- Email: [email protected]
Discrete-time adaptive control for deterministic time-varying systems
0005-1098/8453,00 + 0.00 Pergamon Press Ltd. © 1984 International Federation of Automatic Control
Automatica, Vol. 20, No. 3, pp. 309-319, 1984 Printed in Great Britain
Discrete-time Adaptive Control for Deterministic Time-varying Systems* XIE XIANYAt and ROBIN J. EVANS:~
An adaptive control algorithm using a restricted complexity time-varying model controls rapidly time-varying, deterministic, discrete systems. Key Words--Adaptive control; convergence; discrete-time systems; least-squares estimation; self-tuning regulators; time-varying systems.
quantities alone for identification problems, and output quantities alone for adaptive control problems. In the practically significant situation of unknown time-varying parameters, there has recently been some progress on the analysis of existing algorithms when used in the time-varying case. Preliminary results by Caines and Dorer (1980) and Evans and Betz (1983) established bounds on the output control error for the case when the parameter variation eventually died out. Johnstone and co-workers (1982) established the exponential convergence of recursive least squares with an exponential forgetting factor in the timeinvariant case and under the condition of persistent excitation. Anderson and Johnstone (1983) showed that exponential convergence of the algorithm in time-invariant case will guarantee tracking error and parameter error boundedness when the plant parameters are actually slowly time varying. Fortescue, Kershenbaum and Yastie (1981) presented an algorithm with variable forgetting factor to avoid the problem of covariance matrix blow-up. Cordero and Mayne (1981) have established the convergence of the algorithm with a variable forgetting factor in the time invariant case. Simulation results show, however, that in the fast time-varying case the performance of existing algorithms is limited. The problem stems from the fact that the existing algorithms are based on a time invariant model. Below we present a series of improvements to the standard least-squares estimator and one-stepahead controller structure which ensure good adaptive controller performance for rapidly varying system parameters. Our first improvement concerns the speed of convergence of the parameter estimation algorithm. We know that the speed of parameter estimator convergence and the rate of time variation allowed are directly related. Thus we develop a modified least squares estimator employing a t~ weighing in the cost and a 'tz resetting'
Abstract--There is currently considerable interest in the application of recently developed self-tuning control algorithms to real control problems. However, there exist few algorithms which will work for other than slow time-varying systems. A new algorithm which is able to successfully control unknown systems with quite rapidly varying parameters is presented. Several simulation results are also presented. A convergence proof in the linearly time-varying parameter case is given. 1. INTRODUCTION
THE self-tuning regulator (Astrrm and co-workers, 1977; Astrrm and Wittenmark, 1973) is now over 10 years old and has established itself as an important and useful tool in modem control applications. The self-tuning regulator is conceptually a very simple device. The unknown plant parameters are estimated on-line by a recursive least-squares algorithm and these estimates are then used in a minimum variance controller. Recently Goodwin, Ramadge and Caines (1980, 1981) established the global convergence of the controlled output for such schemes when applied to minimum phase timeinvariant deterministic systems, and stochastic systems subject to a positive real condition. These convergence results were further extended by Evans, Goodwin and Betz (1981), to classes of nonlinear systems and systems whose output is corrupted by unknown but bounded noise. Anderson and Johnson (1982) presented conditions for exponential convergence of adaptive identification and control algorithms and showed that persistent excitation conditions involving input and output quantities can be converted to ones involving input * Received 7 January 1983; revised 7 July 1983; revised 19 December 1983. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor P. C. Parks under the direction of editor B. D. O. Anderson. t Department of Computer Science, Shanghai University of Science and Technology, Shanghai, China. Currently on leave at the Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308, Australia. :~Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308, Australia. 309
310
X. XIANYA and R . J . EVANS
procedure. We show by simulation this algorithm converges faster than standard least squares. Secondly we employ a P(t) matrix resetting procedures based on the current value of~VP~. This ensures that the estimation algorithm stays awake. Finally, and most importantly of all, we model the unknown p a r a m e t e r variations as an offset linear ramp. As will be d e m o n s t r a t e d below this allows good control performance even when the u n k n o w n p a r a m e t e r s are varying sinusoidally and with step changes. The remainder of this p a p e r is arranged as follows: Section 2 describes the new algorithm in detail. Section 3 presents performance simulations and Section 4 discusses theoretical aspects of convergence.
O(t) = O(t -
1)
P(t - 2)~b(t - 1)e(t) + (l/t2) 2 + ~b(t - 1)xP(t - 2)~b(t - 1) P(t-
1)=P(t-2) P(t - 2)~b(t - 1Rb(t - 1)TP(t - 2) (l/t2) 2 + ~b(t - 1)Tp(t -- 2)~b(t -- 1)"
At the start of computation, t = tl = t2 = 1, P ( - 1 ) = P0 = K0l, and ~la(0) = dz~(0) = "'" = d,l(0) =/~11(0) =/~21(0) . . . . . /~,.1(0) = 0, where Ko > 0 and I is an 2(m + n) × 2(m + n) identity matrix. As c o m p u t a t i o n proceeds we determine tx by t ~ = t - k T w h e n kT
2. ADAPTIVE CONTROL ALGORITHM T h r o u g h o u t this p a p e r we will assume that the plant to be controlled can be described by a timevarying A R M A model of k n o w n order and is stably invertible. Following standard notation we write y(t + 1) = c q ( t ) y ( t ) + ~ 2 ( t ) y ( t - 1) + ' " + %(t)y(t
-
=
0
~,o(kT) = £ t i o ( k T - )
+ T£t,,(kT-)
£ql(kT)
= ~l(kT-),
i = 1.....
fljo(kT)
= fl;o(kT-)
+ Tfl;,(KT-)
fli,(kT)
=/~:x(kT-),
P(kT-
n
(5)
j = 1. . . . . m
1)= aoKolorP(kT-
1)= KiP(kT-
-
1)
where k T - means the time just before resetting and 0 < ao ~< 1 aoKoI > P ( k T - - 1) or Ka > 1. While t2 increases step by step with t until the following inequality is satisfied
n + 1) + f l l ( t ) u ( t )
+ [32(t)u(t -- 1) + -'" + fl,,(t)u(t -- m + 1)
(4)
(1)
4}(t - 1)xP(t - 2)~b(t - 1) ~< e
(6)
where y and u are the plant output and control variable, respectively, and oq(t), i = 1 . . . . . n, flj(t), j = 1. . . . , m are the u n k n o w n time-varying parameters. However, n, m are assumed known. Define the 2(m + n ) × 1 estimated p a r a m e t e r vector
where e is a small positive scalar, typically e = 0.1. When (6) is satisfied, the following resetting is carried out
O(t) T = ( ~ l O , d 2 o . . . . . ~.o,
and then continue c o m p u t a t i o n as in (1) with t2 again increasing step by step. An alternative algorithm which also performs well is to let t2 = 1 for all t. This is the ordinary least squares if no resetting is made. This algorithm does not converge quite as fast as when t2 is variable (as will be shown below), however, it is still satisfactory for control of most time-varying systems and is preferable in the noisy case. Clearly the basic idea of the above described technique is to assume that the u n k n o w n parameters vary as
/~1 O, ~20 ..... flmO, ~11, ~21 ..... ~nl, J~l 1, J~21 .... ,~m,) and the vector
2(m + n)×
1 measurable
(2)
information
q~(t)x = (y(t), y(t - 1) . . . . . y(t -- n + 1), u(t), u(t -- 1) . . . . . u(t -- m + 1), t l y ( t ) , t l y ( t -- 1) . . . . . t l y ( t -- n + 1), t l u ( t ) , t l u ( t -- 1) . . . . . t l u ( t -- m + 1)).
(3)
Given the desired output at time t, y * ( t ) , the adaptive control algorithm can now be written as y * ( t ) = O(t -
1)x~(t - 1)
e(t) = y ( t ) - O(t -
1)r4~(t - 1)
t2 = 1 P(t-2)=aoKol
(7) or
P(t-2)=KIP(t--2)
, i ( t ) = Ctio + (t - to)Cqa + (t
-
to)2Cq2
+
"'"
(8) flj(t) = fljo + (t -- to)fljx + (t -- to)2flj2 + "'"
Then from (1), we have
y(t + 1) = OTck(t)
(9)
Discrete-time adaptive control for deterministic time-varying systems
where D is the dimension of ~(t) and that y * ( t ) be made sufficiently rich (Xie and Evans, 1983a). In algorithm (4) t2 is a variable and we call that algorithm modified least squares. It is derived from optimization of the following cost functional
where 0=
( 0 C 1 0 , ' ' ' , 0~n0, i l l 0 , - ' ' ,
flmO, 0 ~ 1 1 , ' ' ' ~ n l ,
f i l l . . . . tirol, 0C12,'-" (~n2, i l l 2 . . . . tim2"" .)T
dp(t) = ( y ( t ) . . . . y ( t -
1), u ( t ) . . . . . u ( t - m + 1),
n +
(t -
to)y(t),...,(t
-
(t -
t o ) U ( t ) . . . . (t -
(t -
to)2y(t),...,
(t -
to)2U(t -
n +
1),
m +
1),
to)y(t to)U(t -
(t -
J
1 N
=
2,=~1 ?(y(0 - 4,(t
-
1)T0) 2 +
1
to)ZU(t) .... , (0 -- 00)Tpo
1),...)V
m +
311
(12)
1(0 -- 00)
(10) and has the form
(9) has the same form as that in the time-invariant case, thus we can extend all the sophisticated techniques of time-invariant deterministic adaptive control systems into the time-varying case. For simplicity, and especially for reduction of computational effort we only consider a first order approximation to the unknown parameters, thus
O(t)
=
O(t
-
1)
P(t - 2)~(t - 1)e(t) + (l/t) 2 + ~(t - 1)Tp(t -- 2)~(t -- l) (13) P(t - 1) = p(t - 2) P(t - 2)~(t - IRb(t - 1)TP(t - 2) 0 / 0 2 + 4p(t - 1)ZP(t -- 2)qb(t -- 1)"
O~i(t) = O~iO "k- (t -- t 0 ) ~ i l
(11) flj(t) = fljO + (t
- -
to)~jl
The derivation and convergence of (13) with a one-step-ahead control strategy can be proven in exactly the same way as for ordinary squares (see Goodwin, Ramadge and Caines, 1980) and is left to the reader. When t ~ ~ this algorithm reduces to the orthogonalized projection algorithm, which performs poorly in the noisy case (Goodwin and Sin, 1983). When t --, oc, qb(t - 1)Tp(t -- 2)4~(t -- 1) -, 0 resulting in numerical problems, consequently the second resetting (7) is required. The convergence of this algorithm with resetting when used in the time-invariant system can again be proven without difficulty. Simulation results clearly indicate that the modified least squares (M.L.) algorithm has faster convergence than ordinary least squares (O.L.), as shown in Table 1, where the desired output y * ( t ) is a
and substitute tl for (t - to) yielding (2), (3). If we use the onqine ordinary least-squares estimation algorithm and a one-step-ahead control strategy, then we get the adaptive control algorithm (4) with t z = 1. However, the performance of (11) is unsatisfactory when ta = t - to becomes large, thus we need to reset when q reaches T, where T is chosen large enough to allow reasonable convergence of the parameters but the variation of the real parameters is still small enough. Also, the resetting is needed to avoid the algorithm gain tending to zero and to keep 4~(t) bounded. It appears to be difficult to compute the optimal T, however it is easy to choose T in practice. We recommend that (cf. Fig. 7) T > D, in the purely deterministic case,
TABLE 1. COMPARISON BETWEEN MODIFIED LEAST-SQUARESALGORITHM (M.L.) AND ORDINARYLEAST-SQUARESALGORITHM (O.L.) WITH AND WITHOUT P(t - 2) RESETTING. WHEN RESETTING P(t - 2) = 10P(t- - 2). t
0
1
2
3
4
5
6
7
5.22015
2.49208
1.20017
0.25710E-I
0.15021E-2
0.14549E-2
0.I0440E-2
0.68671E-3
5.22015
2.49208
1.20017
0.25710E-I
0.15021E-2
0.14129E-2
0.I0353E-2
0.78208E-3
rese~ing
5.22015
2.49208
1.20026
0.32437E-I
0.51079E-2
0.49647E-2
0.39956E-2
0.35578E-2
no-resegclng
5.22015
2.49208
1.20026
0.32437E-I
0.51079E-2
0.49547E-2
0.48191E-2
0.46976E-2
ii
12
13
14
15
reaetclng M.L. no-resetting
t
O.L.
