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In the paper the sliding-mode observer for nonlin- ear system is proposed. The observer is based on the equivalent control method. Comparing with the known.
TPIO = 16:30 Sliding-Mode Observers Based on Equivalent Control Method S.V. Drakunov Department of Electrical Engineering The Ohio State University 2015 Neil Ave.,Columbus, Ohio 43210,USA form. The presented algorithm although similar in this case doesnot require state transformations.

Abstract In the paper the sliding-mode observer for nonlinear system is proposed. The observer is based on the equivalent control method. Comparing with the known output global linearization approach it is given in terms of of the system variables and doesnot require nonlinear state transformation. Several examples are presented which illustrates the proposed method.

11.

Observer design.

It will be considered the case of scalar observations ( m= 1). Let's introduce the following notations: H ( 2 ) = col(hl(z), . . ., hn(Z)),

I.

Introduction

The problem of state estimation for nonlinear system

i = f(4, where

t

(4)

(1)

E R" by output vector measurements y = h(z)E

hi(.)

f(,,), = ah:-l(z) at ~

(6)

i = 1, ..., n. Actually hi+l(x) is ith Lie (directional) derivative of the function h ( z ) along the trajectories of the system

R"

was considered by many authors (see State-of-the-Art

Survey by Misava and Hedrick [3]). For exactly known smooth nonlinearities f and h the method of global linearization permits by using nonlinear state transformation E = T(z) (3) to obtain a system in the new variables z. E R" which is linear with respect to z . Then the standard linear poleplacement technique can be used for designing observer. For successful1 application of this method the nonlinear transformation T as well as its inverse T-' must be determined. The main idea of the observer design in the presented paper is in using equivalent control method to obtain additional information from the system by equivalent values of discontinuous function in sliding-mode. For linear case in [l], [4] a sliding-mode observer was considered which is based on sequential application the state transformations reducing the system to the block

CH3229-2/92/0000-2368$1.OO 0 1992 IEEE

hi(.) = L;-'h(z). (7) From (1) and (5),(6) we have that if ~ ( t is) a solution of (1) then

i = 1, ..., n - 1. The assumptions imposed on the functions f(t) and h ( z ) are as follows: A s s u m p t i o n 1. Functions f(t) and h ( z ) are smooth enough in order all the partial derivatives introduced above exist and are continuous. A s s u m p t i o n 2. For a given domain XO C Rn of initial conditions of the system ( l ) , which is assumed to be bounded, all the solutions of the system (1) belong to the open one-component domain X C R", for all 0 5 t < 00. Jacobian of the continuous map

H : R"

c-*

R"

(9)

f o r i = 1, ..., n, If lh2(t(t))15 m l ( i ( t ) )then in the first equation (16) sliding mode occurs since v l ( t ) = h l ( t ( t ) )and hence v r ( t ) - h l ( i ( t ) )= e l ( t ) . During the sliding mode e l @ ) = 0 and according to the equivalent control method (ml (i)sgn(v1 ( t ) h ~ ( i ( t ) )= ) ~h,(z(t)) , or as it follows from (13) v z ( t ) = h,(z(t)). Again if the condition Ihs(+(t))l5 mz(;i.(t)) is satisfied sliding mode occurs in the second equation and we will have e2(t) = 0 and so on. When the sliding mode occurs in the last equation (16) we have e ( t ) = 0. The conditions for sliding mode to exist in every equation (16) are

H(t) = col(hl(z),. . . , h n ( t ) ) , is nondegenerating in X i.e.:

for some 6

> 0 and every

t

E X.

From Assumption 2 it follows that the map H is an injection. So it is one to one correspondence from the domain X to H(X). This assumption is actually an observability condition for (1),(2). To estimate system (1) state variables by measurements (2) we will use an observer of the form:

Ihi+l(t(t))l

.

=

dH(i)

-l

(7 M(li.)sgn(V(t) ) -H(3)),

Vl(t)

= Y(t),

Xo follows that all solutions

=

(12)

= (mi(i)sgn(vi(t)-hi(i(t))),,,i = 1 , . . . ,n-1, (13) and matrix M is n x n diagonal matrix with positive elements v1+1(t)

M ( i ) = diag(ml(i),. . . , m n ( i ) ) .

(17)

From boundedness of the initial condition region z ( t ) started in X O are uniformely bounded on any finite time interval [0, TI. Therefore by increasing mi any desired time t l < T of convergence li. to t can be achieved. After the moment t l since i ( t ) = t ( t ) this inequality can also be maintained by appropriate choice of mi.

(11)

where V ( t ) = c o l ( v l ( t ) ,. . ., v n ( t ) ) , Sgn(2) col(sgn(a),. . . , sgn(tn))

5 mi(S(t)).

Remark. The choice of matrix M is defined by the region of initial conditions X O for the system (1) and the upper estimates of hi.

(14)

References

It will be shown that by suitable choice of M observer (1 1) converges in any prescribed finite time interval.

[l] Drakunov, S.V., Izosimov D.B., Luk’yanov A.G., Utkin V.A. and Utkin V.I., The Block Control Principle, Part II, Automation and Remote Control, V. 51, no. 6 , 1 9 9 0 , ~737-746. ~.

Theorem 1. Under Assumptions 1 and 2 for any

tl > 0 there exists a diagonal matrix M ( i ) such that i ( t ) 5 z ( t ) for t 2 t l .

[a]

Proof. Since the map H : X H R” is an injection it is sufficient to show that e ( t ) = H(z(t)) - H ( i ( t ) ) converges to zero in finite time. From (1),(2) and (11) it follows that

Krener, A.J. and Respondek, W . , Nonlinear observers with Linearitable Error Dynamics, SIAM J . Control and Optimization, V.23,no.2,March 1985.

[3] Misawa, E.A. and Hedrick,J.K., Nonlinear Observers A Slate-of-lhe-ad Survey, ASME J. Dynamic Syst. Measurement Contr.,v.lll ,No.3,1989. [4] Utkin V.I., Sliding Modes and Their Applicatin In Variable Structure Systems. Moscow: MIR, 1978.

[5] Utkin V.I. A Method of Separation of Motion f o r Designing a Stale Identifier, in: Problems of Control of Multyvariable Systems, Nauka, Moscow, 1983, (in Russian).

H(li.(t))) = d H ( t ( t ) ) - M ( i ) s g n ( V ( t )- H ( i ( t ) ) ) . dt

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