860) (11)

leads to 1198611 = 0 and C1 = Bw(k + A) =Bradic2radic(2A + k)(k + A)(2A(A + k) + B2) which are indepen-dent of 119886 Then oslash1 is affine to 119886 Thus119872 = 1198621 120579 = 0 and

oslash1 (119905) = 1198861198600 +119872 cos (119908119905)1 minus oslash1 (119905) = 1 +119872119908 sin (119908119905)

1198600 = 119860 (119860 + 119896) + 11986122 119872 = 119861119908 (119896 + 119860)

(12)

Now we study the conditions on the parameters 119860 and 119861in order to obtain oslash1(119905) gt 0 and (1 minus oslash1(119905)) gt 0 Defining


119863 = 119861(2119860 + 119896)1198600 and considering the value of frequency(11) then

(1 minus oslash1 (119905)) gt 0 lArrrArr 1 gt 119872119908 lArrrArr 1 gt 119863 (13)

Solving the last equation we obtain the minimum valuethat must be taken by 119860 to satisfy (1 minus oslash1(119905)) gt 0 for a givenvalue of 119861 We arrive to

119860 gt 119860119898 = minus1198962 + 119861 + 12radic1198962 + 21198612 (14)

and then if 119860 gt 119860119898 we have (1 minus oslash1(119905)) gt 0 On the otherhand defining 119865 = radic119861119863(119860 + 119896)1198600 we obtain that

oslash1 (119905) gt 0 lArrrArr 1198861198600 gt 119872 lArrrArr 119886 gt 119865 (15)

and then taking119860 = 119860119898+120575 for some 120575 gt 0 then1198600(120575)119865(120575)and119863(120575) are functions of 120575

Observing that by design it is possible to consider that119886 gt 119886119898119894119899 for some known constant 119886119898119894119899 which correspondsto some given 119877119898119886119909(119886119898119894119899 = (1119877119898119886119909)radic119871119862) and observingthat 119886119898119894119899 also can be considered as a design parameter if wetake

119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(16)

then we have the following

oslash1 (119905) gt 0 lArrrArr 119886119898119894119899 gt 119865 (120575) 120575 gt 0 (17)

Considering 119863(120575) we have that 119863(0) = 1 andlim120575997888rarrinfin119863(120575) = 0 and119889119863 (120575)119889120575 = minus2119861

1198600 (120575)2 [(119860119898 + 120575 +1198962)2 + 11989624 minus 119861

2

2 ]lt 0

(18)

By definition119860119898 gt 119861 and then119863(120575) is a decreasing functionof 120575 For 119865(120575) we have the following

119865 (0) = radic119861 (119860119898 + 119896)119860119898 (119860119898 + 119896) + 11986122 =

radic119861 (119860119898 + 119896)119861 (2119860119898 + 119896)

lim120575997888rarrinfin

119865 (120575) = 0(19)

where (19) is a decreasing function of 120575 as well becauseradic119861119863(120575) and radic(119860(120575) + 119896)1198600(120575) are also decreasing finallytaking 119886119898119894119899 = 119865(0) then (13) and (15) hold for any 120575 gt 03 Main Result on Energy Considerations

In this section some conditions are given in order to obtain animplementable controller that is 0 le 119906 le 1 considering that

the DC energymust be lowThe condition for implementablecontroller in [9] is as follows

inf (119896 + 119891) ( 119891 + 119886119891) ge sup ( 119891 + 119886119891) (20)

And a less restrictive condition given in [11] is as follows

min oslash1min (1 minus oslash1)ge (119886max 1 minus oslash1) (max 1 minus oslash1 minus 119896) (21)

The problem in both cases is that the maximum of onesignal is compared with the minimum of other signal whichgives a restrictive conditions on the DC component of thereference signal 119891(119905) losing more DC energy than necessaryConsidering this problem taking into account that in steadystate the input is given by

1199060 (119905) = 1 minus oslash1119896 + 120574 (22)

and also considering that to obtain physical implementationwe need that 1199060(119905) gt 0 and 1199060(119905) le 1 forall119905 which implies that(1 minus oslash1) le (119896 + 120574) then the first harmonic of 120574(t) is given as119891(119905) = 119860 + 119861 sin(119908119905) and (1 minus oslash1) = 1 + 119872119908 sin(119908119905) has thesame phase then it is possible to relax the above conditions((20) and (21))

