Robustness analysis of discrete-time adaptivecontrol systems using input-output stability theorya tutorial

W.R. Cluett, PhDS.L. Shah, PhDD.G. Fisher, PhD

Indexing terms: Control systems, Stability, Adaptive control

Abstract: This paper presents a summary andconsolidation of stability and robustness resultsbased on input-output theory for discrete adaptivecontrol systems. The objective of this paper is toclarify the techniques involved in applying thisstability approach to the adaptive controlproblem. It is intended that this tutorial mayprovide a basis for continuing work in this area.

1 Introduction

Certainly the earliest attempts to analyse the robustnessof discrete-time adaptive control systems were performedby Lim [1], and Gawthrop and Lim [2]. This workfocused on the robustness of the Clarke and Gawthrop[3] version of the self-tuning controller (STC) in the pre-sence of unmodelled plant dynamics. Stability conditionswere derived for the adaptive control system, in terms ofthe design parameters available with the STC algorithm.The stability approach taken in References 1 and 2closely followed the analysis performed by Gawthrop [4]on the stability and convergence of a self-tuning control-ler without unmodelled dynamics. In Reference 4 theadaptive control system was cast into an error feedbacksystem and the stability of this system was analysed usinginput-output theory. Ortega, Praly and Landau [5] haveextended the robustness analysis of References 1 and 2 toinclude normalised signals in the least squares parameteradaptation algorithm which remove the plant signalboundedness assumption used in References 1, 2 and 4.

The results in References 1, 2 and 5 represent some ofthe earliest work on the robustness of discrete-timeadaptive controllers to unmodelled dynamics. This paperpresents a critical review of this work, and extends theresults in such a way as to unify the results in References1, 2 and 5. In Section 2 the mathematical notation andtheory used in the paper are summarised. In Section 3, ageneric adaptive control system is cast into an error feed-back system which is then suitable for stability analysisusing the theory developed in the previous section.Section 4 examines the nonzero initial conditions charac-teristic of adaptive control systems, and their effect on the

Paper 5906D (C8), first received 6th November 1986 and in revisedform 23rd September 1987Dr. Cluett is with the Department of Chemical Engineering, Universityof Toronto, Toronto, Canada M5S 1A4Dr. Shah and Dr. Fisher are with the Department of Chemical Engin-eering, University of Alberta, Edmonton, Canada T6G 2G6

stability results in References 1, 2 and 5. The vanishingradius problem associated with applying conic sectortheory to adaptive schemes based on least squares isillustrated in Section 5, along with the role of normal-isation in overcoming this problem. In Section 6 an inter-pretation of the role of the key technical lemma inReference 11 is given, as it relates to the stability ofadaptive control systems using conic sector theory. Also,some insight is provided with respect to the choice ofnormalisation factor made in Reference 5. The L2

analysis is extended to the Lw case in Section 7 whereboth a least squares and a constant gain algorithm areexamined. Section 8 presents an augmented plant repre-sentation, which extends the control law used in Refer-ence 5 to inlcude the controller weighting polynomialsconsidered in Reference 1. Some overall guidelines andinsights based on this tutorial are presented in the finalSection.

2 Mathematical preliminaries for stabilityanalysis

This section describes the mathematical notation usedthroughout this paper. The main concern is with discretesignals which are infinite sequences of real numbers.Each signal may be considered a vector of infinite dimen-sion and represents an element of a set known as a linearvector space.

Norms: Norms may be thought of as a measure of thesize of a vector. Let E be the linear vector space. The zerovector in E is denoted by <f>. The function p:E->R + (theset of positive real numbers) is a norm on E only if

(a) x e E and x # <f> implies p(x) > 0(b) p(<xx) = | a | p(x) Va € R, Vx e E(c) p{x + y) ^ p(x) + p{y) Vx,yeE

Given the linear space E and a norm p on E, the pair (E,p) is called a normed vector space.L2-norm: Let x = (xu x2, • ..)• The L2-norm of x isdefined as

1/2

L^-norm: Letdefined as

x = (xu x2, ...). The L^-norm of x is

If these norms exist, the corresponding normed vectorspaces are called L2 and L^, respectively. The extension

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988 133

of a space L, denoted by Le, is the space consisting ofthose elements x whose truncations lie in L, e.g. xbelongs to the extended space L2e if

( T •) 1/2

=i

VT e Z+ (the set of positive integers)

Operator: An operator is a mapping of normed vectorspaces.

Passivity: Define the scalar inner product <.|.> of twoinfinite sequences x and y as

An operator H: x-> y where x, y e L2e is passive only ifthere exists some constant /? such that

Conic sector: An operator H: x-* y where x, y e L2e is:

(a) inside the cone (C, R) if

<y - (C - R)x | y - (C + R)x>T ^ 0 VTeZ +

(b) outside the cone (C, #) if

<y-(C-R)x\y-(C + R)x}T^0 VTeZ +

(c) strictly inside the cone (C, R) if for some £ > 0

<y-(C-R)x\y-(C + R)x>T

<-e\\(x,y)\\l VTEZ+

(d) strictly outside the cone (C, R) if for some e > 0

<y-(C-R)x\y-(C + R)x>T

>e\\(x,y)\\2 VTeZ+

where \\(x, y)\\2 = (\\x\\2 + \\y\\2T)

Dissipativeness: An operator H:x-* y where x, y e L2e isweakly (Q, S, R) dissipative only if there exists a constantj? such that

r + <« | Ru}T + P>0 VTeZ +

With fl = 0, H is called dissipative.The following theorem is a necessary extension to the

conic sector stability of Safanov [6] for discrete-timeadaptive control systems.

