robustness of luenberger observers: linear systems stabilized via non-linear control

Automatica, Vol. 22, No, 4, pp. 413 42L 1986 Printed in Great Britain.

0005 1098/86 $3.00 + 0.00 Pergamon Journals Ltd.

,~', 1986 International Federation of Automatic Control

Robustness of Luenberger Observers: Linear Systems Stabilized via Non-linear Control*

B. ROSS B A R M I S H t and ALBERTO R. G A L I M I D I t

Robust stabilization of uncertain systems via observers is possible provided the uncertain parameters do not exceed a certain computable threshold.

Key Words--Robust stabilization; observers; state estimation; uncertain systems; Lyapunov functions; non-linear control.

Abstract--Given a dynamical system whose state equations include time-varying uncertain parameters, it is often desirable to design a state feedback controller leading to uniform asymptotic stability of a given equilibrium point. If, however, the controller operates on some estimate of the state, instead of the true state itself, it is of interest to know whether the desired stability will be preserved, e.g. suppose that the measured output is processed by a Luenberger observer. This paper concentrates on the scenario above and in addition, our analysis permits the controller to be non-linear. As a first step, inequalities are developed which have implications on the system's robustness; that is, when the uncertain parameters satisfy these inequalities, it becomes possible to separately design controller and observer. This amounts to an extension of the classical separation theorem to the case when the controller is non-linear. It is also of interest to note that the approach given here guarantees stability for some non-zero range of admissible parameter variations. This is achieved by introducing a certain "tuning parameter" into the Lyapunov function which is used to assure the stability of the combined plant-observer-controller system.

1. INTRODUCTION

IN RECENT years, a number of new methods have been developed for synthesizing controllers which lead to asymptotic stability of the state of an uncertain dynamical system; e.g. see Gu tman and Leitmann (1976), Leitmann (1978, 1979), Patel et al. (1977) and their bibliographies. These so-called uncertain dynamical systems are typically described by differential equations which contain parameters whose values are imprecisely known. It is only assumed that bounds are available for the time variations of these parameters; no assumptions are made concerning their probabilistic behaviour.

* Received 9 November 1984; revised 27 August 1985. 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 E. Kreindler under the direction of Editor H. Kwakernaak. This work was supported in part by the National Science Foundation under Grant ECS-8311290 and in part by Rochester Gas and Electric Corporation.

t The authors were formerly with the Department of Electrical Engineering, University of Rochester, Rochester, NY 14627, U.S.A. They are now with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.

The controllers developed in the references cited above are predicated upon the availability of the full state x(t). In order to develop a more practical controller, the intention here is to remove the assumption that the state of the plant is fully accessible. Instead, the controller will be required to operate on some estimate 2(0 of the state x(t). This estimate is obtained through the implemen- tation of a Luenberger observer. Previous work within this context has dealt with robustness issues for a linearly controlled plant; e.g., see Dora to and Menga (1974), Thau and Kestenbaum (1974), Stefani (1982) and Breinl and Muller (1982). In Leitmann and Breinl (1983), saturating linear control is permitted. The problem of designing sensitivity-reducing compensators using an estimated state for control purposes is treated in Cruz and Krogh (1978) and so-called zero-sensitivity observers are also designed in Furuta et al. (1976).

In Dora to and Menga (1974) the problem is posed as that of minimizing a modified quadratic performance index which includes a term related to the estimation error. An algorithm is given to find the desired linear control and observer gain matrices. Convergence properties of the algorithm are not analyzed and no a priori conditions are given under which the existence of a suitable observer-based controller can be assured.

Of the cited references, the work in Thau and Kestenbaum (1974) is most closely related to this paper. In this, the authors give a measure of robustness for an uncertain system stabilized via an observer and a linear controller. The key differences between the results reported therein and the ones given here are the following:

(i) Non-linear control laws are considered here, whereas only linear controllers are considered in Thau and Kestenbaum (1974). This distinction is important because it has recently been dem- onstrated (see Petersen, 1985) that there are uncertain linear systems which are not stabilizable via

413

AUT 22:4-B

414 B. ROSS BARMISH and A. R. GALIMIDI

linear control but are stabilizable via non-linear control.

(ii) The Lyapunov function and full state controller design techniques used here take advantage of a priori bounds on the uncertain parameters. In contrast, the Lyapunov candidate and controller/observer pair in Thau and Kestenbaum (1974) are obtained by using only the nominal matrices.

The results in Leitmann and Breinl (1983), Stefani (1982) and Furuta et al. (1976) are also related to the current work.* These papers consider a linear plant with uncertainty in the parameters of the model matrices. A main objective is to describe a class of "structural" perturbations under which the observer will remain totally insensitive; i.e. for this class of perturbations in the model matrices, the error between the true and estimated state remains invariant.

This paper can be compared and contrasted with Leitmann and Breinl (1983), Stefani (1982) and Furuta et al. (1976):

(i) The plant need not be controlled linearly as in Stefani (1982) and Furuta et al. (1976); allowances for a class of non-linearities are made.

