Observability for deterministic systems and high-gain observers

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1 Observability for deterministic systems and high-gain observers design. Part 1. March 29, 2011

2 Introduction and problem description Definition of observability Consequences of instantaneous observability ms

3 Systems considered We consider a model described by a differential equation in the form of ẋ = f (x, t), y = h(x, t), with x in R n, y the measured output in R p. For all x 0 in a given open set denoted O of R n, a unique solution starting from x 0 at t = t 0 denoted X (x 0, t 0, t). The time domain in which this solution is in O is an open time interval containing t 0 and is denoted ]σ O,t 0 (x 0 ), σ + O,t 0 (x 0 )[.

4 Some other cases We may consider also: 1. The time invariant case ẋ = f (x), y = h(x). 2. The controlled case (in Part 2) ẋ = f (x, u(t)), y = h(x, u(t)).

5 An induced output mapping For all initial point, we have an output trajectory P O,t0 : O C 0 ([0, σ + t (x 0,O 0)[, R p ) x 0 (t h(x (x 0, t 0, t)))

6 The estimation problem Construct an inverse to this mapping! For all t > 0, we wish to construct a map: ( y[0,t] (s), u [t0,t](s) ) x 0 R n. Note that by uniqueness of solutions, once x 0 is known, the entire trajectory can be found by integration. Consequently, it is equivalent with knowing X (x 0, t 0, t).

7 State observer An observer is an Algorithm! Knowing the existence of a solution is good, but we need to COMPUTE it. Exact estimation for all t is too difficult! We can compute an estimate ˆX (t) such that ˆX (t) X (x 0, t 0, t) goes to zero as long as the trajectory exists and for all initial conditions (x 0, t 0 ). In this lecture we give a particular solution based on an observability property.

8 Definition of observability Consequences of instantaneous observability Introduction and problem description Definition of observability Consequences of instantaneous observability ms

9 Definition of observability Consequences of instantaneous observability Distinguishability definition Definition (Distinguishability) Two initial states x a and x b both in O an open set of R n are said to be indistinguishable in O at t 0 if the output time functions of the solution starting from these points at t = t 0 are the same as long as both solutions remain { in O. In other words, } if we denote σ min = min σ + O,t 0 (x a ), σ + O,t 0 (x b ) then we have h(x (x a, t 0, t)) = h(x (x b, t 0, t)), t [t 0, σ min ). Otherwise, they are said to be distinguishable in O at t 0.

10 Definition of observability Consequences of instantaneous observability Observability definition Definition (Observability) The system is said to be observable in O if all x a and x b in O are distinguishable at t 0 in O for all t 0 in R. When O is a neighborhood of a certain point x 0 in R n then the system is said to be locally observable at x 0. When O = R n, the system is said to be globally observable.

11 Definition of observability Consequences of instantaneous observability Instantaneous observability Definition (Instantaneous observability) The system is locally instantaneously observable at x 0 if there exists O an open neighborhood such that for all open subset I O containing x 0 the system is observable in I. If for all x 0 in R n the system is instantaneously observable at x 0, the system is locally instantaneously observable. If O = R n then the system is globally instantaneously observable. x a and x b in { O with x a x b then } for all t 0 and all t 0 < δ < min σ + t (x 0,O a), σ + t (x 0,O b) ) there exists t 0 < t d < δ such that h(x (x a, t d, t 0 )) h(x (x b, t d, t 0 )).

12 Definition of observability Consequences of instantaneous observability Globally instantaneously observable Globally observable Locally instantaneously observable Locally observable

13 Definition of observability Consequences of instantaneous observability Introduction and problem description Definition of observability Consequences of instantaneous observability ms

14 Definition of observability Consequences of instantaneous observability Assume the system is analytic and time invariant. The time function y(x, t) = h(x (x, t)), is an analytic function of the time. Consequently, for all x 0 there exists an open set O 0 such that for all t in (σ O 0 (x 0 ), σ + O 0 (x 0 )) and all x in O 0, y(x, t) = h 1 (x) + + j=1 h j (x) tj j! with, h 1 (x) = h(x), h j+1 (x) = j y (x, 0). tj

15 Definition of observability Consequences of instantaneous observability Let x a and x b two initial points in O 0 giving rise to the same output function in O 0. Then we have for small t, Hence we have, h 1 (x a ) h 1 (x b ) + + j=1 [h j (x a ) h j (x b )] tj j! = 0. h j (x a ) h j (x b ) = 0, j. We see that to check observability, we need to check injectivity of a mapping denoted: x H (x) = (h 1 (x), h 2 (x),..., ) The state is living in an n-dimensional space! we may have some hopes that injectivity of the map may be obtained without checking to all coefficients.

16 Definition of observability Consequences of instantaneous observability Successive time derivative of the output Given an integer k, we introduce the mapping H k : R n R m which to a point x in R n gives the output and its first k 1 successive time derivatives H k (x) = h 1 (x). h k (x), h 1 (x) = h(x), h j (x) = L f h j 1 (x). Definition (Differential observability of order k) The system satisfies the differential observability property of order k in O if the mapping H k is injective in O.

17 Definition of observability Consequences of instantaneous observability Rank condition Lemma (Sufficient condition for instantaneous observability) Given x 0 in R n, if there exists k > 0 such that the rank of the matrix H k x (x 0) is n (the dimension of x) then the system is locally differentially observable of order k and consequently is locally instantaneously observable at x 0.

18 Definition of observability Consequences of instantaneous observability From local to global Lemma (Global differential observability property) Assume the functions f and h are analytic and p = 1 (only one output). Let O be a connected open subset of R n such that 1. The system is analytic and observable in O; 2. The matrix Hn x (x) is full rank for all x in O. Then the function H n is injective in O and is instantaneously observable in O.

