Adaptive Control System Part 4 pps

Hence, a leakage term is addedinto the adaptation law as follows: Since the nominal plant is relative degree three, we choose the following steps to design the adaptive variable structur

(ii) the auxiliary errors eai; i ˆ 1; ; , converge to zero in ®nite time;

(iii) the output tracking errors e0 will converge to a residual set exponentiallywhose size depends on the design parameter 

Remark 4.2: It is well known that the chattering behaviour will be observed inthe input channel due to variable structure control, which causes theimplementation problem in practical design A remedy to the undesirablephenomenon is to introduce the boundary layer concept Take the case ofrelative degree one, for example, the practical redesign of the proposedadaptive variable structure controller by using boundary layer design is nowstated as follows:

for some small  > 0 Note that …e0† is now a continuous function However,

one can expect that the boundary layer design will result in bounded trackingerror, i.e e0 cannot be guaranteed to converge to zero This causes theparameter drift in parameter adaptation law Hence, a leakage term is addedinto the adaptation law as follows:

Since the nominal plant is relative degree three, we choose the following steps

to design the adaptive variable structure controller:

56 Adaptive variable structure control

reference model and reference input:

Three simulation cases are studied extensively in this example in order to verify

Adaptive Control Systems 57

all the theoretical results and corresponding claims All the cases will assumethat there are initial output error yp…0† ym…0† ˆ 4.

(1) In the ®rst case, we arbitrarily choose the initial control parameters as

j…0† ˆ 0:1; j ˆ 1; ; 6

j…0† ˆ 0:1; j ˆ 1; 2

j ˆ gjˆ 0:1 As shown in Figure 3.1 (the

time trajectories of yp and ym), the global stability, robustness, andasymptotic tracking performance are achieved

(2) In the second case, we want to demonstrate the e€ectiveness of a properchoice of j…0† and j…0† and repeat the previous simulation case by

increasing the values of the controller parameters to be

j…0† ˆ 1; j ˆ 1; ; 6

j…0† ˆ 1; j ˆ 1; 2

The better transient and tracking performance between ypand ymcan now

be observed in Figure 3.2

58 Adaptive variable structure control

(3) As commented in Remark 3.2, if there is no easy way to estimate thesuitable initial control parameters j…0† and j…0† like those in the second

simulation case, it is suggested to use large adaptation gains in order toincrease the adaptation rate of control parameters such that the nicetransient and tracking performance as described in case 2 can be retained

to some extent Hence, in this case, we use the initial control parameters

jˆ gjˆ 1 The

expected results are now shown in Figure 3.3, where rapid increase ofcontrol parameters do lead to satisfactory transient and trackingperformance

3.6Conclusion

In this chapter, a new adaptive variable structure scheme is proposed for modelreference adaptive control problems for plants with unmodelled dynamic andoutput disturbance The main contribution of the chapter is the completeversion of adaptive variable structure design for solving the robustness andperformance of the traditional MRAC problem with arbitrary relative degree

A detailed analysis of the closed-loop stability and tracking performance isgiven It is shown that without any persistent excitation the output trackingerror can be driven to zero for relative degree-one plants and driven to a smallresidual set asymptotically for plants with any higher relative degree.Furthermore, under suitable choice of initial conditions on control parameters,the tracking performance can be improved, which are hardly achievable by thetraditional MRAC schemes, especially for plants with uncertainties

Adaptive Control Systems 59

Lemma A Consider the controller design in Theorem 3.1 or 4.1 If the controlparameters j…t†; j ˆ 1; ; 2n; 1…t† and 2…t† are uniformly bounded 8t, then

there exists > 0 such that up…t† satis®es

k…up†tk 1  k…e0†tk 1 ‡  …A:1†

with some positive constant  > 0

Proof Consider the plant (3.1) which is rewritten as follows:

y…t† do…t† ˆ P…s†…1 ‡ Pu…s††‰upŠ…t† …A:2† Let f …s† be the Hurwitz polynomial with degree n  such that f …s†P…s† is

