Variable structure control

based decentralised adaptive

G. Feng Y.A. Jiang

Inderrng fwms Aduptire control. Derenrrulised control. Stability. Vuriuble structure sjsrems

Abstract: A new scheme for decentralised adapt- ive control is proposed. This scheme is based on a variable structure adaptive controller and a pro- portional controller. The adaptive variable struc- ture component of this scheme is used to compensate uncertain interconnections among the subsystems and to ensure global stability of the overall system. The design of the adaptive control- ler is totally model free. The simulation results are also presented to demonstrate the performance of the closed loop control system.

1 Introduction

A number of approaches have been developed for decen- tralised adaptive control of large-scale systems during the last decade. The large-scale systems can usually be con- sidered to be composed of a set of small interconnected subsystems. The essential uncertainty of each subsystem is encountered in the interactions among the subsystems. Therefore, the major concern in decentralised adaptive control for large-scale systems is how to deal with those uncertain interactions. Many papers have been devoted to accounting for uncertain interactions among the sub- systems 11-8. 10-123.

Earlier versions of the decentralised adaptive control methods are based on the applications of model reference adapative controllers [9] to the control of unknown sub- systems [6-81. The standard M-matrix conditions of the bounds on the interconnections have been used to estab- lish the stability of the overall system. Both the sub- systems and interconnection parameters need not be known. A problem with these design methods is that the positive definiteness of the M-matrix cannot easily be checked a priori. since the entries in the M-matrix depend on the unknown system parameters [IO]. Another decen- tralised adaptive control method is based on high gain stabilisation technique 13, 51.

Most of these works are based on the assumption that the interconnections are constant, slowly varying. or bounded by known or unknown first-order polynomials in states [3, 8, 111. However, in practice, there do exist large-scale systems whose interconnections among sub- systems are of high order, as indicated in Reference 12. Higher-order interconnections can potentially render a system unstable if these interconnections are not explic- itly catered for. Recently. the authors 1121 proposed an

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IEE. 1995 Paper 1883D (CX), received 24th August 1994 The authors are with the Department of Systems and Control. School of Electrical Engineering, University of New South Wales, Kensington, NSW 2033. Australia

1t.E Proc.-Conrrol Theory App i . . Vol. 142. No. 3, September 1995

adaptive decentralised control method which is based on a high gain stabilisation technique. They used the adapt- ive controller with explicit higher order polynomials of the states to guarantee the stability of the overall system when the higher order interconnections are present. One major disadvantage with the adaptive control scheme in Reference 12 is that the convergence rate of the adaptive system would be very slow because the number of param- eters to be estimated could be quite large, which is also common for ordinary adaptive control design, and the control performance could be poor as shown in the simu- lated two inverted pendulums example.

In this paper, we develop a new simple decentralised adaptive control scheme for large-scale systems with higher order interconnections. The basic idea is to use a variable structure adaptive controller to guarantee the system stability and to account for the uncertain higher order interactions. The new scheme has a number of advantages over the scheme in Reference 12. For instance, the design of the adaptive controller in this paper is model free. The number of parameters to be updated is only one, and thus the convergence rate of the adaptive system is expected to be faster. It is also demon- strated through a numerical example that much better control performance could be obtained with our method.

2 Problem formulation

Let us consider a large-scale system which is decomposed into N subsystems. Each subsystem Si, i E N , is rep- resented [12] as:

(1) where xit) E R"' and ui(t) E R are the state vector and the control input of Si at time t , respectively, Ai E R"' x R"' is the unknown constant matrix, and Bi E R"' is the unknown constant vector. z i t , x): R x R"+ R, is the unknown function which represents the interconnections among the subsystems, where n = ni denotes the order of the overall system.

Remark I: As indicated in Reference 12, the above description of the large-scale system implies that the interconnections satisfy the matching conditions [3, 123, that is, all the interconnections should fall into the range space of the control vector Bi. These matching conditions restrict the structure of interconnections of the system, and thus the class of large-scale systems to be concerned. However, many mechanical systems do belong to this class of systems 1121.