Now in order to consider the conditions for the controlinput that is 0 le 119906(119905) le 1 we will study their dynamicswhich can be taken from 119906 = (1minus oslash1)119911 and written as follows

= minusoslash1119911 + (1 minus oslash1) = 119886119906 minus oslash1(1 minus oslash1)119906 minus

119886119896(1 minus oslash1)1199062 minus oslash1(1 minus oslash1)119906

3(23)

It follows that if (1 minus oslash1) gt 0 forall119905 and 119906(0) gt 0 then 119906(119905) gt0 forall119905 because the solution of the linear system is

119871 = 119886119906119871 minus oslash1(1 minus oslash1)119906119871 (24)

and the linear part of (23) is

119906119871 = ( 1 minus oslash1 (119905)1 minus oslash1 (0)) 119890119886119905119906119871 (0) (25)

and is an unstable solution that is the equilibrium point 119906 =0 of (23) is unstable Also this implies in steady state that ifoslash1 gt 0 then 1199060oslash1 = 120574 + 119886120574 gt 0 (see [12])

At this point we make the change of variable V(119905) =1 minus 119906(119905) in order to find the conditions under which 119906(119905) le1With this change of variable it is clear that V(119905) lt 1 because119906(119905) gt 0 Then if we can show that V(119905) ge 0 this implies that119906(119905) le 1 Writing the dynamics of V(119905) we foundV = (1 minus V) (Φ119905 minus 119886119896

1 minus 1206011 V minusoslash1(1 minus 1206011)V (2 minus V)) (26)

withΦ119905 = oslash1(1 minus oslash1) minus 119886 + 119886119896(1 minus oslash1) + oslash1(1 minus oslash1)


In the dynamics it can be seen that if Φ119905 ge 0 then V ge 0to see this fact we only need to observe that for any V lt 0 wehave that V gt 0 and then there does not exist a minimum forV lt 0 this implies that the minimum exists only for V ge 0Now considering the condition Φ119905 ge 0 and (1 minus oslash1) gt 0 thisis equivalent to

(1 minus oslash1)Φ119905 = oslash1 minus 119886 (1 minus oslash1) + 119886119896 + oslash1 ge 0 (27)

and taking the definition for oslash1 we can write

oslash1 = 1198861198600 +119872 cos (119908119905)1 minus oslash1 = 1 + 119908119872 sin (119908119905)

oslash1 = minus1199082 cos (119908119905) = minus1199082 (oslash1 minus 1198861198600)(28)

and the condition was transformed on the following

11988611990821198600 + 119886119896 minus 119886 (1 minus oslash1) + (1 minus 1199082) oslash1 ge 0 (29)

At this point if119908 le 1 then (1minus1199082)oslash1 ge 0 to obtain119908 le 1we use (11) and using the fact that 119908 is a decreasing functionof 120575 we obtain the values of 119861 that must be chosen to obtain119908 le 1 for any 120575 ge 0 Then for 119896 = 0 we must choose 119861 ge(radic1 + 1radic2)minus1 = 07653669 and for 119896 = 1 the value is givenfor one of the roots of the equation 1199094+21199093minus41199092minus2119909+2 = 0and the value of 119861must satisfy 119861 ge 05794246 Now it is onlyneeded to fulfill the following

(1 minus oslash1) minus 11990821198600 minus 119896 le 0 (30)

Using (1 minus oslash1) le 1 +119872119908 = 1 + 119863 it is possible to write1 +119872119908 minus 11990821198600 minus 119896 le 0 lArrrArr 119896 + 2119860 + 119896(119896 + 119860) minus 1 ge 119863 (31)

which is satisfied for any 120575 ge 0 because119863 le 1 le 119896 + 119860(119896 + 119860) for 119896 = 0 or 119896 = 1 (32)

and then with these values of 119861 119886119898119894119899 119860 = 119860119898 +120575 and 120575 gt 0it is ensured that 0 lt 119906(119905) le 1 It is important to note thatthe parameter A is the bias in the reference signal and has asmaller value than that reported in [12 14] In this sense thisis the reason for why we state that the energy consumption isreduced

