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Adaptive Control of Fast Time-Varying Systems
Copyright © IFAC Identification and System Parameter Estim ation . Budapest. Hungary 1991
ADAPTIVE CONTROL OF FAST TIME-VARYING SYSTEMS Zheng Li and H. Unbehauen Department of Electrical Engineering, Automatic Control Laboratory, Ruhr-University Bochum , Posifach 102148, D-4630 Bochum I, Germany
Abstract In this paper. the state of the art and a perspective about adaptive control of fast time-varying systems are given. The emphasis is on the prediction of fast time-varying parameters . It is shown that the techniques for tracking and predicting fast time-varying parameters are based on the knowledge ar.JOut variation in the parameters . A unification of some of the existing tracking techniques for fast time-varying parameters is made . Under the assumption that the plant parameters can be exactly described by a known state-space model, the global stability and convergence of the closed-loop adaptive systems having unknown time-varying p.arameters are established. Keywords Adaptive control; convergence analysis ; self-tuning regulators; til~-varying systems.
The adaptive schemes for time-varying systems can be applied to nonlinear systems . Kung and Womack (1984a, b) considered the systems depicted by a cascade connection of a linear time-invariant system and a preload nonlinearity or a two-segment piecewice-linear asymmetric nonlineari~y . The systems were considered cascade pa1rs of an unknown linear time-invariant system and a known time-varying gain. They developed adaptive schemes for the two cascade pairs and established global stability and convergence for the closed-loop systems .
INTRODUCT ION In 1982. Caines and Chen gave a counter example in which the impossibility of stabilizing a system with a zero mean uncorrelated gain parameter process was proved . Generally , the systems with arbitrary parameter variation can not be dealt with. We consider the nth-order standard time-varying systems y(t+l)+al (t)y(t)+az (t)y(t-l)+ .. . +am(t)y(t-m+l1=am.l(tlu(t)
Goodwin and Chan (1983) modeled the periodically time-varying disturbances with known order and unknown frequencies by sine waves . An adaptive control algorithm can be obtained by turning the disturbances to the observable and uncontrollable models of the systems. For the systems with sine parameters Ohkawa (1985) developed a model reference adaptive control algorithm . Li (1986b) suggested to do adaptive control for stochastic discrete periodically time-varying systems by a bank of adaptive algorithms. It was shown that the adaptive system was globally stable and convergent .
+am.2 (tju(t-l)+ ... +an(t)u(t-n+m+l)
identification;
(l)
After k sampling periods, there are k measurement equation available for parameter estimation . However, there are kn values of the unknown parameters in the equations . The unknowns are much more than the equations and can not be determined from them. Fast til~ variation means that the parameters undergo large changes frequently . If the measured data can not be utilized to predict the variation, all the measurements are of no use for adaptation. Due to the above reasons. all the adaptive control schemes for time-varying systems are based on additional knowledge about parameter variations. The knowledge is not only important to the estimation but also necessary to the prediction of fast time-varying parameters
In 1984, Xie and Evans suggested to approximate the time-varying parameters by polynomials of time . They developed an adaptive control algorithm based on the first-order Taylor expansion. A remarkable character of this scheme is that the expansion point can be adjusted on-line 50 that the modeling error can be controlled on-line . Three years later, the scheme was generalized by Li (1987a) to the case of arbitrary order Taylor polynomials in the time variable . Hatko and Nemec (1988) compared, with respect to the adaptive control of a robot arm, the standard least-squares algorithm with fixed forgetting factor and the algorithm which models the parameters by the first order Taylor polynomials. It was found that better results were obtained by the latter when the parameters varied rapidly .
In application. there are cases where the working point of the plant changes frequently . The plant can be modelled at each working point with a linear time - invariant submodel. The difference between the suggested schemes is in the techniques developed to determine the working point . Millnert (1982) assumed that the changing time was a Markov chain and developed an adaptive algorithm based on kalman filter . Jedner and Unbehauen (1988) assumed the new information in the least-squares algorithm was a Gaussian process and determined the working point by chi square-test. It was shown that with such a scheme the number of the submodels could be changed on-line .
In view of the insufficiency of the by the information provided standard time-varying ARMA model, Li (1987b) the measurable auxiliary introduced
441
variable from gain scheduling into the self-tuning control . The auxilliary variable is a measurable variable, which correlates well with the dynamic changes. Since the real control systems are nonlinear, variations in the parameters of the linearized model are often due to the changes of the working point. If the variables , which determine the working point, can be measured, they can be used as auxiliary variables and as information besides the input and output for parameter estimation . It was shown that the sensitivity could be greatly reduced by the utilization of the auxiliary variables . Li (1986a) also suggested a scheme to utilize the prior knowledge of the parameter variation . The knowledge was expressed as a time-varying matrix . It was shown that it could be incorporated into the form of an adaptive control algorithm .
y(t), ~(t) and !?(t) are the known parameter input vector, unknown parameter state vector and unknown parameter vector, respectively . ~(t) is unknown because the initial state ~(to) is unknown. Plant (2) can then be written in a vector form y(t+d)=dT(t)!?(t)
(5)
where d(t) is the data vector. For simplicity, we only consider the situation where y(t)=O . The systems having finite working point (Millnert, 1982, Jedner and Unbehauen, 1988) can be described by y(t+d)=dT (t)!?i i=l, 2, ... , N. For such a system, the state space becomes
(6) model
For the design of controllers for adaptive control of time-varying systems, Tsaklis and Ioannou (1989) have shown that the standard model reference controller, used for linear time-invariant systems, could not guarantee zero tracking error in general when the plant was time-varying . A new model reference controller was then proposed by replacing the adjustable parameters from the output matrix of the auxiliary filter to the input matrix. A pole placement controller has also been developed (1988) . It was shown that zero tracking error and closed-loop stability could been established if the two controllers were combined with a parameter estimator, which could track the time-varying parameters well .
where 1 is the identity matrix with proper dimension. At each sampling period only one qi .(t.) , i=-l, 2, .... , N, is 1, the others are O. qi (t), i=l, 2, ... , N, are determined by the techniques such as Markov chain scheme (Millnert 1982) or chi square-test (Jedner and Unbehauen , 1988) .
UNIFICATION
For periodically time-varying systems (Li , 1986), we can let
We consider the following time-varying predictor model
A(t+1, T)=l and ~(t)=diag(ql(t)l,
. .. , qN ( t l l )
SISO qi (t)=
y(t+d)=al (rt )y(t)+a2 (rt )y(t-1)+ .. . +am(rt)y(t-m+1)+am+l(rt)u(t)
.. . +an(rt)u(t-n+m+1)
ql (t)=
t=KN+i
0,
else .
[
1,
u(t»O,
0,
else.
