Observer Design for Linear Systems with Unknown Inputs - IEEE Xplore

483

TECHNICAL NOTES AND CORRESPONDENCE

for evaluating

REFERENCES [I] D. G. Luenberger, “Obsening the state of a Linear system” IEEE Tram.MiL Electron., vol. MIL-8, pp. 74-80, Apr. 1964. 121 -, “Observers for multivariable systems,’’ IEEE Trans. Auromr. Conrr., VOL AC-II. pp. 190-197, Apr. 1966. 131 W. M.Wonhamand A. S . Morse, “Feedbackinvariants of linear multivariable systems,” Auromarica, vol. 8, pp. 93-100, Jan. 1972. [4] T. E. Fortmann and D. Williamso& “Design of low-order observers for linear feedback control laws,” IEEE Tram Auromt. Confr., vol. AC-17, pp. 301-308, June 1972. [5] J. R. Roman and T. E. Bullock, “Design of minimal orderstable observers for linear functions of the state via realization theory,” IEEE Trans. A u r o m f . Conrr., vol. AC-20, pp. 613422. Oct. 1975. 16) J. B. Moore and G . F. Ledwich, “Minimalorder observers for estimating linear functions of a state Yector.” IEEE Tram. Auromr. Conrr., vol. AC-20, pp. 6 2 3 4 3 2 , Oct. 1975. [7] R. Guidotzi and G . Marro, “On Wonham sfabilizability condition in the synthesis of observers for unknown-input system%” IEEE Trans. Automaf. Confr., vol. AC-16, pp. 499-500, Oct. 1971. [E31 G . Hostetter and J. S . ,Meditch “Observing systems with unmeasurable inputs,” IEEE Tram. Auromar. Conrr., vol. AC-18, pp. 307-308, June 1973. [9] S. H. Wang, E. J. Davison. and P. Dorato, “Observing the states of systems with unmeasurable disturbance&” IEEE Tram. A u r o m . Conrr., vol. AC-20, .. !JP. 716717, Oct. 1975. D.G.Retallack, “Transfer-function matrix approach to observer design,” Proc. IEE, vol. 117, pp. 115>1155, June 1970. S. H. Wang and E. J. Lkvlson, “A minimization algorithm for the design of linear multivariable systems,” in 1973 Proc. Joinr Automufir Confrol Conf., Columbus OH., June 1973. -, “A minimization algorithm for the design of linear multivariable systems,” IEEE Trans. Auromr. Contr.. vol. AC-18, pp. 22&225, June 1973. G. D. Fomey, Jr., “Minimal bases of rational vector spaces, with applications to multivariable linear systems,” S I A M J. Confr., vol. 13, pp. 493-520, May 1975. M. K. Sain, “A free-modular algorithm for m i n i m a l design of linear multivariable systems”, in Proc. IFAC 6th World Cong., Part IB, pp. 9.1-1-9.1-7, Aug. 1975. -, ‘The growing algebraic presence in systems engineering: an introduction,” Proc. IEEE, vol. 6 4 , pp. 9 6 1 1 I, Jan. 1976. B. D. 0. Andenon, N. K. Bose, and E. I. Jury, “Output feedback stabilization and related problems-solutions via decision methods,” IEEE Trans. Auromt. Conlr., vol. AC-20, pp. 53-56, Feb. 1975. A. Tmski, A Decision Mefhod for Elemenfay Algebra and Geometry. Berkeley, CA.: Univ. California Press, 1951. S . H. Wang and E. J. Davison, “On the stabilization of decentralized control systems,” IEEE Trans. Automu. Confr., vol. AC-18, pp. 473478, O c t 1973.

w(t)=H’x(t).

(3)

In (1)-(3), x , u, 5, y , z, and w(G) are n, r, s, m, q, and p vectors, respectively, and 5 is an unmeasurable disturbance. In theexisting literature on the problem [2H5], and references cited therein, the observer (2) is referred to as an unknown input observer. A trial and error approach is indicated in [2] for the case of minimal-order state observers. The problem is solved (for state observers) in [3]under the assumption that 5 satisfy a known differential equation. In [4] and [5], some structural aspects of unknown input observers are derived for the special case H ’ = Z, using geometric concepts.The present note also uses the geometric notions available in [6],the main contribution being a fairly complete theory of synthesis. 11.

PROBLEM

FORMULATION

We shall first fix a suitable definition of observer action by partitioning the complex plane C into disjoint, symmetric subsets C,, Cb,with o(A) the spectrum of A satisfying ‘ J ( A )c c b

(4)

C, c C - (open LHP) and Cg sufficiently to the left V ’ x ( t ) if and only if

(5)

of o ( A ) . Then we say that (2) obserces

lim [ z ( t ) - V ‘ x ( t ) ] = O

r--t w

Vx@z,,u(.),~()

(6)

and the exponents of convergence in (6) lie in Cg.It follows easily from thedifferential equation for e ( t ) = z ( r ) - V ’ x ( r ) that, if (2) observes V‘x(r)

+ LC’=O

V ’ AN V ‘ -

(74

G = V‘B

(7b)

O = V’D

(74

and o ( N ) c cg.

Observer Design for Linear Systems with Unknown Inputs S. P. BHA’ITACHARYYA

Also, if (2) observes V‘x(t), then

Abshact-Results from the geometric theory of linear systems are utilized to present a constroctive solution to the problem of designing a Luenberger observer to evaluate a given set of linear functions of the state of a linear system subject to unknown or disturbance inputs.

I. INTRODUCTION

(la) ( lb)

y(t)=C’x(t)

r > O x(0)=xo

i(t)=Nz(t)+Ly(t)+Cu(t) iqt)=Ey(t)+Fz(t)

with exponents of convergence in C, if and only if EC‘+

F”= H‘.