4 The Adaptive Controller

Now to accomplish the control objective which consists inthe state 119909 997888rarr oslash1(119905) and 119910 997888rarr 119891(119905) = 119860 + 119861 sin(119908119905) in anapproximated way an adaptive controller is proposed For agiven 119861 gt 119861119898119894119899 and 119886119898119894119899 = 119865(0) with 119886 ge 119886119898119894119899 we determinethe amplitude 119860 and the frequency 119908 for 120575 gt 0 Consider afixed 119877119898119886119909 which corresponds to the 119886119898119894119899 then 119871 and 119862 aregiven by

119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(33)

Table 1 Comparison of the conditions for the value A of thereference signal 119891(119905)Reference Condition[11] 119860 ge 369537[14] 119860 ge 26476New condition 119860 = 1466

as an example for given 119896 = 1 119861 = 1 119891119903 = 60119867119911 120575 =01 when 119877119898119886119909 = 500Ω we obtain 119860 = 1466 119908 =06224613 119886119898119894119899 = 0412156 119871 = 34026119898119867 and 119862 =8012120583119865 In Table 1 we compare the conditions on the valueof the parameter A of the reference signal 119891(119905) with the samevalue of 119861 = 1

From the parameters 119860 119861 and 119908 the following canbe calculated oslash1 = 1198861198600 + 119872 cos(119908119905) 1 minus oslash1 = 1 +119872119908 sin(119908119905) Considering 119886 = 119886119898119894119899 + 119886119901 oslash1119898 = 1198861198981198941198991198600 +119872 cos(119908119905) oslash1= 1198861199011198600 + oslash1119898 (6) is written as

= 1198651119911 (119911 119905) + 1198861199011198652119911 (119911 119905)1198651119911 (119911 119905) = 119886119898119894119899119911 (1 minus 119896119911) minus 1199113oslash1119898 (1 minus oslash1)1198652119911 (119911 119905) = 119911 (1 minus 119896119911) minus 11986001199113 (1 minus oslash1)

(34)

where 119886119901 gt 0 implies that the system is stable Now considerthat the voltage and the current are available for feedbackthen the main result is given in the following theorem

Theorem 2 Consider 119886 = 119886119898119894119899 + 119886119901 with 119886119901 ge 0 oslash1 = 1198861198600 +119872 cos(119908120591) where1198600 = 119860(119860+119896)+11986122119872 = 119861119908(119896+119860) 1199082 =(2119860 + 119896)1198600(119896 + 119860) Assume that the pair 119860 119861 satisfy 119860 =119860119898 + 120575 119861 gt 119861119898119894119899 = (07653669)(1 minus 119896) + 119896(05794246) and120575 gt 0 if the tracking adaptive controller is given by

119909 = 1 minus (119910 + 119896) 119906 + 1198921 (119909 minus 119909)119910 = minus119886119898119894119899119910 minus 119886119901119910 + 119909119906 + 1198922 (119910 minus 119910)119886119901 = minus1198923119910 (119910 minus 119910)119911 = 1198651119911 ( 119905) + 1003816100381610038161003816100381611988611990110038161003816100381610038161003816 1198652119911 ( 119905)119906 = (1 minus oslash1)

(35)

and (0) gt 0 then 119909 997888rarr oslash1(119905) 119910 997888rarr 120574(119905) 119886119901 997888rarr 119886119901 119909 997888rarr119909 119910 997888rarr 119910 and 997888rarr 119911 globallyProof Note that is bounded taking 119881119911() = 22 we get

119911 () = (119886119898119894119899 + 1003816100381610038161003816100381611988611990110038161003816100381610038161003816) 2 (1 minus 119896)minus 4 (1003816100381610038161003816100381611988611990110038161003816100381610038161003816 1198600 + oslash1119898) (1 minus oslash1)

(36)

which is negative for 2 gt (119886119898119894119899 + |119886119901|)(1 minus 119896)(|119886119901|1198600 +oslash1119898)(1minusoslash1) and a finite value always exists because if |119886119901| = 0then it is negative for 2 gt 119886119898119894119899(1 minus 119896)oslash1119898(1 minus oslash1) and if|119886119901| 997888rarr infin then it is negative for 2 gt (1 minus 119896)1198600(1 minus oslash1)


u

u

x

yConverter

Controller

x = 1 minus (k + y)uy = minusay + xu

x = 1 minus (y + k)u + g1(x minus x)

y = minusaminy minus apy + xu + g2(y minus y)

ap = minusg3y(y minus y)

z = F1z(z t) + | |ap F2z(z t)

u = (1 minus oslash1)z

1 minus oslash1(t) = 1 + Mwsin(wt)

