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Observer Design for a General Class of Nonlinear Systems

The paper deals with the observer design problem for a wide class of triangular nonlinear time-varying systems. The results of the present work generalize previous results in the literature dealing with the observer design problem for triangular systems and particularly, constitute generalizations of those obtained in a recent authors work concerning a class of triangular systems, whose dynamics have p-normal form.


INTRODUCTION
We derive sufficient conditions for the solvability of the observer design problem for timevarying single output systems of the forṁ x i = f i (t, x 1 , ..., x i+1 ), i = 1, ..., n − 1 x n = f n (t, x 1 , ..., x n ), (1.1a) where the functions f i (·), i = 1, 2, ..., n, are continuous and (locally) Lipschitz. It is known from [5] that every single output control system which has a uniform canonical flag ([5, Chapter 2, Definition 2.1]) can be locally transformed in the above canonical form (1.1) for each fixed input. We also mention the works [4] and [8] where algebraic type necessary and sufficient conditions are established for feedback equivalence between a single input system and a triangular system whose dynamics have p-normal form. In our recent work [3], the observer design problem is studied for a subclass of systems (1.1) whose dynamics have p-normal form. The result of present work constitutes a generalization of previous results in the literature dealing with the observer design problem for triangular systems (see for instance [1], [5], [6] and relative references therein) and particularly generalizes the main result of the recent author's work in [3].
The paper is organized as follows. We first provide notations and various concepts, including the concept of the switching observer that has been originally introduced in [2], for general time-varying systems:ẋ = f (t, x), (t, x) ∈ R ≥0 × R n (1.2a) where y(·) plays the role of output. We then provide the precise statement of our main result (Theorem 1.1) concerning solvability of the observer design problem for (1.1). Section II contains some preliminary results concerning solvability of the observer design problem for the case (1.2) with linear output (Propositions 2.1 and 2.2). In Section III we use the results of Section II, in order to prove our main result.
Notations and definitions: We adopt the following notations. For a given vector x ∈ R n , x denotes its transpose and |x| its Euclidean norm. We use the notation |A| := max{|Ax| : x ∈ R n ; |x| = 1} for the induced norm of a matrix A ∈ R m×n . By N we denote the class of all increasing C 0 functions φ : R ≥0 → R ≥0 . For given R > 0, denote by B R the closed ball of radius R > 0, centered at 0 ∈ R n . Consider a pair of metric spaces X 1 , X 2 and a set-valued map X 1 x → Q(x) ⊂ X 2 . We say that Q(·) satisfies the Compactness Property (CP), if for every sequence (x ν ) ν∈N ⊂ X 1 and (q ν ) ν∈N ⊂ X 2 with x ν → x ∈ X 1 and q ν ∈ Q(x ν ), there exist a subsequence (x ν k ) k∈N and q ∈ Q(x) such that q ν k → q. We also invoke the well known fact, see [7], that the time-varying system (1.2a) is forward complete, if and only if there exists a function β ∈ N N such that the solution x(·) := x(·, t 0 , x 0 ) of (1.2a) initiated from x 0 at time t = t 0 satisfies: provided that the dynamics of (1.2a) are C 0 and Lipschitz on x ∈ R n . It turns out that, under these regularity assumptions plus forward completeness for (1.2a), for each t 0 ≥ 0 and x 0 ∈ R n the corresponding output y(t) = h(t, x(t, t 0 , x 0 )) of (1.2) is defined for all t ≥ t 0 . For each t 0 ≥ 0 and nonempty subset M of R n , we may consider the set O(t 0 , M ) of all outputs of (1.2), corresponding to initial state x 0 ∈ M and initial time t 0 ≥ 0: For given ∅ = M ⊂ R n , we say that the Observer Design Problem (ODP) is solvable for (1.2) with respect to M , if for every t 0 ≥ 0 there exist a continuous map G := G t0 (t, z, w) : [t 0 , ∞) × R n × Rn → R n and a nonempty setM ⊂ R n such that for every z 0 ∈M and output y ∈ O(t 0 , M ) the corresponding trajectory z(·) := z(·, t 0 , z 0 ; y); z(t 0 ) = z 0 of the observeṙ z = G(t, z, y) exists for all t ≥ t 0 and the error e(·) := x(·) − z(·) between the trajectory x(·) := x(·, t 0 , x 0 ), x 0 ∈ M of (1.2a) and the trajectory z(·) := z(·, t 0 , z 0 ; y) of the observer satisfies: lim t→∞ e(t) = 0 (1.5) We say that the Switching Observer Design Problem (SODP) is solvable for (1.2) with respect to M , if for every t 0 ≥ 0 there exist a strictly increasing sequence of times (t k ) k∈N with t 1 = t 0 and lim k→∞ t k = ∞, a sequence of continuous mappings and a nonempty setM ⊂ R n such that the solution z k (·) of the systemż with initial z(t k−1 ) ∈M and output y ∈ O(t 0 , M ), is defined for every t ∈ [t k−1 , t k+1 ] and in such a way that, if we consider the piecewise continuous map Z : where for each k ∈ N the map z k (·) denotes the solution of (1.6), then the error e(·) := x(·) − Z(·) between the trajectory x(·) := x(·, t 0 , x 0 ), of (1.2a) and Z(·) satisfies (1.5).
Our main result is the following theorem.
(ii) if in addition we assume that it is a priori known, that the initial states of (1.1) belong to a given ball B R of radius R > 0 centered at zero 0 ∈ R n , then the ODP is solvable for (1.1) with respect to B R . Theorem 1.1 constitutes a generalization of Theorem 1.1 in [3] for systems (1.1), whose dynamics are C 1 and have the particular form f i (t, where the constants m i , i = 1, ..., n − 1 are odd integers and the functions a i (·, ·), i = 1, ..., n − 1 satisfy the condition |a i (t, y)| > 0, ∀t ∈ R ≥0 , y ∈ R.