~(t)
=diag (Col, Co2, .. . , Con)
A(t, s)=diag(AI(t , s) , A2(t, s), ... , An(t, s»
(3 )
where
where
Coi = ( 1, 0, .. . , 0),
(rt),
.. . , an(rt»T
(9 )
When an auxiliary variable is available and the parameters are approximated by algebraic polynomials (Li, 1987b), the state space model is
~(t)=~(t)~(t)+ll(t)y(t)
!?(t)= (al (rt), a2
(8 )
The other definitions remain the same .
t)~(t)+a(t)y(t)
~(to )=~o,
1,
For the systems with two-segment piecewise-linear asymmetric nonlinearity (Kung and Womack, 1984b), we can let N=2 and
(2 )
where u(t), yet) and rt are the directly measurable plant input, output and auxiliary variable, respectively. This model loses no generality because it can be obtained by successive elimination of the terms of y(t+d-1), y(t+d-2), y(t+1) in the ARMA model. In (2) d is the time delay and ai (rt), i= 1, 2, . . . . . , n, are functions of the auxiliary variables . rt can be viewed as an intermediate function between the parameters and the time variable . It is assumed that the symbol of am+l(rt) is known and the plant is stably invertable . The plant parameters are described by a known MIMO state space model . ~(t+l)=A(t+l,
!
(7 )
Here Nand K are the period and the natural number respectively, O~i
+am+2 (rt)u(t-1)+
lam+ l(rt)1 >(1" >0
q2 (t)l,
(4 )
A(t) , a(t), ~(t) and U(t) are the known state transition matrix, input matrix, output matrix and input output coupling matrix with proper dimensio~ respectively;
442
(10 )
1, rt-re, ... ,
where Qi+l matrix ,
(n-re) (Ni) INi!
is a
positive
semi-definite
Qi + 1s...Kl 0, 1, ... , (n - re ) (N i -1) I (N i - 1) !
Ai
(t,
(18)
In (18) I is an identi ty matrix . prevent E(Si+1/ti+1-1) from going infinite we set Qi+1=01
s)=
To to (19)
when 0,
0,
0,
0,
~(t)
traceE(si / ti+l-1)2X.
... ,
we define a
i.(so Ito )=x.(so)
( 11)
rt.
E(so/to-1)=Eo
generalized
y(t+d)=hT(t,
In the above algorithm there is theoritically the remote possibility of division by zero in finding u(t). This can be avoid by the projection which was first introduced by Goodwin and Mayne (1987). The time-varying linear transformation is E-1/2 (si/t-1)A(si, to), where E-l / 2 (si / t-1) is defined by
t-l)~(t-1)
t-1)~(t-1)
~(to )=~o
(13 )
where ~o is the initial state . Now we have transformed the unknown time-varying system (2) into a known time-varying state space model (13) with unknown initial state ~o . In system (13) the observation equation represents the dynamics of the system (2), the state transition equation represents the knowledge about the variation in plant parameters and the initial state ~o represents the uncertainty of the parameters. So the pro blem of identification of the unknown parameters in system (2) is transformed to the problem of observing the states in system (13). Letting i(s/t) and i2.(t) be the estimates of ~(s) and ~(t) based on the measurements up to and including time t we know from (3) ~(t)=G.(t)A(t,
known,
the
E(Si It-1) =E1/2 (si/t-1)ET/2 (si / t-l)
pm+1(t) is the (m+1)th element of ~(t). Let F (Si) denote the image of F(to). When ~m+l(t)
(15)
X.(Si+1/t+i+1)=E1/2 (si+1/ti+1-1)
i(s / t) can be estimated from the observation equation of (13) with the least-squares algorithm . Defining two adjusting time sets {Ye}= {to, t1, Iti
Pr (E- 1 /2 (s i + 1 Iti + 1 - 1 ) X.(Si+1/ti+l»
k(t)=E(si / t-2)h(t-d,Si)/[R(t) +hT(t-d, silE(si / t-2)h(t-d, sd]
ANALYSIS
e(t)=y(t)-hT(t-d , si)~(si/t-1)
We will establish global convergence for the self-tuning regulator under the assumption that the plant parameters can be described by a known state-space model. The proof is along the line of Goodwin and Sin (1984). We make the following assumptions about the systems:
X.(Si I t)=£(Si It-1)+k(t)e (t) E(s i It-1) = [l-k(t) hT (t-d , Si) ]E(Si I t-2) (16)
adjusting
A1) d is known. A2) The state space model (3) is known , G.(t) is uniformly bounded and ~(t ) is unknown .
X(Si+l / ti+l)=A(Si+l, Si)l\.(Si/ti+1) E(Si + 1 I ti + 1-1)
A3) The system (2) is stably invertable .
= A ( s i + 1, Si) E ( silt i + 1 - 1 ) A(Si+l , si)T+Qi+l
(25 )
where t+i+l denotes the time just after the projection and Pr(w) denotes the projection of w onto the supporting hyperplane to F(Si+1) at E- 1/2 (s i + 1 It i + 1 - 1) i (s i + 1 I t i + 1 ) .
For ti < ts...ti + 1
When t=ti + 1 , the following algorithm is executed
(24 )
R/I1+ 1 (t) ><:>">0
where
d-step-ahead
y*(t+d)= dT(t)~(t) .
(23)
For each examples mentioned above a reasonable assumption can be made about the prior knowledge about ~(t) or ~(t) for the projection . However. it is difficult to give an assumption that is good in general. The reason is obvious. The prior knowledge may be very different when different time-varying systems are considered . Without loss of generality, we assume that the initial parameter state xo lies in a known convex region F(to) such that
(14)
s)x.(s/t)
(22)
Eos...Kl.
Then substituting (3) and (12) into (5) we have
If 2.(t) is control law is
(21 )
where Eo is a positive definite matrix and (12)
hT(t, t-1)= dT(t)G.(t)A(t, t-1) .
~(t)=A(t,
(20)
The initial condition is
1
0,
i= 1, 2,
To estimate data vector
rt -re
1,
A4) A(t+1 , t) is stable .
(17)
443
A5)
~(to) lies in a known such that
convex
~1
region
"~(Si
Subject to assumption A1)-A5), if the adaptive control algorithm (14)-(25) is applied to system (2), then y(t), u(t) are bounded and Lim[y(t)-y*(t)]=O ti
i(Si It)=i(Si
(39)
t-#oO
Since Hi(Si It) -i(s i It-1)1I
(28 )
~II ~(Si
and a Lyapunov function
2
It-2)II e 2 (t) I[R (t)
+hT(t-d,
V(Si It)=iT (Si It)~-l (Si It-1)i(Si It) (29 ) From (13),
(38)
It-1) 1/ 0<2 < <10
Lime 2 (t)[K+K2h(t-d, Si)2)-1=0
we define
It)-~(Si)
and for
From (34) we have
(27 )
t..., ...
(37 )
From (16) and (22) we know that (36) (37) holds for k=O . It follows that any ti ~t~ti + 1
Theorem 1
For
(sk/tk-1)
~1
(26)
pm+1(t) >0
~
(sk/t-1) =A(Sk, Sh) ~(sh/t-1) AT (Sk, Sh)
si)~(Si/t-2)h(t-d,
Si») (40 )
we have from (34) and (38)
(16) and (17) we have
LimHi(si/t)-i(si / t - 1)U 2 =0.