(ii) The stabilizability of an uncertain system generally depends on the model matrices having the perturbations structured in a specific way; e.g. see Corless and Leitmann (1981). Some of these uncertainty structures are "better" than others from the point of view of observer-based feedback; e.g. see Leitmann and Breinl (1983), Stefani (1982) and Furuta et al. (1976). In this paper structure is not exploited. Instead, norm bounds on the uncertain matrices are developed. In some cases, it is shown that uncertainties can be arbitrarily large. These special cases become analogous to the total insen- sitivity results in Leitmann and Breinl (1983), Stefani (1982) and Furuta et al. (1976).

(iii) Besides considering the estimation errors as in Stefani (1982) and Furuta et al. (1976); attention is alsb focused (as in Leitmann and Breinl (1983)) on the issue of plant stability. Not only is design of a robust observer desired, but also maintainance of the internal stability of the plant, which would be guaranteed if the true state were used for feedback.

(iv) Allowances are made for perturbations in the matrices which describe the measurement equation. These matrices are assumed to be fixed in Leitmann and Breinl (1983), Stcfani (1982) and Furuta et al.

(1976). The observer theory presented in this paper has

two salient features: First, the observer is designed

using the nominal system; that is, as input to thc design, the model obtained by fixing the uncertain parameters at some nominal values is used. Secondly, the observer takes the control law as given; this control law is designed under the hypoth- esis of full state feedback. The physical realization of the control, however, processes the estimated state instead of the true state. When this control law is non-linear, it is of importance to know whether one can "separate" estimation and control. This issue will be one of the focal points of this paper. An illustrative example of an uncertain system stabilized via observer-based non-linear control is given in Section 6.

There is considerable motivation for the treat- ment of a control law which is non-linear; e.g. saturation constraints are present on the controller as in Ryan (1982) and Kosut (1983), or alternatively, the designer may wish to stabilize a system with a minimum norm control as in Petersen (1983). This paper presents a condition on the non-linear control law and the plant, the satisfaction of which assures the uniform asymptotic stability of the combined plant-observer controller system. It is shown that this condition represents an extension of the separation theorem given in Luenberger (1966) to a certain class of cone-bounded control laws. That is, if the condition presented in this paper is satisfied, it becomes possible to separate the controller and observer designs. As an integral part of our approach, inequalities are developed which indicate how large the excursions of the uncertain parameters can be. For admissible uncertainties leading to the satisfaction of these inequalities, the controlled system remains asymptotically stable.

One of the main technical novelties of this paper is the introduction of a certain "tuning parameter". It is used to ensure positivity of the computed bound on the allowable uncertainty variations. Loosely speaking, this positivity guarantees non- emptiness of the robustness re qion in the uncertain parameter space. A class of systems are also described for which one can obtain total insen- sitivity of the observer to a subset of the uncertain parameters; see Section 5.

2. SYSTEM AND ASSUMPTIONS Consider a dynamical system described by state

and output equations

5c(t) = (A + AA(r(t))~x(t) + (B + AB(s(t)))u(t);

q(t) = (C + AC(v(t)))x(t) + (D + AD(w(t)))u(t);

t~ [0, oQ), (Z)

*In Cruz and Krogh (1978) the class of parameter perturbations considered are of the "infinitesimal type"; this paper deals with finite (possibly large) perturbations.

where x( t )~ R" is the state; u(t)~ R m is the control;

q(t) ~ R r is the measured output, r < n; r ( t )~ .~ ~ R ~

is the model uncertainty; s(t)~ 5/~ c R h is the input

Robustness of Luenberger observers 415

connection uncertainty; v( t )e 'U c R ~ is the output distribution uncertainty and w ( t ) e ~ c R a is the direct transmission uncertainty. Fur the rmore , A, B, C and D are cons tant matr ices and AA(.), AB(.), AC(.) and AD(.) are matr ix functions of appropr ia t e dimensions. Hencefor th , when referring to the system (Z) it is unders tood that this system includes the bounding sets :~', ,~, ~" and ~ .

T h r o u g h o u t the remainder of this paper , the following assumpt ions are taken as s tandard. Assumption 1 (continuity). The matr ix functions AA(.), AB(-), AC(.) and AD(.) depend cont inuously on their arguments . Assumption 2 (compactness). The bounding sets ;~, ,~, / and ~ " are compac t sets in the indicated f ini te-dimensional Eucl idean spaces. Assumption 3 (measurability). The uncertainties r(.): [0, ~ ) --+ .~¢, s('): [0, ~ ) --+ .~, v('): [0, oo) ~ ~ , w('): [0, oo)--* ~ " are assumed to be Lebesgue measur- able. Assumption 4 (observability). The nomina l pair (A, C) is observable. Assumption 5 (system wi thout observer). There exists a cont inuous cone-bounded control function p('): R " ~ R m and an n x n positive-definite sym- metr ic matr ix Pc such that the system (Y.) with full state feedback control

u(t) = p(x(t)) (2.1)

AB(s); e.g. see the match ing condi t ion in Corless and Le i tmann (1981).

