Adaptive Control of Discrete-Time Systems Subjected to Bounded Disturbances with Unknown Bounds

Adaptive Control of Discrete-Time Systems Subjected to Bounded Disturbances with Unknown Bounds

Copyright © IFAC System Identification , Kitakyushu, Fukuoka. Japan, 1997 ADAPTIVE CONTROL OF DISCRETE-TIME SYSTEMS SUBJECTED TO BOUNDED DISTURBANCES...

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Copyright © IFAC System Identification , Kitakyushu, Fukuoka. Japan, 1997

ADAPTIVE CONTROL OF DISCRETE-TIME SYSTEMS SUBJECTED TO BOUNDED DISTURBANCES WITH UNKNOWN BOUNDS

L.S.Zhiteckij

Dept. of A uta. Control & Data Processing Systems V M Glushkov Institute of Cybernetics 40, Prospect Akademika Glushkova MSP 252650 Kiev 22 Ukraine

Abstract : The adaptive control algorithms for mInImum and norummmum, linear, time-invariant, discrete-time systems in the presence of bounded disturbances are presented. No infonnation about these bounds is required. It does not require a priory knowledge of bounds on unknown plant parameters includling a lower bound on the absolute value of high frequency gain and its sign . The convergence and stability properties of adaptive laws are established. Simulation results have been given . Keywords: Adaptive control, convergence, discrete time, estimation, stability

I. INTRODUCTION

unknown bounds (Bondarko, 1983; Zhiteckij , 1995). But, knowledge of a region to which belong the unknown plant parameters is required in this case. In a recent paper of Feng (1994) it has been shown that a priori knowledge of a range for each parameter is not needed to ensure the robust stability of discrete-time minimum phase systems. However, a lower bound on the high frequency gain must be known to implement this approach. In the later paper (Suarez et al., 1994), authors exploit the similar approach to design the adaptive control of continuous-time, minimum phase system for the case of unknown bound on disturbance, but they do not require knowledge about the high frequency gain . However, it is not yet clear how these results might be extended to the discrete-time system .

Since 1980s, the significant progress has been made in the identification and control theory by using so-called bounding approaches, see the recent work of Milanese et at. (1996). Within these approaches, various adaptive control schemes have been presented before for discrete-time, minimum and nonminimum phase systems with bounded disturbances. See (Egardt, 1980; Fomin et af. , 1981; Goodwin et al., 1984; Gu and Wang, 1989; Lozano-Leal and Collado, 1989; Martin-Sanchez, 1984; Ortega and Lozano-Leal , 1987; Samson, 1983). To cope with these disturbances, a dead zone has been introduced in the adaptive law . The common feature of most of the adaptive control algorithms is that an upper bound on the disturbance must be explicitly known to choose the appropriate size of the dead zone. With this assumption , the convergence and stability properties of adaptive control algorithms is proved. Unfortunately, knowledge of an upper bound on the unmeasured disturbance can hardly be found in practice.

In this paper, new robust adaptive control algorithms for discrete-time systems in the presence of bounded disturbances with unknown bounds are presented. The proposed approach is based on a combination of some ideas developed in Feng (1994), Lozano-Leal and Goodwin (1985), Wen and Hill (1992) and Zhiteckij (1995) . The key feature of these algorithms is that they are applicable to both minimum and not necessarily minimum phase plants and do not require knowledge of bounds on their parameters. To cope

It turns out that the bounding approaches may also by used for ensuring the stability of discrete-time systems in the presence of bounded disturbances with

311

with the singularity that may appear in the control law for minimum phase system , the adjustable step coefficients in the adaptation algorithms are employed as in Fomin et al. (1981) and Ortega and Lozano-Leal (1987). To avoid such a singularity which may occur in the nonmimimum phase case, an approach similar to the one proposed in Lozano-Leal and Goodwin (1985) is utilized.

Iim sup t --+

ie(t)1=:;

EO

(6)

+<>0

with some finite EO < + 00 . The following cases are studies: first when plant (I) is minimum phase as in Feng (1994), and second when it is not necessarily minimum phase.

2. PROBLEM STATEMENT 3. ADAPTIVE CONTROL ALGORITHM OF MINIMUM PHASE PLANT

It is considered a discrete-time, linear, time-invariant

single-input single-output plant described by For this case, it is assumed that :

A (= - I)y( I) = B(= - I )u( t - k) + I;(t) ,

(A3) the polynomial B(z - I) has the roots inside the unit circle only, i.e .,

(I)

B(z - I)

where y et) and u(t) represent the scalar output and control input, respectively, I;(t) denotes the unmeasurable disturbance, t = I , 2, ... is the discrete time. positive integer k ~ I is the time delay and

0 for all lzl ~ 1.

