A parameter estimation perspective of continuous time model reference adaptive control

Automatica, Vol. 23, No. 1, pp. 57 70, 1987

0005-1098/87 $3.00+0.00 PergamonJournalsLtd. © 1987InternationalFederation of AutomaticControl

Printed in Great Britain.

A Parameter Estimation Perspective of Continuous Time Model Reference Adaptive Control* G. C. G O O D W I N t and D. Q. MAYNE:~

7he convergence proofs for continuous time adaptive control algorithms, decomposed into modules dealing with estimation and control, yield a "key technical lemma" analogous to that used in the study of discrete algorithms. Key W o r d s - - A d a p t i v e control; parameter estimation; model reference control. A b s t r a c t - - T h e problem of adaptive control of continuous time deterministic dynamic systems is re-examined. It is shown that the convergence proofs for these algorithms m a y be decomposed into "modules" dealing with estimation and control, yielding a "key technical lemma" analogous to that used successfully in the study of discrete time systems. The extra freedom provided by the modular structure is used to formulate existing algorithms in a c o m m o n framework and to derive several new algorithms. It is also shown how least squares, as opposed to gradient, estimation can be used in continuous time adaptive control.

the model output (y*) to zero. On the other hand, the discrete time algorithms are usually viewed as a combination of a parameter estimator (driven by a suitable prediction error) and a certainty equivalence control law. The former approach is known to have certain difficulties including the fact that, in general, the error y - y* must be augmented by other terms to guarantee stability, and strict positive real (SPR) conditions on various transfer functions appear as necessary conditions for convergence. Also, it is difficult to see what properties are retained if the control law is changed, e.g. if the input reaches a saturation limit. On the other hand, most of the discrete algorithms retain the key properties of the parameter estimates irrespective of the control signal. This paper shows that these apparent differences can be readily resolved. In particular, an idea originally presented by Egardt (1979) is further developed and a counterpart of the discrete prediction equality is shown to exist, which allows an essentially trivial development of all existing continuous model reference control laws; it is shown that existing continuous time model reference adaptive control laws can be described in a unified framework; that the continuous algorithms can be rearranged so that the properties of the parameter estimator are retained irrespective of the nature of the control law; how parameter estimates of the least squares type can be employed and that "augmented errors", "auxiliary signals" and "SPR" conditions have a simple explanation if the algorithms are considered as parameter estimators. An "algorithm model" or "proof paradigm" is developed for the continuous case, which is directly analogous to the key technical lemma for discrete adaptive control and which exploits the underlying certainty equivalence structure. These insights provide simple guidelines to enable a designer of an adaptive control law to achieve certain objectives which, if satisfied (separately) by

1. INTRODUCTION Tins PAPER has two principal aims. Firstly, it presents a survey/tutorial of continuous time model reference adaptive control achieved by expressing existing algorithms within a c o m m o n framework. This has pedagogical value and makes the literature on continuous time model reference adaptive control much more readily accessible to those who are more familiar with the discrete case. Secondly, the paper develops several new continuous time adaptive control algorithms and shows how other algorithms can be similarly derived. Examination of the literature on continuous time adaptive control (e.g. the well known works of Morse, 1980 and Narendra et al., 1980) reveals that the continuous case differs in several substantive ways from the discrete case as presented, e.g. in /~str6m and Wittenmark (1973) and Goodwin and Sin (1984). A key difference is that the continuous time algorithms are typically developed by generating non-linear (adaptive) feedback laws so as to bring the error between the plant output (y) and

*Received 5 December 1984; revised 13 November 1985; revised 3 March 1986; revised 9 June 1986. 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 T. S/Sderstr6m under the direction of Editor P. C. Parks. t Department of Electrical & Computer Engineering, University of Newcastle, N.S.W. 2308, Australia. $ Department of Electrical Engineering, Imperial College of Science & Technology, Exhibition Road, London, U.K. 57

58

G . C . GOODWIN and D. Q. MAYNE

the estimator and the controller, ensure global asymptotic stability of the resultant closed loop system (plant, esimator and controller). A central theme of the paper is to clarify, for those who are familiar with the discrete time literature, the important contributions made by Parks (1966), Monopoli (1974), Feuer and Morse (1978), Egardt (1979), Morse (1980), Narendra et al. (1980) and others who have worked on the continuous problem. Since Morse (1980) was motivated by Goodwin et al. (1980), this paper, partially motivated as it is by Morse (1980), completes a circle which is not inappropriate for a paper on feedback.

is required that this polynomial be Hurwitz. This corresponds to the usual "minimum phase" assumption employed in model reference control. A general feedback control law of the form: Lu = - - P y + fflr

is assumed. Substituting (2.5) into (2.1) immediately gives the closed loop system as (LA + B P ) y = Bfflr.

A(D)y(t) = B(D)u(t),

(2.1)

where

(2.6)

To achieve a closed loop characteristic polynomial of ExE2B' the following "pole-assignment" identity must be solved for L and P:

2. S T R U C T U R I N G THE SYSTEM E Q U A T I O N S FOR M O D E L REFERENCE C O N T R O L

Consider a continuous time linear dynamic system having single input u(-) and single output y(.). The system could be modelled via a state space description. However, a much clearer development is achieved if polynomial descriptions are used, as in the early work of Monopoli (1974) and later by Egardt (1979). As pointed out by Egardt (1979), the polynomial approach gives a direct parallel to the discrete case. Consider the following model:

(2.5)

L A + B P = ExE2B'.

(2.7)

Since B' is a factor of the right hand side of (2.7) and of B P on the left, L may be taken to have the form FB' so that B' is also a factor of the left hand side. Equation (2.7) then becomes: F B ' A + B'G = E I E 2 B ' ,

(2.8)

where G&bmP.

Cancelling B' in (2.8) gives A(D) & D" + a,_ 1D"- 1 + ... ao

(2.2)

and B(D) ~ b,,,D" + . . . b o ~ b,,B'(D)

(2.3)

and O(.) __a(d/dO('). In the sequel the arguments t and D are omitted when no confusion results. The reference model having input r and output y* is defined as follows: Ey* = / q r ;

E

= E1E2,

(2.4)

where E1 and E 2 are monic polynomials of degree n and p, respectively, and /-7 is a polynomial of degree h < p + m. To achieve model reference control, it is necessary to apply feedback to the system (2.1) so that the closed loop characteristic polynomial is E1E2B'. The polynomial B' is cancelled between numerator and denominator and a new numerator polynomial, H, is substituted, thus giving a closed loop transfer function of H I E as desired. Since B' is cancelled it

F A + G = E1E 2.