( 41)
t~..,
)[(Si+ 1/ti+ 1 )=A(Si+ 1, Si )i(Si Iti+ 1) (30)
For any j and
it: (s j
So
It)
ti < t~ti+1,
=A(s j ,
we define
Si) it: (S i It)
(42) 10«~,
Then subject to constraints It-s any tf
V(Si+1/ti+1)-V(Si/ti+1) = iT (s i + 1 I t i + 1 ) [~- 1 (S i + 1 I t i + 1 - 1 )
for
lIi(Sj It) -i(Sj Is) 11 2 ~(t-s+h-f) (C.lli(Sj Ik) k=JtI
- AT (s i, Si +1
) ~-
1 (s i Iti + 1-1)
-i
h
(s j Ik - 1) 11 2
+
r== 1/ A (s
K=f'l
j ,S k )
11 2
= iT (S i + l i t i + 1 ) {[ A (s i + 1, Si) ~(si/ti+1-1)AT(Si+1,
-[A(Si+ l,Si AT (Si+
1,
)~(Si
Si)+Qi+1]-1
h -I
~(t-s+h-f)
Iti+ 1-1)
(CIIA(Sj ,Sk)1(2 k=
Si) )-l}i(Si+ 11tH
1 )~O
f'l
t'U1
Clli(Sk/k') -ii.(sk/k· -1)112 +
(31)
It ': ttci'1
When pm+1(t)<~ the projection will be applied we have
algorithm
..!.ul
IIA(s j ,Sf) 1\ 2L.....Jli(Sf Ik) -i(Sf Ik-1)1I 2 I< =Stl
i (s i + l i t + i + 1 ) = ~ 1/2 (S i + l i t i + 1- 1) Pr (~- 1 /2 (5 i + l i t
i
(s i + l i t i + 1 )
) -
i
-t
+IIA(Sj , sh)il 2LUi(Sh/k) k~t.'1
+ 1 -1)
_
~ (S i + 1 )
h
-~(sh/k-1)112
+
~IIA(Sj
,Sk)112
1'2 fTI
=~1/2
(Si/t- 1)
Pr (~- 1 /2 (S i + l i t i + 1- 1) i (s i + l i t i + 1 ) (32)
where the projection. have
So V(Si+l/t+i+1)~II~-1/2
~ (s
last term is caused by the From (41), (43) and (35) we
(Si+l/ti+1-1) Limll:[(sj It)-X:(sj Is)1I =0
i + l i t i + 1 ) " 2 = V (Si + 1 I t i + 1)
(33 )
where 11' 11 denotes the Euclidean norm . follows (Goodwin and Sin, 1984)
It
Lime 2 (t) [R(t)+hT (t-d, Si
)~(Si
t .... ~
It can be easily shown that for any t and S if It-sl0<, (44) holds . As soon as (39) and (44) are established the rest of the proof is similar to that of Goodwin and Sin (1984) .
It-2)
We note that global convergence is established in theorem 1 without the persistent excitation . The following theorem deals with the cases where the state space model is unstable .
(34)
h(t-d, Si»)- 1=0 From A4) we know
(35 )
IIA(t, S)II0
A6) ti + 1-ti < K< 00 for any i and there exists i real numberP>O such that
Assuming Qk10l, we know from (18)-(22)
t
"~(sk/tk-1)1I ~1I~(sk-
Lim
We further assume that for k+2 j Qp-O From (Li, we ' kno~' that ~h~n th-l
r:= R- 1 (t) h(j-d, t-d) hT
t"""i=t;"
1/b-1)" 11 A(Sk, Sk-l)/I 2
+/IQkll0«1+K12)
(44 )
(j-d, t-d) LPI (45)
Theorem 2
(36)
Subject to assumption A1), A2), A3), A5) and A6), if the adaptive control algorithm (14)-(25) is applied to system (2), then y(t) and u(t) are bounded and (27) holds
h=k+1, 1987b),
444
~ The key point of this proof is to establish the boundness of the covariance matrix, more precisely
~ A7) implies AS) holds . So theorem 2 holds in this case . Since
(46 )
11 E (s i It- 1) 11 s...K3 < ""
j=
We assume that IIQhll=O, h=f+1, f+2, .. . , k-1, but QfiO or tf=tO. We also assume that QJ
we can always choose a Qi which is bounded below by a constant positive matrix . From theorem 2 we know that d(t) and h(t-d, Si+l) are all bounded. It follows from A7) that
Taking tf as the initial time we have from (Li, Ilt87b, for tf
t ;!,'
h(j-d, Si-l)hT(j-d,Si-I»)-I
E (t- 1) = A (t- d , S h ) E (s h It- 1 ) AT (t- d , s h) (47 )
E-
t·
L= R- I (t)
E- I (s i Iti -1) = [E(s i I ti - 1-1) +
E-
(t-l)=AT (Sh, t-d) AT (Sf, Sh)
1
(s i+ I It i + I
-
1)
= [ A (s i, Si - I ) E (s i-I It i-I)
E-l(Sf/t-1)A(Sf, sh)A(Sh, t-d)
AT(Si, Si - I)+Qi)-1
=AT(Sf , t-d)[E-l(Sf/tf-1)
t· +~R-I(t)h(j-d, si)hT(j-d, Si) i=t/'
t
+LRi tIt.
=
1
(t)h(j-d , Sf )hT (j-d,Sf»)
(57 ) is bounded such that
A(Sf , t-d)=[A(t-d, Sf )E(Sf I tf-1)
for all i .
We
let I < Ns...K '< OO
A (s i-I, Si) E (S i-I I t i-I ) AT (s i , Si - I ) AT (t-d, Sf) )-1
< (N-l)Qi
t
Then
+C
R-l (t)h(j-d,t-d)hT (j-d,t-d) i=t,t' (48)
From A6) we know that there is a N< (Xl such that when tf + Ns...ts...tk t
~ Rj", tIt!
(58 )
lIi(si + I Iti + I )11 2 (N - IUQill-
1+ f)
f >O and a
s...IIA(si+ I, Si HI 2x.T (Si I ti+ I) {[ (N
I (t) h( j-d, t-d) hT (j-d , t-d)
~f'1.12
-1)Qi+Qi) - I+fl.}~(si /ti+l)
(49)
s..XT (Si I ti+ I) [E- 1 (Si I ti-l )
So t
IIE(t-I)11 s...L::[R-l (t)h(j-d, t-d) i =tl" h T (j -.d, t- d) ) - I s...2 p- I
~ + >---R-l(t)h(j-d,Si)hT (j-d,Si»)
) '" t,.'
