New results on general observers for discrete-time nonlinear systems

Applied Mathematics Letters

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Applied Mathematics Letters 17 (2004) 1415-1420

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N e w R e s u l t s on G e n e r a l O b s e r v e r s for D i s c r e t e - T i m e N o n l i n e a r S y s t e m s V. SUNDARAPANDIAN Department of Instrumentation and Control Engineering SRM Institute of Science and Technology Kattankulathur-603 203, Tamil Nadu, India sundarsrm©yahoo, co. i n

(Received and accepted July 2003) A b s t r a c t - - I n this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant. @ 2004 Elsevier Ltd. All rights reserved. Keywords--General tems, Detectability.

observers, Exponential observers, Nonlinear observers, Discrete-time sys-

1. I N T R O D U C T I O N The problem of designing observers for nonlinear systems was introduced by Thau [1]. Over the past three decades, there has been significant attention paid, in control systems literature, to the construction of observers for nonlinear systems [2-9]. In this paper, we extend the definition of observers for linear systems [10] in a natural way to define general observers (asymptotic and exponential) for discrete-time nonlinear autonomous systems. The observer design for discrete-time nonlinear systems is very useful in applications. We establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. As a corollary of this result, we show that for the discrete-time nonlinear systems with real parametric uncertainty whose state equilibrium does not change with the parameter values and whose output function is purely a function of the state, there does not exist any general observer (asymptotic or exponential). Next, for Lyapunov stable discrete-time nonlinear systems, we use the center manifold theory [11] to derive necessary and sufficient conditions for general exponential observers. As an application of

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V. SUNDARAPANDIAN

this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues (the eigenvalues with unit modulus) of the linearization matrix of the state dynamics of the plant. This paper is organized as follows. In Section 2, we give a basic definition of general observers (asymptotic and exponential) for discrete-time nonlinear systems. In Section 3, we present a necessary condition for general asymptotic observers, and as an application of this result, we derive a new result on the existence of general asymptotic observers for discrete-time nonlinear systems with real parametric uncertainty. In Section 4, we present necessary and sufficient conditions for general exponential observers of Lyapunov stable discrete-time nonlinear systems, and we also discuss an interesting application of this result.

2. D E F I N I T I O N OF G E N E R A L O B S E R V E R S FOR DISCRETE-TIME NONLINEAR SYSTEMS In this paper, we consider a nonlinear plant of the form, x(k +

1)

= f (x (k)),

(1)

y (k) = h (x (k)),

where x C ] ~ is the state and y E ]~P the output of plant (1). T h e state x belongs to an open neighborhood X of the origin of ] ~ . We assume t h a t f : X --+ ] ~ is a C 1 map, and also t h a t f ( 0 ) = 0. We also assume t h a t the output mapping h : X --~ ]~P is a C 1 map, and also t h a t h(0) = 0. Let Y ~= h ( X ) . The general observer design problem is to find a nonlinear system, z (k +

1)

-- g (z ( k ) , y (k)),

[z E ~ m ] ,

so t h a t w = q(z) E ] ~ serves as an estimate of the state x of plant (1). Here, g and q are C 1 mappings with g(0, 0) = 0 and q(0) = 0. Explicitly, we have the following definition. DEFINITION 1. Consider a discrete-time nonlinear system described by

z (k + 1) = 9 (z (k), u (k)),

[z e Sm],

(2)

where z is defined in a neighborhood Z of the origin of ]~m and g : Z × Y --~ R m is a C 1 map with g(O, O) = O. Consider also the map q : Z -~ R n defined by = q

(3)

where q is a C 1 map with q(O) = O. Then, candidate system (2) is called a general asymptotic (respectively, general exponential) observer for plant (1) corresponding to (3), ff the following two requirements hold.

(01) I[w(O) = x(O), then, w(k) = x ( k ) , V k e Z+, where Z+ is the set of all positive integers. (02) There exists a neighborhood V of the origin of ] ~ , such that, for all w(O) - x(O) E V, the estimation error w( k ) - x ( k ) tends to zero asymptotically (respectively, exponentially) as k -~ ce. | If a general exponential observer (2) satisfies the additional properties t h a t m = n and q is a C 1 diffeomorphism, then, it is called a full-order general exponential observer. A full-order general exponential observer (2) with the additional p r o p e r t y t h a t q = i d x is called an identity exponential observer which is the same as the standard definition of local exponential observers for nonlinear systems. |

New Results on General Observers

1417 /

The estimation error e is defined by

e

q(z)-x.

Now, we consider the composite system, x ( k + 1) = f ( x ( k ) ) , z (k + 1) = g (z (k)) , h (x (k)) .