A0 = A(A + k) +B2

2M = Bw(k + A)

F1z(z t) = aminz(1 minus kz) minus z3oslash1m(1 minus oslash1)

F2z(z t) = z (1 minus kz) minus A0z3(1 minus oslash1)

oslash1m = aminA0 + Mcos(wt)

w2 =2A + k

A0(k + A)

Figure 2 Block diagram of the closed-loop system

also cannot take the zero value because (119886119898119894119899 + |119886119901|)2 gt 0therefore is bounded Finally 119906 and 119910 are bounded sincefor any bounded input the time response of the system also isbounded

Now consider the dynamical error system from (3) and(35) with 119890119910 = 119910 minus 119910 119890119909 = 119909 minus 119909 119890119886 = 119886119901 minus 119886119901 then

119890119909 = minus119906119890119910 minus 1198921119890119909119890119910 = minus119910119890119886 + 119890119909119906 minus 1198922119890119910119890119886 = 1198923119910119890119910

(37)

and taking 119881(119890) = 11989011990922 + 11989011991022 + 1198901198862(21198923) the timederivative is given by (119890) = minus11989211198901199092 minus 11989221198901199102 and by LaSalletheorem [18] (119890) is a decreasing function of 119905 and 119890119909 997888rarr 0119890119910 997888rarr 0 then 119910119890119886 = 0 which implies that 119890119886 = 0 since119910 = 0 Now when 119886119901 = 119886119901 the dynamics of produces thenecessary input to obtain 119909 997888rarr oslash1(119905) and 119910 997888rarr 120574(119905) whosefirst harmonic is equal to 119891(119905) = 119860 + 119861sin(119908119905) and sincethe function 119881(119890) is radially unbounded the error system isglobally stable and the controller globally stabilizes system(3) which completes the proof

The scheme that describes the closed-loop system isillustrated by means of a block diagram in Figure 2 wherethe controller and the system plant (converter) are intercon-nected

5 Numerical Results

A numerical simulation of the proposed scheme is presentedwhich illustrates that the converter buck-boost (given with119896 = 1) tracks a sinusoidal reference given by 119891(119905) = 119860 +119861sin(119908119905) where the departing parameter is the amplitudeof the reference signal given by 119861 = 1 (known design

parameter)Then the values119908 and119860must be calculated Nowconsider the following parameter values

119891 (119905) = 119860 + 119861 sin (119908119905) oslash1119898 (119905) = 1198600119886119898119894119899 +119872 cos (119908119905)1 minus oslash1 (119905) = 1 + 119908119872 sin (119908119905) 119886119898119894119899 = 0412156119860 = 1466119861 = 1119896 = 1119908 = 062246131198600 = 41152119872119908 = 095548

1198600119886119898119894119899 = 16961119872 = 1535119862 = 8012 120583119865119871 = 34026119898119867

119881119862119862 = 12119891119903 = 60119867119911

119877119898119886119909 = 500Ω

(38)

and the controller parameters are chosen as 1198921 = 1198922 = 1198923 =1 Solving (35) Figure 3 shows the voltage response and the


0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal

Figure 3 Voltage response and reference voltage signal

0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

Figure 4 Current response and reference current signal

0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time

Figure 5 Comparison between (1 minus oslash1) and (119896 + 120574)


0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)

Figure 6 Voltage and current tracking of the sinusoidal signal as given in [14]

reference signal A variation in the parameter R is applied asfollows 120591 lt 01321 (119905 lt 80) 119886119901 = 0 (119877 = 500Ω) and for120591 ge 01321 119886119901 = 1 (119877 = 14593Ω) and such a variation inthe parameter is compensated by the controller

Figure 4 shows the response of the current and thereference signals under the same parameter change onceagain the current tracks the reference with zero error

Figure 5 shows the signals (119896+ 120574) and (1 minus oslash1) for 119896 = 0 itcan be seen that it is not necessary thatmax(1minus oslash1) le min(119896+120574)

In order to indicate the improvement in the design ofthe system we present the performance of the design usedas a base of the present one [14] In Figure 6 the voltageand current as given in [14] are presented note that in theimproved design (Figures 3 and 4) the DC component is lessthan the design presented in Figure 6 as defined in [14] in thissense the new algorithm requires a reduced value for the DCcomponent representing a reduction in the consumption ofDC energy