(7)

Remark 1. In contrast to Feng (1994), where the same assumption (A3) is made, no lower bound on Ibl l in (3) and the sign of b I are required.

A(z- I) and B(=- I) are the scalar polynomials in the

inverse shift operator =- I , written as A(z - I) = I + alz - I + ... + a;: -/ ,

;t

As in Feng (1994), F omin et al. (1981 , Chapter 3) and Goodwin et at. (1984, Section 2) the control law is synthesized by utilizing pole assignment strategy. To

(2)

this end, define the unique polynomials F(= - I) and G(= - I) satisfying the identity

(3)

(8)

The coefficients of A(z - I) and B(= - I) are assumed to be unknown . Moreover, no a priori knowledge of bounds on these coefficients is required.

with F(z - I) monic of degree k - I and G(z - I) of degree 1- I . Using (8), it can be shown that equation (1) results in

The following main assumptions are made: yet + k) = - G(z - I)y(t)

(A 1) Integers I and k are known. (A2) The disturbance i;;(t) is upper bounded:

+ F(z - I)B(z - I)u(t) + F(z - 1)I;(t + k) II;(t) ! =:; 11

for all t,

(4)

or where 11 is unknown .

yet + k) = 8 T ((>(t) + ~(t + k),

As in Fomin et at. (1981 , Chapter 4), Ortega and Lozano-Leal (1987) and Wen and Hill (1992),

where

suppose y 0 is a given set-point for output y . The control objective is to design a control law such that the output error

~(t)

T

= [u(t),

u(t - 1), ... , u(t -1- k + 2),

-yet), -)~t-I) , .. . , -y(t-I+ 1] e(t)

=y 0 -

yet)

(5)

= [u(t), ((>'(t) T] , satisfies

312

(9)

y(t) is chosen as in Fomin et al. (1981 , Chapter 4) and

Ortega and Lozano-Leal (1987) from and ~(t): = F(z- I)~(I). Due to (4), the last tenn of(9) satisfies

(17)

o < y' $ yet) $ y" < 2

so that

(10)

(18) with some finite 'il because the coefficients of F(z - I) are finite.

x(t) is a posItIve variable specified later and 11·11 denotes the Euclidean vector nonn. The initial variables are chosen as follows: e(O) is arbitrary vector, however, the requirement el(O) -:F- 0 must be

According to Feng (1994, Section 2) and Fomin et at. (1981 , Section 4.1.4°), the adaptive control law is defined by u(t)

where

= [y0 -

8Jt) T~(t)] l el(t),

ey) = [el(l), 8Jt) T] T

is

an

satisfied, 'il(0) is any non-negative number; 8 is arbitrary, sufficiently small positive number chosen by the designer.

(11)

estimate

Remark 2 . In contrast to Feng (1994), a projection in estimation algorithm (14) is not employed. Since X(t) > 0, it follows from (15) that 'il(t) is a nondecreasing 1\ function of t (as the similar estimate B(t) in Feng (1994». But, 'il(t) is detennined by another way.

of

unknown 8 in (9). Defining the estimation error

E(t) = e(t - 1) T
The stability and convergence properties of the adaptive control algorithm are summarized in the following theorem.

(12)

and a dead-zone function of E as

Theorem I. Denote by t . a time instant when the }

inequality IE(t)1 $ E 0(1 - I) is violated at t = t for the }

jth time (j = 1,2, .. . ). If X(t) satisfies

(13)

L the parameter estimation algorithm is designed as the gradient procedure with the adjustable dead zone:

8(t - 1) 8(t) =

for 11
8(t_I)_y(tI E(t),

-

and (A2) and (A3) hold, then the closed-loop system consisting of the plant (I) and the adaptive controller (1 I) to (18) has the properties: a) there exists a finite t * and constants

'il(t-I), £ O(t-I»
(14)

l

and 8 *

b) the error e(t) is bounded, therein

in which 'il(l) is updated by the law

'il(t - I) + X(t)

11*

such that 'il(t) == 11* and ey) == e_ * for all t 2': t *;

-

otherwise,

'il(t) =

(19)

f

k)1I = 0,

11
X(t)~oc asj~oo }

I.

i~£(t) > E O(t_l) ,

le( t)1 $

Ti * + 8

for all t 2': t' + k.

(20)

Proof Due to space limitation, an outline of the proof is presented. To prove part a), suppose that {t) is

( 15)

'il(t - 1) otherwise,

infinite sequence. Because of (19) and (15), there exists a time instant to such that E O(t) 2': 'il(t) + 8 'It 2':

where 'il(t) is an estimate of unknown 'il given by the constraint (10),

£ O(t) = 'il(t) + 8

(8 >0),

to. Further, if repeat the proof of Theorem 4. 1.3 of Fomin et at. (1981), then it can establish that in this case there exists a constant Q such that

(16)

313

11
Q

4. ADAPTIVE CONTROL ALGORITHM OF NOT NECESSARIL Y MINIMUM PHASE PLANT

Vt. Now, define a Lyapunov type

function V(t) =

118 - 8(t)112.