Si : ii = Ai xi + Bi ui + Bi zi(t, X)

The authors are grateful to the reviewer for several constructive comments upon which this paperhasbeenimproved.

439

Then, the overall system S, which is composed of the subsystems Si interconnected through the variable I,. can be expressed in a compact form :

S: I = A\- + Bu + Bz( t , .\) (2)

where x(t) E R", u(t) E RI', and z(t. x) E RS are the state vector, control input vector and interconnections of S at time t , respectively. A E R" x R" is the unknown constant block matrix defined by A = diag { A l , A , , . . . , AN), and B E R" is the unknown constant matrix defined by B = diag {El , E , , . . . , E N } .

As in References 3 and 12, we make the following assumptions about the system.

Assumption I : The pair ( A i , Bi) for all i E N , is controll- able. This assumption guarantees a choice of the coordi- nate system such that the matrices of the decoupled systems have the control canonical form as

0 1 _ ' ' 0

A . = [ r ; " ' . . . ! I B i = [ ; \ (3)

- a i , -ai , " ' -aim bi

Assumption 2: The interconnections are bounded by a pth-order polynomial in states, (i.e. there exist unknown nonnegative constants c i j ) such that

(4)

where

W~ = 1 + IIxjlj + IIxjl12 + . . . + I Ix j / ip

and vector norms are chosen as Euclidean and matrix norms are induced ones.

Remark 2: It can be easily seen that assumption 2 in this paper and assumption A2 in Reference 12 are actually equivalent.

The control objective concerned in this paper is either to regulate the state x ( t ) of the overall system S to zero or to force it to track the state of a given reference model. However, the first case can be covered by the second case with suitable choice of reference model and initial condi- tion. Therefore, in the following development, only the model following is discussed.

The reference trajectory is supposed to be generated by a set of linear reference models,

M i : imi = Ami.xmi + Emiri i E N ( 5 ) which have the same dimensions as Si and the stability characteristics we would expect Si to have. It can be seen that A,, should be a stable constant matrix. Therefore, for any symmetric positive definite matrix Q i , there exists a unique symmetric positive definite matrix Pi as the solution of the following Lyapunov matrix equation:

(6) The overall reference model can also be rewritten in a compact form as

(7) where r ( t ) E R v is the reference input vector and A,,, = diag {A,,, A,, , . . . , AmN}. B = diag {E,,,,,, E,, , . . . , E,,}.

For each local model M i the coordinates are chosen such that the pairs (Ami, Bmi) are also in the control canonical form as in eqn. 3. Then, with this choice of

A i i P i + P , Ami = -Qi

M : i, = A,,.x,,, + B,r

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coordinates, it is easily verified that there exist constant vectors k: and constants k& such that

Ami = A i + Bi kTT Bmi = Bi kXi (8) However, these model matching conditions are unknown since the subsystem dynamics are unknown. We will assume that the sign of bi is known and, without loss of generality, we set k& > 0, i E N .

In the next Section, we are going to develop a decen- tralised adaptive controller, which is able to achieve the above control goal with guaranteed stability.

3 New decentralised adaptive control

Defining the error e , ( / ) for each subsystem as ei(t) = x i ( t ) - x,,(t), i E N , the new decentralised adaptive control scheme can be expressed as

(9 ) ui = up; + usi where

. = -Pie :, P i e i pi > 0 Pi

is a proportional controller and uSi is a variable structure controller.

Remark 3: It is obvious that the control design of upi is model free. Although this component is not required for stability of the system, it is believed that its introduction may improve the performance of the overall control system by increasing the damping of the overall system.

Then, we can obtain the closed loop error dynamics:

@i = Ami ei + Bi(ui + z i - kTTxi - k*, 0 ' r i ) (11) with k: and k& defined in eqn. 8.

lowing lemmas can be established.

Lemma I : The term zi - kTTxi - k& ri in eqn. 1 1 can be expressed as

(12)

(13)

Before we present our controller design for u S i , the fol-

Y

l,zi - kTTxi - k&riI; < 1 d i j r j

c j = 1 + IIejlI + /lejliZ + . . . + /lejllP

j = 1 where

with suitable constants d i j s .