(2.9)

Equation (2.9) is the continuous analogue of the "prediction equality" used in the discrete case (AstrSm and Wittenmark, 1973 and Goodwin and Sin, 1984). It is important to note that, although (2.9) has been motivated by pole-assignment arguments, no relative primeness conditions are needed to solve (2.9). Thus, if F has degree p and G has degree n - 1, then (2.9) has a unique solution which can be found simply by equating coefficients. Actually (2.9) is simply a statement of the division algorithm of algebra. As pointed out by Egardt (1979), the above design using the polynomial E1E2B' has several interpretations depending upon the degree, p, chosen for E2. For example, if p is taken to be n - m, then the design corresponds to a Kalman filter plus state variable feedback. Alternatively, if p is taken to be n -- m -- 1, then the design corresponds to a reduced order Leunberger observer plus state variable feedback. A direct form of model reference control is obtained by replacing (2.1) by an equivalent model whose parameters are those of the controller, thus avoiding on-line computation of the control law.

Parameter estimation perspective of adaptive control Multiplying (2.1) by F and substituting F A by (E - G) yields the equivalent system Ey = Su + Gy,

(2.10)

59

A. Natural regression form F o r the moment, consider the special case when p = n - m. Other cases will be treated in the sequel. F o r this case, (2.10) may be written in the form: y = s, fi + Rue. + GyE,

where

(e)

E ~ Ea E2, S _a__F B (q, the degree of S, is p + m; sq = bin). (2.11)

Combining (2.10) with (2.4), the model reference control objective is achieved by the following feedback control law: Su + Gy = Hr.

(2.12)

(3.2)

where E x and E 2 are both monic of degree n and n - m respectively, s, is the leading coefficient in S (actually s. = b.,) and R zx S - SHE1,

(3.3)

so that the degree of R is n -- 1. Also,

In the next section, various algorithms for estimating the coefficients in the polynomials S and G, so that the control law (2.12) can be made adaptive, are described. 3. CONTINUOUS PARAMETER ESTIMATION ALGORITHMS Here the work begun by Egardt (1979) of unifying all existing continuous time adaptive control laws, will be continued. In particular, where the recent algorithms of Morse (1980) and Narendra et al. (1980) fit into the overall picture is shown. Also several new algorithms are described, which do not appear to have been previously considered in the continuous case but which are analogues of algorithms for the discrete case. Finally, it is shown how least squares can be substituted for the usual gradient parameter estimator. The task of the estimator is to obtain estimates of S and G in (2.12). Several possible estimation schemes are examined and their convergence properties analysed. Morse's convention is adopted (Morse, 1980), i.e. one writes y -- z(e) if y = z + e where either: EBe = 0 and (EB) is Hurwitz so that e is a linear combination of decaying exponentials, or e is bounded in magnitude by q, where r/ satisfies EBrl = O. In the continuous case, it is usual to structure the algorithm so that the adaptation mechanism is driven by the output tracking error y - y*. Here the alternative parameter estimation point of view will be adopted and the adaptation mechanism will be driven by a prediction error of the form z - £(0), where z is a function of measurable quantities and 2(0) denotes a predicted value of z based on the estimated parameters 0. Later it is shown that the usual continuous time adaptive control algorithms can be recovered by suitable choice of z. To start with, however, the "natural" choice for z, namely the plant output, y, is used.

(3.1)

f~ _a_(1/EE)U

(3.4)

yy ~ (1/Ex)y

(3.5)

Uy a (1~El) u

(3.6)

YE ~ (1/E)y

(3.7)

UE a_ (1/E)u.

(3.8)

The "natural" parameterization of (3.1) then is y = ~kT0o, (e)

(3.9)

where 0 o ~ (go . . . . . g . - 1 , r o . . . . . r . - 1 , s . ) r

(3.10)

and ~k & (YE . . . . . D"- lyE, UE. . . . . D"- lUE, if). (3.11)

This form is natural in the sense that it follows directly from (3.1), and corresponds to a form widely used in the discrete time literature. It is, for example, the form used in Algorithm 1 in Goodwin et al. (1980). Let 0 be an estimate of 0o, then the predicted output, 9, is defined as j) ~ ~bT0.

(3.12)

The corresponding prediction error is e = y =

~bTO

--~0~,

(e)

(3.13)

where O& 0 - - 00.

(3.14)

The two prototype parameter estimation algorithms driven by the prediction error, e, can now be

60

G . C . GOODWIN and D. Q. MAYNE

introduced, which can be used (possibly with minor modifications) in all of the adaptive control laws. Algorithm 1 (Normalized gradient algorithm).

=~e;

M=c+ipT~b,c>0.

(3.15)

Algorithm 2 (Normalized least squares algorithm). Pipe. = M''

M'

= c + IpXip + @Xpip, c > 0

pipip T p = M' ;

(3.16)

P(O) = k o I

> 0.

(3.17)

To the best of the authors' knowledge only Algorithm 1 has been described, to date, in the literature on continuous time adaptive control. However, experience in the discrete case (Goodwin and Teoh, 1983) indicates that the normalized least squares algorithm has a vastly superior convergence rate. The algorithms share the same set of key properties relevant to adaptive control convergence proofs as shown below. Note that (3.15) can be written in "error form" as

IPC/r

c + IPTC

pipipr~r c + IpTpip + IpTip" (e)

(3.18)

(3.19)

For future use the following quantities are defined: ~/x =

In the discrete case, ~ is generally called the normalized error and ~ is called the a posterior error. The only restriction placed on u(.) is that various quantities exist. In particular, u(.) is assumed to be continuous. The properties of the parameter estimator (irrespective of the precise nature of the control law) are as follows.

e

(c + IpTip)½

and normalized least squares algorithm have the properties: (a)

~, ~ and ~ are uniformly bounded;

(b)

~ and ~ belong to L2.

Proof. (In this and subsequent proofs the exponen-

tially decaying term "e" is omitted. If this term is included its effect on f" can be cancelled by adding

f

°°e(s)2ds to the Lyapunov function,

as

in Morse

(1980) leaving the conclusions unaltered. The Euclidean norm of a vector and the induced Euclidean norm of a matrix are both denoted by IL'II.) (i) Normalized Gradient Algorithm. Consider the non-negative time function V defined by:

v(t) ~ (~) II0(t) ll2.

(3.22)

f'(t) = - ~(t)2,

(3.23)

Clearly

so that V is monotone non-increasing, bounded above by V(0) and below by 0. Hence 11/7(011 is monotone non-increasing and so uniformly bounded by 11/7(0)11. Since the induced norm of IlipipT/(c + IpTip)ll is 1 it follows from (3.18) that O(t) is uniformly bounded. Integrating (3.23) shows that ffo~(S)2ds is uniformly bounded by V(0) so that

(~)

Similarly (3.16) can be written as =

(3.21)

Lemma 3.1. Both the normalized gradient algorithm

This analysis applies to the "pure" least squares algorithm (3.16), (3.17). However in practice it is often desirable to modify (3.17) to prevent P from going to zero. Well known modifications a r e - exponential data weighting (possibly with variable forgetting factors), covariance modification, covariance resetting, constant trace algorithms etc. This analysis covers all of these cases provided the following two constraints are satisfied: (a) whenever P is modified it is increased and (b)there is an upper bound to 2maxP(Or trace P). These constraints are readily achieved for specific algorithms, see for example Cordero and Mayne (1981), variable forgetting factor algorithm; Goodwin and Sin (1984), covariance resetting algorithm; Goodwin and Middleton (1985), constant trace algorithm.