t
D M.L.
* Where
J 1
9
resetting
0.43166E-3
0.35259E-3
0.33159E-3 ] 0.63380E-4
0.28151E-4
1 0.96110E-5
0.82487E-5
0.75626E-5
no-reset clng
0.62546E-3
0.52355E-3
0.52263E-3
0.51833E-3
0.50592E-3
0.49274E-3
0.47466E-3
0.45283E-3
resecclng
0.33079E-2
0.31463E-2
0.30335E-2
0.24941E-2
0.20927E-2
0.20066E-2
0.19400E-2
0.18870E-2
no-resetting
0.45877E-2
0.44877E-2
0.43966E-2
0.43508E-2
0.42379E-2
0.41551E-2
0.40792E-2
0.40089E-2
O.L. -.
i0
I l
8
D ~ II0(t)ll = II0 - 0(t)ll = [(~1 - ~l(t)) 2 + (~2 - ~2(t)) 2 + (fl - ~(t))2] 1/2.
"}"To avoid numerical problems, when t 2 reaches 10 we reset t 2 to 1, while the P matrix is not reset.
312
X. XIANYA and R.J. EVANS (0)
square wave with period of 20 and amplitude of + 1. The plant to be controlled is described by y(t + 1) = 4y(t) + 2y(t - 1) - 3u(t).
In both algorithms we use a time-invariant model. The initial estimates are ~ ( 0 ) = 1.5, ~z(0)= 3.0, /~(0) = - 4 . 0 and y(0) = y ( - 1 ) = 1.0, e = 0.1.
3. S I M U L A T I O N
RESULTS
In this section we present several simulation results which clearly demonstrate the ability of our algorithm to control unknown time-varying systems. In particular we examine step, ramp and sinusoidaUy varying unknown parameters, we study the effects of different resetting procedures, and we also examine the effects of different parameter models. It is important to note that in all the cases presented below a standard deterministic adaptive controller is unable to control the system and rapidly blows up. In all the simulation cases the plant is described by
-I
-2
(b) 30
~
2O
,a-
io
y(t + 1) = cq(t)y(t) + ~t2(t)y(t - 1) + fl(t)u(t).
The desired output y*(t) is assumed to be a square wave with period of 20 and amplitude of + 1. Case 1. In this case one unknown parameter is assumed to be varying linearly, and as expected the algorithm works perfectly. But if ~l(t) increases unboundedly, then u(t) increases unboundedly, too. This case is simply included to demonstrate that the algorithm performs correctly when the unknown parameter variation corresponds to the modelled variation. Figure l(a) and (b) clearly shows the excellent performance obtained. Note that if ~l(t) is assumed time-invariant, the system output is highly oscillatory. Here we have assumed that ~2 and fl are unknown and time-varying, despite that they are actually time-invariant. Case 2. In this case we have a step in the parameter ~l(t) at t = 26 (Fig. 2a and b). As can be seen from Fig. 2(a) considerable output oscillation occurs during the parameter change, however the system quickly settles down to satisfactory control. Case 3. For this case we have ~a varying sinusoidally, ~2 being constant and fl undergoing two step changes followed by a ramp (Fig. 3a and b). As in the Case 2 above, the system output undergoes a severe transient at the instant of fl step changes, but rapidly settles down to control the system. Case 4. Figure 4 examines the performance with different estimation algorithms, all assuming a time° invariant parameter model but the real plant being slowly time-varying. Ordinary least-squares without P(t) resetting is unable to control the system. Ordinary least-squares with P(t) resetting works quite well, while the modified least-squares performs slightly better.
o
A I-
i
i
i
i
i
3 2 I
-I
1"
,oz.
-4(
FIG. 1. (a) and (b). Simulation results of the modified leastsquares algorithm with a time-varying model. OSC, drastic oscillation. The real parameters are el(t) = 4 + 0.5t, e2(t) = 2, B(t) = -3. Initial estimates are ~1o(0) = 1.5, ~2o(0)= 3.0,~o(0) =-4.0, ~11(0)=~21(0)=~1(0)=0 and y(0)=y(-1)= 1. ~= 0.1. Resetting period T= 10. Case 5. The plant has one rapidly varying parameter st(t) and the other two are constant. We consider two different approximation models. The first one is y(t + 1) = (~1o + toql)y(t) + (Ot2o+ to~21)y(t - 1) + (flo + tfll)u(t)
i.e. we regard all the parameters as being timevarying. The second model is simplified to y(t + 1) = (Cqo + t~lx)y(t) + ~2y(t - 1) + flu(t).
Discrete-time adaptive control for deterministic time-varying systems (a)
(o)
2
1
I
7
313
0
I 0 ~
"r
q 0
>,
810
,~o
-I
-I
-2
[ (b)
#,
ill t
6
4
2
II II II
2 i
o
~o
i {,
4
6'0
8'o
•
I00
3
'[
o--
20
I ~)
0 -I
~
I
I
,
20
I
I
30
40
I 50
,,II j|
,I II
I
I
40
60
I
[
80
I00
-.5
1"
_.o
-:5
FIG. 3. (a) and (b). The same as Fig. l(a), except that the real parameters are:
cq(t) = 4 + sin (=t/100), FIG. 2. (a) and (b). The same as Fig. 1(a), except that the real parameters are:
4, 0~
Comparison of the simulation results (Fig. 5a and b) show that if we know a priori that some parameters are actually constant, we should prefer to use the corresponding simplified time-varying model. This not only results in better control performance but also in a considerable reduction of the computational effort involved. Case 6. Finally we consider the performance of our algorithm using the simplified time-varying
¢t2(t) = 2,fl(t) =
25 < t ~<65 65
model with an optimal t~ resetting period T. The performance shown in Fig. 6(a) and (b) is extremely good. Resetting considerations
As already pointed out, there is no straightforward theoretical technique for choosing the resetting period T. However, in the purely deterministic case and provided y*(t) is sufficiently
314
X. XIANYA and R.J. EVANS
g
z
(o)
D
x 12
i
I I I I I~.~1 I I~
7
I
I
II 6o'i I
I I I 8dl I I
I I I ,do I
i¢
2c
•Ooo.a r ~ 30
40
~R,
-2
FIG. 4. Comparison between different algorithms. O, Modified least-squares algorithm, using time-invariant model. When ~Tp~) ~< 0.1, reset P(t - 2) to 10P(t- - 2); x,ordinary leastsquares algorithm, using time-invariant model. When ~TP~b ~< 0.1, reset P(t - 2) to 10P(t- - 2); V], ordinary leastsquares algorithm, using time-invariant model without resetting. The real parameters are ~ ( t ) = 4 + sin (zrt/100), ~z(t)= 2, f l ( t ) = - 3.
( b )~.-~
I~
rich it is recommended that T be chosen to be close to but slightly greater than the dimension of 4)(t). Figure 7 shows the performance measured as both maximum output error emax and average output error e.... for different parameter variation rate and different resetting period T, where em~xand ea~ are defined as ~.
1
100
eave = 9-i- ~
t=10
emax
=
max
ly*(t) - y(t)I
lo<~t~1oo
(14) (15)
y * ( t ) - - y(t). y*(t)
"
0
~,,~.~
I
X
I
I0
II
I
20
30
I
I
50
40
'
¢~
2
o....
~ooo~o k
I I0
0
oo'--1"~oeoKt~x~q?o~o
210
3 0 4 0 1
,~
....
510
T
The plant controlled is assumed to be y(t + 1) = ~l(t)y(t) + ~2y(t - 1) + flu(t) and supposing we have known c(2, fl to be constant and use the simplified time-varying model. In Fig. 7(a) ~t(t) = 4 + sin(nt/20), ~2 = 2, fl=-3 and in Fig. 7(b) ~ 1 ( t ) = 4 + sin(nt/10). ~2=2, fl=-3. In both cases ~ o ( 0 ) = 1 . 5
02,~(0) = O, d2(O)= 3.0, 1~(0)= 4.0, y(O)= y ( - 1 ) = 1.0. The desired output y*(t) is a square wave with period of 20 and amplitude of _ 1. In Fig. 7(a) the optimal choice is clearly seen to be T = 5. In Fig. 7(b) the optimal choice is also T = 5. Either resetting P(t - 1) to be aoKol or to be K ~ P ( t - - 1) will do. We recommend that P(t - 1) be reset to aoKol rather than K ~ P ( t - - 1). This is not only for the reduction of computational effort but also to prevent P(t - 1) becoming indefinite due to round offerror and also to prevent P(t - 1) from 'blow-up' in steady state. In our simulations ao = 0.1, Ko = 100.