Fo r this special case of full state feedback, techniques for cons t ruc t ion of control law p(-) are described in Barmish (1985) and Le i tmann (1979). In the remainder of this paper, however

u(t) = p(~(t)) (2.6)

will be taken, where ~(t) is the state est imate generated by a Luenberger observer. Definition 2.1. Given p('): R" ~ R r" as per Assump- tion 5, let q~(., .): R" × R" ~ R" be defined as

¢(4, z) = p(~ - z) - p(¢). (2.7)

This function q~(',') is cont inuous, and as a consequence of the assumed cone-boundedness of p(.), there will exist constants F ~> 0, a :e [0 ,2F] and b: ~ [0, F] such that*

IIp(~)ll ~ rlt~ll; (2.8)

II~b(4,v)ll < a:ll~ll + b:llTII (2.9)

for all (4, z) E R 2n. An impor t an t special case is obta ined when the

function p(.) satisfies a Lipschitz condition; i.e. there exists a non-negat ive constant G such that

satisfies the following condition: given any admissible uncertainties r(-) and s(.), it follows that

x'{Pc[A + AA(r(t))] + [A + AA(r(t))]'Pc}x

+ 2x'Pc[B + AB(s(t))]p(x) <<. --Ilxll 2 (2.2)

for all pairs (x, t )e R" x [0, oo). Equivalently, if the Lyapunov function is defined

Vc(x)& x'Pcx, (2.3)

the maximal Lyapunov derivative which results is

..~max(X) = max x'{Pc(A + AA(r)) .~cj

+ (A + AA(r))'Pc}x + 2x'Pc(B + AB(s))p(x) (2.4)

and moreover ,

~max(X) ~< --Ilxll 2 (2.5)

for all x E R". More s imply stated, V~(x) can be used to "guaran tee" uniform asympto t ic stability of the origin x = 0.

It should be noted that the satisfaction of this a s sumpt ion is typically accompl ished by the impo- sition of constra ints on the structures of AA(r) and

lip(4) - p(r)ll ~ GII4 - rll (2.10)

for all pairs (4, z) E R 2n. When such is the case, then it is easy to verify that a: in (2.9) can be chosen to be zero. No te that when the feedback law is linear; i.e. p(x) = Kx, then the Lipschitz cons tant can be chosen to be IIKI], and as in (2.9) can again be chosen to be zero, while b: = F = G = 14K II. Notation. In the sequel, the identity matr ix will be denoted by I, and O will denote the matr ix with all its elements equal to zero. For nota t ional convenience, the a rguments of AA(.), AB(.), AC(.) and AD(-) will often be dropped. It will be unders tood, however, that AA, AB, AC and AD are t ime-varying.

3. F U L L O R D E R OBSERVER: S T A N D A R D SET-UP A N D C L O S E D L O O P STATE E Q U A T I O N S

Given the dynamical system (Z) and the complete observabi l i ty of the nomina l pair (A,C) as per Assumpt ion 4, there exists an n × r constant gain matr ix L such that

Ae~= A -- LC (3.1)

* I n the sequel, II'lb will denote the Euclidean norm for real vectors. For a matrix M, [IM[I = )-l/~x [M'M] is taken.

416 B. Ross BARMISH and A. R. GALIMIDI

has a prespecified set of strict left half plane eigenvalues: see Luenberger (1966) for further details. Once u(.) is specified,

p ( t ) & q(t) - Du(t) (3.2)

is defined, and the estimate 2(t) of x(t) will be the solution to the full order observer equation

_?;(t) = Ae2(t) + Lf~(t) + Bu(t) . (3.3)

Suppose that the controller and observer designs are fixed, i.e. the control p(.), (possibly non-linear) and the matrix L have been chosen. The aim is to provide a set of sufficient conditions, whose satisfaction will ensure the asymptotic stability of the origin (0, 0) in (x, 2) space. Such conditions can be viewed as restrictions induced by the observer on the known data consisting of the matrix functions AA(-), AB('), AC(.) and AD(.), the bounding sets ~ , ,9 °, ~,", ~" and the constants a t, and b.r. Note also that these conditions are separate from those required for full state feedback control (Assumption 5). The asymptotic stability of (0, 0) will be guaranteed by constructing a quadratic Lyapunov function for the closed loop system adjoined with the error vector e( t )e R" defined as

e(t) A= x(t) - 2(t). (3.4)

Of course, the stabilizing feedback control u(t)

will have as argument the estimate 2(t), i.e.,

u(t) = p(2(t)) = p(x(t) - e(t))

= p(x(t)) + ~(x( t ) , e(t)) (3.5)

in accordance with Definition 2.1. It follows that the error vector e(t) will be governed by the state equation

O(t) = 2 (0 - ,?c(t)

= (A + A A ) x ( t ) + (B + AB)p(x ( t ) - e(t))

-- A j c ( t ) -- Lf~(t) - Bp(x( t ) -- e(t)). (3.6)

4. STABILITY AND ROBtJSTNESS ()l . 12~

The stability and robustness properties of iEl will be studied by proposing a Lyapunov function V(x,e) and giving two sufficient conditions so lhat

P ] c~max[X,e) A m a x [VV(x,e)]' ~<--,til(x, etii-"

~"<~ "~" ~ (4. t )

for all ( x , e ) e R 2" and some positive constant 2. Given that A~ is a stability matrix, there exists a

unique n x n positive-definite symmetric matrix P~ solving the Lyapunov equation

PoA,, + AI, P o = - 1 . (4.2)

It is worth noting that no uncertain parameters or bounding sets are involved in this equation; it can be said that Po is determined by the nominal pair (A, C) and the observer gain matrix L. On the other hand, Pc in (2.2) (2.4) is typically dependent on the matrix functions AA(.) and AB(.), and on the uncertainty bounding sets .~ and .<t'.