6 Conclusions

In this contribution an adaptive control for tracking ofsinusoidal signals in DC-DC converters was described Thescheme reduces the DC energy consumption since a biasis calculated to be small and preserving the tracking of thereference signal Less restrictive conditions on the currentparameters were provided compared to those reported in[12 14] Also an adaptive control scheme to track the voltageoutput of the power converters was proposed and it wasdemonstrated that it can lead the voltage output signal toa biased sinusoidal reference In this way such a controlleris based on a nonlinear model which serves as a mean toobtain the exact necessary input signal to obtain an exacttracking of the current and its reference Also it was shownthat the adaptive controller is robust against changes in theload resistance

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] D Tan ldquoEmerging system applications and technological trendsin power electronicsrdquo IEEE Power Electronics Magazine pp2329ndash9207 2015

[2] J Fei and T Wang ldquoAdaptive fuzzy-neural-network based onRbfnn control for active power filterrdquo International Journal ofMachine Learning and Cybernetics pp 1ndash12 2018

[3] Y Chu J Fei and S Hou ldquoDynamic global proportionalintegral derivative sliding mode control using radial basisfunction neural compensator for three-phase active powerfilterrdquo Transactions of the Institute of Measurement and Controlvol 40 no 12 pp 3549ndash3559 2018

[4] J Fei and C Lu ldquoAdaptive sliding mode control of dynamicsystems using double loop recurrent neural network structurerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 29 no 4 pp 1275ndash1286 2017

[5] C K Tse ldquoCircuit theory of power factor correction inswitching convertersrdquo International Journal of Circuit Theoryand Applications vol 31 no 2 pp 157ndash198 2003

[6] H Zhang X-K Ma B-L Xue and W-Z Liu ldquoStudy ofintermittent bifurcations and chaos in boost PFC converters bynonlinear discrete modelsrdquo Chaos Solitons amp Fractals vol 23no 2 pp 431ndash444 2005

[7] J Chen F Xue X Zhang C Gong and Y Yan ldquoDesign ofthe DCDC converter for aerodynamic characteristics testingof wind turbinerdquo in Proceedings of the 1st World Non-Grid-Connected Wind Power and Energy Conference WNWEC 2009pp 1ndash6 September 2009


[8] F Blaabjerg M Liserre and K Ma ldquoPower electronics convert-ers for wind turbine systemsrdquo IEEE Transactions on IndustryApplications vol 48 no 2 pp 708ndash719 2012

[9] S SivakumarM J Sathik P SManoj andG Sundararajan ldquoAnassessment onperformance ofDC-DCconverters for renewableenergy applicationsrdquo Renewable amp Sustainable Energy Reviewsvol 58 pp 1475ndash1485 2016

[10] Z Chen ldquoDouble loop control of buck-boost converters forwide range of load resistance and reference voltagerdquo IETControlTheory amp Applications vol 6 no 7 pp 900ndash910 2012

[11] E Fossas-Colet and J M Olm-Miras ldquoAsymptotic tracking inDC-to-DC nonlinear power convertersrdquo Discrete and Continu-ous Dynamical Systems - Series B vol 2 no 2 pp 295ndash307 2002

[12] E Fossas-Colet J M Olm and A S I Zinober ldquoRobustTracking Control of DC-to-DC Nonlinear Power Convertersrsquordquoin Proceedings of the 43rd IEEE Conference on Decision andControl pp 5291ndash5296 2004

[13] Y B Shtessel A S I Zinober and I A Shkolnikov ldquoBoostand buck-boost power converters control via sliding modesusing dynamic sliding manifoldrdquo in Proceedings of the 41st IEEEConference on Decision and Control pp 2456ndash2461 December2002

[14] G Obregon-Pulido E Nuno K Castaneda and A De-Gyves ldquoAn adaptive control to perform tracking in DC to DCpower convertersrdquo in Proceedings of the 2010 7th InternationalConference on Electrical Engineering Computing Science andAutomatic Control CCE 2010 pp 188ndash191 September 2010

[15] H Sira-Ramırez E Fossas and M Fliess ldquoAn algebraic on-line parameter identification approach to uncertain dc-to-acpower conversionrdquo in Proceedings of the 41st IEEE Conferenceon Decision and Control pp 2462ndash2467 December 2002