--q FIG. 5. (a) Comparison between the performance of algorithms using different time-varying model, x,Time-varying model; O, simplified time-varying model. The real parameters are el(t) = 4 + sin (nt/20), e2(t) = 2, fl(t) = - 3 . Resetting period T = 10. (b)The same as those in (a). O, Real parameters.
The whole question of resetting needs further examination. For example in many cases when T is small, resetting procedure (6), (7) is not required and actually will not happen. Our simulation shows, however, that better control performance results if we use both the resetting procedures (5) and (6), (7) when T has to be chosen large.
Discrete-time adaptive control for deterministic time-varying systems
I
4
(a)
315 0.4
(a) o - eav e
I I
i
I I
, I
f I
I I
I t
I I
6 IJ,
!"
I
I
I
I
• - emo x
\ .\ E
0.2
2
0)
I I
\ .... 2
I o {b)
4
6
8
I0
T
4
4
(b)
7
1
0
I
I
I
L
L
20
40
60
80
100
T
o "o
0~ >
../\,,"
2
I 2
0
l 4
I 6
I 8
I0
T
0
2
Id 410
I 60
i 80
i I00
-I
FIG. 7. (a) The control performance versus the resetting period. The real parameters are cq(t)=4+ sin(nt/20), ~t2(t)= 2, fl(t) = -3. (b)The same as (a) except the real parameters are ~q(t) = 4 + sin (nt/10), ~t2(t) = 2, fl(t) = - 3 .
-5 o -4
FIG.6. (a) Simulation results of modified least-squares algorithm using a simplified time-varying model and an optimal resetting period (T = 5). The real parameters are ~tl(t) = 4 + sin (nt/20), ct2(t) = 2, fl(t) = -3. Real line, desired output; O, y(t). (b)The same as those in (a). O, Real parameter; ©,estimates.
be achieved with a b o u n d e d input sequence if plant parameters are known. F o r simplicity, we take t2 = 1 and do not use resetting procedure of (6), (7). That is, we just use ordinary least-squares algorithm together with tl resetting procedure (5). Theorem
the the the the
1
Given the plant 4. CONVERGENCE CONSIDERATIONS We are unable to prove convergence of our algorithm in the general case where the parameters are varying in an arbitrary fashion. In fact, we cannot expect to prove convergence if the real plant does not have exactly the same parameter description as our model. However for this case it m a y be possible to derive a b o u n d on the control error similar to Evans, G o o d w i n and Betz (1981) or A n d e r s o n and Johnstone (1983). Below we present a convergence p r o o f based on the real parameter being linearly time-varyin~ which includes the time-invariant case as a special one. Even in this case we still require a strong assumption like persistent excitation. We also need the assumption that lim,_.~ [_V(t) - y*(t)] = 0 can
y ( t + 1) = ~ l ( t ) y ( t ) + ~,(t)y(t
+ ct2(t)y(t -
1) + " "
n + 1) + f l l ( t ) u ( t )
+ f l , . ( t ) u ( t - m + 1)
+ ...
(16)
and
~l(t) = ~lo(to) + (t - to)~,~
~.(t) = ~.o(to) + (t - to)~.~ ill(t) = fllo(to) + (t - t o ) f i l l
fl,~(t) = fl.o(to) + (t - to)ft.1
(17)
where 0til, fljl (i = 1. . . . . n , j = 1 . . . . . m ) are constant.
316
X. XIANYA and R. J. EVANS
If the algorithm (4), (5) ia applied to plant (16) (17), then {y(t)}, {u(t)}, {~b(t)} are bounded and lim [y(t) - y*(t)] = 0
(18)
t~c~
provided the following assumptions are satisfied (i) Cqo, Cqx, fl~o, flj~ (i = 1..... n, j = 1..... m) are unknown but n, m are known; (ii) the plant (16), (17) is stably invertible and with ly*(t)l ~< ml < o~ the control objective lim,.~ [y(t) - y*(t)] = 0 can be achieved with a bounded input sequence if the plant parameters are known; (iii) there exists a constant integer M such that 1 j+M-1
plI ~ ~
.~. ~(i)~(i)T
(19)
where ~ ( k T ) , flfo(kT), 8ii a n d / ~ are the estimates of Cqo(kr), fljo(kT), ~il and fljl, respectively, just before the resetting time t = (k + 1)T. It is clear that the estimation error will change when the estimated parameters exchange. If before resetting, the errors are ai-o = ~i~( k T) - Cqo(k T) ~il ~ ~il -- ~il
(22) fl]-o -- fl~-o(kT) - fljo(k T)
&, = P;, -
then, after resetting, the errors become Rio = ~ o ( k T ) + T ~
- ~io[(k + 1)T] = ~
+ T~
~il = ~i~ for some Pl > 0 and all j, where
J~j0 = ~jO "Jr- Tfl~
(23)
~p(i)T = [y(i) ..... y(i - n + 1), u(i) . . . . . u ( i - m + 1),(i - j ) y ( i ) ..... (i-j)u(i-
m + 1)].
Proof: Our proof is along the lines of Goodwin, Ramadge and Caines (1980), but with additional considerations to account for resetting. Firstly note that (17) can be written as ~ ( t ) = a t o ( k T ) + (t - kT)~xl
In other words the estimation error vector & 0 - O changes from 0 = (/~, O~)r to = 0T ~ T where O (#I+ 2,, =
.....
.....
(24) 01 =
.....
IV1 . . . . .
Define V(t) = O(t)Tp(t -- 1)- 10(t), then between two adjacent resetting times, we have (Goodwin, Ramadge and Caines, 1980)
% ( 0 = ~.o(kT) + (t - kT)~.a V(t)fl~(t) = flxo(kT) + (t - k T ) f l t l
1)= 0(t - 1 ) r ~ ( t - 1 ) ~ ( t - 1 ) r 0 ( t - 1)
V(t-
1 + ~ ( t -- 1)Tp(t -- 2 ) # ( t -- 1) (25)
tim(t) = flmo(kT) + (t -- kT)flml
e2(t)
where, K = 0, 1.... and a11,0c21, ..., ~,1, fill,.., fl,,1 are constant and ~io[(k + 1)T] = aio(kT) + Tail,i = 1 , . . . , n
(20) fljoE(k + 1)T] = fljo(kr) + r f l j x , j = 1 . . . . . m. When k T < t <~ (k + 1)T, the algorithm estimates aio(kT), fljo(kT) a n d ~ia, fljl. When t = (k + 1)T resetting with (5) is equivalent to starting the recursive least-squares estimation of aio [(k + 1)T], rio [(k + 1)T] and sit, fljl under the initial condition &o[(k + 1)T] = ~ ( k T )
+ T~,,8,
= +i]
(21)
1 + ~(t-
1)vp(t-
2)#(t-
l)
~<0
where e(t) = d~(t - 1)x0(/- 1) = y*(t) - y(t). Thus V(t) is a nonincreasing nonnegative sequence between two adjacent resetting times. Now what will happen when resetting occurs. As mentioned above, after resetting O changes because the estimated parameters have been changed. To ensure V(t) is still nonincreasing we need assumption (19) and we need to choose aoKo properly. When k T < t < (k + 1)T, we have (Goodwin, Ramadge and Caines, 1980) p(t -
1) = p(t - 2) -x + # ( t -
1)#(t -
1) T
t--1
/~j0 [(k + 1)T] = l ~ ( k V ) + T ~ f l , ~ j l = ~f~
= p(kT-
1) -1 + ~
¢~(i)#(i) T
i=kT I
* Note the notations here are different from those in (5), where ~o(KT-) means the estimate of Cho(KT- 1) obtained at time
KT-.
t-1
aoKo + i=kT ~ ~(i)~(i)T.
(26)
Discrete-time adaptive control for deterministic time-varying systems When t = (k + 1)T, just before resetting, using (19) and choosing T = M 1 P[(k + 1)T- - 1] -1 = troKo I +
(k+1)r-t 1 E ¢(i)~(i)T t> i = kT
317
considering (26), we have 0 < P(t - 2) -%
I
0 ~< ~(t - 1)TP(t - 2)~(t - 1) ~< aoKoll~(t + p, 721
(27)
~7ogo
thus e(t)
hence 1 ).m~.{p[(k + 1)T- - 1] -1} ~> a--0-~o + p i T 2
1)112]1/2 = 0. tlira ~ O 0 [1 + aoKoll4~(t (28)
where the )~mi.{p[(K + 1)T- - 1] -1} is the minimum eigenvalue of P[(K + 1)T- - 1] -1 If we choose ao > 0 and Ko > 0 such that Koaopl >i 2 + (1/T 2)
1)115 (33)
Now 11¢(0112 -- Iv(t) 2 + ... + y(t - n + 1)2 + u(t) 2 + " " u ( t - m + 1)2][1 + (t - k T ) z] <-% [y(t) -2 + "'" + y ( t -
n + 1)5
+ u(t) 2 + ... + u2(t - m + 1)][1 + T2].
211 + (l/T2)] 1 I> go/> (1/T2) + KoaoPl > 0
(34)
(29) From assumption (ii), we have
then 211 + (l/T2)]
lu(t)l < m2 + ma max ly(t)[,
211 + (1/T2)]
0 ~ m2 < ~ , 0 < m3 < ~ , Vt.
Koao >1 (1/KoT2) + aoPl >~ [(1/aoKoT 2) + Pl ]ao
/>
Considering e(t) = y*(t) - y(t), and [y*(t)] < ml
2(T 2 + 1)
[y(t)[ < [e(t)[ + ml
(1/o'oKo) + Pl T 2 (because of 0 < ao ~< 1) />
2(T 2 + 1) 2mi,{P[(k + 1)T- - 1] -1}
[using (28)]
thus, from (34), (35), we have I[q~(t)[[ < cl + c2 max [e(t)[, 0 <%cl < oo,0 < c2 < oo.