In a number of approaches to the stabilization of an uncertain dynamical system by full state feedback; e.g. Barmish (1985) and Leitmann (19791, the success of the design hinges on the existence of a matrix Pc and a function p(.) satisfying Assumption 5. Therefore, a natural choice for the Lyapunov candidate V(x , e ) is one that includes the matrices Pc and P0 found separately for controller and observer. Note that when non-linear control is present, this choice is not only natural but also has an advantage over other possible constructions of V(x, e) which do not exploit Assumption 5: namely, conditions can be developed on the known data under which it becomes possible to achieve this desired separation while preserving stability. Non- linear controls for uncertain systems are often required when minimum effort or minimum energy expenditure is desired. More importantly, it has been shown in Petersen (I 985) that some uncertain dynamical systems cannot be "quadratically stabilized" via linear control, but nevertheless, admit a non-linear stabilizing control.

Given the motivation in the preceding paragraph, a candidate Lyapunov function of the form

Regrouping terms in (3.6) and replacing 9(t) using (3.2) yields the overall set of equations for the state and the error. They are

2 (0 = (A + A A ) x ( t ) + (B + AB)p(x( t ) )

+ (B + AB)e~(x(t), e(t));

O(t) = Ae(t ) + (AA L A C ) x ( t )

+ (AB -- LAD)p(x ( t ) - e(t)). (2)

t s, V(x, e) ;~ %CPcx + e Poe (4.3)

is proposed, where a<. > 0 is a "tuning parameter" which will later be specifed. This parameter will be chosen with the goal of making the Lyapunov derivative negative for the combined plant observer-controller. Furthermore, it will be shown that when the controller satisfies a certain "size condition", then the robustness inequalities can always be satisfied for at least some range of perturbation in •A, AB, AC and AD.


Having the closed-loop state equat ions for ('Z) and L y a p u n o v candidate (4.3), let r('), s('), v(') and w(') be admissible uncertainties. Then it follows that the L y a p u n o v derivative is

Condi t ion 1.

1 max IIP~(B -4- AB(s))a/I < ~. (4.8)

5/'

&a(x, e, t) & [V V(x, e)]' = 2o~x'Pc[(A + A A ) x O

+ (B + AB)p(x - e)] + 2e'PoA¢e

+ 2e 'Po[(AA -- L A C ) x + (AB - L A D ) p ( x - e)].

(4.4)

Condit ion 2.

min det ¢(r, s, v, w) > 0. (4.9) ~', 5: ,-f-,-ff-

When Condi t ion 1 holds, notice that Condi t ion 2 will be satisfied if

Recalling (3.5), (4.4) can be rewritten as

L#(x, e, t) = 2~¢x'P¢[(A + A A ) x

+ (AB - L AD)p(x -- e)],

+ 2e'PoA~e + 2e 'Po[ (AA - LAC)x

+ (AB - L A O ) p ( x -- e)], (4.5)

which can be bounded using (2.4), (2.5), (2.8), (2.9) and (4.2) as

oL,~'(x,e,t) <~ - ~ l l x l l 2

+ 2c~llP<(n + An)afl l ' l lx l l 2

Ilell 2 + 211Po(AZ - Laf)ll ' l lxl l ' l lel l

211Po(AB - LADWII'Ilell 2

2~<IIPAB + An)bfll'llxll'llell

21IPo(AB - ZAO)Fll ' l lxll ' l lell . (4.6)

+

+

+

Using (4.6), it

~max(X, e)

~< max

is clear that

I-Ilxll I le l l3¢(r ,s ,v ,w)[l txl l Ilel12' (4.7)

where O(r, s, v, w) is a 2 × 2 matrix having entries

0 1 l ( r , s , v , w ) & c~¢II2P~(B + AB)a:II - ~c;

¢l~12(r,s,v, w ) ~- dPxl(r,s,v, w )

& IIPo(AA -LAC) I I + I IPo(AB- LAD)VII

+ ~,IIP<(B + AB)b/I;

(1) 22(r, s, v, w) A_~ 2IIPo(AB -- LAD)VII - 1.

Therefore, if ~l)(r, s, v, w) remains negative-definite for all admissible r, s, v and w, there will exist 2 > 0 leading to satisfaction of (4.1), and moreover, asymptot ic stability of ('Z) can be guaranteed. Clearly, the following two condit ions are necessary and sufficient for negative-definite invariance of ¢(r, s, v, w):

min {c~¢(1 - 2]lPc(B -4- AB)af l l ) 5:,3t¢

-(1 - 211Po(AB - LAD)VII)}

> max {c~IIP~(B + aB)bf l l + IIPo(AA .@, S : ,~ ' ,W

--LAC)II + ] ]Po(AB- LAD)VII} 2

which in turn is satisfied if

(4.10)

min {c~(1 - 2IIP~(B + AB)afl l) cj,.~¢-

• (1 -- 2HPo(AB -- LAD)VII)} 1/2

> max{~tcllPc(B + AB)bj-} + max {llPo(AA .~' ~t ,.5:, ~v ,'¢¢

- LAC)II + IIPo(AB - LAD)VII}. (4.11)

For (4.11) to hold, notice that it must be the case that

fl(~, ~¢:)~ 1 - 2 max IIPo(AB -- LAD)VII > 0. ~,,¢ (4.12)