Proof: The result can be easily verified by observing the definition of error e j and by noting that ri is the reference input whose norm is bounded, and that llkTrll and k& are bounded constants.

Lemma 2: The following inequality holds: N N Y

;S Ilejll 1 vi < N 1 llejllcj j = 1 j = , j = I

Proof: According to the definition have

lieill < IIejll o u i < v j Vi, j Then, the results follows based inequality,

N N Y

1 s j 1 ti < N 1 s j t j j = 1 j = 1 j = 1

of v j in eqn. 12, we

( 1 5 )

on the Chebyshev

0 < S I < s, < " ' < S H 0 < t , < t , < . ' . < t ,

The two cases will be considered. In the first case, the parameters d i j are supposed to be known, while in the

IEE Proc.-Control Theory App l . , Vol. 142, N o . 5 , September 1995

second case, the parameters d , are assumed to be unknown.

3.1 Known d,, The second term uJt) of the control law in eqn. 9 is based on the variable structure control, that is,

1 sign (EL, P , e , ) U = - -

I1 E:, P, I SL gc,

where g is given by

g = 2N max (llB:iPiJ) max ( d i j ) (17)

With the above decentralised semiadaptive control law, we can establish the following global stability results.

Theorem 1: Under assumptions 1 and 2 with known bounded constants d i j , the decentralised adaptive control system is globally stable in the sense that all the signals in the closed loop system are bounded. And moreover, the tracking error ei will approach to zero as time goes to infinity.

Proof': Define a Lyapunov function candidate as

i. j

Then its derivative along the solutions of eqn. 1 1 can be obtained as

+ 2N max (/lB:iPiJ) max (dij) 1 lleil,ui i . j i = l

(Lemma 2) N

< 1 { - kZi erQi ei - 2pi ( ( E:, Pi ei ( ( i = 1

+ 2B:,Pieiu, + g l l e i / / u i } (19)

I E E Proc.-Control Theory Appl.. Vol. 142, No. j, September 1995

is obviously a bounded constant. Then, by using the control law defined in eqn. 16, we have

N V < 1 { -k&e[Qiei - 2j?i~~B~iPiei1/2}

i= I

N

< ,E { - k t i M 4 2 - 2PillB:i Pil1211eil12) ,= 1

N

< { - ( & A i + 2BillBLi~illZ)ll~il12~ (21) i = l

where Ai = min {eig ( Q J } . It can then be concluded according to Lyapunov theory that e,{i E N ) are bounded and thus all the variables in the closed loop system are bounded. Moreover, the tracking error ei s will approach to zero as time goes to infinity.

Remark 4 : It is noted that, according to eqn. 21, the exponential convergence of I ei I can be proved, and more- over, the sliding surface s = BT Pe can be reached in finite time [14].

3.2 d , unknown In this case, the bound g is also unknown. Therefore, the above control scheme cannot be implemented. Fortu- nately, adaptation can be used in the above algorithm. That is, the second part of the control law will be

1 sign (BLiPie i ) . = - - 9.0.

2 IlBZi Pill and ii is updated by the following law,

ii = +yillei/lui i i (0) = o (23)

With the above decentralised adaptive control law, we can establish the following global stability results.

Theorem 2: Under assumptions 1 and 2 with unknown but bounded constants dij, the decentralised adaptive control system is globally stable in the sense that all the signals in the closed loop system are bounded and the tracking errors ei s will approach to zero as time goes to infinity. Proof: Define a Lyapunov function candidate as

where Si = ii - g.

be obtained as Then its derivative along the solutions of eqn. 11 can

N V < { - k& e'Qi e i + 2B:, P i e i ui

i = 1

+ g/leil/ui + 1leilivi(ii - 9)) N

C

+ 2B:iPieiu,i + lleillvig^i}

< 1 { - k& efQi ei - 2pi/l ELi P i eil/ '}

< i = 1 1 {-k&e'Qiei - 2Bi / l~~ i~ i l1211~ i l12 )

{ - k,*i e[Qi ei - 2pi11 ELi Pi ei/l i = 1

N

i = 1

N

N c { + & A i + 28i11B~i~i/12)l/ei/12~ (25)

i = I

441

where i., = min (eig (Q,)}. Similarly to the proof of Theorem 1, it then can be concluded that e, and Q, (i E N ) are bounded and thus all the variables in the closed loop system are bounded. Moreover, the tracking errors e , s will approach to zero as time goes to infinity.