(50)
When tfs...t
i(si/ti+I)I/A(Si+I, siHI2
E(t-1) 11 s...IIA(t- d, Sf) E(Sf It-I) AT (t-d, Sf )/1 s.../lE.(Sf Itf-1)IIIIA(t-d, Sf
Form (14)- (22), {Za} we have 11
)11
2
~V(si/ti+I)/lA(Si+I,
(51)
~ V (s
A6) and the definition of
+
i ) 11 A (s i+ I, Si) 11 2
s.,.iT (S i It + i ) [ A ( Si, S i-I ) E(si - I / ti - 1)AT(si, Si-I )
E(Sf Itf -1 )1I0{4 <00
IIA(t-d, Sh)IIs...Ks < oo ,
iIt
s1)1I2
for th<
t~th+
+Qi ) - I i (s i I t + i ) 11 A (s i+ I , Si) 11 2
I
~lIi(si /t+i)1I2 11 A(Si+I , si ) 1I2
and 11
~~ n (Qi)
A(t-d, sf)1I s...Ks <00 , for tf < t< tf +N (52)
where A. mi n denotes the minimum eigenvalue of a matrix. Then
Since for th
11 iI: (s i + I I t i + I ) 112 I 'liI: (s i It + i ) 11 2
(53)
s...\tA.(Si ·n, "5.-) /l2 / f[N-l
The rest of the (46) can be established. proof is similar to that of theorem 1 and is omitted . For the exponential convergence we need stronger persistent excitation
+pi\max (Qi) n.- 1max (Qi »)..min (Qi)} (6 0 )
a
Correct Qi can always be found to (55) holds .
ti + I-ti < K< 00 and there A7 ) For any i, exists a real number f >O such that
~ j ; tot'
~R-l(t)h(j-d,
ensure
CONCLUSION
silhT(j-d,si»(l. (54)
The approaches of adaptive control of time-varying systems c a n be divided into two classes. In the first approach, the original robustness of the adaptive algorithms developed for time-invariant systems is utilized or improved to accommodate the adaptive control algorithms to time-varying systems . The current parameter estimates are taken as their predictions in this approach , whi c h makes the adaptive co ntrol algorithms
Theorem 3 Subject to assumption AI), A2), A3), A5) and A7), if the adaptive control algorithm (14)-(25) is applied to system (2), then 1Ix.(Si+l / ti+I)/l 2 /J1i(Si/t+i)112 < 1
(59)
(55)
445
unable to control fast time-varying systems . In the second approach. additional information about the parameter variation is introduced to predict the parameter variation. Describing the plant parameters by a state-space model. we have obtained a unification of some of the adapti ve algorithms . It is shown that a unified approach of the stability and convergence of the adaptive control algorithms is possible by means of the state-space description of the time-varying parameters .
REFERENCES Caines . P . E. and H. F . Chen (1982). On the adaptive control of stochastic systems with random parameters : A counter example . Proc 151 Workshop on Adaptive Control. Florence. Italy. Goodwin. G.C . and S.W. Chan (1983) . Model reference adaptive control of systems having purely deterministic disturbances . Trans Aut Control. AC-28, IEEE pp . 8SS-8S8 . Goodwin. G.C . and Sin . K. S . (1984) . Adaptive filtedng predjctjon aur! contro l Englewood Cliffs . NJ : Pretice Hall . Goodwin. G. C. and D. Q. Mayne (1987) . A parameter estimation perspective of continuous time model reference adaptive control . Automatjca 23 ppS7-70. Jedner . U. and H. Unbehauen (1988). Intelligent adaptive control of a class of time-varying systems. Proc of 12th IMACS World Congress. Vol . 1 . Paris. pp . 377-380 . and B.F . Womack (1984) . Kung. M.C. Discrete time adaptive control of linear system with preload nonlinearaity. Automatjca 20. pp . 477-479. Kung. M. C. and B.F . Womack (1984 ) . Discrete time adaptive control of linear dynamic with a two-segment piecewiselinear asymmetric nonlinearity . IEEE. Trans Aut Control. AC-29 . pp. 170-172. Li. Z. (1986a) . Discrete-time adaptive control for time-va!:'yi.ng systems and a class of gray systems. Prepr IFAC Syrop on Microcomputer ...bpplication in process contro l. Istanbul. ' Turkey. 1986 . Li. Z. (1986b). Discrete tiroe adaptive control for periodically time-varying systems . 2nd IFAC Workshop on Adaptive Systems in Control and Signal Proce?sing. Lund. Sweden pp . 411 415 . Li. Z. (1987a) . Discrete time adaptive time-varying control for linear fast systems. IEEE Trans Aut Contr AC-32. pp.444-447. Li. Z. (1987b). Stochastic adaptive for time-varying systems via an auxiliary variable. Proc of 26th cnc. Los Angeles C A USA. pp.8S0-854. Li. Z. (1988) . A discrete-time adaptive controller for time-varying systems. uQ!:.... of Int Conference Control-.!lJL.. UK. pp.S23 526 . Matko. D. and B. Neroec (1988) . Identification of fast time-varying systems-comparative characters of thre e algorithms. Preprints of 8th IFAC IIFORS Symposium on Identification and System parameter e'3timation Vol. 2. Beij ing. pp. 720-725. Millnert. M. (1982). Identification and control of systems subject to abrupt changes . ~al dissertation. Linkoping University. Sweden. Ohkawa. F. (1985) . Model reference adaptive control system f o r discre te linear time-varying systems with periodically varying parameters and time delay . Int J Control, 44. pp. 171-179. Tsakalis . K.S. and P. A. Ioannou (1988) . Adaptive control of time-varying plants : Simple examples. Int J of adaptive control and signal processing vol.2. pp . 291-310. Tsakalis. K.S. and P.A . Ioannou (1989). Adaptive control of linear time-varying plants : A new model reference controller structure . IEEE Tran, Aut. Contr AC-34. pp . 1038-1046. Xie. X. Y. and R. J . Evans (1984) . Discrete-time adaptive control for deterministic time-varying systems. Automatica. 20. 1984 . pp . 309-319 .
The key difficulty in adaptive control of time-varying systems is the prediction of time variation in plant parameters . The reason is as follows: 1. The information provided by plant input and output is far from being sufficient for tracking the parameters. 2. Unless large frequent changes in the parameters are well predicted. there will be always tracking errors between the reference output and the plant output . 3 . There are no efficient schemes for the identification of fast time-varying systems . Uptodate . only limited success has been achieved in the adaptive control of fast time-varying systems. The reason is that most of the research works are based on the standard time-varying ARMA models. Additional a priori knowledge on the parameter variation is necessary to the adaptive control schemes. However. the knowledge depends on the engineers' insight about the real systems. For different physical systems the knowledge may be quite different . It seems impossible to develop an adaptive control algorithm which is applicable to many types of time-varying systems . It seems to us from this paper that the following topics are interesting and promising for the study of adaptive control of systems with fast time-varying parameters. 1. Developing new models with which the additional information available about the parameter variations can be utilized efficiently. 2 . Combination vf adaptive control schemes and such intelligent control schemes as utilize the additional information about the variation in the parameters. 3. Developing the adaptive control algorithm into which engineers can easily incorporate their knowledge about the plant dynamic variation. 4 . Unlike the time-invariant systems. it is possible that the tracking error always exists in the adaptive systems. So the robust estimation schemes and robust controllers is particularly important for adaptive control of time-varying plant. 5 . Adaptive control varying systems .
for
special
Various controllers for 6. control of time-varying systems.
time-
adaptive
7. Adaptive control of the systems whose parameters are functions of one or several measurable variables . ACKNOWLEDGMENT The work of the first author was supported by the Alexander von Humboldt Foundation and in part by the Chinese Science Foundation .
