ThC07.4
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
OBSERVER DESIGN FOR NONLINEAR SYSTEMS BY USING INPUT-TO-STATE STABILITY A. Alessandri
Abstract— The problem of constructing full-order state observers for a class of systems with Lipschitz nonlinearities is addressed. By performing a suitable decomposition of the estimation error dynamics into cascaded systems, conditions have been found that guarantee the asymptotic stability of the estimation error in the absence of disturbances. These conditions can be conveniently expressed by means of Linear Matrix Inequalities (LMIs). In the presence of system and measurement perturbations, when such noises are regarded as unknown deterministic inputs acting on the error dynamics, the estimator can be designed so as to be Input-to-State Stable (ISS) with respect to the estimation error.
I. I NTRODUCTION There exists a vast literature on state observers and filters for nonlinear systems as they play crucial roles in control theory. The difficulties in dealing with observer problems for general enough nonlinear systems regard the proof of global convergence of the estimation error. Similar issues also concern filtering problems, as nonlinearities are difficult to overcome when one has to derive probabilistic convergence properties. The first attempts to construct observers for nonlinear continuous-time systems were reported in [1], [2]. Later on, special attention was given to approaches that, by performing a state transformation, provide a linear error dynamics in the transformed space [3], [4], [5], [6]. Slidingmode observers were proposed in [7], [8]. A different observer structure was suggested in [9], where the nonlinearity in the error dynamics is compensated for by choosing a sufficiently high gain [10]. Thus, such estimators are usually called high-gain observers and extensively used in output feedback control (see, among others, [11]). A similar approach to high-gain observer design was followed in [12]. Recent researches on this topic have concerned the development of observers that do not require state-space transformation (see, e.g., [13], [14]). In addition, in [15] and [16], the observers are studied under the effect of disturbances by using Input-to-State Stability (ISS), when the noises are seen as the input and the estimation error as the state. In this context, the present work reports new results on the stability of the estimation error of a class of observers for systems with Lipschitz nonlinearities without and with the presence of disturbances. The convergence of the estimation error is established by relying on the ISS properties of cascaded systems (see [17] for an introduction). The resulting state estimators can be designed by solving LMI problems [18]. A. Alessandri is with the Institute of Intelligent Systems for Automation, ISSIA-CNR National Research Council of Italy, Via De Marini 6, 16149 Genova, Italy
[email protected]
0-7803-8682-5/04/$20.00 ©2004 IEEE
The paper is organized as follows. Section II gives some basic definitions and introduces the notation. Section III deals with the description of the proposed observer, its stability analysis in the absence of disturbances, and its design based on LMIs. The case when disturbances act on the dynamics and measurement equations is considered in Section IV. Section V illustrates the results obtained by simulations. Finally, conclusions and prospects for future work are briefly discussed in Section VI. II. N OTATION AND DEFINITIONS The following notation and definitions will be used throughout the paper: – · is the Euclidean norm in Rn , for every n ∈ N+ ; – for a square matrix S , S > 0 ( S < 0 ) means that this matrix is positive definite (negative definite); – I denotes the identity matrix; – λmin (S) and λmax (S) are the minimum and maximum eigenvalues of the symmetric positive or negative definite matrix S, respectively; – the norm of a matrix U is U = λmax ( U T U ) ; if U is symmetric positive definite, U = λmax (U ); – a continuous function α : [0, a) → [0, +∞) is said to belong to class K if it is strictly increasing and α(0) = 0. It is said to belong to class K∞ if a = +∞ and lim α(r) = +∞; r→+∞
– a continuous function β : [0, a) × [0, +∞) → [0, +∞) is said to belong to class KL if, for each fixed s, the mapping β(r, s) belongs to class K with respect to r and, for each fixed r, the mapping β(r, s) is decreasing with respect to s and lim β(r, s) = 0; s→+∞ – the Schur lemma can be expressed as follows: R S > 0 is equivalent to T > 0 , R−S T −1 S T > ST T 0 as well as to R > 0 , T − S T R−1 S > 0, where R , S , and T are matrices of appropriate dimensions. III. O BSERVERS FOR A CLASS OF NOISE - FREE NONLINEAR SYSTEMS
Consider a multivariable nonlinear system represented by x˙ = A x + f (x) y =Cx
,
t≥0
(1)
where x(t) ∈ X ⊆ Rn is the state vector, y(t) ∈ Y ⊆ Rp is the output vector; moreover, the matrices A ∈ Rn×n and C ∈ Rp×n are such that the pair (A, C) is observable. We need to assume that f (·) is locally Lipschitz to ensure the
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uniqueness of the solution of the differential equation in (1). Moreover, we suppose the solution well-defined ∀t ≥ 0. Remark 1. In the following, we shall refer to system equations (1); however, the methods presented later on for observer design can be applied to a more general class of nonlinear observable systems that are diffeomorphic to (1). Specifically, in [19] necessary and sufficient conditions are provided to ensure the existence of a diffeomorphism that transforms quite a general nonlinear system into a system like (1) with A
=
Ai
=
C and
block diag (A1 , · · · , Ap ) ⎛ ⎞ 0 1 0 ⎜ 0 0 1 ⎟ ⎜ ⎟ ⎜ .. .. .. . . ⎟ ⎜ . . . ⎟ ∈ Rni ×ni . ⎜ ⎟ ⎝ 0 0 0 ... 0 1 ⎠ 0 0 0 ... 0 0 block diag (C1 , · · · , Cp ) Ci = (1 0 . . . 0) ∈ Rni
= p
ni = n. Thus, an observer for (1) becomes a
i=1
convergent estimator in the original state space if one uses the inverse of the aforesaid diffeomorphism. A full-order state observer for (1) is given by x ˆ˙ = A x ˆ + f (ˆ x) + L (y − C x ˆ)
,
t≥0
(2)
where x ˆ(t) ∈ Rn is the estimate of x(t) at time t and n×p L∈R is a gain matrix. In order to find a gain L such ˆ(t) that the corresponding estimation error e(t) = x(t) − x is stable, consider the error dynamics e˙ = (A − LC) e + f (x) − f (ˆ x)
,
Theorem 1. Suppose to choose a gain matrix L such that A−LC is Hurwitz and assume that, in spite of x(t) given by (1), the resulting Σ1 for e2 = 0 is globally asymptotically stable in e1 = 0 as an equilibrium point and that there exists a compact set to which e1 (t) belongs when e2 (t) tends to zero. Then e1 (t) converges to zero globally. Proof. As A − LC is Hurwitz, e2 (t) → 0 for any initial condition e2 (0). Since, for assumption, e1 = 0 is an asymptotically stable equilibrium point for Σ1 with an arbitrary large domain of attraction in Rn and e1 (t) belongs to a compact set when e2 (t) → 0, we conclude that e1 (t) → 0 for any initial condition using [22][Theorem 1, p. 313].