The transient bound for the error e(t) can also be found. Taking integration of both sides of eqn. 25, one obtains

rr N

I 0 - Y l 4 Y

Thus,

< V(0)

which is actually the L , performance.

Remark 5 : It should be noted that the design of the control laws defined in eqns. 16 and 22 are actually model free.

Remark 6 : The switching control law defined in eqns. 16 or 22 is discontinuous with respect to its argument, which is known to display chattering phenomena [14]. As sug- gested in Reference 13, this problem can be avoided by the so-called boundary layer approach, that is, sign (x) is replaced by (xjnorm (x) + 6) with 6 being a small positive constant.

4 Simulation example

To demonstrate the performance of the proposed decen- tralised adaptive controller, the numerical example used in Reference 12 will be considered here.

The two inverted pendulums are connected by a moving spring mounted on two carts as shown in Fig. 1.

Fig. 1 Two incerred pendulums on carts

I t is assumed that the pivot position of the moving spring is a function of time which can change along the full length I of the pendulums. The motion of the carts is specified as sinusoidal trajectories. The input to each pen- dulum is the torque ui applied at the pivot point. The control objective is to find suitable ui so that each pendu- lum tracks its own desired reference trajectory while the connected spring and carts are moving.

and xz = ( i2 , I)~)', choosing k = 1 , I = 1, M = rn, the dynamic

Defining the state vectors x, = ($,,

4 4 2

model for the system can be given as

i = 1, 2 i i = A i x i + B i u i + Bizi

where - -

with

y , = sin (2t) y2 = 2 + sin (3 t )

and the uncertainty u(t) = sin (5t). It has been shown 1123 that the interconnections are

bounded by a second-order polynomial in states and the decentralised adaptive controller proposed in Reference 12 can obtain the stability of the system and reasonable control performance.

The reference model in eqn. 5 is chosen in Reference 12

With Q = 41, where I is the identity matrix with suitable dimensions, we can get

Pi=[; ;] . : , P i = [ ; ]

As in Reference 12, the regulation problem will be con- sidered in this example. The design parameters are set as pi = 30, yi = 80, ui = 0.005 (ui is the parameter in the usual a-modification for update law (eqn. (23) [6]), and 6 = 0.05. Then, with the initial conditions e,(O) = (0.5, O)= and e,(O) = (-0.5, O)T, we have the following simulation results.

Fig. 2 shows the results when m = 10 and Fig. 3 shows the results when rn = 50.

In both Figures, a, b, c and d represent position track- ing errors, overall control signals, estimated gain gi s and compensator control signals, respectively. It should be noted that, in c, the estimated g i s are the same for both pendulums.

It can be observed that the closed loop decentralised adaptive control system resulting from the design in this paper is stable in both cases, and the tracking per- formance is much better than that obtained in Reference 12.

5 Conclusions

In this paper, a new decentralised variable structure adaptive control scheme has been developed. Stability analysis is provided for the closed loop adaptive system. The scheme is able to achieve zero tracking error while guaranteeing the stability of the overall system. The feasi- bility and performance of the scheme are demonstrated through a numerical simulation example.

I E E Proc -Control Theory App l . , Vol. 142, N o . 5, September 1995

E 05 0

aJ L L

6 0 - 6

- 0 5 L

,- el 1

- ,,,-

/ \ e 2 1

hu 0 10 20 30

time, s

b

-10

6 3 0 - U

0 10 20 30 tlme, s

c

Fig. 2 Decenrralised adaptice control m = I O

0 10 20 30 time, s

d

c 0 10

Q

0

II F e 2 1

-05’

0 10 20 30 tirne.5

a

0 10 20 30 time+

c

Fig. 3 Decenrralised adaptice control m = 50

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I E E Proc.-Conrrol Theorj Appl.. Vol. 142, No. 5 , September 1995

0 10 20 30 t !me, s

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I I

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d

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