In order to motivate the selection of the positive tuning parameter ~¢<, (4.11) is first rewritten as

#(cQb min {at(1 - 211P~(B + AB)afl l ) ha ,W

• (1 -- 211Po(AB - LAD)NIl)} ~/2 - max {c¢<llPc(B

+ AB)b/I} > max {llPo(AA - LAC)ll ,~,5:, 'V, 'W

+ IIPo(AB - LAD)VII}. (4.13)

We shall choose ~c for two separate cases. Case 1. max IIP<(B + AB)bfll = O. Given that Con-

5¢

dition 1 and (4.12) must hold, let

M ~

m a x ~ , ~ , ~ , ~

{llPo(AA -- LAC)II + IIPo(AB -- LAD)VII} 2

min {(1 - 211Pc(B + AB)afll)(1 -- 211Po(AB -- LAD)VII)} 5a,~t¢"

418 B. ROSS BARMISH and A. R. GAL1MIDI

Thus any' choice of ,:q. > M leads to satisfaction of (4.13). Case 2. max NP,~B + AB)bIII > 0. The optimal

/ /

selection ~ of ~<, will be the one maximizing 14~,.).

It is found by a straightforward differentiation* of

14~<.):

min {(1 - 2IIP<(B + AB)afll)(1 - 211Po(AB - LAD)FII)}

4 m a x I/P<(B + AB)brll 2

(4.14)

With ~ = e~, Condi t ion 2 reduces to the requirement

min {(1 - 2NPJB + AB(s))asH)(1 - 211Po(AB(s) J ' , ' t t

- LAD(w))FN)} > 4 m a x IlP~(B f f

+ AB(s))bsl l . max { N P o ( A A ( r ) - LAC(v))II :~¢,.~ "t, J 4 :

+ IIPofAB(s) - LAD(w)r ' I I } . (4.15)

These findings are summarized by the following theorem. Theorem 4.1. Consider the uncertain dynamical system (Z) satisfying Assumptions 1-5 and assume further that the observer gain matrix L has been selected so that A,, = A - L C has strict left half plane eigenvalues. Then the following pair of conditions are sufficient for the existence o f a L y a p u n o v function + V(x ,e ) for the combined state and error system (Z):

l max IIP<(B + AB(s))asH < ~; (4.16)

¢ /

Remarks and special cases (1) Even for the special case when A B - 0 .

Condi t ion I can be viewed as a restriction on the magnitude of ar which is used in the cone bound on 0(')- We are thus led to the requirement

1 ar < ~(IIP, Brl)-1. (4.18)

This criterion is trivially met by all non-linear control laws that satisfy a Lipschitz condit ion (because a s can be taken to be z.ero in these cases, as noted in Section 2). Hence, (4.18) provides a sufficient condit ion under which the separation proper ty (see Luenberger, 1966) applies to a class of systems with cone-bounded control. A more detailed discussion of the importance of (4.18) will be provided in Appendix A.

(2) When (4.18) holds, Condit ions 1 and 2 guarantee a non-zero measure o f robustness for (E) and its associated controller and observer. More precisely, one can always assure the satisfaction of (4.17) provided that the bounds on the norm of the "A matrices" are sufficiently small. This fact becomes apparent by not ing that if these A matrices are zero, Condi t ion 1 is equivalent to (4.18) and (4.17) holds trivially. Therefore, by a continuity argument, one concludes that (4. l 7) will continue to be satisfied over some range of perturbat ions in AA, AB, A C

and AD. (3) For the special case when linear control is

used; i.e. p(Yc(t)) = K [ x ( t ) - e(t)], (4.5) can again be taken as the starting point and a single condit ion developed for asymptot ic stability:

min (1 - 2 IIPo(AB(s) - L A D ( w ) ) K II) .c/~ # 4 '

> 4 m a x IIPc(B + AB(s))KN max NPo[AA(r) , ~ :~, c p , ~r ,g,

min {(1 - 2NP<(B + AB(s))ayll)(1 - 211Po(AB(s) <9 ¢ , ~ ,

- LAD(w))FN)} > 4 m a x IIP<(B 5~

+ AB(s))brN, max {l lPo(Ah(r) - LAf (v ) ) l l :,.f ,,~/ "t , Tae

+ NPo(AB(s) - L A D ( w ) ) F I t } (4.17)

where the matrix Po has been selected according to (4.2). [ ]

* It can be verified that dZ#(c£)/d~< z is indeed negative for c~< = ~.

i'This Lyapunov function can be used to establish the existence of a 2 < 0 such that 5t"ma,.(x,e ) <~ -.l.llx, e)ll 2 for all (x,e)eR2"; recall (4.1).

- LAC(v ) + (AB(s) - LAD(w))K]N. (4.19)

(4) A special case of (4.19) is obtained when AB(s) ~- 0, AC(v) - 0 and AD(w) - 0. In this case, (4.19) degenerates into

1 max IlPoAA(r)ll < 4l tPcBKIl ' (4.20)

:4?

which indicates how much uncertainty can be tolerated in AA if no other uncertainties are present. Note that one can obtain similar bounds on AB, AC and AD.