446
ADAPTIVE CONTROL OF FAST TIME-VARYING SYSTEMS Zheng Li and H. Unbehauen Department of Electrical Engineering, Automatic Control Laboratory, Ruhr-University Bochum , Posifach 102148, D-4630 Bochum I, Germany
Abstract In this paper. the state of the art and a perspective about adaptive control of fast time-varying systems are given. The emphasis is on the prediction of fast time-varying parameters . It is shown that the techniques for tracking and predicting fast time-varying parameters are based on the knowledge ar.JOut variation in the parameters . A unification of some of the existing tracking techniques for fast time-varying parameters is made . Under the assumption that the plant parameters can be exactly described by a known state-space model, the global stability and convergence of the closed-loop adaptive systems having unknown time-varying p.arameters are established. Keywords Adaptive control; convergence analysis ; self-tuning regulators; til~-varying systems.
The adaptive schemes for time-varying systems can be applied to nonlinear systems . Kung and Womack (1984a, b) considered the systems depicted by a cascade connection of a linear time-invariant system and a preload nonlinearity or a two-segment piecewice-linear asymmetric nonlineari~y . The systems were considered cascade pa1rs of an unknown linear time-invariant system and a known time-varying gain. They developed adaptive schemes for the two cascade pairs and established global stability and convergence for the closed-loop systems .
INTRODUCT ION In 1982. Caines and Chen gave a counter example in which the impossibility of stabilizing a system with a zero mean uncorrelated gain parameter process was proved . Generally , the systems with arbitrary parameter variation can not be dealt with. We consider the nth-order standard time-varying systems y(t+l)+al (t)y(t)+az (t)y(t-l)+ .. . +am(t)y(t-m+l1=am.l(tlu(t)
Goodwin and Chan (1983) modeled the periodically time-varying disturbances with known order and unknown frequencies by sine waves . An adaptive control algorithm can be obtained by turning the disturbances to the observable and uncontrollable models of the systems. For the systems with sine parameters Ohkawa (1985) developed a model reference adaptive control algorithm . Li (1986b) suggested to do adaptive control for stochastic discrete periodically time-varying systems by a bank of adaptive algorithms. It was shown that the adaptive system was globally stable and convergent .
+am.2 (tju(t-l)+ ... +an(t)u(t-n+m+l)
identification;
(l)
After k sampling periods, there are k measurement equation available for parameter estimation . However, there are kn values of the unknown parameters in the equations . The unknowns are much more than the equations and can not be determined from them. Fast til~ variation means that the parameters undergo large changes frequently . If the measured data can not be utilized to predict the variation, all the measurements are of no use for adaptation. Due to the above reasons. all the adaptive control schemes for time-varying systems are based on additional knowledge about parameter variations. The knowledge is not only important to the estimation but also necessary to the prediction of fast time-varying parameters
In 1984, Xie and Evans suggested to approximate the time-varying parameters by polynomials of time . They developed an adaptive control algorithm based on the first-order Taylor expansion. A remarkable character of this scheme is that the expansion point can be adjusted on-line 50 that the modeling error can be controlled on-line . Three years later, the scheme was generalized by Li (1987a) to the case of arbitrary order Taylor polynomials in the time variable . Hatko and Nemec (1988) compared, with respect to the adaptive control of a robot arm, the standard least-squares algorithm with fixed forgetting factor and the algorithm which models the parameters by the first order Taylor polynomials. It was found that better results were obtained by the latter when the parameters varied rapidly .
In application. there are cases where the working point of the plant changes frequently . The plant can be modelled at each working point with a linear time - invariant submodel. The difference between the suggested schemes is in the techniques developed to determine the working point . Millnert (1982) assumed that the changing time was a Markov chain and developed an adaptive algorithm based on kalman filter . Jedner and Unbehauen (1988) assumed the new information in the least-squares algorithm was a Gaussian process and determined the working point by chi square-test. It was shown that with such a scheme the number of the submodels could be changed on-line .
In view of the insufficiency of the by the information provided standard time-varying ARMA model, Li (1987b) the measurable auxiliary introduced
441
variable from gain scheduling into the self-tuning control . The auxilliary variable is a measurable variable, which correlates well with the dynamic changes. Since the real control systems are nonlinear, variations in the parameters of the linearized model are often due to the changes of the working point. If the variables , which determine the working point, can be measured, they can be used as auxiliary variables and as information besides the input and output for parameter estimation . It was shown that the sensitivity could be greatly reduced by the utilization of the auxiliary variables . Li (1986a) also suggested a scheme to utilize the prior knowledge of the parameter variation . The knowledge was expressed as a time-varying matrix . It was shown that it could be incorporated into the form of an adaptive control algorithm .
y(t), ~(t) and !?(t) are the known parameter input vector, unknown parameter state vector and unknown parameter vector, respectively . ~(t) is unknown because the initial state ~(to) is unknown. Plant (2) can then be written in a vector form y(t+d)=dT(t)!?(t)
(5)
where d(t) is the data vector. For simplicity, we only consider the situation where y(t)=O . The systems having finite working point (Millnert, 1982, Jedner and Unbehauen, 1988) can be described by y(t+d)=dT (t)!?i i=l, 2, ... , N. For such a system, the state space becomes
(6) model
For the design of controllers for adaptive control of time-varying systems, Tsaklis and Ioannou (1989) have shown that the standard model reference controller, used for linear time-invariant systems, could not guarantee zero tracking error in general when the plant was time-varying . A new model reference controller was then proposed by replacing the adjustable parameters from the output matrix of the auxiliary filter to the input matrix. A pole placement controller has also been developed (1988) . It was shown that zero tracking error and closed-loop stability could been established if the two controllers were combined with a parameter estimator, which could track the time-varying parameters well .
where 1 is the identity matrix with proper dimension. At each sampling period only one qi .(t.) , i=-l, 2, .... , N, is 1, the others are O. qi (t), i=l, 2, ... , N, are determined by the techniques such as Markov chain scheme (Millnert 1982) or chi square-test (Jedner and Unbehauen , 1988) .
UNIFICATION
For periodically time-varying systems (Li , 1986), we can let
We consider the following time-varying predictor model
A(t+1, T)=l and ~(t)=diag(ql(t)l,
. .. , qN ( t l l )
SISO qi (t)=
y(t+d)=al (rt )y(t)+a2 (rt )y(t-1)+ .. . +am(rt)y(t-m+1)+am+l(rt)u(t)
.. . +an(rt)u(t-n+m+1)
ql (t)=
t=KN+i
0,
else .
[
1,
u(t»O,
0,
else.