PRELIMINARY RESULTS
The proof of our main result concerning the case (1.1), is based on some preliminary results concerning the case of systems (1.2) with linear output: where H : R ≥0 → Rn ×n is C 0 and F : R ≥0 ×R n ×Rn → R n is C 0 and Lipschitz on (x, y) ∈ R n × Rn. We assume that system (2.1a) is forward complete, namely, the solution x(·) := x(·, t 0 , x 0 ) of (2.1a) satisfies (1.3) for certain β ∈ N N , hence, for every R > 0 and t ≥ 0 we can define: where H(·) is given in (2.1b). Obviously, Y R (t) = ∅ for all t ≥ 0 and, if (1.3) holds, the set-valued map [0, ∞) t → Y R (t) ⊂ Rn satisfies the CP and further y(t) ∈ Y R (t), for every t ≥ t 0 ≥ 0 and y ∈ O(t 0 , B R ); (see notations). Also, given integers , m, n,n ∈ N and a map A : R ≥0 × R × R n × R m × Rn → R m×m , we say that A(·, ·, ·, ·, ·) satisfies Property P1, if the following holds: P1. A(t, q, x, e, y) has the form A(t, q, x, e, y) := (A C1 (t, q, x, e 1 , y), A C2 (t, q, x, e 1 , e 2 , y), · · · , A Cm (t, q, x, e 1 , ..., e m , y)) (2.3) where each mapping A Ci : .., e i ) ∈ R i : e i = 0} × Rn and bounded on every compact subset of We make the following hypothesis: There exist a function g ∈ C 1 (R ≥0 ; R) satisfying: such that the following properties hold: A1. For every ξ ≥ 1 there exists a set-valued map with Q R (t) = ∅ for any t ≥ 0, satisfying the CP and such that with Y R (·) as given by (2.2). A2. For every ξ ≥ 1 there exists a set-valued map Q R := Q R,ξ as in Hypothesis A1, in such a way that for every t 0 ≥ 0, a time-varying symmetric matrix P R := P R,ξ,t0 ∈ C 1 ([t 0 , ∞); R n×n ) and a function d R := d R,ξ,t0 ∈ C 0 ([t 0 , ∞); R) can be found, satisfying: with Y R (·) as given by (2.2).
The following result, constitutes a slight generalization of Proposition 2.1 in [3].
The following proposition constitutes a slight generalization of Proposition 2.2 in [3]. It establishes sufficient conditions for the existence of a switching observer exhibiting the state determination of (2.1), without any a priori information concerning the initial condition. We make the following hypothesis: Hypothesis 2.2: There exist constants L > 1, c 1 , c 2 > 0 such that (2.5) holds, an integer ∈ N, a function g ∈ C 1 (R ≥0 ; R) satisfying (2.4) and a map A : R ≥0 ×R ×R n ×R n ×Rn → R n×n satisfying P1, in such a way that for every R > 0 Hypothesis 2.1 is fulfilled, namely, both A1 and A2 hold.

Proposition 2.2:
In addition to the hypothesis of forward completeness for (2.1a), assume that system (2.1) satisfies Hypothesis 2.2. Then the SODP is solvable for (2.1) with respect to R n .
The proofs of Propositions 2.1 and 2.2 and the main result of Theorem 1.1 in the next section, are based on a preliminary technical result (Lemma 2.1 below) which constitutes a slight modification of the corresponding result of Lemma 2.1 in [3]. Let , m, n,n ∈ N and consider a pair (H, A) of mappings: where H(·) is continuous and A(·, ·, ·, ·, ·) satisfies Property P1. We make the following hypothesis: ) and assume that for certain constant R > 0 and for every ξ ≥ 1, there exist a function β R := β R,ξ ∈ N and set-valued mappings (2.13) Lemma 2.1: Consider the pair (H, A) of the time-varying mappings in (2.12) and assume that Hypothesis 2.3 is fulfilled for certain R > 0. Then for every ξ ≥ 1, (2.14) 3. PROOF OF THEOREM 1.1 In this section we apply the results of Section II to prove our main result concerning the solvability of the SODP(ODP) for triangular systems (1.1).
We have shown that all requirements of Claim 1 hold, which, as was pointed out, establishes that for every R > 0 Hypothesis A2 is fulfilled. We therefore conclude, that for system (1.1) Hypothesis 2.2 is satisfied hence, by invoking the result of Proposition 2.2, it follows that the SODP is solvable for (1.1) with respect to R n . The establishment of the second statement of Theorem 1.1, follows directly from Proposition 2.1.
Example: As an illustrative example of Theorem 1.1, consider the two dimensional polynomial system:ẋ It is easy to check that system (3.51) satisfies all conditions of Theorem 1.1, therefore the SODP is solvable for (3.51) with respect to R 2 .