Now, condition (7) is not required. This implies that plant (I) may be nonminimum phase. As in Goodwin and Sin (1981), Lozano-Leal and Goodwin (1985), Wen and Hill (1992) and Zhiteckij (1995), the following assumption will be used:

(21 )

Then , with the arguments as in proving Theorem 2.1.1 a from (Fomin et al., 1981) it can be shown that

(A4) A(:- I) and B(z-I) are relatively prime. (A5) k = I.

V(t) - VC) _ I) ::; - y'(2 - y")Q- 2 Vt"? to. This gives

that {t} is the finite sequence, contradicting the )

hypothesis. Therefore, {t}

is indead finite. This

)

Note that (AS) is needed in order to utilize the property of matrix A(t) given by expression (31) of

implies part a) of the theotem. Using this fact and equations (14), (15) together with (12) and (11) and recalling (5) , thus the validity of part b) is proved.

Wen and Hill (1992) for establishing the convergence of adaptive control law.

Corollary. Under conditions of Theorem I, control objective (6) is achieved for

Proof

Result immediately.

follows

°=11* + O.

£

from

(6)

and

Following the Certainty Equivalence Principle (Astrom and Wittenmark. 1980; Goodwin and Sin, 1981; Lozano-Leal and Goodwin, 1985; Wen and Hill, 1992), the parameters of controller are updated by using the estimate parameter vector

(20) •

8Y)

Remark 3. The variable X(t) may be chosen as

= [al(t), .. . , aft), bl(t), ... , bP)] T obtained

from

estimator (12) to (18) setting k = I. As in Goodwin and Sin (1981) and Zhiteckij (1995), the adaptive control law is chosen according to

x(t) - X == const.

Actually. the validity of such a chooce of X(t) obvious.

IS

Remark 4. The variable X(t) may be chosen as

x== const

l

(I - Z

for 11
XC!) = !fiE(t), 'il(t - I), E o(t - 1»1

2

"--'-"':"':"---'-~--':'-'----~';-----'-'-'---

2

0t

h

. e rw I se.

where G(t,

11
Z-

-I

)u(t)

= Vu(t),

(24)

I) = I + g I (t)z - I + ... + gP)z - I and

F(t,: -I )=fl(t)z-I + ... +fp)z-I are obtained by solving the equatiom

Proof Result can be proved by employing Theorem 4.n.3 given in Section 4.n .5° of Fomin et al. (1981).

(I -z -I)~( A t,z -I)G( t,z -I) +F (t,,: - I)~( B t,z -I )

= 1 (25)

(Details are omitted because of space limitation.) in which A and 13 are the time-varying polynomials induced by an estimate vector

Remark 5. Note that the Lyapunov type function Vet) defined by (21) may increase as long as 'il(t) < 'il. For compasison, another Lyapunov type function

e(t) = 8(t) + aCt). V(I)

= 118 -

2

8(1)11 2 + l'il - 'il(1)1

-

-

-

(26)

(22) It is known (see (Lozano-Leal and Goodwin, 1985», that (32) is solvable if the deteminant of Sy1vester

defined in Feng (1994) is always non increasing. However, there is also no the guarantee that the first term in (22) does not increase as long as 'il(/) < 'il, since just the second term decreases. Thus, in the both cases it is not guaranteed that estimate 8Y) goes to a

matrix M(8) for A(t, z - I) and B(t,

det M(8)"* O.

neighbourhoot of 8_ as long as 1'il(/) - 'ill decreases.

314

Z -

I) is nonzero :

(27)

It can be shown that if det M(8)

=0,

then (27) can

50

always be satisfied by the corresponding choice of a nonzero vector ~(t) in (26). (~(t) is zero only when

30

det M(8) -:;; 0.). The nonzero vector ~(t) may be found

...

~

as in Lozano-Leal and Goodwin (1985) .

';:;

,

...

10

Theorem 2. If Assumptions (A2), (A4) and (A5) are valid, then the adaptive control system consisting of plant (I), controller (23) to (25) and estimator (12) to (18) for k = 1 together with (26), (27), has the properties pointed out in part a) of Theorem I and

...

Q.

g -10 -30

c) {u(t)} and (y(t)} are bounded for all time.

-50~~nn~~nT~nT~~~~~~~

o

50

100

150

200

250

Discrete time t

Proof Because of space limitation, the proof is omitted.