(5) Depending on the observability index (see Luenberger, 1966) of the nominal pair (A,C), at most (r - 1) columns of the gain matrix L can be set to zero while still retaining the proper ty of


arbitrary pole-assignment for Ae. Although this may sometimes imply unduly large gains in the non-zero column of L, there is a potential benefit in this choice. Namely, the uncertainties present in those rows of AC and AD have no bearing on the satisfaction of (4.10). In other words, the system can be rendered totally insensitive to at most (r - 1) rows of AC and AD by choice of L. When the jth column of the gain matrix L is set to zero, the jth component of the vector 33(0 is not fed into the observer dynamics.

5. O B S E R V E R I N S E N S I T I V I T Y T O AA(r)

The stability and robustness properties of the overall plant-control ler-observer system have been investigated in an indirect way by studying the system (X). In this section, a different set of results is presented that can be obtained if one works directly with (Z) rather than (X). The results of this section should be considered as an alternative to the ones given in Section 4. Consider the state equations for the combined system

Yc(t) = (A + AA)x(t) + (B + AB)u(t);

~(t) = A~c(t) + Lp(t) + Bu(t) (£)

further that the observer gain matrix L makes A e = A - LC strictly stable. Then the following pair of conditions are sufficient for the existence of a Lyapunov function* V(x,2) for the combined system ('~)

1 max IIPc(B + AB(s))Fll < ~; (5.3)

oq ~

min (1 - 2 II Pc( B + AB(s))F II) 5~,-W

• (1 - 211PoLAD(w)FII - 2LIPoBFII) > 4max IIPc(n

+ AB(s))Fll' max IIPoL( C + AC(v))l[ (5.4)

where the matrix Po has been selected according to (4.2). []

Remarks and special cases (1) Given that neither (5.3) nor (5.4) contain any

terms in AA(r), one concludes that the observer is indeed insensitive to perturbations in AA when these conditions hold.

(2) For the special case when linear control u(t) = Kfc(t) is used, conditions (5.3) and (5.4) can be replaced by the simpler sufficient conditions:

where A e, )3(0 and u(t) are defined in (3.1), (3.2) and (3.5) respectively; finally, p(.) satisfies Assumption 5 and thus (2.8). There are certain trade-offs involved when working with (~,) in lieu of (,~): on one hand, situations will arise where the observer imposes absolutely no requirements on AA over and above those of the controller. On the other hand, there are systems which do not meet the criteria given in this section, but satisfy Conditions 1 and 2 of Section 4; see Remark 3 following Theorem 5.1.

The candidate Lyapunov function

V(x,~c) = ~x'Pcx + &'Po~¢ (5.1)

is proposed, with ct > 0 to be specified, Pc as in Assumption 5 and P0 as defined in (4.2).

Proceeding analogously to Section 4, note now that the origin (x, ~) = (0, 0) will be an asymptotically stable equilibrium point if

~ m a x ( X , ) ~ ) = m a x [VV(x,~)] ~ ~-~l l (x ,~) l l 2 ~,S¢,'U,W

(5.2)

for all (x, ~) # (0, 0) and some positive constant e. The proof of Theorem 5.1 to follow is entirely

similar to that of Theorem 4.1; it is omitted for the sake of brevity. Theorem 5.1. Consider the uncertain dynamical system (Y.) satisfying Assumptions 1 5 and assume

1 max IIPc(B + AB(s))KII < ~; (5.5)

6~

min {(1 - 211Pc(n + An(s))gll) ' (1 - IlPo(n 6e,$¢

+ LAD(w))K + K'(B + LAD(w))'PoII)

> 4 max IIPc(B + AB(s))K II "max Ib PoL(C + AC(v))II.

(5.6)

(3) It can be shown that the joint satisfaction of (2.5) and (5.3) imply uniform asymptotic stability of the uncontrolled plant. However, for the cases where feedback p(~(.)) is applied, the result of this section can be used to establish uniform asymptotic stability of the resulting closed loop system.

6. I L L U S T R A T I V E E X A M P L E

To illustrate the application of the results in this paper, the stabilization of ship heading is considered as described in Astr6m and Wittenmark (1984). Variations in the parameters entering both the A and B matrices are studied; these uncertainties represent + 100% and _ 5% variations around the nominal values of the parameters in the model matrices. Note also that the system is unstable without control.

* T h i s L y a p u n o v funct ion can be used to es tabl ish the exis tence of an e > 0 to satisfy (5.2).


Proceeding, consider a dynamical system described by

2(0 = [A + AA(r(t))]x(t) + [B + AB(s(t))]u(O;

q(t) = Cx(t),

where

[01 [01 B = 0.0005 ; AB(s )= s ;

C = [1 0]; r e ~ & [--0.001, 0.001J;

s ~ cf zx [ _ 0.000025, 0.000025].

The state vector has components xl( t ) and xz(t ) representing heading (angle) and turning yaw rate, while the control input u(t) is the rudder angle. The measured variable rl(t ) is the ship heading.

Given the set-up above and a stabilizing non- linear full state feedback control law p(.), the objective is to select an observer which guarantees asymptot ic stability when the estimate is used in lieu of the true state.