=

(e + IpTip)"

(3.20)

~ E L 2. Since the norm of Ip(c + IpTip)-½. does not exceed 1 it follows (from (3.18)) that /TEL2 and (from (3.13) and (3.20)) that ~ is uniformly bounded.

(ii) Normalized least squares algorithm. Modifying (3.17) gives the more general form P _

_ pipipTp + Q; c + IpT(p + I)ip

Q _> 0, trace P < ~ .

(3.24)

Parameter estimation perspective of adaptive control The case Q = 0 corresponds to "pure" least squares and this automatically gives P < Po and hence trace P < oo. The case Q _> 0 corresponds to the various modifications to the least squares algorithm discussed earlier. Consider the following non-negative time function V V = /TTp- 1/7.

(3.25)

From (3.16), (3.17a)

~<_ _<

obtained, at least when C has a simple structure. The algorithm is trivial in special cases, e.g. when a lower bound is known for one parameter. The corresponding least squares algorithm becomes the following.

Algorithm 2' (Normalized least squares with projection). A time varying linear transformation, P-~, is introduced where P = p~pr/2. Let (7 denote the image of C under the transformation P-½, define ~' ~ e/(c + ~xp~p + ~x~b), and let

= ~=

_e 2

61

c + tpr(P + I)~b

P~be

c + ffxP~b + ~T~ (= P~bF')ifOeC°

orifOe6(C)and(P~pU)XN <_0 1(3"26a)

--e 2

c + (2m.xP + 1)~PX~P

= P½ Pr[pT/Z~og] otherwise,

--1 [2maxP + 1] ~,2.

Hence V is monotone, non-increasing and bounded from below and therefore converges. Thus ~eL2. Also

V > l]/Tll2,;tminP-1 = I 2 ~ , ~ e ] ll0112. Hence/7 is bounded and this implies ~ is bounded. Next, note that

where C °, 6(C) and N denote, respectively, the interior, boundary and normal vector at 0 of C. Pr[v] denotes the projection of v onto the supporting hyperplane to (~ at P - ½0 (assuming this hyperplane is unique). When there are multiple supporting hyperplanes at 0, an appropriate algorithm is easily obtained, at least when C has a simple structure. Again the algorithm is straightforward in special cases. Properties of the above algorithms are described below.

Lemma 3.1'. The estimators (3.26) and (3.26a) have properties (a) and (b) given in Lemma 3.1 and in addition

~T~= (/7T~O)~PTP2tP(~T/7) [C + ~bT(P + l)lp] 2

(c)

0e C

for all t provided 0o e C.

< [-2maxP]~'2.

Proof. Thus ~ is bounded and Oe L 2. VVV In some cases, it is known a priori that Oo lies in a convex region C (e.g. when one of the parameters is known to be not less than a fixed constant). The above algorithms can be readily modified to include their prior information. The normalized gradient algorithm becomes the following.

(i) Gradient. This projection ensures 0TN < 0 whenever 0E 6(C). Thus (c) follows immediately. To establish (a), (b), note that 0 is modified only when 0e 6(C) and F~TN > 0. Thus consider a time t when these conditions hold. Write

Algorithm 1' (Normalized gradient algorithm with projection).

~ = ~ otjhj + fiN,

= 0 = ~9F

p-1 j=l

if 0e C °,

or if 0e 6(C) and ~,TN < 0;

p-1

where p is the dimension of 0 and (3.26)

= Pr[~(] otherwise: where C °, 6(C) and N denote, respectively, the interior, boundary and normal vector at 0 of C. Pr[v] denotes the projection of v onto the supporting hyperplane to C at 0 (assuming this hyperplane is unique). When there are multiple supporting hyperplanes at 0, an appropriate algorithm is easily

~ ctjhj~ j=l

supporting hyperplane at 0. Now, since ~bXN = flNTN > 0, fl > 0. Also, since Ooe C and O~ 6(C), OTN = ( 0 - o0)TN > 0. Finally, = Pr{f~h] p-1

= ~ ~jhj. j=l

62

G . C . GOODW1N and D. Q. MAYNE where

Since V(g) = (½)I1~112 it follows that

A

z = ff

lT

~b =zx- - L Yr . E . . . . . D n-x"YE, u E. . . . . Dn- lu E,--Y] (3.30)

= { ~ q , ] ~ 0 - flNTO < [~b]T0

(3.29)

since fl > 0 and NT0 _> 0.

Hence, the projection causes I7 to be not m o r e positive than it would be in the absence of projection and thus the same a r g u m e n t as in the p r o o f of L e m m a 3.1 gives properties (a) and (b). (ii) Least squares. Consider as before the case when 0E 6(C) and (PI~g")TN > 0. If V& (½)~rTp- 10 then 1) ___0Tp - 1~ + (½)0T(p- 1)/7 = (p-~O)Tpr[p-~{p~O~,}] + (½)~T(p-~)O < (p-½0)T[p-~r{p~O~,}] + (½)~rT(p-1)~

= 0~q,e' + ½ ~ ( P -

~)~,

the inequality following from an a r g u m e n t similar to the one e m p l o y e d in the p r o o f of L e m m a 3.2. Since in the absence of projections

0o -~ [go . . . . . g , - x , ro . . . . . . _x,(1/s,)]. --

!

t

t

r !

(3.31)

The previously introduced p a r a m e t e r estimators apply equally well to this "inverted" model and the associated properties are retained. C. Tracking error forms Finally, the usual continuous time algorithms as in Egardt (1979), Morse (1980), N a r e n d r a et al. (1980) etc., and a rather general algorithm which includes existing algorithms as special cases are described. Define z as a filtered version of the tracking error y - y*, and relax the condition p = n - m. F r o m (2.10), (2.4):

flAy--y*= (3.32)

1/= ffrp- ~p@~, + ~ f f r ( p - 1)0,

N o w define a filtered error, qy, as:

it follows that the projection causes 17 not to be m o r e positive; the desired result can now be established as for the gradient case. VVV In the above discussion the natural choice for z has been used to form the prediction error, namely z = y. However, z can be any function of the measured variables. T w o other possible choices for z are described below. B. Linear control form It will turn out that, in adaptive control, the coefficient of u(t) (i.e. the p a r a m e t e r b m = s,) plays a unique role. Indeed this was the motivation for introducing the projection scheme in (3.26). An alternative m e t h o d used in discrete time (Goodwin and Sin, 1984; Kreisselmeier and Anderson, 1985) is to "invert" the model so that ti is explicitly defined. Thus (3.1) can be written in the following equivalent way: t7 = (1/s,)y - G'YE -- R'uE,

f|h

/h

where G ' = ~ ) G , R ' = ~ , ~ . ) R .