{[l(0~r, 01)It2 + T2I[(0~, 0T)112}2
>/
{ll0I, Og)ll2 + T211(Og, 0r)112}2 1> (/~i, 0T)p [(k + 1 ) r - - 1 ] - I(0T, 0T) T
lim e(t) = lim [y*(t) - y(t)] = 0
II(01 + TOl, O~)ll~ I> (0T, ()T)p [(k + 1)T- - 1 ] - I(0T, 0T) T thus (rlgo IIOT + TOT2,0~)II2 -%<(/~,01)P[(k + 1)T- - 1]-1(0"~,0~) T
(36)
According to the 'Key Technical Lemma' of Goodwin, Ramadge and Caines (1980), we get
2m~.{P[(k + I ) T - - 1]-~}II(OLOT2)II 2
(30)
where 0 is the (n + m)-dimensional null vector. Equation (30) means, by proper choice of ¢0 and Ko, we can ensure that V(t) is nonincreasing at the resetting time. Thus V(t) is a nonincreasing nonnegative sequence in 0 ~< t < ~ and e2(t)
,=o [1 + ~ ( t -
(35)
1)rP(t - 2)~(t - 1)] < oo
and {IRb(t)l)}, {u(t)}, {y(t)} are bounded. [] Recently, Anderson and Johnstone (1983) proved that if the adaptive control algorithm in the timeinvariant case has exponential parameter convergence, then even if the real plant is slowly timevarying the control performance is still good. Thus, in our case, if we can establish exponential parameter convergence in the linearly time-varying case, then even if the real plant parameters are not exactly linearly time-varying but their first order derivatives vary slowly, then good control performance can be expected. In fact the simulation results in Section 3 have to some extent implied this assertion. Motivated by this we now establish parameter exponential convergence for our algorithm.
(31) which implies
Theorem 2 e(t)
lim ,-,~o [1 + ¢(t - 1)TP(t - 2)#(t - 1)] 1/2 = 0 (32)
Given the plant and assumptions exactly as in Theorem 1, when the algorithm (4), (5) is applied to plant (16), (17), then the parameter estimates converge exponentially to their real values.
318
X. XIANYAand R. J. EVANS
Proof: As in Theorem 1, between two adjacent resetting times V(t) - V(t - 1) <.%0
-
thus V[(K + 1)T-] ~< V ( K T )
(37)
where V [ ( K + 1)T-] is the value of Liapunov function at time t = (K + 1)T just before resetting and V ( K T ) is the value at time t = K T , just after resetting. In other words, from K T to (K + 1)T- no resetting is made. So
[(K + 1)T-] ~< O ( K T ) T p ( K T - 1)-10(KT) (38)
and
y*(t) = O(t - 1)V¢(t -- 1) e(t) = y(t) -- O(t -- 1)T¢(t -- 1)
0 [ ( K + 1)T- ]Tp[(K + 1)T- - 1 - - ' 0
substituting (27) into p ( K T - 1) = aoKoI,
Noting that 0(K T) is the parameter error at time just after resetting, i.e. O ( K T ) = O ( K T ) O(KT), (44) thus implies parameter exponential convergence. In the proof of Theorem 1 and Theorem 2 for simplicity we let t2 = 1 and do not use the resetting procedure (6), (7). In fact it is straightforward to establish the result if we let t2 -= tl and omit the resetting procedure (6), (7). In this case, the algorithm reduces to KT
(38)
0 ( 0 = O(t - 1) +
P(t - 2)q~(t - 1)e(t) (1/tt) 2 + q~(t - 1)xP(t - 2)¢(t - 1)
considering
(4')
P ( t - 1) = P(t - 2) P(t - 2)¢(t - 1)¢(t - 1)xP(t - 2)
+ 1)T- ]112 ~< a@KoII0(KT)II2
110[(K + 1)T- ]11z [IO(KT)[[ 2
-
1 <~ 1 + o o K o p l T z'
O[(K+ 1)T] = (0T + TOT,0T) T II0[(K + 1)T][[ 2 ----11(01 + T02,02)1[ 2
= 2(1 + T2)II0[(K + 1)T-]112. (40) From (39), (40), we have
1
SO
2(1 + T 2) (42)
In other words, if we choose KoaoPl strictly larger than 2 + (1/T 2) then we get (43)
which implies exponentially lira II0(gZ)ll = 0.
1)
~
(i-j)2d~(i)dp(i) T.
(19')
i=j
If we use both the resetting procedures (5)-(7), the proofs of Theorems 1 and 2 will become difficult. Another possibility is to use algorithm (4) with resetting procedure (5) and letting t2 = q. But resetting (5) is made only when the following inequality is satisfied (45)
where cl is a small scalar. In this case, resetting period T is not constant. Remarks
KoaoPa >>-2 + ~-£
K~oo
~_~
trace P(t - 1) -% 0 (41)
We have chosen in (29) that
II0[(K + 1)]TII < 1 IIO(KT)II
2)4~(t -
j+M-1
= 2(1 + T2)J[(O1,02)[I 2
2(1 + T 2)
-
rather than (26). Thus the condition (19) should be changed to pll <~
Tz11(01,02)11z]
1 + ¢roKoPlT 2 ~< 1 + [2 + ( 1 / T 2 ) ] T 2 = 1.
1)XP(t
P(t - 1) -1 = P(t - 2) -1 + t2dp(t - 1)4~(t - 1)x (26')
~< 2 [11(01,0zll z + II(T0z,0)ll:]
II0[(K + 1)TIll: 2(1 + T 2) ]]O(KT)[[ 2 <~ 1 + a o K o p l T 2"
2 + 4~(t -
with resetting procedure (5). In this case (39)
Denote 0[(K + 1)T-] = (01,02) -x T X, where 01,02 ~T ~T are those in (24), then after resetting
~< 2[11(01,0z)11 z +
(l/t0
(44)
1. In the above proofs, the persistent excitation condition (19) is crucial. If (19) is satisfied, then appropriate go and Ko are guaranteed to exist. In the time invariant case, Anderson and Johnson (1982) have shown that in the adaptive control case, persistent excitation of the desired output {y*(t)} ensures that {¢(t)} is persistently exciting. In the time-varying case, however, we do not know if the same assertion is true. Simulation shows that when {y*(t)} is sufficiently rich, the trace of P(t) decreases rapidly, which implies the above assertion is true, at least under some conditions. Moreover, in the special case, when we extend our algorithm to control a known parameter plant with unknown
Discrete-time adaptive control for deterministic time-varying systems deterministic time-varying disturbances, the persistent excitation condition (19) is automatically satisfied (see Xie and Evans, 1983a). 2. Above we have presented several modifications to the discrete-time deterministic adaptive control algorithm to handle fast time-varying parameters. The most important modification is the use of a restricted complexity time-varying model (an offset linear ramp) and the use of resetting procedure (5). These give the algorithm the ability to track timevarying parameters. The modified least-squares estimation algorithm with resetting procedure (6), (7) is used to accelerate the convergence as shown by the simulation results in Table 1 and Fig. 4. This modification to the ordinary least-squares algorithm is similar to the use of a forgetting factor, since in both cases a greater emphasis is placed on the most recent data, hence should be useful for the case when the real parameter is not linearly time-varying. In this paper we use the modified least squares algorithm rather than a forgetting factor. This is because the former yields a reduction in computational effort and has no problem of covariance matrix blow-up. We have also studied a restricted complexity time-varying model combined with a forgetting factor (see Xie and Evans, 1983b). 5. CONCLUSION
This paper has presented an adaptive control strategy which has been found to work extremely well for rapidly time-varying systems. A similar technique has also been developed for control of unknown systems with unknown time-varying deterministic disturbances (Xie and Evans, 1983a). Again the approach allows accurate control despite rapidly varying disturbances. Proofs of output and parameter convergence are presented for t 2 = 1 and t2 - tl, when the parameter variation and model correspond (including the time-invariant parameter as a special case). In order to handle the resetting procedures, a persistent excitation assumption is required. If the real plant parameters do not vary
AUT 20:3-D
319
linearly, but vary smoothly and can be reasonably approximated by a piecewise linear function, then good control is achieved. REFERENCES Anderson, B. D. O. and C. R. Johnson, Jr (1982). Exponential convergence of adaptive identification and control algorithm. Automatica, 18, 1. Anderson, B. D. O. and R. M. Johnstone (1983). Adaptive systems and time varying plants. Int. J. Control, 37, 367. Astrbm, K. J., U. Borrison, L. Ljung and B. Wittenmark (1977). Theory and application of self-tuning regulators. Automatica, 13, 457. AstrSm, K. J. and B. Wittenmark (1973). On self-tuning regulators. Automatica, 9, 185. Caines, P. E. and D. Dorer (1980). Adaptive control of systems subject to a class of random parameter variations and disturbances. Technical report, Department of Electrical Engineering, McGill University. Cordero, A. O. and D. Q. Mayne (1981). Deterministic convergence of a self-tuning regulator with variable forgetting factor. Proc IEE 1, 19-23. Evans, R. J., G. C. Goodwin and R. E. Betz (1981). Discrete time adaptive control for classes of nonlinear systems. In A. Bensonssan and J. L Lions (Eds), Information and Optimization of Systems. Springer, p. 213. Evans, R. J. and R. E. Betz (1983). Further properties of discrete time adaptive control for time-varying systems. Technical report, Department of Electrical and Computer Engineering, University of Newcastle. Fortescue, T. R., L. S. Kershenbaum and B. E. Ydstie (1981). Implementation of self-tuning regulators with variable forgetting factors. Automatica, 17, 831. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1980). Discrete time adaptive control for deterministic systems. IEEE Trans Aut. Control, AC-25, 449. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1981). Discrete time stochastic adaptive control. SIAM J. Control & Optimiz., 19, 829. Goodwin, G. C. and Kwai Sang Sin (1983). Adaptive Filtering, Prediction and Control. Prentice-Hall. Johnstone, R. M., C. R. Johnson, Jr, R. R. Bitmead and B. D. O. Anderson (1982). Exponential convergence of recursive least squares with exponential forgetting factor. Systems & Letters, 2, 77. Xie Xianya and R. J. Evans (1983a). Adaptive control of discrete time deterministic systems with unknown deterministic disturbances. Technical report EE8306, Department of Electrical and Computer Engineering, University of Newcastle. Xie Xianya and R. J. Evans (1983b). Discrete time stochastic adaptive control for time varying systems. Technical report EE8307, Department of Electrical and Computer Engineering, University of Newcastle.