~(t)
=diag (Col, Co2, .. . , Con)
A(t, s)=diag(AI(t , s) , A2(t, s), ... , An(t, s»
(3 )
where
where
Coi = ( 1, 0, .. . , 0),
(rt),
.. . , an(rt»T
(9 )
When an auxiliary variable is available and the parameters are approximated by algebraic polynomials (Li, 1987b), the state space model is
~(t)=~(t)~(t)+ll(t)y(t)
!?(t)= (al (rt), a2
(8 )
The other definitions remain the same .
t)~(t)+a(t)y(t)
~(to )=~o,
1,
For the systems with two-segment piecewise-linear asymmetric nonlinearity (Kung and Womack, 1984b), we can let N=2 and
(2 )
where u(t), yet) and rt are the directly measurable plant input, output and auxiliary variable, respectively. This model loses no generality because it can be obtained by successive elimination of the terms of y(t+d-1), y(t+d-2), y(t+1) in the ARMA model. In (2) d is the time delay and ai (rt), i= 1, 2, . . . . . , n, are functions of the auxiliary variables . rt can be viewed as an intermediate function between the parameters and the time variable . It is assumed that the symbol of am+l(rt) is known and the plant is stably invertable . The plant parameters are described by a known MIMO state space model . ~(t+l)=A(t+l,
!
(7 )
Here Nand K are the period and the natural number respectively, O~i
+am+2 (rt)u(t-1)+
lam+ l(rt)1 >(1" >0
q2 (t)l,
(4 )
A(t) , a(t), ~(t) and U(t) are the known state transition matrix, input matrix, output matrix and input output coupling matrix with proper dimensio~ respectively;
442
(10 )
1, rt-re, ... ,
where Qi+l matrix ,
(n-re) (Ni) INi!
is a
positive
semi-definite
Qi + 1s...Kl 0, 1, ... , (n - re ) (N i -1) I (N i - 1) !
Ai
(t,
(18)
In (18) I is an identi ty matrix . prevent E(Si+1/ti+1-1) from going infinite we set Qi+1=01
s)=
To to (19)
when 0,
0,
0,
0,
~(t)
traceE(si / ti+l-1)2X.
... ,
we define a
i.(so Ito )=x.(so)
( 11)
rt.
E(so/to-1)=Eo
generalized
y(t+d)=hT(t,
In the above algorithm there is theoritically the remote possibility of division by zero in finding u(t). This can be avoid by the projection which was first introduced by Goodwin and Mayne (1987). The time-varying linear transformation is E-1/2 (si/t-1)A(si, to), where E-l / 2 (si / t-1) is defined by
t-l)~(t-1)
t-1)~(t-1)
~(to )=~o
(13 )
where ~o is the initial state . Now we have transformed the unknown time-varying system (2) into a known time-varying state space model (13) with unknown initial state ~o . In system (13) the observation equation represents the dynamics of the system (2), the state transition equation represents the knowledge about the variation in plant parameters and the initial state ~o represents the uncertainty of the parameters. So the pro blem of identification of the unknown parameters in system (2) is transformed to the problem of observing the states in system (13). Letting i(s/t) and i2.(t) be the estimates of ~(s) and ~(t) based on the measurements up to and including time t we know from (3) ~(t)=G.(t)A(t,
known,
the
E(Si It-1) =E1/2 (si/t-1)ET/2 (si / t-l)
pm+1(t) is the (m+1)th element of ~(t). Let F (Si) denote the image of F(to). When ~m+l(t)
(15)
X.(Si+1/t+i+1)=E1/2 (si+1/ti+1-1)
i(s / t) can be estimated from the observation equation of (13) with the least-squares algorithm . Defining two adjusting time sets {Ye}= {to, t1, Iti
Pr (E- 1 /2 (s i + 1 Iti + 1 - 1 ) X.(Si+1/ti+l»
k(t)=E(si / t-2)h(t-d,Si)/[R(t) +hT(t-d, silE(si / t-2)h(t-d, sd]
ANALYSIS
e(t)=y(t)-hT(t-d , si)~(si/t-1)
We will establish global convergence for the self-tuning regulator under the assumption that the plant parameters can be described by a known state-space model. The proof is along the line of Goodwin and Sin (1984). We make the following assumptions about the systems:
X.(Si I t)=£(Si It-1)+k(t)e (t) E(s i It-1) = [l-k(t) hT (t-d , Si) ]E(Si I t-2) (16)
adjusting
A1) d is known. A2) The state space model (3) is known , G.(t) is uniformly bounded and ~(t ) is unknown .
X(Si+l / ti+l)=A(Si+l, Si)l\.(Si/ti+1) E(Si + 1 I ti + 1-1)
A3) The system (2) is stably invertable .
= A ( s i + 1, Si) E ( silt i + 1 - 1 ) A(Si+l , si)T+Qi+l
(25 )
where t+i+l denotes the time just after the projection and Pr(w) denotes the projection of w onto the supporting hyperplane to F(Si+1) at E- 1/2 (s i + 1 It i + 1 - 1) i (s i + 1 I t i + 1 ) .
For ti < ts...ti + 1
When t=ti + 1 , the following algorithm is executed
(24 )
R/I1+ 1 (t) ><:>">0
where
d-step-ahead
y*(t+d)= dT(t)~(t) .
(23)
For each examples mentioned above a reasonable assumption can be made about the prior knowledge about ~(t) or ~(t) for the projection . However. it is difficult to give an assumption that is good in general. The reason is obvious. The prior knowledge may be very different when different time-varying systems are considered . Without loss of generality, we assume that the initial parameter state xo lies in a known convex region F(to) such that
(14)
s)x.(s/t)
(22)
Eos...Kl.
Then substituting (3) and (12) into (5) we have
If 2.(t) is control law is
(21 )
where Eo is a positive definite matrix and (12)
hT(t, t-1)= dT(t)G.(t)A(t, t-1) .
~(t)=A(t,
(20)
The initial condition is
1
0,
i= 1, 2,
To estimate data vector
rt -re
1,
A4) A(t+1 , t) is stable .
(17)
443
A5)
~(to) lies in a known such that
convex
~1
region
"~(Si
Subject to assumption A1)-A5), if the adaptive control algorithm (14)-(25) is applied to system (2), then y(t), u(t) are bounded and Lim[y(t)-y*(t)]=O ti
i(Si It)=i(Si
(39)
t-#oO
Since Hi(Si It) -i(s i It-1)1I
(28 )
~II ~(Si
and a Lyapunov function
2
It-2)II e 2 (t) I[R (t)
+hT(t-d,
V(Si It)=iT (Si It)~-l (Si It-1)i(Si It) (29 ) From (13),
(38)
It-1) 1/ 0<2 < <10
Lime 2 (t)[K+K2h(t-d, Si)2)-1=0
we define
It)-~(Si)
and for
From (34) we have
(27 )
t..., ...
(37 )
From (16) and (22) we know that (36) (37) holds for k=O . It follows that any ti ~t~ti + 1
Theorem 1
For
(sk/tk-1)
~1
(26)
pm+1(t) >
~
(sk/t-1) =A(Sk, Sh) ~(sh/t-1) AT (Sk, Sh)
si)~(Si/t-2)h(t-d,
Si») (40 )
we have from (34) and (38)
(16) and (17) we have
LimHi(si/t)-i(si / t - 1)U 2 =0.