proper, and hence, f 1…s†P 1…s† is proper stable since P…s† is minimum phase

by assumption (A3) Then

y…t† do…t† ˆ P…s†f …s†f 1…s†…1 ‡ Pu…s††‰upŠ…t† …A:3†

which implies that

f 1…s†P 1…s†‰y doŠ…t† f 1…s†Pu…s†‰upŠ…t† ˆ f 1…s†‰upŠ…t† ˆ 4u …t† …A:4†

Since f 1…s†P 1…s† and f 1…s†Pu…s† are proper or strictly proper stable, we can

®nd by small gain theorem [7] that there exists > 0 such that

k…u  †tk 1  k…yp†tk 1 ‡   k…e0†tk 1 ‡  …A:5† for some suitably de®ned  > 0 and for all  2 ‰0;   Š Now if we can show that

k…up†tk 1  k…u  †tk 1 ‡  …A:6†

for some  > 0, then (A.1) is achieved By using Lemma 2.8 in [19], the keypoint to show the boundedness between up and u in (A.6) is the growingbehaviour of signal up The above statement can be stated more precisely asfollows: if up satis®es the following requirement

jup…t1†j  cjup…t1‡ T†j …A:7†

where t1 and t1‡ T are the time instants de®ned as

‰t1; t1 pj ˆ k…up†tk 1 g …A:8† and c is a constant 2 …0; 1†, then upwill be bounded by u, i.e (A.6) is achieved

Now in order to establish (A.7) and (A.8), let …Ap; Bp; Cp† and …; B† be the state space realizations of P…s†…1 ‡ Pu…s†† and a…s† …s† respectively Also

60 Adaptive variable structure control

xp

w1

w2m

264

37

375

Since do is uniformly bounded, we can easily show according to the controldesign (3.16) or (3.24) that there exists  such that

j _Sj  k…S†tk 1 ‡ 

This means that S is regular [21] so that xp; w1; w2; m; yp and up will grow atmost exponentially fast (if unbounded), which in turn guarantees (A.7) and(A.8) by Lemma 2.8 in [19] This completes our proof Q.E.D

[3] Chien, C J and Fu, L C., (1993) `An Adaptive Variable Structure Control for aClass of Nonlinear Systems', Syst Contr Lett., Vol 21, No 1, 49±57

[4] Chien, C J and Fu, L C., (1994) `An Adaptive Variable Structure Control of FastTime-varying Plants', Control Theory and Advanced Technology, Vol 10, No 4,part I, 593±620

[5] Chien, C J., Sun, K S., Wu, A C and Fu, L C., (1996) `A Robust MRAC UsingVariable Structure Design for Multivariable Plants', Automatica, Vol 32, No 6,833±848

[6] Datta, A and Ioannou, P A., (1991) `Performance Improvement versus RobustStability in Model Reference Adaptive Control', Proc CDC, 748±753

[7] Desoer, C A and Vidyasagar, M., (1975) Feedback Systems: Input±OutputProperties, Academic Press, NY

[8] Filippov, A F., (1964) `Di€erential Equations with Discontinuous Right-handSide', Amer Math Soc Transl., Vol 42, 199±231

[9] Fu, L C., (1991) `A Robust Model Reference Adaptive Control Using VariableStructure Adaptation for a Class of Plants', Int J Control, Vol 53, 1359±1375.[10] Fu, L C., (1992) `A New Robust Model Reference Adaptive Control UsingVariable Structure Design for Plants with Relative Degree Two', Automatica,Vol 28, No 5, 911±926

[11] Hsu, L and Costa, R R., (1989) `Variable Structure Model Reference AdaptiveControl Using Only Input and Output Measurement: Part 1', Int J Control, Vol

49, 339±419

Adaptive Control Systems 61

[12] Hsu, L., (1990) `Variable Structure Model-Reference Adaptive Control MRAC) Using Only Input Output Measurements: the General Case', IEEETrans Automatic Control, Vol 35, 1238±1243.