Theorem 2.1: Consider the following feedback system

e(k) = u(k) - x{k)

y(k) = Hxe{k)

x(k) = H2y(k) (1)

with Hu H2: L2e-> L2e and x(k), y(k), e(k) e L2e andu(k) € L2.

If(a) Hl: e(k) -*• y(k) satisfies

Z ly(k)2 + ae(k)y(k) + /fe(fc)2] & -y (2)

(b) H2: y{k) -* x(k) satisfies

Uix(k)2 - <xx(k)y(k)k = 0

^-Mm, mm (3)for some a, j5, y, rj, eR and y, r\ > 0, then the closed loopsignals x{k), y(k) e L2.

Proof: From eqn. 2 and using e(k) = u(k) — x(k)

£ [fiMJc)2 - <xx(k)y(k) •fc = O

- 2f}u(k)x(k) + )?u(/c)2] > - y (4)

Combining eqns. 3 and 4

Z tuJc = O

-y (5)

Using the Schwartz inequality

ri\\(x(k),y(k))\\2-\a\- \\u(k)\\ • \\y(k)\\N

-2\fS\- \\u(k)\\ • | |x(fc) | |N^y + | ^ | - \\u(k)\\2 (6)

Assume \\(x(k), y(k))\\ j , -*• oo as N -> oo. Therefore fromeqn. 6

This is a contradiction. Therefore \\(x(k), y{k))\\^ isbounded (i.e. x(k), y(k) e L2).

A corollary of this theorem is presented which will beuseful in the analysis of certain adaptive control systems.

Corollary 2.1: Consider the feedback system of eqn. 1. If(a) Hl: e(k) -* y(k) satisfies

Z le{k)y{k) + de(k)2/2-] > -y<t = 0

(b) H2: y(k) -> x{k) satisfies

[ox{k)2l2 - x -r,\\(x(k), y(k))\\2

(7)

(8)

for some a, y, rj > 0, then the closed loop signals x(k),y(k)eL2.

Proof: The proof follows the approach taken forTheorem 2.1.

3 Problem formulation

In this Section, a generic direct (or implicit) discrete-timeadaptive controller is presented. The algorithm is thencast into an error feedback system which is required forapplication of the input-output theorems presented inSection 2.

Process representation:

where A and B are polynomials in the backward shiftoperator q~l, d is the delay and u, y, £ represent theinput, output and disturbance sequences.

Control law:

yr(k + d) = S(k)u(k) + R(k)y(k) (10)

where S and R are polynomials in q ~x of degrees ns andnr with time-varying coefficients, and yr is the referencesignal, known d steps ahead. This control law may alsobe written in vector notation as

with

134

(11)

= M k ) . . . u ( k - ns), y(k) . . . y ( k - n r ) ] (12)

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

Parameter estimation:

6(k) = 0(k -d) + P(k)<f>(k - d)e{k)

eik) = y(k) - yr(k)

where P(k) represents the gain of the estimator.

yr (k+d)

(13)

(14)

Dl Lawu(k)

ParameterEstimator

Processy(k)

Reformulation into an error feedback system:Define a set of tuned parameters

0*' = [S* £*] (15)

Define the polynomial C in terms of the tuned para-meters

q~dR*B (16)

The process (eqn. 9) may now be expressed in terms ofthe tuned parameters as

Cy(k) = B[S*u(k - d) + R*y(k - d)~] + S*£(k) (17)

y(k) = H2 . 0*'<J>(fc - d) + C-lS*£(k) (18)

where H2 = C~lB. From eqn. 18, H2 represents thetransfer function relating yr(k) to y(k) with a fixed controllaw (eqn. 10), i.e. 0(k) = 0*. The error feedback systemmay now be written as

e(k) = y(k) - yr(k)

= -H2[0{k -d)- 0*y<f>(k - d)

(19)

where

= [<§(k -d)- O*y<f>(k - d)

= 0{k - d)'^k - d)

(20)

(21)

(22)

(23)

e(k)*

Hx: e(k) -*• ^(k) denotes an operator defined by the par-ticular parameter adaptation algorithm. The adaptivecontrol problem may now be analysed as a feedbacksystem using input-outout stability theory.

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

4 Nonzero initial conditions

In the stability analyses of References 4 and 5 input-output (I/O) properties are derived for the H Y operator inthe error feedback system based on a least squaresparameter adaptation algorithm. This section shows thatthis operator satisfies a weakly dissipative condition asopposed to an exterior conic or 'outside the cone' condi-tion due to the presence of nonzero errors in the initialparameter estimates.

Consider the following update mechanism for theparameter estimates

6(k) = 0(k-d) + P(k)<f>(k - d)e(k) (24)

For a least squares type of algorithm, P(k) is a time-varying matrix adjusted according to

Pik)'1 = pP{k - d)~1 + (f>{k - d)<f>(k - d)' (25)

where j3 (0 < /? ̂ 1) is a forgetting factor. A usefulrelationship between P(k) and P(k — d) is presented inlemma 4.1.

Lemma 4.1: If P{k)~l is updated according to eqn. 25then

(f>(k - d)tP(k)(f>(k -d) =- dfPjk - d)<t>{k - d)

(k- d)'P(k - d)<t>(k - d)

(26)

Proof: P(k) can be found by applying the matrix inver-sion Lemma to eqn. 25 followed by post and premultipli-cation by (f>(k — d).