(e)

= -g

q

~,

o

o]

+ N y - ~;r ,(~)

Then (3.27)is ex-

where N', D' are monic polynomials of order ny(> p + m) with D' Hurwitz. (N' and D' need not be coprime.) F o r future use D' = D'~D'2,

(3.34)

where D'I,D'2 have degree (n I - p -- m) and (p + m), respectively. N o t e that D~ has the same degree as S. Equation (3.33) can then be written as N'I(sqD'2+S--sqD~) u G H ] "~ = 7 o' + ~y - -yr N' = ~ [SqU' + Ru o + Gy o -- Hro],

(Q

(3.33)

(3.27)

pressed in regression form as: z = ~kT0o,

r/y A

(3.28)

(5)

(5) (3.35)

Parameter estimation perspective of adaptive control

63

This suggests the following parameter estimator:

where q = p + m as in (2.11) and

u' LX 1

= --U

(3.36)

[~0q]= Flu0, + ~kxO] e'

1, 1 uo = ~Tu

(3.37)

where F is the algorithm gain. F o r example, the gradient algorithm is achieved with

Yo = ~ Y

(3.38)

Di

F - - -

(3.49)

1

(3.50)

c + 070 .

(3.39)

ro - ~ r

R -= zx S

, - - s q D 2.

(3.40)

The algorithm (3.49), (3.50) has similar properties to those described in earlier sections, as shown below. Lemma 3.2. The estimator defined by (3.49), (3.50)

Note that R has degree p + m - 1 and is defined slightly differently from the R used in early sections. Factoring sq from (3.35) gives

N,[u' + R' uo + G' yo

(~)

qf = s q - -

El_

has the following properties: (a)

/7 and ~ are uniformly bounded;

(b)

0 and ~ are in

L 2.

Proof. This is basically the same as the proof of

L e m m a 3.1 using the non-negative function

(3.41) V(t) ~ ½[Sq/7(t)TO(t) + gq(t)2].

where

R'/'R,G' sq

(3.51)

"R sq

(3.42)

N o t e that 0 uniformly bounded cannot be established here. (More will be said on this point later.)

As originally suggested by Egardt (1979) an advantageous choice in (3.41) is to set n: = 2n - m, p = n - m and

If in the above analysis u was replaced by the special control -q~X0 this would yield

vvv

C.1. The Egardt/Morse algorithm

N' = D ' = E.

(3.43)

This choice removes the filter N ' / E from (3.41). Then, if z is defined as q:, (3.41) can then be written in regression form as

(3.52)

D1 where c~ T =/x [ Y D E . . . D n - l y D 2 ,

z = sq[u' + ~,r0o],

(e)

(3.44)

UDz . . . D q -

lUD2 ,

__ H r D z ]

1 Yo2 & ~2 y

where

1

z& q:

(3.45)

~ ~=(Yo. . . D" - ly o, uo . . . . . D q- tuo, -- Hro)

go .... g, - 1 ro .... rq_ 1,

•

II D 2 ~_~ - -

D,2u

r

A

1

o,= Dr"

(3.46)

The predicted value of z can be computed from = gq[u' + 0T0].

Substituting (3.52) into e = z - :~ and using the fact that ~O= (I/D~ qb) gives the following alternative expression for e which is valid iff u = - q~r0:

(3.47) 1

The prediction error e & z - ~ is then e = -sq[~,T/7] - gq[u' + o r 0 ] .

(e)

(3.48)

T

(3.53)

This form of the error was first introduced in the

64

G . C . GOODWIN and D. Q. MAYNE

continuous time adaptive control literature (e.g. Monopoli, 1974) when it was found that simply using y - y* to derive the algorithm did not guarantee stability in general. The error given in (3.53) has been called an "augmented error" in the literature since y - y* has been "augmented" by the term T

1

E lE

ny=2n-m--2,

q=p+m=n--I D~ = L

(3.54)

(a monic polynomial of degree n - m - 1)

/7

The predicted value of z is

= gq

C.2. The Monopoli/Narendra et al. algorithm Returning to (3.41), the following choices are made (Narendra et al., 1980): 1,

(3.60)

X

The authors prefer to use the alternative form for e given earlier namely e = z - ~, where z = y - y* and ~ = gq[u' + Oxo'], since (a) this clearly shows that the algorithm is a simple prediction error algorithm without the added mystery of "augmented errors" and (b) should u not be equal to -~bv0 (for example when the input saturates at a hard limit) then the properties of the parameter estimator will be retained if e = z - ~ is used, whereas if (3.53) is used these properties will no longer hold. This means if (3.53) were to be used, the algorithm would almost certainly fail to work in practice due to the generation of totally erroneous parameter estimates.

p=n--m-

I~T= Iyl) ..... Dn-lyo, uo ..... Dq-luo,--L].

(3.55)

(a monic polynomial of degree n - 1 s i n c e / 4 = hqD q + ...ho)

u ' + ~bT0]

(3.61)

and the prediction error is

e=z--~. The algorithm is 0 = ~ = -~kel;

gq = - ( u ' +

~0T0)el, (3.62)

where

el& e -

~-{4, ~ e l }.

(3.63)

N o t e that if N'/E were 1, then ea would simply be ~ as used in (3.15), (3.21) etc. In fact the presence of N'/E in (3.64) and hence (3.62) leads to assumptions that L must be chosen so that the transfer function N'/E is strictly positive real ( S P R ) - - s e e section 3.9 of G o o d w i n and Sin (1984) for discussion of the corresponding discrete time case. The basic idea is that the filter N'/E must be such that, on average, it does not reverse the direction of the parameter update. This is trivially achieved if N'/E = 1 but N'/E being S P R suffices in g e n e r a l - - s e e N a r e n d r a et al. (1980). The special case when n - m = 1 has received particular attention in recent literature in robust adaptive control (,l~9~ut and Friedlander, 1984; G o m a r t and Caines,

(3.56) 4. T O O L S F O R P R O V I N G G L O B A L STABILITY

N' = D'

LH hq

(a monic polynomial of degree 2n - m - 2).

(3.57)

As in (3.41),

rl~= y -- y*

The algorithm model has several components; the system being controlled, the estimator and the controller. The system is, in the simplest case (see (3.9)) described by the equation y = 0T00,

sqL/7 [ ,

= hqE L u + R ' u o + G ' y o - ~ / T r o ] (3.58) ' , = sq ~ - [u + OV0o]& z,

IN 1

1

(e)

1

(3.59)

where u' = ~-,u, u o = w,,u, Yo = ~,,Y as before and /-G D1 D,

(e)

(4.1)

where, from (3.11), q, is a vector c o m p o s e d of the filtered input and outputs and their derivatives up to the required degree and 0o is a vector of u n k n o w n parameters. e lms ~ e g m° r~;~]~8 sl~l~u~st1~mate 0 of 0o; it is a oTyhr{a~[~ e = y -- ~9T0 = --~kT0 (e).