Controller without observer (fidl state feedback) The full state feedback controller can be designed

using the theory in Barmish (1982,1985). This controller is taken as given in the observer design; can stability still be guaranteed even though the state is unavailable? Indeed, using Barmish (1985), the matrix

and the non-l inear control law

min (1 - 211PoAB(s)I'[])> 4 max H P,.(B + AB(s))h I I: 2/ /

• max(HPoAAlr)l[ + [IPoAB(s)FIi)~ (6.2)

In view of (6.2), it is desirable to choose an observer gain matrix L leading to a small value of IIPoH* Suppose that the desired observer eigenvalues are 21.2 = - 2 ___ j3, These eigenvalues are attained with observer gain matrix L = [4 13]' which yields (see (3.1) and (4.2))

i -4 11 Ae = -- 13 --0.001 ;

F 1.749 - 0 . 4 9 9 7 ~ P0 = [_-0.4997 0.288 J ' (6.3)

Verification of the conditions for stability When Theorem 4.1 is applied to the combined

p lan t -observer -con t ro l l e r system, (4.16) is satisfied with ay = 0 because the control p(x) satisfies a Lipschitz condition. Straightforward computa t ions yield

min (1 - 211PoAB(s) FI]) ~ 0.9457;

max IJPc(B + AB(s))bfl ~ 8.003;

max [IIPoAA(r)II + IIPoAB(s)FII] ~ 0.0276.

Hence, it is now easily verified that the stability requirements in Theorem 4.1 are satisfied. To further illustrate, the dynamics of(Z) were simulated from the initial condit ion of x ( 0 ) = [1 3]' and 2(0) = [3 - 1]'. This was accomplished by imple- menting a R u n g e - K u t t a algori thm with variable stepsize on a D E C VAX-11/785 computer. Rep- resentative projections of the (x, 2) trajectories onto the (X1,X2) and ( 2 1 , 2 2 ) planes are shown in Fig. 1. The temporal evolutions of xl, x2, 21 and -'?2 are

p ( x ) =

--max[0.39x~ +2 .39x~ + 2 x 1 x 2 , 0]

0.000475(x 1 -I- 3X 2)

0

--max[O.39x12 + 2.378x 2 + 1.988xlx 2,

0.000475(X 1 + 3X2)

o]

ifx2(X 1 + 3X2) > 0

if X~ + 3X 2 = O,

ifx2(X 1 + 3X2) < 0

or x 2 = 0,

(6.1)

are obtained, which lead to the satisfaction of Assumpt ion 5. It can readily be shown that the control law p(') is cont inuous and satisfies a Lip- schitz condi t ion° such as (2.10). Hence, F = bf ~ 1880 can be computed and ar = 0 chosen.

Full order observer Condi t ion 2 of Theorem 4.1 simplifies to

shown in Figs 2 and 3. For simulation purposes, the uncertain parameters r(t) and s(t) were taken to be sinusoidal functions of time with frequencies of 2 and 7 time units, respectively.

* In fact, a simple optimization procedure can be devised to obtain a stable observer with the goal of minimizing IIPoll- Basically, it involves a search for the entries of L subject to an upper bound.


I I I -3 .6 -2 .4 - I . 2

x 2 ( f ) 6.0 ;2It)

4.8

3.6 ( X I p X 2 )

2.4--

1.2

~ i}2 xl ( t )

x ~ ( t )

FIG. 1. Projection of trajectories on to (xl ,x2) and (:~1, Z2) planes.

-3

-4

3 6 I g

I

12

FIG. 2. Tempora l evolutions of x 1 and :~1.

A

A

3,2/" 2 . 4 -

1.6

0 8

0

-0 .8

- I .6

- 2 . 4

- 3 2

~ s ( t )

"~2 (t)

i 12

FIG. 3. Tempora l evolutions of x 2 and 2 2.


7. C O N C L U D I N G R E M AR KS

In this paper, sufficient conditions were given for the separate design of controller and observer for an uncertain dynamical system. When non-linear control was required, an extension of the classical separation property was obtained. In this area of future research, it would be of interest to focus attention on the "optimal selection" of the observer gain matrix L. Can one maximize the likelihood that Conditions 1 and 2 will be satisfied by choice of L? While large gains in the entries of L may make the observer faster and IIP0]l smaller, they may also increase the sensitivity to AC and AD.

The allowable uncertainties were described in this paper in terms of norms; i.e., no emphasis is placed on which particular entries of AA, AB, AC and AD were perturbed. In many cases, however, these matrices can be endowed with a special structure; e.g. if AA(.) is a companion form realization, then only the last row of this matrix would be uncertain. By exploiting structure, it may be possible to develop a theory which permits "larger" sets ~ , ,9 ~, ~/o and ~" when separately designing observer and controller. Another problem of interest would be to investigate the extent to which it is possible to extend the separation property to non-linear control schemes that violate (4.16). Finally, the reader is reminded that the Lyapunov function was chosen here to be block diagonal. If one does not insist on separation between controller and observer designs, it may be possible to suc- cessfully exploit a Lyapunov function containing cross terms in x and e.

Acknowledgements The authors express their appreciation to Drs I. R. Petersen and M. Corless for their detailed technical comments. The numerous suggestions made by Professors G. Leitmann and E. Ryan were also most helpful.

REFERENCES

Astr6m, K. J. and B. Winenmark (1984). Computer Controlled Systems. Prentice-Hall, Englewood Cliffs, New Jersey.

Barmish, B. R. (1982). Fundamenta l issues in guidance and control of uncertain systems. Proc. Am. Control Conf., Arling- ton.