Both Gawthrop [4] and Ortega, Praly and Landau[5] in their analyses of similar parameter adaptationschemes concluded that the operator Hx: e(k) -*• ¥(&)defined by eqns. 24 and 25 is outside the cone (—1,(1 - OLSY12) where

<f>(k - dfP{k - d)<t>{k - d)

- dfP{k - d)<f>{k - d)(27)

However from the definition of a sector, it may be shownthat Hx is only outside the cone (—1, (1 — ffLS)

l/2) if theinitial parameter errors are equal to zero (i.e. 6(k) = 0Vfc < 0). In principle, this is possible if the elements of thevector 0* in eqn. 15 may be considered arbitrary.However, both Reference 4 and Reference 5 require that0* have specific properties (i.e. equal to the true processparameters or to some vector of stabilising parameters,respectively). As a result, the initial parameter errors are,in general, nonzero and hence Hl does not satisfy theexterior conic condition. In Lemma 4.2 it is shown thatthe operator Hx is weakly dissipative.

Lemma 4.2: The operator Hx: e(/c)-> ^(k) defined byeqns. 24 and 25 is weakly (1,1, aLS) dissipative for all aLS

satisfying eqn. 27.

(28)

(29)

Proof: Consider the quadratic function

V(k) = 0(k)tP(k)-10(k)

= [6(k -d) + P{k)<t>{k - d)e(k)y •

[_6(k -d) + P{k)<f>(k - d)e(m

By replacing P(k)~l by eqn. 25 gives

V(k) = pV{k-d) + V{k)2 + 2V(k)e{k)

+ <j>{k - d)'P(k)4>(k - d)e(k)2 (30)

135

Summing from 0 to AT

LV(k) -V(k-d) - dj\ =k = 0 k = 0

- d)e(k)2] (31)

Since (1 — ft) 3* 0 by definition and using the result oflemma 4.1, it follows that

+ TV{k)e{k) + aLSe{k)2l > - £ V(k) (32)k=-l

Theorem 2.1 is an L2 stability result which accommo-dates the I/O properties of the parameter adaptationalgorithm presented in eqns. 24 and 25, i.e. from lemma4.2, Hx satisfies (a) of theorem 2.1 with a = 2 and /? =aLS. According to Safanov [6], if condition (b) ofTheorem 2.1 is satisfied then H2

l is strictly inside thecone (Cl9 Rj) where Cx = 1 and Rt = (1 — aLS)

112 whichis equivalent to H2 being strictly inside the cone (C2, R2)where C2 = ols a n d R2 = aEsO- ~ aLsY12-

Remark 4.1: The error feedback systems developed inReferences 4 and 5 are similar to eqn. 1 and the conditionon H2 in theorem 2.1 is identical to the condition on thecorresponding H2 in their L2 stability results. Therefore,it may be concluded that rigorous treatment of thenonzero initial conditions of the parameter adaptationalgorithms does not change the main results in Reference4 or 5.

5 The vanishing radius problem

This section describes the vanishing radius problemwhere the radius of the allowable cone for the H2 oper-ator approaches zero. One approach to avoiding thisproblem is through the use of a normalised regressorvector in the parameter adaptation algorithm whichenables the user to select the radius of the H2 cone.

In References 1, 2 and 5 the operator H2 containsinformation on the unmodelled dynamics. For the casewhere no unmodelled dynamics are present (i.e. themodel order is equivalent to the process order) H2

reduces to a scalar. For example, H2 = 1 when nounmodelled dynamics are present, in Reference 5. FromReference 6, H2 = 1 is strictly inside any cone (C2, R2) if

'LS < 1 (33)

Gawthrop [4] assumed a priori that the regressor <j> isbounded, in order to show that there exists some valuefor aLS which satisfies eqn. 33. In Reference 5 it is statedthat this assumption of a bounded regressor leaves theresults of References 1, 2 and 4 incomplete. The difficultymay be seen in eqn. 27 where P is a positive definitematrix and it appears that an unbounded <f> prevents sel-ection of a value for aLS which is strictly less than unity.Gawthrop [7] has subsequently pointed out that the reg-ressor does not need to be assumed bounded but in fact amuch weaker condition on <j> is required.

The behaviour of the allowable cone for H2 as aLS -*• 1is referred to as the vanishing radius problem, i.e. theconic region is reduced to a cone centered at unity with aradius approaching zero. This problem arises in much ofthe early literature dealing with the robustness problemin discrete-time based on conic sector theory [1, 2, 8, 9].In more recent work [5] the vanishing radius problemhas been addressed. Certainly one means of ensuring anonzero radius for the H2 cone is to use a vector <J> in the

adaptation scheme which remains bounded. Ortega,Praly and Landau [5] considered normalising the vector<j> for this reason. The normalisation factor selected inReference 5 was first introduced by Egardt [10] and is ofthe form

p(k) = w{k - 1) + max (I <|>(/c - d) |2, p)

p > 0, ti G (0, 1) (34)

and the normalised regressor has been defined as

r(k-d) = p(kri'2<Kk-d) (35)

It may be easily shown that | <£n(fc - d) \2 < 1 Vfc, i.e.

\<t>{k-d)\2

\<t>"(k-d)\2 = < 1

(36)

In Reference 5 a normalised tracking error and a normal-ised *P(/c) have also been defined as

en(k) = p(kyll2e(k) (37)

The parameter adaptation scheme in eqns. 24 and 25 isnow expressed in terms of the normalised variables.

6(k) = 6(k-d) + P{k)<f>n(k - d)e\k) (39)

P{k)~l = pP(k -d)'1 + (f>n{k - d)<f>n{k - df (40)

Consider the above algorithm with f$ = 1. From eqn. 40it follows that XmaxP(k) ^ XmaxP(k - d). Therefore if Xx =XmaxP(0), where P(0) is the initial value for the time-varying matrix P(k) as selected by the user, thenXmaxP(k) ^ X^k.