([13] Hsu, L and Lizarralde, F., (1992) `Redesign and Stability Analysis of I/O MRAC Systems', Proc American Control Conference, 2725±2729

VS-[14] Hsu, L., de Araujo A D and Costa, R R., (1994) `Analysis and Design of I/OBased Variable Structure Adaptive Control', IEEE Trans Automatic Control, Vol

[18] Narendra, K S and Annaswamy, A M., (1987) `A New Adaptive Law for RobustAdaptation Without Persistent Excitation', IEEE Trans Automatic Control, Vol

32, 134±145

[19] Narendra, K S and Valavani, L., (1989) Stable Adaptive Systems, Prentice-Hall.[20] Narendra, K S and BoÆskovicÂ, J D., (1992) `A Combined Direct, Indirect andVariable Structure Method for Robust Adaptive Control', IEEE Trans AutomaticControl, Vol 37, 262±268

[21] Sastry, S S and Bodson, M., (1989) Adaptive Control: Stability, Convergence, andRobustness, Prentice-Hall, Englewood Cli€s, NJ

[22] Sun, J., (1991) `A Modi®ed Model Reference Adaptive Control Scheme forImproved Transient Performance', Proc American Control Conference, 150±155.[23] Wu, A C., Fu, L C and Hsu, C F., (1992) `Robust MRAC for Plants withArbitrary Relative Degree Using a Variable Structure Design', Proc AmericanControl Conference, 2735±2739

[24] Wu, A C and Fu, L C., (1994) `New Decentralized MRAC Algorithms for Scale Uncertain Dynamic Systems', Proc American Control Conference,3389±3393

Large-62 Adaptive variable structure control

of the controllability/observability matrix's estimate As compared to previousstudies in the subject the controller proposed here does not require the frequentintroduction of periodic n-length sequences of zero inputs Therefore the newcontroller is such that the system is almost always operating in closed loopwhich should lead to better performance characteristics.

4.1 Introduction

The problem of adaptive control of possibly nonminimum phase systems hasreceived several solutions over the past decade These solutions can be dividedinto several di€erent categories depending on the a priori knowledge on theplant, and on whether persistent excitation can be added into the system or not.Schemes based on persistent excitation were proposed in [1], [11] amongothers This approach has been thoroughly studied and is based on the factthat convergence of the estimates to the true plant parameter values isguaranteed when the plant input and output are rich enough Stability of the

closed loop system follows from the unbiased convergence of the estimates.The external persistent excitation signal is then required to be always present,therefore the plant output cannot exactly converge to its desired value because

of the external dither signal This diculty has been removed in [2] using a excitation technique In this approach excitation is introduced periodicallyduring some pre-speci®ed intervals as long as the plant state and/or outputhave not reached their desired values Once the control objectives areaccomplished the excitation is automatically removed Stability of these type

self-of schemes is guaranteed in spite self-of the fact that convergence self-of the parameterestimates to their true values is not assured This technique has also beenextended to the case of systems for which only an upper bound on the plantorder is known in [12] for the discrete-time case and in [13] for the continuous-time case

Since adding extra perturbations into the system is not always feasible ordesirable, other adaptive techniques not resorting to persistent exciting signalshave been developed Di€erent strategies have been proposed depending on theavailable information on the system

When the parameters are known to belong to given intervals or convex setsinside the controllable regions in the parameter space, the schemes in [9] or [10]can be used, respectively These controllers require a priori knowledge of suchcontrollable regions An alternative method proposes the use of a pre-speci®ednumber of di€erent controllers together with a switching strategy to commuteamong them (see [8]) This method o€ers an interesting solution for the caseswhen the number of possible controllers in the set is ®nite and available In thegeneral case the required number of controllers may be large so as to guaranteethat the set contains a stabilizing controller