Barmish, B. R. (1985). Necessary and sufficient conditions for quadratic stabilizability of an uncertain linear system. J. Opt. Theory Appl., 46.

Breinl, W. and P. C. Muller (1982). Ein Para- meterunempfindlichen Zustandsbeobachter und seine Anwen- dung bei einem Tragregelsystem eines Magnet- schwebefahrzeugs. Regelungstechnik, 30.

Chen, C. T. (1970). Introduction to Linear System Theory. Holt, Rinehart and Winston, New York.

Corless, M. and G. Lei tmann (1981). Cont inuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamical systems. IEEE Trans. Aut. Control, AC-26.

Cruz, J. B., Jr. and B. Krogh (1978). Design of sensitivity- reducing compensators using observers. IEEE Trans. Aut. Control, AC-23.

Dorato, P. and G. Menga (1974). Observer feedback for uncertain systems. Proc. IEEE Conf. Decis. Control, Las Vegas.

Furuta, K., S. Hara and S. Mori (1976). A class of systems with the same observer. IEEE Trans. Aut. Control, AC-21.

Gutman, S. and G. Lei tmann (1976). Stabilizing feedback control for dynamical systems with bounded uncertainty. Proc. IEEE Con[~ Decis. Control.

Kosut, R. K. (1983). Design of linear systems with saturating linear control and bounded states. IEEE Trans. Aut. Control, AC-28.

Leitmann, G. (1978). Guaranteed ultimate boundedness for a class of uncertain linear dynamical systems. IEEE Trans. Aut. Control, AC-23.

Leitmann, G. (1979). Guaranteed asymptotic stability for some linear systems with bounded uncertainties. J. Dyn. Syst. Meas. Control, 101.

Leitmann, G. and W. Breinl (1983). Zustandsri ickfuhrung fiir dynamische Systeme mit Parameterunsicherheiten. Regelungstechnik, 31.

Luenberger, D. G. (1966). Observers for multivariable systems. IEEE Trans. Aut. Control, AC-II .

Patel, R. V., M. Toda and B. Sridhar (1977). Robustness of linear quadratic state feedback designs in the presence of system uncertainty. IEEE Trans. Aut. Control, AC-22.

Petersen, I. R. (1983). Investigation of control structure in the stabilization of uncertain dynamical systems. Ph.D. disser- tation, University of Rochester.

Petersen, I. R. (1985). Quadratic stabilizability of uncertain linear systems: Existence of a non-linear stabilizing control does not imply existence of a linear stabilizing control. IEEE Trans. Aut. Control, AC-30.

Ryan, E. P. (1982). Optimal feedback control of saturating systems. Int. J. Control, 35.

Stefani, R. T. (1982). Reducing the sensitivity to parameter variations of a minimum-order reduced-order observer. Int. J. Control, 35.

Thau, F. E. and A. Kes tenbaum (1974). The effect of modeling errors of linear state reconstructors and regulators. J. Dyn. Syst., Meas. Control, 96.

APPENDIX A

Separation theorem for non-linear control Let (Zd) be the deterministic system obtained when all the

uncertain matrices in (Y) are zero; i.e.

Yc(t) - Ax(t) + Bu(t); (Zd)

q(t) = Cx(t) + Du(t).

Assumptions 4 and 5 are also taken as standard. When a Luenberger observer is designed for (En), it is of

interest to know whether one can use the estimate 2(t) as input to the controller and still guarantee the asymptotic stability of the origin (0, 0) in (x, 2) space. It is already well known that this property does indeed hold when the control p(-) satisfies a Lipschitz condition; e.g. see Luenberger (1966) and Chen (1970) for a specialization to linear controls. But there are cone- bounded non-linear control laws to which the results of Luen- berger (1966) do not apply, and some other criterion is needed to establish stability. In this case, the Lyapunov candidate

V(x,e) = ~cx'Pcx + e'Poe (A.1)

is proposed, with Pc as per Assumption 5, % > 0 unspecified as yet, P0 as in (4.2) and the error*, e(t), as defined by (3.4).

Given the scenario above, the results of Section 4 specialize to the following theorem. Theorem A.1. Consider the deterministic dynamical system (Za) satisfying Assumptions 4 and 5 and assume further that the observer gain matrix L has been selected so that A e = A - LC has strict left plane eigenvalues. In addition, suppose ay E [0, 2F] as per Assumption 5. Then the following condition is sufficient

* Note that the stability properties of the origin (0,0) in (x, ~) space correspond to those of the origin (0, 0) in (x, e) space.


for global uniform asymptotic stability of the combined state and error system (E):

a s < (211P~BII)- '. (A.2)

Remarks and special cases (1) As noted in Section 2, all control laws satisfying a Lipschitz

condition will also satisfy the inequality (4.16) of Theorem 4.1.

Of course, (A.2) is simply a special case of this inequality. Given that all linear control laws do satisfy a Lipschitz condition, one can simply take a s = 0 in this case. When the controller is non- linear, then inequality (A.2) describes a class of "tolerable non- linearities" allowed in the controller structure so that the separation theorem applies. It is of interest to note that all multi-input saturating linear controls also satisfy a Lipschitz condition, and therefore, separation condition (A.2) will be trivially satisfied by these controls as well.

robustness of luenberger observers: linear systems stabilized via non-linear control

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