(4.2)

Parameter estimation perspective of adaptive control Referring to (3.9), the system model can alternatively be written as Ezy = dpXOo and correspondingly the optimal model reference control law can be written as

(4.3)

(])TOo = __/tr E 1'

where q5& (yf . . . . . D " - l y f , uy ..... Dn-auf, u)T

(4.4)

Oo ~ (go ..... g , - 1, ro . . . . . r, _

(4.5)

1' Sn) T

and where Yl, ul are as in (3.5), (3.6). Thus q5 and qs are related by

= E~(~b).

(4.6)

In the adaptive case, the unknown 0o is simply replaced by the appropriate estimate 0 to give the following certainty equivalence law

if)TO =

/~r

E2y* = - Ez"

65

Comment. Note the feedback structure in the result: depends on ~ and ~, depends on e. However, the whole point of the result is that condition (a) can be established independently of(b), i.e. independently of the choice of control law. The estimation error e drives a dynamic system producing ~, as its output; the properties of the system depend on the controller and are used in establishing condition (b) only. With these results it is usually a simple matter to deduce global convergence for appropriate discrete time adaptive control laws. An analogous result for continuous time systems is required. In fact such a result appears implicitly in Morse (1980) and Narendra et al. (1980) where it is shown that in continuous time a growth condition is required on qs and its derivatives up to order n - m - 1. This result is highlighted because it shows clearly what must be established to prove stability. Note that the case n - m = 1 is particularly simple because no derivatives of qJ are involved. This partially explains why the case n - m = 1 was resolved much earlier in the literature than the general case. Let H(t)~ R ("-") ×dim(q¢)be defined as follows:

]

I ~,(t)~

(4.7)

H(t)~= D~(t)T The elements of this structure which are identified as being relevant for stability analysis are the prediction error e and the regression vector ~O.The essential properties of e depend purely on the estimator and are independent of the controller. The properties of ~b depend on both the estimator and the controller. Simple conditions on e and ~ which guarantee stability of the adaptive system are needed. As a guide to such a result, recall the "key technical lemma" in Goodwin et al. (1980), for discrete time systems (all functions of t are temporarily sequences). Stability lemma for discrete time systems (a) Suppose the estimator is such that the normalized estimation error ~ & e/[c + CTff]½ lies in l 2. (b) Suppose the controller is such that the regression vector qs satisfies the growth condition

D,-"-I@(t)T Lemma 4.1. (Stability lemma for continuous time systems). Suppose the estimator is such that the following hypothesis is satisfied: (El)

(C1)

o

Then (i) (ii)

IIq/(t)l[ is uniformly bounded, and e ( t ) ~ O as t - - * ~ .

AHT 23:1-E

e/[c

+

I//T,~/]½ lies in L 2.

There exists a constant d E (0, ~ ) such that Itg(t)lJ 2 <_ d + d

Ifg(s)ll2~(s)2ds.

Then (i) H is uniformly bounded. If, in addition, the following hypothesis is satisfied (EC1)

II@(t)ll2 ~ d + a ~ 11@(0)12~(0 2.

~

Suppose the controller is such that the following growth hypothesis is satisfied:

II~J(t)ll z ~ d + d(max {le(r)121 r mI-0, t]}) for all t > 0, which implies when ~ ~ 12 that

(4.8)

The variable d is uniformly bounded and

e~L2. Then

(ii) e(t)-~O

as

t--,~.

Proof. (i) From (C1) and Gronwall's lemma (e.g. Desoer and Vidyasager, 1975): IIH(t)ll 2 < dexp

d~(s)Eds .

66

G . C . GOODWIN and D. Q. MAYNE Since ~ e L 2 by (E 1) it follows that H is uniformly bounded.

and that ~ = 1/E2(~bTO) is the solution of:

(ii) Standard convergence results are used--see, for example, Desoer and Vidyasagar (1975, p. 232). If d is uniformly bounded then e is uniformly continuous. If e~Lp (1 < p < ~ ) and if e is uniformly continuous, then e(t) ~ 0 as t ~ ~ . The details of how to apply the above results vary slightly depending on the specified algorithm. For example, if the "natural" form of the normalized gradient algorithm (3.15), or the normalized least squares algorithm (3.16), (3.17) is used then from Lemmas 3.1, 3.1' (in addition to (El)) ~, ~ and ~ are uniformly bounded and ~ L 2 . The proof then proceeds by verifying (C1) and hence that H is uniformly bounded. Conditions (EC1) then follow fairly easily and therefore e(t) ~ O. When the tracking error algorithms are used, e.g. (3.49), (3.50), then from Lemma 3.3 (in addition to El) ~r, ~ are uniformly bounded and O~L2. Thus is no longer uniformly bounded as before. The pattern of proof then proceeds by establishing that H is uniformly bounded, and deducing that q~ and hence ti are uniformly .bounded. It can then be shown that u, e and ~ are uniformly bounded. Finally it can be shown that condition (EC1) is satisfied, therefore e(t) ~ O. An example of the former analysis will be presented later. An illustration of the latter analysis can be found in Morse (1980). The stability Lemma 4.1 is the major tool for unifying existing continuous time model reference adaptive control laws and establishing global stability of new algorithms. However, most algorithms must have a bound on the "swapping" error caused when a term like 1/E2(t~T~) (in which 1/E 2 operates on the product ~bVO) is replaced by a term like (1/E24>)TO (in which l/E2 operates only on ~b). That this error is small when 0 changes slowly is intuitively obvious and is made precise in the following result. Firstly, note that ~b = (1/E2)q~ satisfies the following set of equations:

Iit = A H + bc~T ~1 T ~ -

cTH,

(4.9)

(4.10)

where A is in controller canonical form, p~(s) = Ez(s), b = ep and c = el. (Here, and elsewhere, e3 denotes a null vector save for a 1 in the ]th entry.) Note that

1

T (4.11)

= Ag + b(q~Tg)

(4.12)

= cXg.

(4.13)

Hence H~r - g can be computed and this automatically yields:

( ~ q ~ ) T o - - ~(dpT~) = cT[H~-- g].

(4.14)

Lemma 4.2 (Morse (1980)) (The swapping lemma). HO-- g = e a' * H~

(e)

and

Ez ]

(¢x~) = cXEHO _ g] = cTem , H{f

(~)

Proof. d H o = AHO + b~X~ + H~ d ~ g = Ag + b(fiO. Hence

dEH• - g] = AEHO -- g] + H~ Since pA(S) = E2(s) is Hurwitz, the desired result follows. VVV The above result can also be expressed in transfer function form as: 1

T

_~_ . . . E ~ - r a E o n - r a -

I(~T)~]},

where E~, E2... E~-" are the Tchirnhausen polynomials (Goodwin and Sin, 1984) associated with E 2. One further result is needed.

Lemma 4.3 (Morse (1980)). Suppose the weighting function W is exponentially stable. (a) If u is uniformly bounded, then W . u is uniformly bounded. (b) If W is also strictly proper and u is piecewise constant then there exists a d < ~ such that II(W*u)(t)ll 2 ~ d + d

(c)

;o

Ilu(s)ll2ds.