(4)

Next, we state a simple lemma, which provides a geometric characterization of Condition (O1) in Definition 1, and which may be easily established, as in [12]. LEMMA 1. The following statements axe equivalent. (a) Condition (01) in Definition 1 holds for composite system (1,2). (b) The submanifold defined via q(z) = x is invariant under the flow of composite system (4). | 3. A N E C E S S A R Y CONDITION FOR GENERAL ASYMPTOTIC OBSERVERS In this section, we first prove a necessary condition for plant (1) to have general asymptotic observers. THEOREM 1. A necessary condition for the existence of a general asymptotic observer for discretetime plant (1) is that plant (1) is asymptotically detectable, i.e., any state trajectory x(k ) of the plant dynamics in (1) with small initial condition Xo, satisfying h(x(k) ) -- O, must be, such that x ( k ) ~ O,

as k --* c~.

PROOF. Let (2) be a general asymptotic observer for plant (1). Then, Requirements (O1) and (02) in Definition 1 hold. Now, let x(k) be any state t r a j e c t o r y of plant dynamics with small initial condition x0, satisfying y(k) -= h(x(k)) =- O. Then, the observer dynamics reduces to

z (k + 1) = g (z (k), 0).

(5)

Taking zo = 0, it is immediate from (5) that z(k) = z(k; zo) - O. Hence, w(k) = q(z(k)) =- O. By Condition (O2), we know that the estimation error trajectory e(k) = w(k) - x(k) tends to zero as k ~ oc. Since w(k) - 0, it is now immediate that x(k) --* 0, as k --* ec. This completes the proof. | Using Theorem 1, we can prove the following result, which says that there is no local asymptotic observer for the discrete-time nonlinear plant,

(k + 1) = F (k + 1) =

(k),

(k)),

(k),

(6)

y (k) = h ( x ( k ) ) , if the equilibrium x = 0 does not change with the disturbance or the real parametric uncertainty A, i.e., if F(0, ,k) _= 0. (Note that in this model [13], we consider both x and A as states, so that the estimation is carried out for both x and A.) THEOREM 2. Suppose that plant (6) satisfies the assumption, F (0,)~) ---0, i.e., x = 0 is an equilibrium of the state dynamics, for all values of the disturbance A, and also that the output function y is purely a function of x, i.e., it has the form, y = h (x). Then, t.here is no general asymptotic observer for plant (6).

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V. SUNDARAPANDIAN

PROOF. This is an immediate consequence of Theorem I. We show that plant (6) is not asymptotically detectable. This is easily seen by taking x(0) = x0 = 0 and A(0) = A0 ¢ 0. Then, we have

y (k) = h (x (k)) - 0

and

x (k) - 0,

but A(k) _-- A0 ¢ 0. Hence, plant (6) is not asymptotically detectable. From Theorem 1, we deduce that there is no local asymptotic observer for plant (6). This completes the proof. | 4. N E C E S S A R Y FOR GENERAL

AND SUFFICIENT EXPONENTIAL

CONDITIONS OBSERVERS

In this section, we establish a basic theorem that completely characterizes the existence of general exponential observers of form (2) for Lyapunov stable discrete-time nonlinear plants of form (i). For this purpose, we define

C = Dh(O)

and

A = Df(O),

(7)

i.e., (C, A) is the system linearization pair for the given nonlinear plant (1). Also, define E=~zOg (0, 0)

and

0g (0, 0) K = ~yy

(s)

Now, we state and prove the following result giving a characterization of the general exponential observers for Lyapunov stable discrete-time nonlinear systems. THEOREM 3. Suppose that the plant dynamics in (1) is Lyapunov stable at x = O. Then, system (2) is a general exponential observer for plant (1) with respect to (3) if, and only if, (a) the submanifold defined via q(z) = x is invariant under the flow of the composite system (4); (b) the dynamics

z (k + 1) = g (z (k), 0)

(9)

are 1ocally exponentially stable at z = 0. PROOF. Necessity. Suppose that system (2) is a local exponential observer for plant (1). Then, Conditions (O1) and (02) in Definition 1 are readily satisfied. By Lemma 1, Condition (O1) implies Condition (a). To see that Condition (b) also holds, take x(0) = 0. Then, x(k) - O, and y(k) = h(x(k)) -- 0, for all k E Z+. Now, the dynamics for z become

z (k + 1) =

(z (k), y (k)) = g (z (k), o).