( 41)
t~..,
)[(Si+ 1/ti+ 1 )=A(Si+ 1, Si )i(Si Iti+ 1) (30)
For any j and
it: (s j
So
It)
ti < t~ti+1,
=A(s j ,
we define
Si) it: (S i It)
(42) 10«~,
Then subject to constraints It-s any tf
V(Si+1/ti+1)-V(Si/ti+1) = iT (s i + 1 I t i + 1 ) [~- 1 (S i + 1 I t i + 1 - 1 )
for
lIi(Sj It) -i(Sj Is) 11 2 ~(t-s+h-f) (C.lli(Sj Ik) k=JtI
- AT (s i, Si +1
) ~-
1 (s i Iti + 1-1)
-i
h
(s j Ik - 1) 11 2
+
r== 1/ A (s
K=f'l
j ,S k )
11 2
= iT (S i + l i t i + 1 ) {[ A (s i + 1, Si) ~(si/ti+1-1)AT(Si+1,
-[A(Si+ l,Si AT (Si+
1,
)~(Si
Si)+Qi+1]-1
h -I
~(t-s+h-f)
Iti+ 1-1)
(CIIA(Sj ,Sk)1(2 k=
Si) )-l}i(Si+ 11tH
1 )~O
f'l
t'U1
Clli(Sk/k') -ii.(sk/k· -1)112 +
(31)
It ': ttci'1
When pm+1(t)<~ the projection will be applied we have
algorithm
..!.ul
IIA(s j ,Sf) 1\ 2L.....Jli(Sf Ik) -i(Sf Ik-1)1I 2 I< =Stl
i (s i + l i t + i + 1 ) = ~ 1/2 (S i + l i t i + 1- 1) Pr (~- 1 /2 (5 i + l i t
i
(s i + l i t i + 1 )
) -
i
-t
+IIA(Sj , sh)il 2LUi(Sh/k) k~t.'1
+ 1 -1)
_
~ (S i + 1 )
h
-~(sh/k-1)112
+
~IIA(Sj
,Sk)112
1'2 fTI
=~1/2
(Si/t- 1)
Pr (~- 1 /2 (S i + l i t i + 1- 1) i (s i + l i t i + 1 ) (32)
where the projection. have
So V(Si+l/t+i+1)~II~-1/2
~ (s
last term is caused by the From (41), (43) and (35) we
(Si+l/ti+1-1) Limll:[(sj It)-X:(sj Is)1I =0
i + l i t i + 1 ) " 2 = V (Si + 1 I t i + 1)
(33 )
where 11' 11 denotes the Euclidean norm . follows (Goodwin and Sin, 1984)
It
Lime 2 (t) [R(t)+hT (t-d, Si
)~(Si
t .... ~
It can be easily shown that for any t and S if It-sl0<, (44) holds . As soon as (39) and (44) are established the rest of the proof is similar to that of Goodwin and Sin (1984) .
It-2)
We note that global convergence is established in theorem 1 without the persistent excitation . The following theorem deals with the cases where the state space model is unstable .
(34)
h(t-d, Si»)- 1=0 From A4) we know
(35 )
IIA(t, S)II0
A6) ti + 1-ti < K< 00 for any i and there exists i real numberP>O such that
Assuming Qk10l, we know from (18)-(22)
t
"~(sk/tk-1)1I ~1I~(sk-
Lim
We further assume that for k+2 j Qp-O From (Li, we ' kno~' that ~h~n th-l
r:= R- 1 (t) h(j-d, t-d) hT
t"""i=t;"
1/b-1)" 11 A(Sk, Sk-l)/I 2
+/IQkll0«1+K12)
(44 )
(j-d, t-d) LPI (45)
Theorem 2
(36)
Subject to assumption A1), A2), A3), A5) and A6), if the adaptive control algorithm (14)-(25) is applied to system (2), then y(t) and u(t) are bounded and (27) holds
h=k+1, 1987b),
444
~ The key point of this proof is to establish the boundness of the covariance matrix, more precisely
~ A7) implies AS) holds . So theorem 2 holds in this case . Since
(46 )
11 E (s i It- 1) 11 s...K3 < ""
j=
We assume that IIQhll=O, h=f+1, f+2, .. . , k-1, but QfiO or tf=tO. We also assume that QJ
we can always choose a Qi which is bounded below by a constant positive matrix . From theorem 2 we know that d(t) and h(t-d, Si+l) are all bounded. It follows from A7) that
Taking tf as the initial time we have from (Li, Ilt87b, for tf
t ;!,'
h(j-d, Si-l)hT(j-d,Si-I»)-I
E (t- 1) = A (t- d , S h ) E (s h It- 1 ) AT (t- d , s h) (47 )
E-
t·
L= R- I (t)
E- I (s i Iti -1) = [E(s i I ti - 1-1) +
E-
(t-l)=AT (Sh, t-d) AT (Sf, Sh)
1
(s i+ I It i + I
-
1)
= [ A (s i, Si - I ) E (s i-I It i-I)
E-l(Sf/t-1)A(Sf, sh)A(Sh, t-d)
AT(Si, Si - I)+Qi)-1
=AT(Sf , t-d)[E-l(Sf/tf-1)
t· +~R-I(t)h(j-d, si)hT(j-d, Si) i=t/'
t
+LRi tIt.
=
1
(t)h(j-d , Sf )hT (j-d,Sf»)
(57 ) is bounded such that
A(Sf , t-d)=[A(t-d, Sf )E(Sf I tf-1)
for all i .
We
let I < Ns...K '< OO
A (s i-I, Si) E (S i-I I t i-I ) AT (s i , Si - I ) AT (t-d, Sf) )-1
< (N-l)Qi
t
Then
+C
R-l (t)h(j-d,t-d)hT (j-d,t-d) i=t,t' (48)
From A6) we know that there is a N< (Xl such that when tf + Ns...ts...tk t
~ Rj", tIt!
(58 )
lIi(si + I Iti + I )11 2 (N - IUQill-
1+ f)
f >O and a
s...IIA(si+ I, Si HI 2x.T (Si I ti+ I) {[ (N
I (t) h( j-d, t-d) hT (j-d , t-d)
~f'1.12
-1)Qi+Qi) - I+fl.}~(si /ti+l)
(49)
s..XT (Si I ti+ I) [E- 1 (Si I ti-l )
So t
IIE(t-I)11 s...L::[R-l (t)h(j-d, t-d) i =tl" h T (j -.d, t- d) ) - I s...2 p- I
~ + >---R-l(t)h(j-d,Si)hT (j-d,Si»)
) '" t,.'