Lemma 5.1: The operator H\: en(k) -+ ¥n(/c) defined byeqns 39 and 40 with /? = 1 is weakly (1,1, aLS) dissipativefor all oLS satisfying

oLS > XJ{\ + AJ (41)

Proof: The proof follows the approach taken for lemma4.2.

Remark 5.1: The value for Xx is selected by the user. Aslong as k1 is finite aLS may be selected strictly less thanunity according to eqn. 41. Therefore the vanishingradius problem is overcome by normalisation.

6 Choice of the normalisation factor

The use of normalisation gives rise to the problem ofensuring stability of the unnormalised signals. Thissection gives an interpretation of the role of Goodwin'skey technical lemma (KTL) in solving this problem forthe case when no unmodelled dynamics are present. Alsosome insight into Ortega's choice of normalisation factoris provided.

Although the use of the normalised variables in theparameter adaptation algorithm solves the vanishingradius problem, it has created another problem. Theorem2.1, under the normalised system, only guarantees L2 sta-bility of the normalised variables. Stability of the normal-ised signals (e.g. e"(k)) does not necessarily imply stabilityof the unnormalised signals (e.g. e(k)) which is what is ofinterest. This problem was solved in Reference 5 for aparticular choice of normalisation factor (eqn. 34) usingthe Bellman-Gronwall lemma. However, this problem

136 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

was first solved by Goodwin, Ramadge and Caines [11]for a different choice of normalisation factor using theirkey technical lemma.

6.1 Role of the KTLThe result presented in Corollary 2.1 and the KTL arecombined in this Section to provide an alternative proofof stability for projection algorithm 1 of Reference 11. Indoing so, it is demonstrated that the role of the KTL is toenable stability of the unnormalised signals to be derivedfrom the stability of the normalised signals.

For projection algorithm 1 the problem formulation isas in Section 3 with £(k) = 0 and the parameter estimatorgiven by

0(k) = 0(k -a{k)<t>{k - d)e{k)

\+<f>{k- df<j>{k - d)(42)

where a{k) e (0, 2). Define the quadratic function V(k) =0(k)'0(k)/2. Let the normalisation factor be selected as

p{k) = a(k)~ l(l + <j>(k - df<t>{k - d)) (43)

Substituting eqn. 42 into V(k) gives

V(k) -V{k-d) = p(kylx¥(k)e(k) + p(ky2

x <f>(k - df<t>{k - d)e(k)2/2 (44)

Define a(k) as

a{k) = p(k) ~l<l>(k- df<j>{k - d) (45)

and let the normalised signals e"(k) and ¥"(&) be definedas in eqns. 37 and 38 with p(k) given by eqn. 43. There-fore eqn. 44 may be rewritten as

a(k)en(k)2/2 (46)V{k) -V(k-d) =

Summing eqn. 46 from 0 to N gives

£ V¥n(k)e"(k) + a{k)e\kfll-] > - £ V(k)k=O k=-l

(47)

Therefore H\: e\k) -• ¥"(£) as defined by eqn. 42 satisfiesproperty (a) of Corollary 2.1 with a > a(k) V/c. In the casewhere there are no unmodelled dynamics there exists a 0*such that

B = AS* + q-"R*B (48)

which implies that H2 = 1 and e(k)* = 0. Therefore pro-jection algorithm 1 may be cast into the feedback systemof eqn. 1 in terms of the normalised variables, i.e.

¥n(/c) = H\e"{k)

en(k)= -Hn2V

(49)

(50)

with H" satisfying eqn. 47 and H2 = H"2 = 1. L2 stabilityof e"(k) and ¥"(&) is guaranteed if H2 satisfies condition(b) of Corollary 2.1 (eqn. 8). Eqn. 8 may be rearranged as

- (2/a)x(k)y(k)-] ^ - (51)

where a > 0 and rj' = 2rj/a. According to Reference 6,H2- y{k) -*• x(k) is strictly inside the cone {&~l, a~l) if thecondition (eqn. 51) is satisfied. From eqns. 43 and 45

- df<t>(k - d)\+<f>(k- df4>{k - d)

< a(k) < 2 (since a(k) e (0, 2)) (52)

Therefore it is possible to select a a such that<j(k) <a<2 and then L2 stability of eqns. 49 and 50

follows becuase H"2 = 1 is strictly inside the cone {aa'1). L2 stability of the normalised signals implies that

e"(k)2 =e(k)2

p(k) -d)

-> 0 as k -*• oo (53)

The KTL from Reference 11 may now be used in con-junction with eqn. 53 and a stable-inverse plant assump-tion to prove that {||#(fc)||} is bounded and e(k) e L2 (i.e.stability of the unnormalised error signal).

Remark 6.1: One of the conditions in the KTL is theuniform boundedness condition (UBC), i.e. for a generalp(k) of the form

\<Kk-d)\: (54)

byik) and b2(k) must be finite. For example, since b^k) =b2(k) < oo in eqn. 43, the UBC is satisfied for this nor-malisation factor. However for p(k) as selected in Refer-ence 5

b^k) = np(k — 1) (55)

In this case, the UBC is not satisfied (i.e. p{k) is boundedonly if <f> is assumed bounded) and hence the KTLcannot be used in Reference 5 to prove that e(k) e L2.

6.2 Extension to the mismatch caseOrtega, Praly and Landau [5] have extended the resultsof Section 6.1 to include process-model order mismatchand bounded disturbances to the plant output. This is theformulation presented in Section 3. In Reference 5 a sta-bilisability assumption on 0* is introduced, i.e. thereexists a nonempty set 0LS such that

0LS = [0*: C(q) # 0, V | q \ > fi1'2} # 0 (56)

where /x e (0, 1). This set 0LS defines the fixed parametercontrollers which produce closed loop poles within a diskof radius /i1/2.