[16] H Sira-Ramırez ldquoDC-to-AC power conversion on a boost con-verterrdquo International Journal of Robust and Nonlinear Controlvol 11 no 6 pp 589ndash600 2001

[17] R O Caceres and I Barbi ldquoA boost DC-AC converter Analysisdesign and experimentationrdquo IEEE Transactions on PowerElectronics vol 1 no 14 pp 134ndash141 1999

[18] H K KhalilNonlinear Systems PrenticeHall 3rd edition 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


such dynamics is required to be stable in order for the systemoutput to track the given reference In order to overcome thisdifficulty the indirect control method is proposed where analternative system output is considered and for such a systemthe internal dynamics is stable However other problemsarise for instance the determination of the reference forthe new system output and the control synthesis which arecommonly studied within the field of power converters Oneof them is the current mode regulation which is justified in[11] which provides the manner to find an approach to theunstable periodic solution required for the tracking problemThis solution is obtained using the harmonic balancemethod[12] On the other hand a boost topology is used in [11]in order to obtain a system that provides an output voltagegreater than the input Nevertheless it has been shown thatto control the boost and buck-boost converters the indirectcontrol of the output voltage is required [17]

In this note we propose an adaptive method applied toa boost and buck-boost topologies to achieve tracking ofa sinusoidal voltage where the reference signal is given byf (119905) = 119860 + 119861 sin(119908119905) The procedure consists in designingan adaptive controller Besides it is required to propose astable nonlinear system in order to obtain the control inputsrequired to practically track the reference signal The maincontribution consists in finding the conditions under whichthe controller is implementable and with the capability ofreducing the necessary DC energy imposed by the conditionsgiven in [11 14] In this sense the aim of the proposal is toreduce the use of theDC component in the tracking signals asestablished in [11 14] this represents a reduction in the energyrequired to generate or to track a sinusoidal signal besidesthe error in the tracking signals is practically zero and the DCcomponent is almost the minimum required which is a greatadvantage in the tracking of sinusoidal signals using DC-DCconverters

The paper is organized as follows Section 2 presents themodel of the converters and some considerations on electricalconditions In Section 3 an analysis is performed in orderto obtain conditions under which the design loses less DCenergy than that given in [6 9] Section 4 presents the devel-opment of the adaptive scheme to track practically the voltagereference In Section 5 the proposed result is corroboratedvia numerical simulations Finally the contribution is closedwith some conclusions on the tracking voltage problem inDC-DC power converters

2 Model Description

The mathematical equations which describe the boost andbuck-boost converters are given by

119871119889119868119871119889120591 = (119878119908 minus 1) (119881119862 + 119896119881119862119862) + 119881119862119862119862119889119881119862119889120591 = (1 minus 119878119908) 119868119871 minus 119881119862119877

(1)

where if k=0 we have the boost converter (Figure 1(a))otherwise if k=1 we have the buck-boost converter (Fig-ure 1(b)) VC IL and VCC are the capacitor voltage inductor

current and input voltage respectively The switching signalSw is the control signal which takes values from the set S =0 1 120591 is the nonscaled time The next change of variable isproposed in order to simplify the analysis [12]

119909 = 119868119871119881119862119862radic119871119862

119910 = 119881119862119881119862119862 119905 = 120591

2radic119871119862119886 = 1119877radic 119871119862119906 = 1 minus 119878119908

(2)

Then the dynamical model takes the form bellows

= 1 minus (119896 + 119910) 119906119910 = minus119886119910 + 119909119906 (3)

Therefore the problem consists in finding a control signalinput such that the solution 119910(119905) 997888rarr 119891(119905) gt 0 where thereference 119891(119905) is a positive signal

Multiplying (3) by 119909 and (119896+119910) the following differentialequation is obtained

(119896 + 119910) ( 119910 + 119886119910) = 119909 (1 minus ) (4)

When 119910(119905) reaches the steady state 119910(119905) = 119891(119905) given bya periodic orbit the differential equation (119896 + 119891)( 119891 + 119886119891) =119909(1minus) has a unique solution given by an unstable limit cycleoslash(119905) when 119892(119905) = (119896 + 119891)( 119891 + 119886119891) gt 0 [11] Thus to obtainconvergence of the solution 119909 997888rarr oslash1(119905) an approximation ofoslash1(119905) gt 0 is proposed and is considered as the 1st harmonicof the Fourier series of oslash(119905) From this it is clear that theseconverters can track a sinusoidal signal