If W is also strictly proper, u I is uniformly bounded and u 2 is piecewise constant then

Parameter estimation perspective of adaptive control there exists a d < ~ such that IIW*(ul + u2)(t)ll 2 < d + d

Iluz(s)[12ds.

Proof. Parts (a) and (b) are straightforward and are proved in Morse (1980). Part (c) follows from the trivial observation that

shall take bm to be positive without loss of generality.) Also note that bm = s, where s, is the leading coefficient in S. (A.4) The reference model is stable and the plant is minimum phase, i.e. EB is Hurwitz. Summarizing some of the key equations, the system is expressed in the natural regression form as:

E y = Su + Gy

IIW*(u, + u2)ll 2 = II(W* u 0 + (W*u2)ll 2 211W* u, II2 + 211W*u21r2. 5.

(e)

(5.1)

or equivalently, y = ~//T00.

GLOBAL STABILITY O F ADAPTIVE C O N T R O L SYSTEMS

The tools of section 4 can be used to establish global convergence of continuous time model reference adaptive control algorithms. The tools could be used to establish convergence of the Edgardt/ Morse algorithm or the Monopoli/q~larendra et al. algorithm. However, this would amount to a simple reworking of proofs already available elsewhere in the literature. Instead, in this section they will be used to establish global convergence for what is believed to be a new continuous time model reference adaptive control algorithm. This algorithm uses the "natural" parameterization given in (3.9) and p = n - m is taken. As far as the parameter estimator is concerned, the results apply equally to either of the following algorithms: (i) the normalized gradient algorithm (3.15), or (ii) the normalized least squares algorithm (3.16), (3.17). The fact that either estimator can be used is made obvious by the modularization of the proof and the fact that the properties established in Lemma 3.1 for the two algorithms are identical. Assume that the sign and a lower bound, Smin, are known for the leading coefficient in B, i.e. b,, = s,. The projection facility explained in (3.26) or (3.26a) will be used to ensure that 8,(t) > Smin for all t whilst retaining all the properties in Lemmas (3.1) and (3.1'). Earlier papers use related assumptions, e.g. the sign of b,, is assumed known in Morse (1980) and Narendra et al. (1980). More recent work (e.g. Mudgett and Morse, 1985) has shown that the sign information can also be done away with by use of high gain strategies. The standing assumptions for the rest of this section are as follows. (A.1) The order of the plant, n, is known (actually this can be an overbound on the order since A, B can have stable common factors). (A.2) The relative degree, n - m, is known. (A.3) The sign and a lower bound, Smi,, are known for the leading coefficient, b,,, in B. (We

67

(/~)

(5.2)

The controller is implicitly defined by the certainty equivalence law (4.7), i.e.

E 2 y * = R(~---~)rA

Rr:=~u:+

C,y:(5.3)

or equivalently, Hry = q~r0.

(5.4)

The main result is the following.

Theorem 5.1. Consider the system (2.1) subject to assumptions (A.1)-(A.4). If the normalized gradient parameter estimation algorithm (3.26) (or equivalently the normalized least squares algorithm (3.26a), (3.17)) is combined with the certainty equivalence control law (5.3), then (1)

H, Or, ~, e, ~, 4, ~b and u are uniformly bounded.

e(t)~O as t ~ o % (3) ly(t)--y*(t)J-~O as Proof. See Appendix A.

(2)

t-~oo.

6. C O N C L U S I O N S

The continuous time model reference adaptive control problem has been formulated as a combination of parameter estimation and certainty equivalence control. Thus parallels have been drawn between results on discrete time model reference control and continuous time model reference control and existing algorithms for the continuous case unified. The algorithms have been shown to be essentially identical for the discrete and continuous cases and, apart from certain technical difficulties in the continuous case, the methods of analysis follow the same pattern. The paper has developed a continuous analogue of the discrete "key technical lemma" and has used the "convergence tool" to establish global convergence for a new continuous time adaptive control algorithm using a least squares parameter estimator. It is important in practice that the properties of

68

G.C.

GOODWIN a n d D . Q . MAYNE

the parameter estimator be retained if the input goes into saturation. This has been shown to be possible for all algorithms studied, provided they are suitably arranged, but it is not true of many algorithms as described in the literature on continuous time adaptive control•

B

D i- ~B ~i = - - - ~ - - [ r Uf

REFERENCES Astrrm, K. J. and B. Wittenmark (1973). O n self-tuning regulators. Automatica, 9, 159-199. Desoer, C. A. and M. Vidyasagar (1975). Feedback Systems: Input-Output Properties. Academic Press, New York. Egardt, B. (1979). Unification of some continuous-time adaptive control schemes. IEEE Trans. Aut. Control, AC-24, 588-592. Feuer, A. and A. S. Morse (1978). Adaptive control of single input single output linear systems. IEEE Trans. Aut. Control, AC-23, 557-570. Gomart, O. and P. E. Caines (1984). Robust adaptive control of time varying systems. McGill University Technical Report. Goodwin, G. C. and R. H. Middleton (1985). Continuous and discrete adaptive control. In C. T. Leondes (Ed.), Control and Dynamic Systems, Vol. XXXIV. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1980). Discrete time multivariable adaptive control. IEEE Trans. Aut. Control, AC-25, 449-456. Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, New Jersey. Goodwin, G. C. and E. K. Teoh (1983). Adaptive control of a class of linear time varying systems. Presented at the IFAC Workshop on Adaptive Systems in Control and Signal Processing, San Francisco, June 1983. Kosut, R. L. and B. Friedlander (1982). Performance robustness properties of adaptive control systems. IEEE Trans. Aut. Control (to appear). Kreisselmeier, G. and B. D. O. Anderson (1985). Robust model reference adaptive control. D F V L R Technical Report. Monopoli, R. V. (1974). Model reference adaptive control with an augmented error. IEEE Trans. Aut. Control, AC-19, 474484. Morse, A. S. (1980). Global stability of parameter-adaptive control systems. IEEE Trans. Aut. Control, AC-25, 433-439. Mudgett, D. R. and A. S. Morse (1985). Adaptive stabilization of linear systems with u n k n o w n high frequency gains. IEEE Trans. Aut. Control, AC-30, 549-554. Narendra, K. S., Y. H. Lin and L. S. Valavani (1980). Stable adaptive controller design, Part II: Proof of stability. IEEE Trans. Aut. Control, AC-25, 243-247. Parks, P. C. (1966). Lyapunov redesign of model reference adaptive control systems. IEEE Trans. Aut. Control, AC-11, 362-367.

i=1 ..... n

A

= ~1-~7_ ~bT~

D i- ~A _ ~b,+, = - - ~ - [ r - -

Acknowledgements--The authors acknowledge helpful discussions with R. Middleton who also suggested the form of the normalization necessary to extend the theory to the pure least squares algorithm.