In general, very little is known about the structure of the admissible regions

in the parameter space This explains the diculties encountered in the search

of adaptive controllers not relying on exciting signals and using the order of theplant as the only a priori information In this line of research a di€erentapproach to avoid singularities in adaptive control has been proposed in [6]which only requires the order of the plant as available information Themethod consists of an appropriate modi®cation to the parameter estimates

so that, while retaining all their convergence properties, they are brought to theadmissible area The extension of this scheme to the stochastic case has beencarried out in [7] The extension of this technique to the minimum phasemultivariable case can be found in [4] This method does not require any apriori knowledge on the structure of the controllable region It can also beviewed as the solution of a least-squares parameter estimation problem subject

to the constraint that the estimates belong to the admissible area Theadmissible area is de®ned here as those points in the parameter space whosecorresponding Sylvester resultant matrix is nonsingular The main drawback ofthe scheme presented in [6] is that the number of directions to be explored in

64 Indirect adaptive periodic control

the search for an appropriate modi®cation becomes very large as the order ofthe system increases This is due essentially to the fact that the determinant ofthe Sylvester resultant matrix is a very complex function of the parameters.The method based on parameter modi®cation has also been previously used

in [3] for a particular lifting plant representation The plant descriptionproposed in [3] has more parameters than the original plant, but has thevery appealing feature of explicitly depending on the system's controllabilitymatrix Indeed, one of the matrix coecients in the new parametrization turnsout to be precisely the controllability matrix times the observability matrix.Therefore, the estimates modi®cation can actually be computed straightfor-wardly without having to explore a large number of possible directions as is thecase in [6] It actually requires one on-line computation involving a polardecomposition of the estimate of the controllability matrix However, noindication was given in [3] on how this computation can be e€ectively carriedout Recently, [14] presented an interesting direct adaptive control scheme forthe same class of liftings proposed in [3] As pointed out in [14] the polardecomposition can be written in terms of a singular value decomposition (SVD)which is more widely known Methods to perform SVD are readily available.This puts the adaptive periodic controllers in [3] and [14] into a much betterperspective At this point it should be highlighted that even if persistentexcitation is allowed into the system, the presented adaptive control schemeso€er a better performance during the transient period by avoiding singularities.The adaptive controller proposed in [3] and [14] is based on a periodiccontroller A dead-beat controller is used in one half of the cycle and the input

is identically zero during the other half of the cycle Therefore a weakness ofthis type of controllers is that the system is left in open loop half of the time Inthis chapter we propose a solution to this problem

As compared to [3] and [14] the controller proposed here does not require thefrequent introduction of periodic n-length sequences of zero inputs The newcontrol strategy is a periodic controller calculated every n-steps, n being theorder of the system For technical reasons we still have to introduce a periodicsequence of zero inputs but the periodicity can be arbitrarily large As a resultthe new controller is such that the system is almost always operating in closedloop which should lead to better performance characteristics

where x is the …n  1† state vector, u, y are respectively the plant's input and

output and A; b; cT are matrices of appropriate dimensions Signals v0v00can beidenti®ed as perturbations belonging to a class, which will be discussed later.The state part of equation (4.1) is iterated n times:

5 G 2 R

nn Yt ˆ

yt n

yt 1

26

37

Since the state is not supposed to be measurable, we will obtain an expressionthat depends only on the system's output and input Introducing (4.4) into theabove it follows

t‡2n is another noise term Therefore the state xt‡n ; t ˆ 0; n; 2n; is

bounded by the (bounded) noise In the ideal case xt‡nwill be identically zero.From (4.3), it is clear that Ut is bounded and becomes zero in the ideal case.Remark 2.1 Dead-beat control can induce large control inputs For asmoother performance, one can use the following control strategy:

BUtˆ DYt B0Ut n‡ C Y ‰ t GUt nŠ …4:18†

where C has all its eigenvalues inside the unit disc The closed loop system is inthis case:

Yt‡2n GUt‡n ˆ C Y ‰ t GUt nŠ ‡ N t‡2n …4:19†

In the ideal case the LHS of the above converges to zero and so does the state

Adaptive Control Systems 67

49 , 339? ?41 9

Adaptive Control Systems 61