(10)

By Condition (02) in Definition 1, it is immediate that e(k) = q(z(k)) - x(k) = q(z(k)) --~ 0 exponentially, as k ~ oc, for all small initial conditions zo. Hence, we must have z(k) -~ 0 exponentially, as k --* oo, for the solution trajectory z(k) of the dynamics (10). Hence, we conclude that the dynamics (10) is locally exponentially stable at z = 0. Thus, we have established the necessity of Conditions (a) and (b). Sufficiency. Suppose that Conditions (a) and (b) hold for plants (1) and (2). Since Condition (a) implies Condition (Ot) in Definition 1 by Lemma 1, it suffices to show that Condition (02) in Definition 1 also holds. By hypotheses, the equilibrium e = 0 of dynamics (9) is locally exponentially stable, and the equilibrium x = 0 of the plant dynamics in (1) is Lyapunov stable. Hence, matrix E = o°-~(0, 0) must be convergent, i.e., it has all the eigenvalues inside the open unit disc of the complex plane, and matrix A = Df(O) must have all eigenvalues in the closed unit disc of the complex plane. We have two cases to consider.

New Results on General Observers

1419

i

C A S E I : A is C O N V E R G E N T .

By Hartman-Grobman Theorem for maps, it follows that composite system (4) is locally topologically conjugate to the system

Ax(k),

x (k + 1) :

z (k + Note

1) =

(11)

E z (k) + K C x (k).

that, [ A eig K C

0 ] = e i g ( A ) Ueig(E). E

Since both A and E are convergent matrices, it is immediate that x(k) and e(k) tend to zero exponentially, as k -4 oo. Hence, it follows trivially that e(k) = q(z(k)) - x(k) --~ O, as k -4 ec, for all small initial conditions z(0) and x(0). CASE II : A Is NOT CONVERGENT. Without loss of generality, we can assume that the plant dynamics in (1) has the form,

x2 (k + 1)

L A2x2 (k) + ¢2 (Xl ( k ) , X 2 (It)) J '

f2 (Xl ( k ) , X 2 (It))

(12)

where xl E R ~1, x2 E R n~ (nl + n2 = n), A1 is an nl x nl matrix having all eigenvalues with unit modulus, A2 is a n2 x n2 convergent matrix, and ¢1 and ¢2 are C 1 functions vanishing at (Xl, x2) = (0, 0) together with all their first-order partial derivatives. Now, x = 0 is a Lyapunov stable equilibrium of x(k + 1) = f ( x ( k ) ) . Also, z = 0 is a locally exponential stability equilibrium of dynamics (9). Hence, by a total stability result, it follows that: (x, z) = (0, 0) is a Lyapunov stable equilibrium of composite system (4) (by its triangular structure). Also, by the center manifold theorem for maps [11], we know t h a t composite system (4) has a local center manifold at (x, z) = (0, 0), the graph of a C 1 map,

= zr (xl) --

~2 (Xl)J "

Since composite system (4) is Lyapunov stable at (x, z) = (0, 0), we know that center manifold (13) is unique [14]. Since q(z) = x is an invariant manifold for composite system (4), it is immediate that along the center manifold, we have q (7r2 (Xl)) =

71"1 (Xl)

•

By the principle of asymptotic phase in the center manifold theory, there exists a neighbourhood V of (x, z) = (0, 0), such that, for all (x(0), z(0)) e V, we have

z (k) -

(k)) j

-

[ z (0) -

(xl (0))

'

for some positive constants M and a with 0 < a < 1. Hence, it is immediate that ext)onentially, as k -4 oc.

Z (k) - 4 71"2 (X 1 ( k ) ) ,

(16)

From (14) and (16), it follows that

[

zl(k)

q (Z (It)) --+ [71.1 (X 1 (]~))

] ,

exponentially, as k ---* e~.

(17)

1420

V. SUNDARAPANDIAN

From (15) and (17), it follows that

q (z (k))

[Xl (k)]

Ix2 (k) '

exponentially, as k -~ cx).

Thus, Condition (O2) also holds. This completes the proof.

(18)

|

REMARK 1. There is a necessary condition for general exponential observers of form (2) for Lyapunov stable discrete-time plants of form (1) which is implicitly contained in the proof of Theorem 3. This necessary condition is that dim (z) > nl, i.e., the dimension of the state of the candidate observer cannot be lower than the number of critical eigenvalues of the linearization matrix A = Dr(O) of the plant dynamics in (1). This necessary condition is a simple application of equation (14) in the proof of Theorem 3,

q(Tr2 (Xl)) :

[x l 7rl(Xl )

•

Xl

For every small xl E I~nl, we know that vector [rl(x~)] has a pre-image 7r2(x1) under the mapping, q. Hence, it is immediate that the dimension of the domain of the mapping, q, cannot be lower than the dimension of state xl. |

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