¢~T~,

(A.5) (A.6)

q~r0],

i = 1. . . . . n

E x A [ f _ ~bwff], q=~ 2n + 1 dp~ = EB

(A.7) (A.8)

= (1/E2)u, ~b, = ti; so q~ = E2~bq = u. Hence, for i = 1,...,2n + 1, ~b, = Wi * I f - q~'r~,

(A.9)

where W~is the weighting function of a time invariant exponentially stable system. For i = 1 to 2n, W~ is strictly proper. m a y be decomposed as follows:

= (1/s,) + Wg',

(A.10)

where Wq' is strictly proper. (The (1/s,) arises from the fact that E~, A, and E are monic.) To establish the growth condition given in L e m m a 4.1 on H, note from (A.9), (4.9), (4.10) that

¢, = cTeA,b, Wi, (~ _ $T~ for i = 1,..., q. Since eA' and W~ are exponentially stable and f is (assumed) uniformly bounded, the component Wi • f of ~'i due to f is uniformly bounded. Using Morse's convention (y = z (/a) if each element of y - z is uniformly bounded)

~, = creA'b * Wi * (-- d~r~), (Iz) i = 1,..., q. Since the two weighting functions are time invariant:

~ki = Wi * cTeatb, (-- ~bT~), (/~) i = 1. . . . . q. Since ere'% realizes (l/E2) the (swapping) lemma 4.2 is used to obtain:

= W,,(--q~;O) + W,*cTeA'*ng = W,,e + ~ , H ~ ,

0~)

where ~ & W~• c r e A' is exponentially stable and strictly proper, i = 1,...,q. Let F denote the set of real valued functions satisfying

y(t)2 <_d + d £e(s) 2 + IIH~(s)ll2ds APPENDIX

Proof of Theorem 5.1

for some d < o0. Since (a + b)2 < 2a 2 + 2b2, if?l,?2 E F so does ?x + ?2. The first step to establishing the growth condition is:

The closed loop equation:

Lemma A.I. II~011e F , e e F .

Eyy = Suy + Gy: = / T r r -- q~rg

(A.1)

Proof. Let Then

is obtained, so that with f & (H/E1)r:

ElBa_ y = ~ f f k r -- qST0 -]

(A.2)

- 0TO]

(A.3)

= --rE~A ~ r EB

_

~b~~ W~* e; ~b'l' ~ ~ * H~. ~'i = ~'~ + ¢'~"

Since W~ is exponentially stable and strictly proper for i = 1. . . . . 2n and since g,~ = W~, e, then L e m m a 4.3 gives (for some d < oo), the inequality (~k~)2 <_d + d [ e 2 d s

,¢0

Parameter

estimation

perspective

of adaptive

control

69

= - w~ • [eTA ~- tl:l~ -- crAJe m * n ~ ]

so t h a t ¢" e F, i = 1. . . . . 2n.

= -- W~* [ c r a ~- ~D[n~] -- e r a ~- ~ n ~ -- d A ~ e ~' * n ~ ] for/= 1,...,q. F o r i = 1 , . . . , q - 1, Wi is e x p o n e n t i a l l y stable a n d strictly proper, a n d therefore satisfies

Similarly

~ " = W~* H ~

satisfies

Wi(t) = c~eA,'bt,

(¢,")~_
IIH~ll2ds,

do

where A t e ~<* + ") ×(" + -) is in c o m p a n i o n form, pAt(s) = E(s)B(s) and b a = e,+ m. F o r later use W~ is defined by

i.e. ~b;'eF, i = 1. . . . . 2n.

Hence ~l,~,(t)~ cT A,eAt*b,.

~ q e F , i = 1. . . . . 2n.

Hence, for i = 1 , . . . , q -- 1:

N e x t c o n s i d e r ~,q.

¢~ =

W~,e + ~ , n O

Wi * eTA J- 1 D [ H ~ = ~o Wi(t -- T)cTA J- 1D[H(z)~(z)]dz

= (lls.)e + W~'*e + W~*HO --

(using (A.IO)) = WI(0)cTAj - 'H(t)~(t) -- Wi(t)cTA j - 'n(0)O(0)

~bTO~I- Wq'*e + I ~ * H O Sn

+ ( ~ 9 * c T A j - 'HO = Wi(0)[DJ- l~k]g -- Wi(t)cXAj - tH(0)~0) + (~*cTAJ-IHO.

= -- qQs" -- ~'TOe + W~'*e + W~* H~, Sn Sn where ~bz = [ O ~ , O j ;

Since, II~rll is uniformly b o u n d e d , the first term is in F b y hypothesis. The second t e r m is u n i f o r m l y b o u n d e d because Wi is e x p o n e n t i a l l y bounded. Because cTA ~- 1H = e~H = D i - t~k, the t h i r d t e r m is in F by L e m m a 4.3. Hence, Wj •cTA j - 1DEH0~ e F. The r e m a i n i n g t e r m s also lie in F. H e n c e , / T $ ~ ~ F, i = 1. . . . . q - 1, i.e. ]lD/$e II ¢ F. N o w consider

~r = [ ~ , ~ ] .

Hence

Sn q

Sn

Since ~kl. . . . . $ ~ e F, IlCell e F. Since ~ is u n i f o r m l y b o u n d e d a n d s. > s=tn > 0, - ~ k ~ e / s . ~ F. Also, by L e m m a 4.3 I,V~'• e a n d W~ • H ~ r lie in F. H e n c e ~ds. ~b~~ F. Since ~. > Sml., Sa ~ F. Since t . . . . . ~bqe F so does I1~ II- Since e = - s T ~ and ff is uniformly b o u n d e d , e ~ F. [] F o r the case n -- m = 1, the a b o v e result is sufficient t o s h o w t h a t H satisfies (C 1). F o r the general case b o u n d s are also needed o n the g r o w t h of the derivatives of ~k. P r o c e e d i n g by i n d u c t i o n shows t h a t if IID~blI~F so does IID~+ t~b II- This requires the following result from M o r s e (1980). L e m m a A.2. F o r a n y v e e t o r f i n R " - ' , (fTH)~, the ith c o m p o n e n t of the vector f X H satisfies, for i = 1 ...q: ( f T n ) , = -- W/* [fTH~J -- fTeAt * H~].

Recall t h a t W~ is p r o p e r b u t n o t strictly proper. T h u s

w~ = O/sO + w;, where W~' is strictly p r o p e r a n d is given b y Wa'(t) = c~e&'b t. By L e m m a A.2 (/~¢0) = [ c r A J n ] 0 = -- Wq • [cTAJH~ -- f T e m * H 6 ;

(It)

fr =cTA j

= -- I-1/s~](D/~k)r~ r + (1/sn)W~s * H ~ + g~ Proof. ( f T H)i = fTeA'b * dp, where W~f ~ f r e a t is e x p o n e n t i a l l y stable a n d

¢, = W , * ( - C ~ .

O0 gj & -- W~'EcTA J H ~ -- f r ea* * H~].