Automatica, Vol. 20, No. 3, pp. 309-319, 1984 Printed in Great Britain
Discrete-time Adaptive Control for Deterministic Time-varying Systems* XIE XIANYAt and ROBIN J. EVANS:~
An adaptive control algorithm using a restricted complexity time-varying model controls rapidly time-varying, deterministic, discrete systems. Key Words--Adaptive control; convergence; discrete-time systems; least-squares estimation; self-tuning regulators; time-varying systems.
quantities alone for identification problems, and output quantities alone for adaptive control problems. In the practically significant situation of unknown time-varying parameters, there has recently been some progress on the analysis of existing algorithms when used in the time-varying case. Preliminary results by Caines and Dorer (1980) and Evans and Betz (1983) established bounds on the output control error for the case when the parameter variation eventually died out. Johnstone and co-workers (1982) established the exponential convergence of recursive least squares with an exponential forgetting factor in the timeinvariant case and under the condition of persistent excitation. Anderson and Johnstone (1983) showed that exponential convergence of the algorithm in time-invariant case will guarantee tracking error and parameter error boundedness when the plant parameters are actually slowly time varying. Fortescue, Kershenbaum and Yastie (1981) presented an algorithm with variable forgetting factor to avoid the problem of covariance matrix blow-up. Cordero and Mayne (1981) have established the convergence of the algorithm with a variable forgetting factor in the time invariant case. Simulation results show, however, that in the fast time-varying case the performance of existing algorithms is limited. The problem stems from the fact that the existing algorithms are based on a time invariant model. Below we present a series of improvements to the standard least-squares estimator and one-stepahead controller structure which ensure good adaptive controller performance for rapidly varying system parameters. Our first improvement concerns the speed of convergence of the parameter estimation algorithm. We know that the speed of parameter estimator convergence and the rate of time variation allowed are directly related. Thus we develop a modified least squares estimator employing a t~ weighing in the cost and a 'tz resetting'
Abstract--There is currently considerable interest in the application of recently developed self-tuning control algorithms to real control problems. However, there exist few algorithms which will work for other than slow time-varying systems. A new algorithm which is able to successfully control unknown systems with quite rapidly varying parameters is presented. Several simulation results are also presented. A convergence proof in the linearly time-varying parameter case is given. 1. INTRODUCTION
THE self-tuning regulator (Astrrm and co-workers, 1977; Astrrm and Wittenmark, 1973) is now over 10 years old and has established itself as an important and useful tool in modem control applications. The self-tuning regulator is conceptually a very simple device. The unknown plant parameters are estimated on-line by a recursive least-squares algorithm and these estimates are then used in a minimum variance controller. Recently Goodwin, Ramadge and Caines (1980, 1981) established the global convergence of the controlled output for such schemes when applied to minimum phase timeinvariant deterministic systems, and stochastic systems subject to a positive real condition. These convergence results were further extended by Evans, Goodwin and Betz (1981), to classes of nonlinear systems and systems whose output is corrupted by unknown but bounded noise. Anderson and Johnson (1982) presented conditions for exponential convergence of adaptive identification and control algorithms and showed that persistent excitation conditions involving input and output quantities can be converted to ones involving input * Received 7 January 1983; revised 7 July 1983; revised 19 December 1983. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor P. C. Parks under the direction of editor B. D. O. Anderson. t Department of Computer Science, Shanghai University of Science and Technology, Shanghai, China. Currently on leave at the Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308, Australia. :~Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308, Australia. 309
310
X. XIANYA and R . J . EVANS
procedure. We show by simulation this algorithm converges faster than standard least squares. Secondly we employ a P(t) matrix resetting procedures based on the current value of~VP~. This ensures that the estimation algorithm stays awake. Finally, and most importantly of all, we model the unknown p a r a m e t e r variations as an offset linear ramp. As will be d e m o n s t r a t e d below this allows good control performance even when the u n k n o w n p a r a m e t e r s are varying sinusoidally and with step changes. The remainder of this p a p e r is arranged as follows: Section 2 describes the new algorithm in detail. Section 3 presents performance simulations and Section 4 discusses theoretical aspects of convergence.
O(t) = O(t -
1)
P(t - 2)~b(t - 1)e(t) + (l/t2) 2 + ~b(t - 1)xP(t - 2)~b(t - 1) P(t-
1)=P(t-2) P(t - 2)~b(t - 1Rb(t - 1)TP(t - 2) (l/t2) 2 + ~b(t - 1)Tp(t -- 2)~b(t -- 1)"
At the start of computation, t = tl = t2 = 1, P ( - 1 ) = P0 = K0l, and ~la(0) = dz~(0) = "'" = d,l(0) =/~11(0) =/~21(0) . . . . . /~,.1(0) = 0, where Ko > 0 and I is an 2(m + n) × 2(m + n) identity matrix. As c o m p u t a t i o n proceeds we determine tx by t ~ = t - k T w h e n kT
2. ADAPTIVE CONTROL ALGORITHM T h r o u g h o u t this p a p e r we will assume that the plant to be controlled can be described by a timevarying A R M A model of k n o w n order and is stably invertible. Following standard notation we write y(t + 1) = c q ( t ) y ( t ) + ~ 2 ( t ) y ( t - 1) + ' " + %(t)y(t
-
=
0
~,o(kT) = £ t i o ( k T - )
+ T£t,,(kT-)
£ql(kT)
= ~l(kT-),
i = 1.....
fljo(kT)
= fl;o(kT-)
+ Tfl;,(KT-)
fli,(kT)
=/~:x(kT-),
P(kT-
n
(5)
j = 1. . . . . m
1)= aoKolorP(kT-
1)= KiP(kT-
-
1)
where k T - means the time just before resetting and 0 < ao ~< 1 aoKoI > P ( k T - - 1) or Ka > 1. While t2 increases step by step with t until the following inequality is satisfied
n + 1) + f l l ( t ) u ( t )
+ [32(t)u(t -- 1) + -'" + fl,,(t)u(t -- m + 1)
(4)
(1)
4}(t - 1)xP(t - 2)~b(t - 1) ~< e
(6)
where y and u are the plant output and control variable, respectively, and oq(t), i = 1 . . . . . n, flj(t), j = 1. . . . , m are the u n k n o w n time-varying parameters. However, n, m are assumed known. Define the 2(m + n ) × 1 estimated p a r a m e t e r vector
where e is a small positive scalar, typically e = 0.1. When (6) is satisfied, the following resetting is carried out
O(t) T = ( ~ l O , d 2 o . . . . . ~.o,
and then continue c o m p u t a t i o n as in (1) with t2 again increasing step by step. An alternative algorithm which also performs well is to let t2 = 1 for all t. This is the ordinary least squares if no resetting is made. This algorithm does not converge quite as fast as when t2 is variable (as will be shown below), however, it is still satisfactory for control of most time-varying systems and is preferable in the noisy case. Clearly the basic idea of the above described technique is to assume that the u n k n o w n parameters vary as
/~1 O, ~20 ..... flmO, ~11, ~21 ..... ~nl, J~l 1, J~21 .... ,~m,) and the vector
2(m + n)×
1 measurable
(2)
information
q~(t)x = (y(t), y(t - 1) . . . . . y(t -- n + 1), u(t), u(t -- 1) . . . . . u(t -- m + 1), t l y ( t ) , t l y ( t -- 1) . . . . . t l y ( t -- n + 1), t l u ( t ) , t l u ( t -- 1) . . . . . t l u ( t -- m + 1)).
(3)
Given the desired output at time t, y * ( t ) , the adaptive control algorithm can now be written as y * ( t ) = O(t -
1)x~(t - 1)
e(t) = y ( t ) - O(t -
1)r4~(t - 1)
t2 = 1 P(t-2)=aoKol
(7) or
P(t-2)=KIP(t--2)
, i ( t ) = Ctio + (t - to)Cqa + (t
-
to)2Cq2
+
"'"
(8) flj(t) = fljo + (t -- to)fljx + (t -- to)2flj2 + "'"
Then from (1), we have
y(t + 1) = OTck(t)
(9)
Discrete-time adaptive control for deterministic time-varying systems
where D is the dimension of ~(t) and that y * ( t ) be made sufficiently rich (Xie and Evans, 1983a). In algorithm (4) t2 is a variable and we call that algorithm modified least squares. It is derived from optimization of the following cost functional
where 0=
( 0 C 1 0 , ' ' ' , 0~n0, i l l 0 , - ' ' ,
flmO, 0 ~ 1 1 , ' ' ' ~ n l ,
f i l l . . . . tirol, 0C12,'-" (~n2, i l l 2 . . . . tim2"" .)T
dp(t) = ( y ( t ) . . . . y ( t -
1), u ( t ) . . . . . u ( t - m + 1),
n +
(t -
to)y(t),...,(t
-
(t -
t o ) U ( t ) . . . . (t -
(t -
to)2y(t),...,
(t -
to)2U(t -
n +
1),
m +
1),
to)y(t to)U(t -
(t -
J
1 N
=
2,=~1 ?(y(0 - 4,(t
-
1)T0) 2 +
1
to)ZU(t) .... , (0 -- 00)Tpo
1),...)V
m +
311
(12)
1(0 -- 00)
(10) and has the form
(9) has the same form as that in the time-invariant case, thus we can extend all the sophisticated techniques of time-invariant deterministic adaptive control systems into the time-varying case. For simplicity, and especially for reduction of computational effort we only consider a first order approximation to the unknown parameters, thus
O(t)
=
O(t
-
1)
P(t - 2)~(t - 1)e(t) + (l/t) 2 + ~(t - 1)Tp(t -- 2)~(t -- l) (13) P(t - 1) = p(t - 2) P(t - 2)~(t - IRb(t - 1)TP(t - 2) 0 / 0 2 + 4p(t - 1)ZP(t -- 2)qb(t -- 1)"
O~i(t) = O~iO "k- (t -- t 0 ) ~ i l
(11) flj(t) = fljO + (t
- -
to)~jl
The derivation and convergence of (13) with a one-step-ahead control strategy can be proven in exactly the same way as for ordinary squares (see Goodwin, Ramadge and Caines, 1980) and is left to the reader. When t ~ ~ this algorithm reduces to the orthogonalized projection algorithm, which performs poorly in the noisy case (Goodwin and Sin, 1983). When t --, oc, qb(t - 1)Tp(t -- 2)4~(t -- 1) -, 0 resulting in numerical problems, consequently the second resetting (7) is required. The convergence of this algorithm with resetting when used in the time-invariant system can again be proven without difficulty. Simulation results clearly indicate that the modified least squares (M.L.) algorithm has faster convergence than ordinary least squares (O.L.), as shown in Table 1, where the desired output y * ( t ) is a
and substitute tl for (t - to) yielding (2), (3). If we use the onqine ordinary least-squares estimation algorithm and a one-step-ahead control strategy, then we get the adaptive control algorithm (4) with t z = 1. However, the performance of (11) is unsatisfactory when ta = t - to becomes large, thus we need to reset when q reaches T, where T is chosen large enough to allow reasonable convergence of the parameters but the variation of the real parameters is still small enough. Also, the resetting is needed to avoid the algorithm gain tending to zero and to keep 4~(t) bounded. It appears to be difficult to compute the optimal T, however it is easy to choose T in practice. We recommend that (cf. Fig. 7) T > D, in the purely deterministic case,
TABLE 1. COMPARISON BETWEEN MODIFIED LEAST-SQUARESALGORITHM (M.L.) AND ORDINARYLEAST-SQUARESALGORITHM (O.L.) WITH AND WITHOUT P(t - 2) RESETTING. WHEN RESETTING P(t - 2) = 10P(t- - 2). t
0
1
2
3
4
5
6
7
5.22015
2.49208
1.20017
0.25710E-I
0.15021E-2
0.14549E-2
0.I0440E-2
0.68671E-3
5.22015
2.49208
1.20017
0.25710E-I
0.15021E-2
0.14129E-2
0.I0353E-2
0.78208E-3
rese~ing
5.22015
2.49208
1.20026
0.32437E-I
0.51079E-2
0.49647E-2
0.39956E-2
0.35578E-2
no-resegclng
5.22015
2.49208
1.20026
0.32437E-I
0.51079E-2
0.49547E-2
0.48191E-2
0.46976E-2
ii
12
13
14
15
reaetclng M.L. no-resetting
t
O.L.