(50)
When tfs...t
i(si/ti+I)I/A(Si+I, siHI2
E(t-1) 11 s...IIA(t- d, Sf) E(Sf It-I) AT (t-d, Sf )/1 s.../lE.(Sf Itf-1)IIIIA(t-d, Sf
Form (14)- (22), {Za} we have 11
)11
2
~V(si/ti+I)/lA(Si+I,
(51)
~ V (s
A6) and the definition of
+
i ) 11 A (s i+ I, Si) 11 2
s.,.iT (S i It + i ) [ A ( Si, S i-I ) E(si - I / ti - 1)AT(si, Si-I )
E(Sf Itf -1 )1I0{4 <00
IIA(t-d, Sh)IIs...Ks < oo ,
iIt
s1)1I2
for th<
t~th+
+Qi ) - I i (s i I t + i ) 11 A (s i+ I , Si) 11 2
I
~lIi(si /t+i)1I2 11 A(Si+I , si ) 1I2
and 11
~~ n (Qi)
A(t-d, sf)1I s...Ks <00 , for tf < t< tf +N (52)
where A. mi n denotes the minimum eigenvalue of a matrix. Then
Since for th
11 iI: (s i + I I t i + I ) 112 I 'liI: (s i It + i ) 11 2
(53)
s...\tA.(Si ·n, "5.-) /l2 / f[N-l
The rest of the (46) can be established. proof is similar to that of theorem 1 and is omitted . For the exponential convergence we need stronger persistent excitation
+pi\max (Qi) n.- 1max (Qi »)..min (Qi)} (6 0 )
a
Correct Qi can always be found to (55) holds .
ti + I-ti < K< 00 and there A7 ) For any i, exists a real number f >O such that
~ j ; tot'
~R-l(t)h(j-d,
ensure
CONCLUSION
silhT(j-d,si»(l. (54)
The approaches of adaptive control of time-varying systems c a n be divided into two classes. In the first approach, the original robustness of the adaptive algorithms developed for time-invariant systems is utilized or improved to accommodate the adaptive control algorithms to time-varying systems . The current parameter estimates are taken as their predictions in this approach , whi c h makes the adaptive co ntrol algorithms
Theorem 3 Subject to assumption AI), A2), A3), A5) and A7), if the adaptive control algorithm (14)-(25) is applied to system (2), then 1Ix.(Si+l / ti+I)/l 2 /J1i(Si/t+i)112 < 1
(59)
(55)
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unable to control fast time-varying systems . In the second approach. additional information about the parameter variation is introduced to predict the parameter variation. Describing the plant parameters by a state-space model. we have obtained a unification of some of the adapti ve algorithms . It is shown that a unified approach of the stability and convergence of the adaptive control algorithms is possible by means of the state-space description of the time-varying parameters .
REFERENCES Caines . P . E. and H. F . Chen (1982). On the adaptive control of stochastic systems with random parameters : A counter example . Proc 151 Workshop on Adaptive Control. Florence. Italy. Goodwin. G.C . and S.W. Chan (1983) . Model reference adaptive control of systems having purely deterministic disturbances . Trans Aut Control. AC-28, IEEE pp . 8SS-8S8 . Goodwin. G.C . and Sin . K. S . (1984) . Adaptive filtedng predjctjon aur! contro l Englewood Cliffs . NJ : Pretice Hall . Goodwin. G. C. and D. Q. Mayne (1987) . A parameter estimation perspective of continuous time model reference adaptive control . Automatjca 23 ppS7-70. Jedner . U. and H. Unbehauen (1988). Intelligent adaptive control of a class of time-varying systems. Proc of 12th IMACS World Congress. Vol . 1 . Paris. pp . 377-380 . and B.F . Womack (1984) . Kung. M.C. Discrete time adaptive control of linear system with preload nonlinearaity. Automatjca 20. pp . 477-479. Kung. M. C. and B.F . Womack (1984 ) . Discrete time adaptive control of linear dynamic with a two-segment piecewiselinear asymmetric nonlinearity . IEEE. Trans Aut Control. AC-29 . pp. 170-172. Li. Z. (1986a) . Discrete-time adaptive control for time-va!:'yi.ng systems and a class of gray systems. Prepr IFAC Syrop on Microcomputer ...bpplication in process contro l. Istanbul. ' Turkey. 1986 . Li. Z. (1986b). Discrete tiroe adaptive control for periodically time-varying systems . 2nd IFAC Workshop on Adaptive Systems in Control and Signal Proce?sing. Lund. Sweden pp . 411 415 . Li. Z. (1987a) . Discrete time adaptive time-varying control for linear fast systems. IEEE Trans Aut Contr AC-32. pp.444-447. Li. Z. (1987b). Stochastic adaptive for time-varying systems via an auxiliary variable. Proc of 26th cnc. Los Angeles C A USA. pp.8S0-854. Li. Z. (1988) . A discrete-time adaptive controller for time-varying systems. uQ!:.... of Int Conference Control-.!lJL.. UK. pp.S23 526 . Matko. D. and B. Neroec (1988) . Identification of fast time-varying systems-comparative characters of thre e algorithms. Preprints of 8th IFAC IIFORS Symposium on Identification and System parameter e'3timation Vol. 2. Beij ing. pp. 720-725. Millnert. M. (1982). Identification and control of systems subject to abrupt changes . ~al dissertation. Linkoping University. Sweden. Ohkawa. F. (1985) . Model reference adaptive control system f o r discre te linear time-varying systems with periodically varying parameters and time delay . Int J Control, 44. pp. 171-179. Tsakalis . K.S. and P. A. Ioannou (1988) . Adaptive control of time-varying plants : Simple examples. Int J of adaptive control and signal processing vol.2. pp . 291-310. Tsakalis. K.S. and P.A . Ioannou (1989). Adaptive control of linear time-varying plants : A new model reference controller structure . IEEE Tran, Aut. Contr AC-34. pp . 1038-1046. Xie. X. Y. and R. J . Evans (1984) . Discrete-time adaptive control for deterministic time-varying systems. Automatica. 20. 1984 . pp . 309-319 .
The key difficulty in adaptive control of time-varying systems is the prediction of time variation in plant parameters . The reason is as follows: 1. The information provided by plant input and output is far from being sufficient for tracking the parameters. 2. Unless large frequent changes in the parameters are well predicted. there will be always tracking errors between the reference output and the plant output . 3 . There are no efficient schemes for the identification of fast time-varying systems . Uptodate . only limited success has been achieved in the adaptive control of fast time-varying systems. The reason is that most of the research works are based on the standard time-varying ARMA models. Additional a priori knowledge on the parameter variation is necessary to the adaptive control schemes. However. the knowledge depends on the engineers' insight about the real systems. For different physical systems the knowledge may be quite different . It seems impossible to develop an adaptive control algorithm which is applicable to many types of time-varying systems . It seems to us from this paper that the following topics are interesting and promising for the study of adaptive control of systems with fast time-varying parameters. 1. Developing new models with which the additional information available about the parameter variations can be utilized efficiently. 2 . Combination vf adaptive control schemes and such intelligent control schemes as utilize the additional information about the variation in the parameters. 3. Developing the adaptive control algorithm into which engineers can easily incorporate their knowledge about the plant dynamic variation. 4 . Unlike the time-invariant systems. it is possible that the tracking error always exists in the adaptive systems. So the robust estimation schemes and robust controllers is particularly important for adaptive control of time-varying plant. 5 . Adaptive control varying systems .
for
special
Various controllers for 6. control of time-varying systems.
time-
adaptive
7. Adaptive control of the systems whose parameters are functions of one or several measurable variables . ACKNOWLEDGMENT The work of the first author was supported by the Alexander von Humboldt Foundation and in part by the Chinese Science Foundation .
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