The I/O properties of H\ are derived in Reference 5for two parameter adaptation algorithms: 1) a constantgain (CG) algorithm, and 2) a regularised least squares(RLS) algorithm. The CG scheme is of the form

= 0(k - d) +fP(k - d)e"(k) feR, / > 0 (57)

If eqns. 35 and 37 are substituted into eqn. 57, then

6(k) = 6(k-d)f<Kk - d)e(k)

(58)

which differs from the projection algorithm used in Refer-ence 11 where

= 0(k-d) a{k)<t>{k - d)e{k)(59)

The RLS scheme in Reference 5 is given by

= 0{k -d) + P{k)4>n{k - d)e\k) (60)

P(k) = (1 - A0Mi

P(k - d)<f>"(k - d)<f>n(k - dfPjk - d)

X + <}>n{k - d)'P(k - d)4>\k - d)X0I (61)

where XQ < ) n , X are strictly positive scalars and theeigenvalues of P(k) are contained in the interval [Ao, X{].The L2 stability analysis of the RLS scheme may goalong without the regularisation (i.e. Xo = 0) which

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988 137

reduces the P(k) update to

P(k) = P(k-d)-P(k - d)<f>n(k - d)<f>n{k - dfPjk - d)

X + <t>n{k - d)'P(k - d)<f>"{k - d)

(62)

It is proven in Reference 5 that for the CG algorithmdefined by eqn. 57, (//] + (TCG/2) is passive for all aCG > f.The proof follows exactly the analysis required to deriveeqn. 47 for the projection algorithm. The operator H\defined by eqns. 60 and 62 is weakly (1, X, X2 • cRLS) dis-ipative for aRLS satisfying

(63)

where Xx = XmaxP(0).

Remark 6.2: Normalised variables were introduced in theleast squares algorithm to avoid the vanishing radiusproblem. However several variations of the normalisationfactor in eqn. 34 (e.g. eqn. 43) would also be suitable, i.e.the only requirement of the normalisation factor is thatthe norm of <j>n in eqn. 35 be bounded. This certainlyraises the question as to why the normalisation factor ineqn. 34 was selected by the authors of Reference 5.

6.3 Some insight into Ortega's choiceAn important result in Reference 5 is the derivation of aconicity condition on H2 which ensures that the conicitycondition on H"2 is satisfied. This result is importantbecause conditions on H2 in eqn. 18 are more useful tothe designer than conditions on H"2. (For an applicationof the results in Reference 5 see Reference 12.) It is some-what hidden in the derivation of this result why othernormalisation factors such as eqn. 43 are not as suitablefor this conic sector type of stability analysis as the oneselected in Reference 5.

In general, H2 ̂ H2 where H2: x(k) -> y(k) andH"2: x"{k) -*• y"(k) with the normalised signals defined as ineqn. 37. The implications of this are that conditions onH2 from the L2 stability analysis of the normalised errorsystem cannot be interpreted directly as conditions onH2. However a special case is when H2 = c, where c is ascalar. Then

y(k) = c • x(k)

y"(k) = H"2 • xn(k)

Multiplying eqn. 64 through by p(k)~112

yn(k) = c • xn(k)

gives

(64)

(65)

(66)

From comparing eqns. 65 and 66 it may be concludedthat Hn

2 = H2 = c. However, this is only the case whenH2 contains no dynamics. From the definition of H2 inSection 3 it is clear that H2 is, in general, a rational func-tion in the backward shift operator q~l. For example, let

Therefore

y(k) = x(k) + y(k - 1)

Now assume that H2 = H2. Therefore

/(/c) = xn{k) + y"(k - 1)

Multiplying eqn. 69 through by p(k)1'2 gives

y(k) = x(k) + p(kY'2p(k - ly^yik - 1)

(67)

(68)

(69)

(70)

eqns. 68 and 70 are equivalent if p(k) = p{k — 1).However, from inspection of eqn. 34 it may be seen that

138

this is not true in general and therefore H2^ H2. In Ref-erence 5 this difficulty was resolved by deriving conicconditions on H2 which ensure the conditions on H2.This derivation is presented here to illustrate moreclearly the role of the normalisation factor.

Lemma 6.1 (see also Lemma 5.1 in Reference 5): Considerthe operator H: x(k) -• y(k). If H[(jill2

qyl] is inside the

cone (C, R), then H": xn(k) -• y"(k) with the normalisedvariables defined using eqn. 34 is inside the same cone (C,R).

Proof: Define

W(k) = (y(k) - Cx(k))2 - (Rx(k))2

Wl(k) = (y"(k) - Cx\k))2

- (Rx"(k))2 = p(fc)-yfe)

Taking the sum from 0 to N of w^k) gives

(71)

(72)

(73)

N

Ej=o

• - k w(k)

(74)

which may be verified by straightforward expansion.Multiplying eqn. 34 through by /i~(fc + 1) gives

-d)\2, p) (75)

Therefore, n~(k + l)p(k) is nondecreasing (or nk + 1p(k)~1 isnonincreasing) and hence in eqn. 74

/ * / + 1 p O + i r 1 - M / P O r 1 < 0 Y/ (76)

Now express H: x(k) -*• y(k) as a general rational functionin q'1, i.e.