As it was mentioned previously the boost and buck-boost converters require indirect control of the output voltage[11] in order to avoid instabilities in the internal dynamicsConsidering the signal oslash1(119905) as the reference for x(t) themaintracking problem is defined to control the system in orderto obtain a zero error tracking between the reference andthe output using a smooth control From (4) and considering119909(119905) = oslash1(119905) we obtain

119910 = minus119886119910 + oslash1 (1 minus oslash1)119896 + 119910 (5)

and now with the change 119911119910 = (119896 + 119910)minus1 it can be seen thatsystem (5) has a stable limit cycle 120574(119905) as the dynamics of 119911119910(119905)is

119910 = 119886119911119910 minus 1198861198961199111199102 minus 1199111199103oslash1 (1 minus oslash1) (6)

where we can see that if oslash1(119905) gt 0 and (1 minus oslash1) gt 0 thenfor any initial condition 119911119910(0) gt 0 the solution 119911119910 of system


LD

C R

+

minussw６

Ｃ

ＣＣ２

(a)

sw

６

L

D

C R

+

minus

ＣＣＣ２

(b)




(7)



(8)





oslash1 = 1198861198600 + 1198611 sin (119908119905) + 1198621 cos (119908119905) (9)

1198611 = 119872 sin (120579) = 119886119861 [(2119860 + 119896) minus 11990821198600 (119896 + 119860)]1198862119860201199082 + 11198621 = 119872 cos (120579) = 119908119861 [11988621198600 (2119860 + 119896) + (119896 + 119860)]1198862119860201199082 + 11198600 = 119860 (119860 + 119896) + 11986122

(10)





1199082 = 2119860 + 1198961198600 (119896 + 119860) (11)



1198600 = 119860 (119860 + 119896) + 11986122 119872 = 119861119908 (119896 + 119860)

(12)






119860 gt 119860119898 = minus1198962 + 119861 + 12radic1198962 + 21198612 (14)





119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(16)




1198600 (120575)2 [(119860119898 + 120575 +1198962)2 + 11989624 minus 119861

2

2 ]lt 0

(18)


119865 (0) = radic119861 (119860119898 + 119896)119860119898 (119860119898 + 119896) + 11986122 =

radic119861 (119860119898 + 119896)119861 (2119860119898 + 119896)


119865 (120575) = 0(19)




inf (119896 + 119891) ( 119891 + 119886119891) ge sup ( 119891 + 119886119891) (20)




1199060 (119905) = 1 minus oslash1119896 + 120574 (22)





3(23)
























119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(33)





(34)




(35)



(36)



u

u

x

yConverter

Controller





z = F1z(z t) + | |ap F2z(z t)



A0 = A(A + k) +B2

2M = Bw(k + A)




w2 =2A + k

A0(k + A)





(37)



5 Numerical Results




1198600119886119898119894119899 = 16961119872 = 1535119862 = 8012 120583119865119871 = 34026119898119867

119881119862119862 = 12119891119903 = 60119867119911

119877119898119886119909 = 500Ω

(38)



0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal


0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals


0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time



0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018









LD

C R

+

minussw６

Ｃ

ＣＣ２

(a)

sw

６

L

D

C R

+

minus

ＣＣＣ２

(b)




(7)



(8)





oslash1 = 1198861198600 + 1198611 sin (119908119905) + 1198621 cos (119908119905) (9)

1198611 = 119872 sin (120579) = 119886119861 [(2119860 + 119896) minus 11990821198600 (119896 + 119860)]1198862119860201199082 + 11198621 = 119872 cos (120579) = 119908119861 [11988621198600 (2119860 + 119896) + (119896 + 119860)]1198862119860201199082 + 11198600 = 119860 (119860 + 119896) + 11986122

(10)





1199082 = 2119860 + 1198961198600 (119896 + 119860) (11)



1198600 = 119860 (119860 + 119896) + 11986122 119872 = 119861119908 (119896 + 119860)

(12)






119860 gt 119860119898 = minus1198962 + 119861 + 12radic1198962 + 21198612 (14)





119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(16)




1198600 (120575)2 [(119860119898 + 120575 +1198962)2 + 11989624 minus 119861

2

2 ]lt 0

(18)