Fig. I. Plant output y(t) 5. SIMULA TION RESULTS

3.0

r ~

2.0

.,

...

b

1.0

Cl>

To demonstrate a power of the proposed adaptive control algorithm, the simulations were performed for data taken from Zhiteckij (1995): I = 2, a I = -2.0, a 2 = 3.0, b I = 1.0 and b2 = -1.25 . In this case, the plant

t)

C2 ( 2

has two unstable poles anddone unstable zero. The signal y 0 was given as: y 0 = 10.0, i;;(t) is a pseudorandom variable from [-0.5, 0.5]. The initial estimates were chosen as: al(O) = -1.5, aiO) = 2.5,

(t) 1.

ID

S

.,

:;

0.0

f;Il

-1.0

bl(O)

Jl 1.

b (t)

~

a

-2.0

= 1.5 and b2(0) = -0.75 . The variable X(t) was

given as X

= 0.05.

2

3

The transient and steady-state behavior of the adaptive system is depicted in Figures 1-3. They show how the parameter estimates converge and the control objective is achieved. It is observed that when

(t) 1

ii"'I"",""""',""""',""""',"""""

o

50

100

150

200

250

Discrete time t

E O(t);::: 0.51, then the estimates a I(t), ait), b (t) , and l

bit) do not practically change, while the size

Fig. 2. Parameter estimates al(t), ait), bl(t), bit)

E o(t)

of

dead zone increases.

6. CONCLUSION 0.7

0.0 , j [ j

o

11 i , Ii

i

I , .. 11 i IJ I , , . . , , , , "

SO

100

, , , • "

i "

150

Discrete time t

i iI I ,

200

In summary, the main contributions of this paper are: I) a modified estimatiom procedure for on-line determining the size of dead zone in the adaptive law; 2) the convergence and stability results. A practical significance of these results is that they can be applied to a widely class of control system with minimum prior information about the plant and environment parameters.

t I , . IT

250

Fig. 3. Estimate E o(t)

315

Lozano-Leal, R. and 1. Collado (1989) . Adaptive control for systems with bounded disturbances. IEEE Trans. Automatic Control, AC-34, 225-228 . Martin-Sanchez, J.M . (1984). A globally stable APCS in the presence of bounded noises and disturbances. IEEE Trans. Automatic Control, AC-29,461-464 . Milanese, M., 1.P.Norton, H.Piet-Lahanier and E.Waiter (1996) (Eds.) Bounding Approaches to System Identification, Plenum Press, New York, London. Ortega, R. and R. Lozano-Leal (1987). A note on direct adaptive control of systems with bounded disturbances. Automatica, 23,253-254. Samson, C. (1983). Stability analysis of adaptively controlled systems subject to bounded disturbances . Automatica, 19, 81-86. Suarez,D.A., R.G. Moctezuma and R. Lozano (1994) . Robust continuous-time indirect adaptive control algorithm . In: Preprints of 10th IFAC Symposium on System Identification, Vo1.1, pp. 365-370. Wen , C. and DJ. Hill (\ 992) . Global boundedness of discrete-time adaptive control just using Automatica, 28, estimator projection. 1143-1157. Zhiteckij, L.S. (1995). Adaptive control of nonminimum phase systems in the presence of bounded disturbance with unknown bound. In : Proceedings of 3rd European Control Conference, pp. 891-896.

REFERENCES Astrom, KJ . and B.Winenmark (\ 980). Self-tuning controller based on pole-zero placement. Proc lEE, 127,120-130. Bondarko, V.A. (\ 983). Adaptive suboptimal control of solution of of linear difference equations. Dokl. AN SSSR, 270, no 2, 301- 303 (in Russian). Egardt, B. (1980). Stability analysis of discrete-time adaptive control schemes. IEEE Trans. Automatic Control, AC-25, 710-716. Feng, G. (\994). A robust discrete-time direct adaptive control algorithm. Systems and Control Letters, 22, 203-208. Fomin, V.N., A.L. Fradkov and V.A . Yakubovich (1981). Adaptive Control of Dynamic Objects. Nauka, Moscow (in Russian). Goodwin,G.C. and K.S .Sin (1981). Adaptive control of nonminimum phase systems. IEEE Trans. Automatic Control, AC-26, 478-483 . Goodwin,G.C. , DJ. Hill and M. Palaniswami (\ 984). A perspective on convergence of adaptive control algorithms. Automatica, 20, 519-531. Gu, X.Y. and W. Wang (1989). On the stability of self-tuning controller in the presence of bounded disturbances. IEEE Trans. Automatic Control, AC-34, 211-214. Lozano-Leal, R. and G .c. Goodwin (1985). A globally convergent adaptive pole placement algorithm with a persistency of excitation requirement. IEEE Trans. Automatic Control, AC-30, 795-799.

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