••• + anq(77)

bo + b1q-1 + --- + bmq-m

Therefore x(k) and y(k) are related as follows

boy(k) + ••• + bmy(k - m) = aox(k) + • • • + anx(k - ri)

(78)

Multiplying eqn. 78 through by n~k'2 gives

= aofi-kl2x(k) + •••+ anfi-

k'2x(k - n) (79)

Define

x'(k) = n~kl2x{k) (80)

y'(k) = >i-kl2y(k) (81)

Substituting eqns. 80 and 81 into eqn. 79 allows us towrite

x y\k -m) = a0x'(k) +••• + ann-"l2x'(k - ri) (82)

which may be expressed as

Hln1i2q)-^x\k) (83)

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Therefore i 1] is inside the cone (C, R), then

Z [(/(*) - Cx'(/c))2 - (Rx'(fe))2]

which implies that

Z ti~kw(k) ^ 0

(84)

(85)

From combining eqns. 76 and 85 with eqn. 74, it may beconcluded that

N

E>fc = O

0 (86)

Therefore Hn is also inside the cone (C, R).Specific properties of the selected normalisation factor

come into the proof of Lemma 6.1 only at eqn. 76. Let usselect a normalisation factor similar to eqn. 43 and see ifit also satisfies a condition similar to eqn. 76. Let

p(k) = a(k)~1(l + | <j>(k — d)\2) (87)

Since fi no longer appears in this factor the summation ineqn. 73 is written without \i as

(88)

Zfc = O

- z | Y z w(/c)j = ol_\* = o

' - P O T 1 ) ] (89)

For the choice of p(k) in eqn. 87

PU + D"1 " PUT1 = a(j + 1)(1 + \#j - d + I ) ! 2 ) ' 1

/-<*) I2)"1 (90)

The left hand side of eqn. 90 is less than or equal to zeroif

1*0-\4>{j-d)Y

(91)

If eqn. 91 is satisfied and Z*=o w(k) ^ 0, thenZ*=o w^k) ^ 0 and therefore H inside the cone (C, R)would imply H" inside the same cone. (A similar analysiswas performed in Reference 8 for a generalised leastmean squares algorithm.) This approach does give someguidelines for selecting the parameter a(k). Howeverensuring an inequality such as expr. 91 may violate otherrestrictions on a\k) (e.g. a(k) e (0, 2)).

Remark 6.3: Some motivation for using a normalisationfactor of the form in eqn. 34 over eqn. 87 is now clearer.From the proof of Lemma 6.1, expr. 76 is guaranteedwithout any effect on the choice of design parameterssuch as adaptation gain, i.e. any choice of \i e (0, 1) willlead to eqn. 76. On the other hand, ensuring eqn. 91would require careful selection of a(k) at each samplinginstant.

7 Extension to a L-inf inity result

This section describes the techniques and some of the dif-ficulties associated with extending the L2 analysis ofTheorem 2.1 to the L^ case. In Section 7.1, the I/Oproperties of the operator based on a least squares algo-rithm are presented using exponentially weighted signals.Section 7.2 illustrates the importance of using exponen-tially weighted signals in achieving an L^ result. Section7.3 shows that the constant gain algorithm does not leadto a stability result for the Lx case.

All of the previous results discussed in this paper havebeen based on the L2 stability result of Theorem 2.1. Forthe error system developed in Reference 5 the L2 stabilitytheorem restricts e(k)* e L2 which in general requires yrand £ to eventually decay to zero (see eqn. 23). The morepractical case is to consider bounded reference signalsand disturbances (i.e. yr, £ e Lffl).

Both Reference 1 and Reference 5 considered exponen-tially weighted signals for their respective L^ analyses.For instance, the exponentially weighted counterpart ofx(k) may be defined as x(k) where

x(k) = ak • x(k) a > 0 (92)

The next step is to derive I/O properties of the parameteradaptation schemes based on these exponentiallyweighted signals.

7.1 Least squares algorithm

Lemma 7.1: Consider the operator H\: e\k) — *¥n(k)defined by eqns. 39 and_ 40. Define its exponentiallyweighted counterpart as HI: e\k) -> ¥"(£). H\ is weakly(1,1, trLS) dissipative for <rLS satisfying

*i) (93)

where X^ = /lmax P(k) Vfc if (1 - pa2d) ^ 0.

Proof: The proof follows the same approach as that inLemma 4.2.

As pointed out in Reference 1, if ^ in eqn. 40 is chosenstrictly less than unity then there exists an a > 1 suchthat (1 - p<x2d) ^ 0 . If a < 1 then (1 - p2d) ^ 0 for0 < jS ^ 1. However, if /? < 1, then some difficulty ariseswith guaranteeing boundedness of Xu the maximumeigenvalue of P(k). From eqn. 40, it is clear that when theplant is at steady-state (i.e. <l>n(k — d) ca 0) the eigenvaluesof P(k) will be unbounded (i.e. A1 -*• oo) which will cause avanishing radius (i.e. <rLS -* 1). In order to prevent a van-ishing radius, the algorithm must be modified in such away as to impose limits on the eigenvalues of P(k). Thismay be accomplished via a covariance resetting feature,based on a check of the eigenvalues of P(k) at each sam-pling instant. For instance, if at some instant k, Xmax P(k)exceeds some designer selected bound, ku then P{k) wouldbe reset such that Amax P(k) < Xu. This feature remains tobe formally included in the derivation of the I/O proper-ties of H\ in Lemma 7.1.

Another way to avoid the vanishing radius problem inthis case is to use a least squares algorithm which forcesan upper bound on Xv For example the RLS scheme ineqns. 60 and 61 ensures that the eigenvalues of P{k) arecontained in the user selected interval \_X0, AJ.

7.2 L-inf inity extensionThe L^ extension of the L2 results in References 1 and 5are rederived to emphasize the role of the exponentiallyweighted signals in obtaining a Lw result.

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988 139

Lemma 7.2 (see also lemma 4.2 in Reference 5): Considerthe error feedback system defined in Section 3 with theleast squares adaptation algorithm presented in eqns. 39and 40. If Hn

2 = akH2a~k is strictly inside the coneC^LS1 , ^Ls'a - <7LS)1/2) with a > 1 and (1 - 0a2d) > 0,then Vik) e L^.