Unknown Input Obsemr Design Problem

i(t)=Ax(t)+Bu(t)+D{(t)

ofLuenberger designing the observer

Vxo,zo,u(.),~(.)

Therefore, we formulate our problem as follows.

Consider the linear time-invariant system

and problem the

lim [ G ( t ) - w ( t ) ] = O f+W

Given (A,& D, C’,H’, Cg), find, if theyexist, q (the observer order), B E { N , L, G,E, F } (the observer parameters) and V‘ such that (7)and (8) are satisfied. 111. MAINRESULT

111

(2a) (2b)

Manuscript received July 8. 1977. Thjs u.ork was supported by FINEP. The author is with the Department of Electrical Engineering,Universidade Federal do Rio de Janeiro, Rio de Janeiro, B r d .

If T is a given p X q real matrix ET=Im T cRP will denote the ~ denote the subspace spanned by the columns of T, and Ker T c R will null space of T. From (7a) it follows, with this notation. that

YrcKerD’ and (8) implies that

0018-9286/78/0600-0483$00.75 01978 IEEE

IEEE T R A N S A ~ O N ON S AUTOMATIC

484

EIcCC+Y.

Exanple

This leads to the following. 0 1 01 1 0 0 0

A = [ ,0

Lemma I

C‘=(l

If there exists an unknown input observer, then

XCC+Y*

cornor, VOL

AC-23, NO. 3, JUNE

1978

I:[=. .=[;I 0

1 H‘=(1 1)

0)

c*=c-

It is easily seen that T = K e r D ’ and that (15) holds. With

where Y* is the unique maximal element of

Y={YIA‘YcY+C, YcKerD‘). To proceed, define an observer to be internal& nonredundant if rankV‘=q and externul& nonredundant if

and (8) is satisfied with E=O,F=(l

rank[ Cy:]=rank~’+rank~‘.

IV. CONCLUDING REMARKS

If (lla) or (llb) is satisfied it is easily shown that

L=

0).

V‘L

hasasolution for every L, and we assume,henceforth, that the observer is either internally or externally nonredundant. Now

Although the assumption (1 1) is always made in observer theory, it is worthinvestigating the structural constraints imposed by it for the problem treated here. The design of minimal-order and robust [7l unknown-input observers are also interesting unsolved problems. REFERENCES

( A ’ - CL’)Y= YN‘.

(13)

Frzm (13), N‘ is the restriction of (A‘- CE’) to Im V; i.e., N ’ = ( A ’ CL’)IY. This motivates us to introduce the following result proved in [6, pp.1201. Lemma 2

The class of subspaces

“T,={113L‘suchthat(A’-CL’)YcVcKerD’u,(A’-C~IV)cC,) (14)

[I] D. G . Luenberger, “An introduction to observers,’’ IEEE Tram. Aufomal. Contr., vol. 16, pp. 591971. 121 S. H. Wan& E. J. Davison, and P. Dorato, “Observing the states of systems with unmeasurabledisturbances,” IEEE Tram Awomat.Contr., vol. 20, pp. 716-717, 1975. [3] G . Hostetter and J. S. Meditch, Wlbserving systems withunmeasurableinputs,” IEEE Trans. Automat. Corn., vol. AC-18, pp. 307-308, June 1973. [4] G . B a d e and G . Marro “On the observability of Linear time-invariant systems with unknown inputs,” J. Opfimiz. Themy Appl., vol. 3, pp. 410-415, June 1969. [5] R Guidorzi and G. Marro, “On Wonham stability condition in the synthesis of observers for unknown-input system$” IEEE Tram Automar. Contr., vol. AC-16, pp. 499-500, Oct. 1971. (61 W. M. Wonham, Linear Multiwriable Confrol: A Geomefric Apprwch (Lecture Notes in Economies and Mathematical Systems). Berlin: Springer, 1974. 171 S. P. Bhattacharyya, ‘Thestructure of robustobserver$” IEEE Trans. Aulomot. Contr., vol AC-21, pp. 581-594, Aug. 1976.

has a unique maximal element V.: The construction of V ; and a suitable E is given in [6, pp. 1201 and is omitted here. The main result can now be stated as follows. Theorem

There exists a solution to the unknown input observer design problem, subject to (1 la) or (1 lb), if and only if

(15)

3CC6?+??.

Proofi Necessity is clear from the preceding discussion and the fact that u ( N ) = u(N’). For sufficiency, start by computing V,: the maximal element of (14), and a suitable L*, satisfying (A’ - CL*’JV;) c Ck using the procedure given in [6, pp. 1201. Let v* be an n X q* matnx with q* = dim.Yi and Im Y* = V .: Now calculate L* from L* = p*’p

and N* from

Application of Equivalent Control Method to the Systems with Large Feedback Gain

v. UTKIN Absrmct-Diierential equations of slow motion Zn the systems with a feedback gain are obtained using the equivalent control method developed for the analysis of sliding mode in systems ~ t discontinuoas h controls [l]. large

Given a control system x=f(X,f)+B(x,l)u

(A‘-cL*‘)P=Y*N*’.

u=ks(x)

Calculate E * , P satisfymg

(1)

where X J E R ’, k is a scalar, the function s ( x ) is continuously differentiable, det GB # 0, G =(&/ax).

CP‘+ PP’= H and evaluate

G* = V ’ B Then B = (N*, L*, G*,E*, E * ) is the desired observer.

Manuscript received July 23, 1977. The author was with the Department of Electrical Engineering, Universiiy of Illinois, Urbana, IL 61801. He is now with the Institute of Control Sciences Moscow, USSR.

0018-9286/78/~0484$4%0.75 01978 IEEE