Theorem 1 is in quite a general form, which is not wellsuited to being applied. To allow a more convenient way of using such results, we need to assume the following. Assumption 1. The function f : X → Rn is Lipschitz in x, i.e., there exists kf > 0 such that f (x1 ) − f (x2 ) ≤ kf x1 − x2 , ∀x1 , x2 ∈ X
t≥0
and decompose e(t) into two components, e1 (t) ∈ Rn and e2 (t) ∈ Rn , such that e(t) = e1 (t) + e2 (t) and Σ1 : Σ2 :
In general, even if Σ1 is asymptotically stable with an input equal to zero, the asymptotic stability of Σ2 is not sufficient to guarantee the global stability of the overall system. The stability of cascaded systems is related to the Converging-Input-Bounded-State (CIBS) property [20], [21], [22]. If Σ1 with a zero input and Σ2 are globally asymptotically stable and Σ1 is CIBS, then the cascade of Σ1 and Σ2 is globally asymptotically stable. The foregoing can be summarized as follows.
e˙ 1 = (A − LC) e1 + f (x) − f (x − e1 − e2 ) e˙ 2 = (A − LC) e2
(3)
where e1 (0) = 0, e2 (0) = e(0), and t ≥ 0. Hence the stability of the observer can be studied by considering the cascade of the subsystems Σ1 and Σ2 , as shown in Fig. 1.
.
Before proving Theorem 2, which results form the specific application of Theorem 1 to systems like (1) that satisfies Assumption 1, we need some technical lemmas. Lemma 1. Let Assumption 1 hold and consider e˙ 1 = (A − LC) e1 + f (x) − f (x − e1 − e2 ) , t ≥ 0 (4) where e1 (t) ∈ Rn with e1 (0) = 0 and e2 (t) ∈ Rn are regarded as state and input vectors, respectively. Then, in spite of x(t) given by (1), if e2 (t) is bounded and one can find α > 0, a gain matrix L, and a symmetric positive definite matrix P such that P (A − LC)T P + P (A − LC) + α kf2 I < 0, P −αI (5)
Fig. 1.
Cascade of systems describing the error dynamics.
there exists a compact set K such that e1 (t) ∈ K, ∀t ≥ 0.
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Proof. Consider the Lyapunov function V = eT 1 P e1 . The derivative of V is T V˙ = eT 1 (A − LC) P + P (A − LC) e1 T
+ 2 [f (x) − f (x − e1 − e2 )] P e1 .
T
2 [f (x) − f (x − e1 − e2 )] P e1
T ≤ 2 [f (x) − f (x − e1 − e2 )] P e1 2f (x) − f (x − e1 − e2 ) P e1 2kf e1 + e2 P e1
≤ =
2kf (e1 + e2 ) P e1 2kf e1 P e1 + 2kf e2 P e1
Using the Young inequality, we have 1 T 2 2kf e1 P e1 ≤ e1 α kf I + P P e1 α where α is any positive real constant and hence from (6) we have V˙ ≤ −eT 1 Qe1 + 2kf e2 P e1 ≤ −eT 1 Qe1 + 2kf e2 P e1
T
2 [f (x) − f (x − e1 )] P e1 1 2 α k ≤ eT I + P P e1 1 f α
(6)
The second term in the right-hand side of (6) can be upper bounded as follows:
≤ ≤
As in the proof of Lemma 1, the Young inequality yields
(7)
where 1 PP . α Using the Schur lemma, Q < 0 is equivalent to (5). As, for some M > 0, e2 (t) ≤ M ∀t ≥ 0 and (5) ensures that Q is negative definite, (7) yields
for any positive real constant α, thus obtaining from (11) T 2 V˙ ≤ eT 1 (A − LC) P + P (A − LC) + α kf I 1 + P P e1 . α Therefore, the dynamics of the estimation error is stable if there exists P = P T > 0 such that 1 (A − LC)T P + P (A − LC) + α kf2 I + P P < 0 . α Using the Schur lemma in the last inequality, V˙ turns out to be negative definite by imposing (10).
Theorem 2. Consider observer (2) for system (1) and let Assumption 1 hold; then, if, in spite of x(t) given by (1), there exist α > 0, a gain matrix L, and a symmetric positive definite matrix P such that (A − LC)T P + P (A − LC) + α kf2 I P 0, a gain matrix L, and a symmetric positive definite matrix P such that P (A − LC)T P + P (A − LC) + α kf2 I