t
D M.L.
* Where
J 1
9
resetting
0.43166E-3
0.35259E-3
0.33159E-3 ] 0.63380E-4
0.28151E-4
1 0.96110E-5
0.82487E-5
0.75626E-5
no-reset clng
0.62546E-3
0.52355E-3
0.52263E-3
0.51833E-3
0.50592E-3
0.49274E-3
0.47466E-3
0.45283E-3
resecclng
0.33079E-2
0.31463E-2
0.30335E-2
0.24941E-2
0.20927E-2
0.20066E-2
0.19400E-2
0.18870E-2
no-resetting
0.45877E-2
0.44877E-2
0.43966E-2
0.43508E-2
0.42379E-2
0.41551E-2
0.40792E-2
0.40089E-2
O.L. -.
i0
I l
8
D ~ II0(t)ll = II0 - 0(t)ll = [(~1 - ~l(t)) 2 + (~2 - ~2(t)) 2 + (fl - ~(t))2] 1/2.
"}"To avoid numerical problems, when t 2 reaches 10 we reset t 2 to 1, while the P matrix is not reset.
312
X. XIANYA and R.J. EVANS (0)
square wave with period of 20 and amplitude of + 1. The plant to be controlled is described by y(t + 1) = 4y(t) + 2y(t - 1) - 3u(t).
In both algorithms we use a time-invariant model. The initial estimates are ~ ( 0 ) = 1.5, ~z(0)= 3.0, /~(0) = - 4 . 0 and y(0) = y ( - 1 ) = 1.0, e = 0.1.
3. S I M U L A T I O N
RESULTS
In this section we present several simulation results which clearly demonstrate the ability of our algorithm to control unknown time-varying systems. In particular we examine step, ramp and sinusoidaUy varying unknown parameters, we study the effects of different resetting procedures, and we also examine the effects of different parameter models. It is important to note that in all the cases presented below a standard deterministic adaptive controller is unable to control the system and rapidly blows up. In all the simulation cases the plant is described by
-I
-2
(b) 30
~
2O
,a-
io
y(t + 1) = cq(t)y(t) + ~t2(t)y(t - 1) + fl(t)u(t).
The desired output y*(t) is assumed to be a square wave with period of 20 and amplitude of + 1. Case 1. In this case one unknown parameter is assumed to be varying linearly, and as expected the algorithm works perfectly. But if ~l(t) increases unboundedly, then u(t) increases unboundedly, too. This case is simply included to demonstrate that the algorithm performs correctly when the unknown parameter variation corresponds to the modelled variation. Figure l(a) and (b) clearly shows the excellent performance obtained. Note that if ~l(t) is assumed time-invariant, the system output is highly oscillatory. Here we have assumed that ~2 and fl are unknown and time-varying, despite that they are actually time-invariant. Case 2. In this case we have a step in the parameter ~l(t) at t = 26 (Fig. 2a and b). As can be seen from Fig. 2(a) considerable output oscillation occurs during the parameter change, however the system quickly settles down to satisfactory control. Case 3. For this case we have ~a varying sinusoidally, ~2 being constant and fl undergoing two step changes followed by a ramp (Fig. 3a and b). As in the Case 2 above, the system output undergoes a severe transient at the instant of fl step changes, but rapidly settles down to control the system. Case 4. Figure 4 examines the performance with different estimation algorithms, all assuming a time° invariant parameter model but the real plant being slowly time-varying. Ordinary least-squares without P(t) resetting is unable to control the system. Ordinary least-squares with P(t) resetting works quite well, while the modified least-squares performs slightly better.
o
A I-
i
i
i
i
i
3 2 I
-I
1"
,oz.
-4(
FIG. 1. (a) and (b). Simulation results of the modified leastsquares algorithm with a time-varying model. OSC, drastic oscillation. The real parameters are el(t) = 4 + 0.5t, e2(t) = 2, B(t) = -3. Initial estimates are ~1o(0) = 1.5, ~2o(0)= 3.0,~o(0) =-4.0, ~11(0)=~21(0)=~1(0)=0 and y(0)=y(-1)= 1. ~= 0.1. Resetting period T= 10. Case 5. The plant has one rapidly varying parameter st(t) and the other two are constant. We consider two different approximation models. The first one is y(t + 1) = (~1o + toql)y(t) + (Ot2o+ to~21)y(t - 1) + (flo + tfll)u(t)
i.e. we regard all the parameters as being timevarying. The second model is simplified to y(t + 1) = (Cqo + t~lx)y(t) + ~2y(t - 1) + flu(t).
Discrete-time adaptive control for deterministic time-varying systems (a)
(o)
2
1
I
7
313
0
I 0 ~
"r
q 0
>,
810
,~o
-I
-I
-2
[ (b)
#,
ill t
6
4
2
II II II
2 i
o
~o
i {,
4
6'0
8'o
•
I00
3
'[
o--
20
I ~)
0 -I
~
I
I
,
20
I
I
30
40
I 50
,,II j|
,I II
I
I
40
60
I
[
80
I00
-.5
1"
_.o
-:5
FIG. 3. (a) and (b). The same as Fig. l(a), except that the real parameters are:
cq(t) = 4 + sin (=t/100), FIG. 2. (a) and (b). The same as Fig. 1(a), except that the real parameters are:
4, 0~
Comparison of the simulation results (Fig. 5a and b) show that if we know a priori that some parameters are actually constant, we should prefer to use the corresponding simplified time-varying model. This not only results in better control performance but also in a considerable reduction of the computational effort involved. Case 6. Finally we consider the performance of our algorithm using the simplified time-varying
¢t2(t) = 2,fl(t) =
25 < t ~<65 65
model with an optimal t~ resetting period T. The performance shown in Fig. 6(a) and (b) is extremely good. Resetting considerations
As already pointed out, there is no straightforward theoretical technique for choosing the resetting period T. However, in the purely deterministic case and provided y*(t) is sufficiently
314
X. XIANYA and R.J. EVANS
g
z
(o)
D
x 12
i
I I I I I~.~1 I I~
7
I
I
II 6o'i I
I I I 8dl I I
I I I ,do I
i¢
2c
•Ooo.a r ~ 30
40
~R,
-2
FIG. 4. Comparison between different algorithms. O, Modified least-squares algorithm, using time-invariant model. When ~Tp~) ~< 0.1, reset P(t - 2) to 10P(t- - 2); x,ordinary leastsquares algorithm, using time-invariant model. When ~TP~b ~< 0.1, reset P(t - 2) to 10P(t- - 2); V], ordinary leastsquares algorithm, using time-invariant model without resetting. The real parameters are ~ ( t ) = 4 + sin (zrt/100), ~z(t)= 2, f l ( t ) = - 3.
( b )~.-~
I~
rich it is recommended that T be chosen to be close to but slightly greater than the dimension of 4)(t). Figure 7 shows the performance measured as both maximum output error emax and average output error e.... for different parameter variation rate and different resetting period T, where em~xand ea~ are defined as ~.
1
100
eave = 9-i- ~
t=10
emax
=
max
ly*(t) - y(t)I
lo<~t~1oo
(14) (15)
y * ( t ) - - y(t). y*(t)
"
0
~,,~.~
I
X
I
I0
II
I
20
30
I
I
50
40
'
¢~
2
o....
~ooo~o k
I I0
0
oo'--1"~oeoKt~x~q?o~o
210
3 0 4 0 1
,~
....
510
T
The plant controlled is assumed to be y(t + 1) = ~l(t)y(t) + ~2y(t - 1) + flu(t) and supposing we have known c(2, fl to be constant and use the simplified time-varying model. In Fig. 7(a) ~t(t) = 4 + sin(nt/20), ~2 = 2, fl=-3 and in Fig. 7(b) ~ 1 ( t ) = 4 + sin(nt/10). ~2=2, fl=-3. In both cases ~ o ( 0 ) = 1 . 5
02,~(0) = O, d2(O)= 3.0, 1~(0)= 4.0, y(O)= y ( - 1 ) = 1.0. The desired output y*(t) is a square wave with period of 20 and amplitude of _ 1. In Fig. 7(a) the optimal choice is clearly seen to be T = 5. In Fig. 7(b) the optimal choice is also T = 5. Either resetting P(t - 1) to be aoKol or to be K ~ P ( t - - 1) will do. We recommend that P(t - 1) be reset to aoKol rather than K ~ P ( t - - 1). This is not only for the reduction of computational effort but also to prevent P(t - 1) becoming indefinite due to round offerror and also to prevent P(t - 1) from 'blow-up' in steady state. In our simulations ao = 0.1, Ko = 100.