Hence Since ( f v H) ~ = f T ea, b , Wi , ( _ Ca'~)

cTAJH~ = (DJ~kT)~

= W~ * f r e a ' b * ( - d#'rO) = - W~, f V g ,

(it)

where g = eA'b * (djro). (it) The result follows from the s w a p p i n g L e m m a 4.2. The next result can n o w be established.

= (DJ~g)Sn "4- (DJ~IR)T~R

it follows t h a t []

L e m m a A.3. IID'¢II e F , i = 0,1 . . . . . n -- m -- 1. Proof. By L e m m a A.1, liD°ell e F . A s s u m e t h a t II/~-X¢ll e F , where j < n -- m - 1 a n d show t h a t IIDJ¢ll e F . The fact t h a t in (4.10), (4.11) c = et, b = e , _ = a n d A is in c o m p a n i o n form so t h a t cTA = e~, e r a ~ = cTA A = er2A = e~ . . . . . yielding eTA ~- t = e~ for j = 1. . . . . n -- m will be used. Also, since H = A H + bdpr it follows t h a t e ~ A H = e~/~ so t h a t c r A I - t A H = c r A I H = c r A J - t f f l for j = l . . . . . n - - m - - 1 . Also / ~ , = e ~ + t H = crASH, j = O, .... n -- m -- 1. By L e m m a A.2, for a n y j _< n -- m -- 1 Dl~i = (cTAJH)i = -- W i * [cTAJH~ -- c'rAle At * n ~ ]

[s~](DJ~,) = --(l/s~)[(Di~e)T~le] + (1/s.)WqI*

"~ +

g~.

It has been p r o v e d a b o v e t h a t II~RII is in F; since ~e is u n i f o r m l y bounded, the first t e r m is in F. The second t e r m clearly lies in F. T h e t h i r d term, gj, has the s a m e form as DJ~q, i < q -- 1 a n d hence lies in F. Because (s,/~) is uniformly b o u n d e d it follows t h a t D J ~ ~ F. Hence IlD~¢ll E r . By i n d u c t i o n IID~' l l ¢ F, i = O, 1. . . . . n -- m -- 1. N e x t n o t e t h a t

70

G . C . GOODWIN a n d D. Q. MAYNE n-m

IIHII<

~

1

n~m-I

IIDJ'@II s o t h a t

IIHII2_<2 ("-m~

j=o

~ j=o

F o r i = l , . . . , q -- 1

[ID/@II2.

F r o m L e m m a A.3, there exists a d~ e(0, oo) such that

c~i = - Z i [ c r A ~- 1D(H~) - crA• I H ~ = --(ZiD)[cTAS-IHO]

IlO~@(t)[I2 < d, + d, J~oe(s) 2 + IIH(s)~(s)l 12ds for j = 0 , 1 . . . . . n - m - 1 . Hence there exists a d 2 e(d~, o~) such that

cTASH~]

+ ZIEcTAS-'H~-

(#)

cTAsH~].

(#)

Since (for, i = l , . . . , q - 1) ZiD is proper and exponentially stable and H is uniformly boutaded, (ZiD)[cTA~-1H~] is uniformly bounded. Also since H~" and H(7 are uniformly bounded, Z~[cT A ~-~1H~ -- cTA~H(7] is uniformly bounded. Hence q~l... ~bq_ 1 are uniformly bounded. When i = q

IlH(t)ll 2 _< d2 + d2 fo~S)~[1 + II@(s)ll2]ds + dz~ollH(s)~(s)llZds.

(.)

¢ ~ ~T~_ Z;(¢'O) Since

@=

q

cTH, d 3 E:(d2, o0) can be chosen to satisfy

IlH(t)ll2 < d 3 + d3 F e 2 d s + o0

= b~¢ Sn

d3 ~0 IIHIIZ[e2 + II~llZ]ds"

Since ~ e L ~ , and I1~11-< II@/(c + @T@)½1[le-'l < [~1 there exists a d ~ ( d ~ , oo) such that

Sn

¢ ~ o ,, Sn

where ~bR & ((bl ... @q_ 1)T, Ok __a(/71.. ' t7~_ 0 T. Hence

Sn q

[IH(t)ll 2 ~ d4 + d4fo [[Hl[2~Zds. Hence H is uniformly b o u n d e d ((El) and (C1) in L e m m a 4.1 are satisfied). It has t h u s been established that H, 9, ~ a n d ~ are uniformly bounded. The uniform b o u n d e d n e s s of e and ~ follows from the uniform b o u n d e d n e s s of ~ a n d ~b (H is uniformly b o u n d e d iff D/@ is uniformly bounded, j = 0 . . . . . n - m - 1). The uniform b o u n d e d n e s s of q~ and, hence, of u = ~q is established next. N o t e t h a t ~)i = Z i [ f - ~)TO]

vi-lB EB

(2)

i= 1,...,n

Z.+ i = D i - t A EB '

i= 1,...,n

If

n-m=

1

then

A

is

a

scalar,

H=@

and

= 151 = A H + ~bT. Since H and ~b are uniformly b o u n d e d

so are that d= e = (c

@ and ~. If n - m > I, then H includes @ a n d ~ so b o t h tertns are uniformly bounded. Hence ~T(7 - @TiT is uniformly bounded. Since + @v@)~ it follows (from the b o u n d e d n e s s of ~k) that e e L 2 (and is uniformly bounded). Hence e(t)--,0 as t --', o0.

i = 1 . . . . . q,

'

Sn

Since ~bI . . . ¢ q - 1 and 0 1 ...(Iq_ 1 are uniformly bounded, so is ~b~gR. Also, Z~(~bT~ is uniformly b o u n d e d using the s a m e a r g u m e n t employed for ~bi = Zi(-~bv~), i = 1 . . . q - 1. Hence g,/s, dpq is uniformly bounded. Since g, is b o u n d e d below by Smi,, it follows that q~q is uniformly bounded. Hence q~ and u = ~bq are uniformly bounded.

where Z~ is defined in (A.5), (A.7) and (A.8) i.e.

Zi=

(~)

z~(¢,o),

(3)

E(y - y*) = E y -- E1E2y*

so that 1

Z

E~A q=--~-ff,

Y _ y. = @1-0o_ ~(¢T~) i=q. = @1"00 -

@TO

Clearly Z~ is strictly proper for i = 1 ... q - 1 a n d Z~ is proper (Zq = (1/s,) + Z'q where Z~ is strictly proper). Since A a R ~×~ is in c o m p a n i o n form, s z~ n - m, c = e~ and b = e~ it follows that = - e - c're A' * H~.

(e)

D(cT A ~- ~H ) = cT A ~- l ffl = cT A S H + ~ v

so that

Since ~ = c + @@-~e where @ is uniformly b o u n d e d and since (OXO = c T A ' - ~I?t~ -- cTA ' H O = crA * *D(H~) -- c r A ' - ~ n ~ - cTA'H~.

e(t) ~ 0 it follows that ~(t) ~ 0 as t --* oc. Since H is uniformly bounded, it follows that l y(t) - y*(t)[ - , 0 as t ~ o~. []