119865 (0) = radic119861 (119860119898 + 119896)119860119898 (119860119898 + 119896) + 11986122 =

radic119861 (119860119898 + 119896)119861 (2119860119898 + 119896)


119865 (120575) = 0(19)




inf (119896 + 119891) ( 119891 + 119886119891) ge sup ( 119891 + 119886119891) (20)




1199060 (119905) = 1 minus oslash1119896 + 120574 (22)





3(23)
























119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(33)





(34)




(35)



(36)



u

u

x

yConverter

Controller





z = F1z(z t) + | |ap F2z(z t)



A0 = A(A + k) +B2

2M = Bw(k + A)




w2 =2A + k

A0(k + A)





(37)



5 Numerical Results




1198600119886119898119894119899 = 16961119872 = 1535119862 = 8012 120583119865119871 = 34026119898119867

119881119862119862 = 12119891119903 = 60119867119911

119877119898119886119909 = 500Ω

(38)



0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal


0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals


0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time



0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018












119860 gt 119860119898 = minus1198962 + 119861 + 12radic1198962 + 21198612 (14)





119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(16)




1198600 (120575)2 [(119860119898 + 120575 +1198962)2 + 11989624 minus 119861

2

2 ]lt 0

(18)


119865 (0) = radic119861 (119860119898 + 119896)119860119898 (119860119898 + 119896) + 11986122 =

radic119861 (119860119898 + 119896)119861 (2119860119898 + 119896)


119865 (120575) = 0(19)




inf (119896 + 119891) ( 119891 + 119886119891) ge sup ( 119891 + 119886119891) (20)




1199060 (119905) = 1 minus oslash1119896 + 120574 (22)





3(23)
























119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(33)





(34)




(35)



(36)



u

u

x

yConverter

Controller





z = F1z(z t) + | |ap F2z(z t)



A0 = A(A + k) +B2

2M = Bw(k + A)




w2 =2A + k

A0(k + A)





(37)



5 Numerical Results




1198600119886119898119894119899 = 16961119872 = 1535119862 = 8012 120583119865119871 = 34026119898119867

119881119862119862 = 12119891119903 = 60119867119911

119877119898119886119909 = 500Ω

(38)



0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal


0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals


0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time



0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018























119862 = 1199082120587119891119903119877119898119886119909119886119898119894119899 119871 = 1199081198771198981198861199091198861198981198941198992120587119891119903

(33)





(34)




(35)



(36)



u

u

x

yConverter

Controller





z = F1z(z t) + | |ap F2z(z t)



A0 = A(A + k) +B2

2M = Bw(k + A)




w2 =2A + k

A0(k + A)





(37)



5 Numerical Results




1198600119886119898119894119899 = 16961119872 = 1535119862 = 8012 120583119865119871 = 34026119898119867

119881119862119862 = 12119891119903 = 60119867119911

119877119898119886119909 = 500Ω

(38)



0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal


0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals


0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time



0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018









u

u

x

yConverter

Controller





z = F1z(z t) + | |ap F2z(z t)



A0 = A(A + k) +B2

2M = Bw(k + A)




w2 =2A + k

A0(k + A)





(37)



5 Numerical Results




1198600119886119898119894119899 = 16961119872 = 1535119862 = 8012 120583119865119871 = 34026119898119867

119881119862119862 = 12119891119903 = 60119867119911

119877119898119886119909 = 500Ω

(38)



0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal


0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals


0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time



0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018









0 005 01 015 02 025 03 035 040

5

10

15

20

25

30

35

Time

Volta

ge R

espo

nse a

nd R

efer

ence

Sig

nal


0 005 01 015 02 025 03 035 04minus01

0

01

02

03

04

Time

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals


0

05

1

15

2

25

3

35

Sign

als

005 01 015 02 025 03 035 04 045 050Time



0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018









0

10

20

30

40

50

60Vo

ltage

Res

pons

e and

Ref

eren

ce S

igna

l

005 01 015 02 0250Time

(a)

0

02

04

06

08

1

12

Curr

ent R

espo

nse a

nd R

efer

ence

Sig

nals

005 01 015 02 0250Time

(b)






6 Conclusions


Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018



























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Energy Considerations for Tracking in DC to DC Power ...downloads.hindawi.com/journals/mpe/2019/1098243.pdf · ResearchArticle Energy Considerations for Tracking in DC to DC Power