--q FIG. 5. (a) Comparison between the performance of algorithms using different time-varying model, x,Time-varying model; O, simplified time-varying model. The real parameters are el(t) = 4 + sin (nt/20), e2(t) = 2, fl(t) = - 3 . Resetting period T = 10. (b)The same as those in (a). O, Real parameters.
The whole question of resetting needs further examination. For example in many cases when T is small, resetting procedure (6), (7) is not required and actually will not happen. Our simulation shows, however, that better control performance results if we use both the resetting procedures (5) and (6), (7) when T has to be chosen large.
Discrete-time adaptive control for deterministic time-varying systems
I
4
(a)
315 0.4
(a) o - eav e
I I
i
I I
, I
f I
I I
I t
I I
6 IJ,
!"
I
I
I
I
• - emo x
\ .\ E
0.2
2
0)
I I
\ .... 2
I o {b)
4
6
8
I0
T
4
4
(b)
7
1
0
I
I
I
L
L
20
40
60
80
100
T
o "o
0~ >
../\,,"
2
I 2
0
l 4
I 6
I 8
I0
T
0
2
Id 410
I 60
i 80
i I00
-I
FIG. 7. (a) The control performance versus the resetting period. The real parameters are cq(t)=4+ sin(nt/20), ~t2(t)= 2, fl(t) = -3. (b)The same as (a) except the real parameters are ~q(t) = 4 + sin (nt/10), ~t2(t) = 2, fl(t) = - 3 .
-5 o -4
FIG.6. (a) Simulation results of modified least-squares algorithm using a simplified time-varying model and an optimal resetting period (T = 5). The real parameters are ~tl(t) = 4 + sin (nt/20), ct2(t) = 2, fl(t) = -3. Real line, desired output; O, y(t). (b)The same as those in (a). O, Real parameter; ©,estimates.
be achieved with a b o u n d e d input sequence if plant parameters are known. F o r simplicity, we take t2 = 1 and do not use resetting procedure of (6), (7). That is, we just use ordinary least-squares algorithm together with tl resetting procedure (5). Theorem
the the the the
1
Given the plant 4. CONVERGENCE CONSIDERATIONS We are unable to prove convergence of our algorithm in the general case where the parameters are varying in an arbitrary fashion. In fact, we cannot expect to prove convergence if the real plant does not have exactly the same parameter description as our model. However for this case it m a y be possible to derive a b o u n d on the control error similar to Evans, G o o d w i n and Betz (1981) or A n d e r s o n and Johnstone (1983). Below we present a convergence p r o o f based on the real parameter being linearly time-varyin~ which includes the time-invariant case as a special one. Even in this case we still require a strong assumption like persistent excitation. We also need the assumption that lim,_.~ [_V(t) - y*(t)] = 0 can
y ( t + 1) = ~ l ( t ) y ( t ) + ~,(t)y(t
+ ct2(t)y(t -
1) + " "
n + 1) + f l l ( t ) u ( t )
+ f l , . ( t ) u ( t - m + 1)
+ ...
(16)
and
~l(t) = ~lo(to) + (t - to)~,~
~.(t) = ~.o(to) + (t - to)~.~ ill(t) = fllo(to) + (t - t o ) f i l l
fl,~(t) = fl.o(to) + (t - to)ft.1
(17)
where 0til, fljl (i = 1. . . . . n , j = 1 . . . . . m ) are constant.
316
X. XIANYA and R. J. EVANS
If the algorithm (4), (5) ia applied to plant (16) (17), then {y(t)}, {u(t)}, {~b(t)} are bounded and lim [y(t) - y*(t)] = 0
(18)
t~c~
provided the following assumptions are satisfied (i) Cqo, Cqx, fl~o, flj~ (i = 1..... n, j = 1..... m) are unknown but n, m are known; (ii) the plant (16), (17) is stably invertible and with ly*(t)l ~< ml < o~ the control objective lim,.~ [y(t) - y*(t)] = 0 can be achieved with a bounded input sequence if the plant parameters are known; (iii) there exists a constant integer M such that 1 j+M-1
plI ~ ~
.~. ~(i)~(i)T
(19)
where ~ ( k T ) , flfo(kT), 8ii a n d / ~ are the estimates of Cqo(kr), fljo(kT), ~il and fljl, respectively, just before the resetting time t = (k + 1)T. It is clear that the estimation error will change when the estimated parameters exchange. If before resetting, the errors are ai-o = ~i~( k T) - Cqo(k T) ~il ~ ~il -- ~il
(22) fl]-o -- fl~-o(kT) - fljo(k T)
&, = P;, -
then, after resetting, the errors become Rio = ~ o ( k T ) + T ~
- ~io[(k + 1)T] = ~
+ T~
~il = ~i~ for some Pl > 0 and all j, where
J~j0 = ~jO "Jr- Tfl~
(23)
~p(i)T = [y(i) ..... y(i - n + 1), u(i) . . . . . u ( i - m + 1),(i - j ) y ( i ) ..... (i-j)u(i-
m + 1)].
Proof: Our proof is along the lines of Goodwin, Ramadge and Caines (1980), but with additional considerations to account for resetting. Firstly note that (17) can be written as ~ ( t ) = a t o ( k T ) + (t - kT)~xl
In other words the estimation error vector & 0 - O changes from 0 = (/~, O~)r to = 0T ~ T where O (#I+ 2,, =
.....
.....
(24) 01 =
.....
IV1 . . . . .
Define V(t) = O(t)Tp(t -- 1)- 10(t), then between two adjacent resetting times, we have (Goodwin, Ramadge and Caines, 1980)
% ( 0 = ~.o(kT) + (t - kT)~.a V(t)fl~(t) = flxo(kT) + (t - k T ) f l t l
1)= 0(t - 1 ) r ~ ( t - 1 ) ~ ( t - 1 ) r 0 ( t - 1)
V(t-
1 + ~ ( t -- 1)Tp(t -- 2 ) # ( t -- 1) (25)
tim(t) = flmo(kT) + (t -- kT)flml
e2(t)
where, K = 0, 1.... and a11,0c21, ..., ~,1, fill,.., fl,,1 are constant and ~io[(k + 1)T] = aio(kT) + Tail,i = 1 , . . . , n
(20) fljoE(k + 1)T] = fljo(kr) + r f l j x , j = 1 . . . . . m. When k T < t <~ (k + 1)T, the algorithm estimates aio(kT), fljo(kT) a n d ~ia, fljl. When t = (k + 1)T resetting with (5) is equivalent to starting the recursive least-squares estimation of aio [(k + 1)T], rio [(k + 1)T] and sit, fljl under the initial condition &o[(k + 1)T] = ~ ( k T )
+ T~,,8,
= +i]
(21)
1 + ~(t-
1)vp(t-
2)#(t-
l)
~<0
where e(t) = d~(t - 1)x0(/- 1) = y*(t) - y(t). Thus V(t) is a nonincreasing nonnegative sequence between two adjacent resetting times. Now what will happen when resetting occurs. As mentioned above, after resetting O changes because the estimated parameters have been changed. To ensure V(t) is still nonincreasing we need assumption (19) and we need to choose aoKo properly. When k T < t < (k + 1)T, we have (Goodwin, Ramadge and Caines, 1980) p(t -
1) = p(t - 2) -x + # ( t -
1)#(t -
1) T
t--1
/~j0 [(k + 1)T] = l ~ ( k V ) + T ~ f l , ~ j l = ~f~
= p(kT-
1) -1 + ~
¢~(i)#(i) T
i=kT I
* Note the notations here are different from those in (5), where ~o(KT-) means the estimate of Cho(KT- 1) obtained at time
KT-.
t-1
aoKo + i=kT ~ ~(i)~(i)T.
(26)
Discrete-time adaptive control for deterministic time-varying systems When t = (k + 1)T, just before resetting, using (19) and choosing T = M 1 P[(k + 1)T- - 1] -1 = troKo I +
(k+1)r-t 1 E ¢(i)~(i)T t> i = kT
317
considering (26), we have 0 < P(t - 2) -%
I
0 ~< ~(t - 1)TP(t - 2)~(t - 1) ~< aoKoll~(t + p, 721
(27)
~7ogo
thus e(t)
hence 1 ).m~.{p[(k + 1)T- - 1] -1} ~> a--0-~o + p i T 2
1)112]1/2 = 0. tlira ~ O 0 [1 + aoKoll4~(t (28)
where the )~mi.{p[(K + 1)T- - 1] -1} is the minimum eigenvalue of P[(K + 1)T- - 1] -1 If we choose ao > 0 and Ko > 0 such that Koaopl >i 2 + (1/T 2)
1)115 (33)
Now 11¢(0112 -- Iv(t) 2 + ... + y(t - n + 1)2 + u(t) 2 + " " u ( t - m + 1)2][1 + (t - k T ) z] <-% [y(t) -2 + "'" + y ( t -
n + 1)5
+ u(t) 2 + ... + u2(t - m + 1)][1 + T2].
211 + (l/T2)] 1 I> go/> (1/T2) + KoaoPl > 0
(34)
(29) From assumption (ii), we have
then 211 + (l/T2)]
lu(t)l < m2 + ma max ly(t)[,
211 + (1/T2)]
0 ~ m2 < ~ , 0 < m3 < ~ , Vt.
Koao >1 (1/KoT2) + aoPl >~ [(1/aoKoT 2) + Pl ]ao
/>
Considering e(t) = y*(t) - y(t), and [y*(t)] < ml
2(T 2 + 1)
[y(t)[ < [e(t)[ + ml
(1/o'oKo) + Pl T 2 (because of 0 < ao ~< 1) />
2(T 2 + 1) 2mi,{P[(k + 1)T- - 1] -1}
[using (28)]
thus, from (34), (35), we have I[q~(t)[[ < cl + c2 max [e(t)[, 0 <%cl < oo,0 < c2 < oo.