Proof: L2 stability of map e*(k)* -* ^"(k) is guaranteedfrom lemma 7.1 and theorem 2.1, i.e. there exists aK < co such that

K

By definition

\\V\k)\\2N =

fc =

k = 0

*N\iin( \T\\2(aNx¥"(N))2 (last term in series)

and

\\en(k)*\\2N =

= Y(aken(k)*

\\e"(kyAT

Z«2

Since a >N 2N

a = "I ~-2 ^ 1 " ^k=o 1 — a 1 — a

(94)

(95)

(96)

(97)

(98)

(99)

From combining eqns. 94, 96, 98 and 99, it may be con-cluded that

*¥"(N)K

\*2| (100)

where p ^ p(fc) Vfe from eqn. 34.The importance of a being greater than unity in the

proof of lemma 7.2 is evident from eqn. 99. If a ^ 1, thenthe upper bound in eqn. 100 is no longer valid, i.e.

OL — a-2 ,2JV

1 - a " 2 ^ 1 - a " 2

For this case, combining eqns. 94, 96, 98 and 101 gives

K (\-<x2N+1

P

(101)

(102)

Hk)*2\\

and multiplying eqn. 102 through by a 2N results in

^ P V 1 - a2

In this case with a < 1

<x~2N -* oo as N -+ oo

(103)

(104)

Therefore the upper bound on ^"(k) in eqn. 103 isunbounded and hence it cannot be concluded that

Remark 7.1: Exponential weighting allows for the deriva-tion of a L^ result from the L2 result in theorem 2.1.However the value of a must be selected strictly greaterthan unity in order to achieve this result.

7.3 Constant gain algorithmOrtega, Praly and Landau [5] did not carry out their L^analysis on the constant gain algorithm in eqn. 57. In thefollowing lemma, the I/O properties of the exponentiallyweighted operator H\ are derived for the constant gainalgorithm.

Lemma 7.3: Consider the operator H\: en(k) -• ^"(k)defined by eqn._ 57. Define its exponentially weightedcounterpart as H\: en(k) -• Tn(fc). (H\ + aCG/2) is passiveforal l<7C G>/ifa2 d2d

Proof: The proof follows the analysis required to deriveeqn. 47 for the projection algorithm.

If a2d < 1 then a < 1. The next step would be to con-tinue with the L^ extension as done in lemma 7.2 for theleast squares algorithm. However as demonstrated in thediscussion immediately following lemma 7.2, if a < 1 itcannot be shown that ¥"(£) e L^ and the L^ stabilityresult does not follow.

Remark 7.2: The constant gain algorithm does notqualify for the Lw extension of lemma 7.2.

8 A robust control law

This section describes an augmented plant representationwhich enables control weighting to be incorporated intothe adaptive control law.

The majority of the discussion to this point in thispaper has dealt with aspects of the parameter adaptationalgorithms. The other key element of an adaptive controlsystem is the control law itself. One of the important fea-tures in the work of Reference 1 is that the robustnessanalysis incorporates weighting polynomials in thecontrol law which are characteristic of the Clarke andGawthrop [3] self-tuning controller. The objective of thissection is to extend the controller structure presented inSection 3 to include the same weighting polynomials con-sidered in Reference 1.

Assume that the plant may be described by eqn. 9 (set£(fc) = 0 for clarity). Define an augmented plant withoutput z, and input u, where

z(k) = P(q-l)y(k) + q-dQ(q-l)u(k) (105)

and P and Q are weighting polynomials in the delayoperator q~1. Substituting eqn. 9 into eqn. 105 gives

Az(k) = q~\PB + QA)u(k) (106)

and by defining B' = PB + QA, eqn. 106 becomes

Az{k) = q-dB'u{k) (107)

The tracking error may now be defined as

e(k) = z(k) - yJLk) (108)

The regulator structure for the augmented system isbased on the same control law in eqn. 10 and is given by

yr(k + d) = S{k)u(k) + R(k)z(k)

= 0\k)'<Kk) (109)

The polynomial C is defined in terms of tuned para-meters 0*' = IS* i?*] where

C = AS* + q~dR*B' (110)

Adding q~dR*B'z(k) to both sides of eqn. 107 multipliedby S* gives

AS*z(k) + q-dR*B'z(k)

= B'S*u(k -d) + B'R*z(k - d) (111)

140 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

Substituting eqn. 110 into eqn. I l l and rearranging

z(k) = C~lB' • [S*u(k -d) + R*z(k - d)~] (112)

= H2- 0*'<f>(k-d) (113)

where H2 = C~x B'. From eqn. 113, an error feedbacksystem based on this augmented system may be derived.

= z(k)-yr(k)

+ H2yr(k)-yr(k) (114)

= -H2{6(k - df4>{k -d)- 0*f<i>(k - d))

+ (H2-l)yr(k) (115)

e{k) = -H2V(k) + e(k)* (116)

and

^(k) = Hie(k) (117)

where e{k)* = (H2 - l)yr(k) and V(k) is defined in eqn. 22.

Remark 8.1: Eqns. 116 and 117 now define an errorsystem which extends the adaptive controller to include Pand Q weighting polynomials. The stability analysisbased on Theorem 2.1 may now proceed from this point,where the adaptation algorithms discussed in the earliersections would now be used to estimate the controllerparameters in eqn. 109.

9 Concluding remarks

(a) The early stability results [1, 2, 4] have used aboundedness assumption to avoid a vanishing radius.Although this assumption is not necessary in their work,the concept of normalisation introduced in Reference 5gives the designer the ability to determine the size of theallowable cone (and hence the robustness of the adaptivecontroller) and removes any of its dependence on thebehaviour of the regressor sequence.