{[l(0~r, 01)It2 + T2I[(0~, 0T)112}2
>/
{ll0I, Og)ll2 + T211(Og, 0r)112}2 1> (/~i, 0T)p [(k + 1 ) r - - 1 ] - I(0T, 0T) T
lim e(t) = lim [y*(t) - y(t)] = 0
II(01 + TOl, O~)ll~ I> (0T, ()T)p [(k + 1)T- - 1 ] - I(0T, 0T) T thus (rlgo IIOT + TOT2,0~)II2 -%<(/~,01)P[(k + 1)T- - 1]-1(0"~,0~) T
(36)
According to the 'Key Technical Lemma' of Goodwin, Ramadge and Caines (1980), we get
2m~.{P[(k + I ) T - - 1]-~}II(OLOT2)II 2
(30)
where 0 is the (n + m)-dimensional null vector. Equation (30) means, by proper choice of ¢0 and Ko, we can ensure that V(t) is nonincreasing at the resetting time. Thus V(t) is a nonincreasing nonnegative sequence in 0 ~< t < ~ and e2(t)
,=o [1 + ~ ( t -
(35)
1)rP(t - 2)~(t - 1)] < oo
and {IRb(t)l)}, {u(t)}, {y(t)} are bounded. [] Recently, Anderson and Johnstone (1983) proved that if the adaptive control algorithm in the timeinvariant case has exponential parameter convergence, then even if the real plant is slowly timevarying the control performance is still good. Thus, in our case, if we can establish exponential parameter convergence in the linearly time-varying case, then even if the real plant parameters are not exactly linearly time-varying but their first order derivatives vary slowly, then good control performance can be expected. In fact the simulation results in Section 3 have to some extent implied this assertion. Motivated by this we now establish parameter exponential convergence for our algorithm.
(31) which implies
Theorem 2 e(t)
lim ,-,~o [1 + ¢(t - 1)TP(t - 2)#(t - 1)] 1/2 = 0 (32)
Given the plant and assumptions exactly as in Theorem 1, when the algorithm (4), (5) is applied to plant (16), (17), then the parameter estimates converge exponentially to their real values.
318
X. XIANYAand R. J. EVANS
Proof: As in Theorem 1, between two adjacent resetting times V(t) - V(t - 1) <.%0
-
thus V[(K + 1)T-] ~< V ( K T )
(37)
where V [ ( K + 1)T-] is the value of Liapunov function at time t = (K + 1)T just before resetting and V ( K T ) is the value at time t = K T , just after resetting. In other words, from K T to (K + 1)T- no resetting is made. So
[(K + 1)T-] ~< O ( K T ) T p ( K T - 1)-10(KT) (38)
and
y*(t) = O(t - 1)V¢(t -- 1) e(t) = y(t) -- O(t -- 1)T¢(t -- 1)
0 [ ( K + 1)T- ]Tp[(K + 1)T- - 1 - - ' 0
substituting (27) into p ( K T - 1) = aoKoI,
Noting that 0(K T) is the parameter error at time just after resetting, i.e. O ( K T ) = O ( K T ) O(KT), (44) thus implies parameter exponential convergence. In the proof of Theorem 1 and Theorem 2 for simplicity we let t2 = 1 and do not use the resetting procedure (6), (7). In fact it is straightforward to establish the result if we let t2 -= tl and omit the resetting procedure (6), (7). In this case, the algorithm reduces to KT
(38)
0 ( 0 = O(t - 1) +
P(t - 2)q~(t - 1)e(t) (1/tt) 2 + q~(t - 1)xP(t - 2)¢(t - 1)
considering
(4')
P ( t - 1) = P(t - 2) P(t - 2)¢(t - 1)¢(t - 1)xP(t - 2)
+ 1)T- ]112 ~< a@KoII0(KT)II2
110[(K + 1)T- ]11z [IO(KT)[[ 2
-
1 <~ 1 + o o K o p l T z'
O[(K+ 1)T] = (0T + TOT,0T) T II0[(K + 1)T][[ 2 ----11(01 + T02,02)1[ 2
= 2(1 + T2)II0[(K + 1)T-]112. (40) From (39), (40), we have
1
SO
2(1 + T 2) (42)
In other words, if we choose KoaoPl strictly larger than 2 + (1/T 2) then we get (43)
which implies exponentially lira II0(gZ)ll = 0.
1)
~
(i-j)2d~(i)dp(i) T.
(19')
i=j
If we use both the resetting procedures (5)-(7), the proofs of Theorems 1 and 2 will become difficult. Another possibility is to use algorithm (4) with resetting procedure (5) and letting t2 = q. But resetting (5) is made only when the following inequality is satisfied (45)
where cl is a small scalar. In this case, resetting period T is not constant. Remarks
KoaoPa >>-2 + ~-£
K~oo
~_~
trace P(t - 1) -%
We have chosen in (29) that
II0[(K + 1)]TII < 1 IIO(KT)II
2)4~(t -
j+M-1
= 2(1 + T2)J[(O1,02)[I 2
2(1 + T 2)
-
rather than (26). Thus the condition (19) should be changed to pll <~
Tz11(01,02)11z]
1 + ¢roKoPlT 2 ~< 1 + [2 + ( 1 / T 2 ) ] T 2 = 1.
1)XP(t
P(t - 1) -1 = P(t - 2) -1 + t2dp(t - 1)4~(t - 1)x (26')
~< 2 [11(01,0zll z + II(T0z,0)ll:]
II0[(K + 1)TIll: 2(1 + T 2) ]]O(KT)[[ 2 <~ 1 + a o K o p l T 2"
2 + 4~(t -
with resetting procedure (5). In this case (39)
Denote 0[(K + 1)T-] = (01,02) -x T X, where 01,02 ~T ~T are those in (24), then after resetting
~< 2[11(01,0z)11 z +
(l/t0
(44)
1. In the above proofs, the persistent excitation condition (19) is crucial. If (19) is satisfied, then appropriate go and Ko are guaranteed to exist. In the time invariant case, Anderson and Johnson (1982) have shown that in the adaptive control case, persistent excitation of the desired output {y*(t)} ensures that {¢(t)} is persistently exciting. In the time-varying case, however, we do not know if the same assertion is true. Simulation shows that when {y*(t)} is sufficiently rich, the trace of P(t) decreases rapidly, which implies the above assertion is true, at least under some conditions. Moreover, in the special case, when we extend our algorithm to control a known parameter plant with unknown
Discrete-time adaptive control for deterministic time-varying systems deterministic time-varying disturbances, the persistent excitation condition (19) is automatically satisfied (see Xie and Evans, 1983a). 2. Above we have presented several modifications to the discrete-time deterministic adaptive control algorithm to handle fast time-varying parameters. The most important modification is the use of a restricted complexity time-varying model (an offset linear ramp) and the use of resetting procedure (5). These give the algorithm the ability to track timevarying parameters. The modified least-squares estimation algorithm with resetting procedure (6), (7) is used to accelerate the convergence as shown by the simulation results in Table 1 and Fig. 4. This modification to the ordinary least-squares algorithm is similar to the use of a forgetting factor, since in both cases a greater emphasis is placed on the most recent data, hence should be useful for the case when the real parameter is not linearly time-varying. In this paper we use the modified least squares algorithm rather than a forgetting factor. This is because the former yields a reduction in computational effort and has no problem of covariance matrix blow-up. We have also studied a restricted complexity time-varying model combined with a forgetting factor (see Xie and Evans, 1983b). 5. CONCLUSION
This paper has presented an adaptive control strategy which has been found to work extremely well for rapidly time-varying systems. A similar technique has also been developed for control of unknown systems with unknown time-varying deterministic disturbances (Xie and Evans, 1983a). Again the approach allows accurate control despite rapidly varying disturbances. Proofs of output and parameter convergence are presented for t 2 = 1 and t2 - tl, when the parameter variation and model correspond (including the time-invariant parameter as a special case). In order to handle the resetting procedures, a persistent excitation assumption is required. If the real plant parameters do not vary
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linearly, but vary smoothly and can be reasonably approximated by a piecewise linear function, then good control is achieved. REFERENCES Anderson, B. D. O. and C. R. Johnson, Jr (1982). Exponential convergence of adaptive identification and control algorithm. Automatica, 18, 1. Anderson, B. D. O. and R. M. Johnstone (1983). Adaptive systems and time varying plants. Int. J. Control, 37, 367. Astrbm, K. J., U. Borrison, L. Ljung and B. Wittenmark (1977). Theory and application of self-tuning regulators. Automatica, 13, 457. AstrSm, K. J. and B. Wittenmark (1973). On self-tuning regulators. Automatica, 9, 185. Caines, P. E. and D. Dorer (1980). Adaptive control of systems subject to a class of random parameter variations and disturbances. Technical report, Department of Electrical Engineering, McGill University. Cordero, A. O. and D. Q. Mayne (1981). Deterministic convergence of a self-tuning regulator with variable forgetting factor. Proc IEE 1, 19-23. Evans, R. J., G. C. Goodwin and R. E. Betz (1981). Discrete time adaptive control for classes of nonlinear systems. In A. Bensonssan and J. L Lions (Eds), Information and Optimization of Systems. Springer, p. 213. Evans, R. J. and R. E. Betz (1983). Further properties of discrete time adaptive control for time-varying systems. Technical report, Department of Electrical and Computer Engineering, University of Newcastle. Fortescue, T. R., L. S. Kershenbaum and B. E. Ydstie (1981). Implementation of self-tuning regulators with variable forgetting factors. Automatica, 17, 831. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1980). Discrete time adaptive control for deterministic systems. IEEE Trans Aut. Control, AC-25, 449. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1981). Discrete time stochastic adaptive control. SIAM J. Control & Optimiz., 19, 829. Goodwin, G. C. and Kwai Sang Sin (1983). Adaptive Filtering, Prediction and Control. Prentice-Hall. Johnstone, R. M., C. R. Johnson, Jr, R. R. Bitmead and B. D. O. Anderson (1982). Exponential convergence of recursive least squares with exponential forgetting factor. Systems & Letters, 2, 77. Xie Xianya and R. J. Evans (1983a). Adaptive control of discrete time deterministic systems with unknown deterministic disturbances. Technical report EE8306, Department of Electrical and Computer Engineering, University of Newcastle. Xie Xianya and R. J. Evans (1983b). Discrete time stochastic adaptive control for time varying systems. Technical report EE8307, Department of Electrical and Computer Engineering, University of Newcastle.