(b) The normalisation factor used in Reference 5 con-tains an additional design parameter \i. The authors ofReference 5 point out that the choice of /J, is a compro-mise between alertness of the parameter estimator androbustness of the adaptive controller. For instance, alarge value of n(n -*• 1) is desirable from a robustnessstandpoint because this leads to a smaller H2 (see Lemma6.1). However, the same large value for n makes the nor-malisation factor p(k) in eqn. 34, and hence the adapta-tion scheme, very 'sluggish' to changes in the regressorsequence. (The connection between size of H2, its allow-able cone, and robustness is made clear in Reference 12.)

(c) A great many algorithms exist for estimation of thecontroller parameters. However, most schemes arevariants of the two algorithms examined in this paper;the constant gain and the least squares. The L2 stabilityresults based on Theorem 2.1 lead to the following allow-able cones for H2:

CG (eqn. 57): cone (a"1, a"1), a > f

LS (eqns. 39, 40): cone (p~\ a~\l- a)112),

In order to improve robustness to unmodelled dynamicsit is desirable to have a 'large' allowable cone. This inturn means that the adaptation gain of the respectiveparameter estimation schemes ( / o r Xt = lmaxP(k)) mustbe kept small. It is well known that least squares schemes

provide faster convergence than constant gain schemes.However a large value for Al should be avoided whenusing least squares in the presence of unmodelleddynamics because this would lead to a small allowableH2 cone and hence a reduction in robustness.

id) This paper has discussed only algorithms with 'con-tinuous' adaptation, i.e. the controller parameters areupdated each and every sampling instant k. Cluett et al.[13], Goodwin et al. [14] and Kreisselmeier and Ander-son [15] have combined the concept of normalisationwith a dead zone mechanism which turns parameteradaptation on or off based on the magnitude of a nor-malised error signal. Praly [16] combined normalisationwith a projection which keeps the parameters within adesigner chosen sphere. Ioannou and Tsakalis [17] havealso used a normalising signal with a modified parameteradaptation law which employs a switching-^ term. Otherthan Praly's work [16], which is based on conic sectorconditions, the results in References 13-15, and 17 makeuse of somewhat less conventional stability theory. Theadvantage of using the conic sector theory discussed inthis paper is that the sector conditions have explicit fre-quency domain circle-type interpretations as explainedby Zames [18] and Safanov [6] and as illustrated in Ref-erences 1, 2 and 12. Therefore further work might pursueincorporating some of these other modifications such asdead zones and the a term into the conic sector frame-work.

10 References

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2 GAWTHROP, P.J., and LIM, K.W.: 'Robustness of self-tuning con-trollers', IEE Proc. D Control Theory & Appl., 1982,129, pp. 21-29

3 CLARKE, D.W., and GAWTHROP, P.J.: 'Self-tuning controller',Proc. IEE, 1975, 122, pp. 929-934

4 GAWTHROP, P.J.: 'On the stability and convergence of a self-tuning controller', Int. J. Control, 1980, 31, pp. 973-998

5 ORTEGA, R., PRALY, L., and LANDAU, I.D.: 'Robustness ofdiscrete-time direct adaptive controllers', IEEE Trans., 1985, AC-30,pp. 1179-1187

6 SAFANOV, M.G.: 'Stability and robustness of multivariable feed-back systems' (MIT Press, 1980)

7 GAWTHROP, P.J.: Private correspondence, April, 19868 ORTEGA, R., and LANDAU, I.D.: 'On the model-process mis-

match tolerance of various parameter adaptation algorithms indirect control schemes: a sectoricity approach', IF AC WorkshopAdapt. Syst., 1983, San Francisco, USA

9 ORTEGA, R., and LANDAU, I.D.: 'On the design of robustly per-forming adaptive controllers for partially modelled systems', Proc.22nd IEEE CDC, 1983, pp. 967-971

10 EGARDT, B.: 'Stability of adaptive controllers' (Springer-Verlag,1979)

11 GOODWIN, G.C., RAMADGE, P.J., and CAINES, P.E.: 'Discrete-time multivariable adaptive control', IEEE Trans., 1980, AC-25, pp.449-456

12 CLUETT, W.R., SHAH, S.L., and FISHER, D.G.: 'Robust design ofadaptive control systems using conic sector theory', Automatica,1987, 23, pp. 221-224

13 CLUETT, W.R., MARTIN-SANCHEZ, J.M., SHAH, S.L., andFISHER, D.G.: 'Stable discrete-time adaptive control in the pre-sence of unmodeled dynamics', IEEE Trans., 1988, AC-33

14 GOODWIN, G.C., HILL, D.J., MAYNE, D.Q., and MIDDLE-TON, R.H.: 'Adaptive robust control (convergence, stability andperformance)', Proc. 25th IEEE CDC, 1986, pp. 468-473

15 KREISSELMEIER, G., and ANDERSON, B.D.O.: 'Robust modelreference adaptive control', IEEE Trans., 1986, AC-31, pp. 127-133

16 PRALY, L.: 'Robust model reference adaptive control, part 1: sta-bility analysis', Proc. 23rd IEEE CDC, 1984, pp. 1009-1014

17 IOANNOU, P.A., and TSAKALIS, K.: 'A robust discrete-timeadaptive controller', Proc. 25th IEEE CDC, 1986, pp. 838-843

18 ZAMES, G.: 'On the input-output stability of time-varying nonlin-ear feedback systems — part II: conditions involving circles in thefrequency plane and sector nonlinearities', IEEE Trans., 1966,AC-11, pp. 465-476

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