19 Definition of observability Consequences of instantaneous observability Necessity Theorem (Necessity of the rank condition almost everywhere) If the functions f and h are C and if the system is instantaneously observable in an open set O then the matrix Hn x (x 0) has its rank equal to n almost everywhere 1 in O. From this analysis: As we have seen the existence of a positive integer k such that the mapping H k is injective is a sufficient and an almost necessary condition for the instantaneous observability! 1 In an open and dense subset of O.

20 Definition of observability Consequences of instantaneous observability In Conclusion It gives some hint on how to solve the estimation problem: y(t) Estimate the k 1 first time derivatives of the output at each time: Ĥ k (x) = ŷ(t), ẏ(t) ÿ(t)..., y k 1 (t) Apply τ such that τ(h k (x)) = x ) ˆx(t) = τ (ŷ(t), ẏ(t) ÿ(t)..., y k 1 (t)

21 Definition of observability Consequences of instantaneous observability Some comment on this approach However each step of this estimation strategy give rise to some difficulties: 1. Computing the left inverse of a given map H k may be difficult and most of the time rely on optimization procedure which may give some local minima. Indeed, a general expression for a left inverse can simply be given as ˆx = τ(ξ) = Argmin x O ξ H k (x) 2. However, in some example, this approach can give explicite solution. 2. It is very hard to compute the successive time derivatives of the output (especially when there are disturbances in the output).

22 Introduction and problem description Definition of observability Consequences of instantaneous observability ms

23 The framework 1. Only one output. 2. We assume also that there exists an integer m 0 such that the mapping H m is an injective function from an open subset O of R n toward H m (O).

24 Model of the output derivatives Along the solution of the system we have, { { H m (x) = A H m (x) + BL f h m (x) with A = , B =. 0 1.

25 Model of the output derivatives Injectivity in x of the mapping H m means the existence of a a function ϕ m : H m (O) R such that ϕ m (H m (x)) = L f h m (x), x O. Assume we are able to continuously extend ϕ m outside H m (O). ξ = H m (x) is an invariant manifold of the system ẋ = f (x), ξ = Aξ + Bϕm (ξ), Also, ξ 1 = h(x) = y.

26 Canonical observable systems We can focus on estimating the state of the system based on the knowledge of ξ 1. ξ = Aξ + Bϕ m (ξ), y = ξ 1, Given ξ in R m, we denote the solutions of this system initiated from ξ at time t = 0 by Ξ(ξ, t).

27 Introduction and problem description Definition of observability Consequences of instantaneous observability ms

28 Theorem (High-Observer for OCF systems ) Consider system in canonical observability form. Assume that ϕ m (ξ a ) ϕ m (ξ b ) c L ξ a ξ b (ξ a, ξ b ) R m. Then there exists a vector K in( R m and two positive) real number L min > 1 and c 0 such that the solutions Ξ(ξ, t), ˆΞ((ξ, ˆξ), t) of the system { ξ = Aξ + Bϕm (ξ), ˆξ = Aˆξ + Bϕ m (ˆξ) + L(L)K(ˆξ 1 ξ 1 ). where L(L) = Diag(L,..., L m ) are complete in positive time and satisfy for all (ξ 0, ˆξ 0 ) in R 2m and L > L min, Ξ(ξ, t) ˆΞ((ξ, ˆξ), t) 2 c 0 exp( (L L min )(t))l 2m 2 ξ ˆξ 2, t 0.

29 Exact estimation but only asymptotically. However, if ξ is in a known bounded set, then ɛ and t e > 0, we can select L such that Ξ(ξ, t) ˆΞ((ξ, ˆξ), t) ɛ, t > t e. If we know a bounded set in which the trajectory Ξ(ξ, t) remains, it is always possible to modify the model outside this set in order to guarantee a global Lipschitz property. as L increases, the bound on the estimation we get around t 0 goes to infinity. This is the well known picking phenomena.

30 Introduction and problem description Definition of observability Consequences of instantaneous observability ms

31 The system is : ξ = A ξ + Bϕ m (ξ) with A = Which can be rewritten : , B = ξ = A ξ }{{} Chain of integrator part + B ϕ(ξ) }{{} Nonlinearities = Disturbances The idea of the high-gain design can be decomposed into two steps: 1. In a first step we synthesize a robust observer for a linear system. 2. Amplify the convergence and robustness to deal with ϕ m (ξ).

32 Part1: Design of an observer for the linear part Consider ξ = Aξ + v, y = Cξ + w, with C = (1, 0..., 0) and where v and w are disturbances. The couple (A, C) is observable. There exists K such that (A + KC) P + P(A + KC) I d.

33 Part1: Design of an observer for the linear part Lemma (Robust observer for the linear system) There exists positive real numbers c 1, c 2, c 3, c 4, c 5 and c 6 such that for all ξ and ˆξ, if we denote e(t) = Ξ(t) ˆΞ(t) where (Ξ(t), ˆΞ(t)) is a solution of the system ξ = Aξ, ˆξ = Aˆξ + K(C ˆξ Cx) the following two inequalities are satisfied: and { { e(t) Pe(t) c 1 e (t)pe(t) + c 2 w(t) 2 + c 3 v(t) 2 e(t) 2 c 4 exp( c 1 t) e(0) + 1 c 1 { max c5 w(s) 2 + c 6 v(s) 2} 0 s t

34 Part2: Amplification of the robustness and convergence Given L we introduce the matrix L(L) defined as L(L) = Diag(L,..., L m ) With this matrix the idea is to modify the observer as follows ˆξ = Aˆξ + L(L)K(C ˆξ y),

35 Lemma (High-gain Amplification) Assume L > 1, for all ξ and ˆξ both in R m, the time function e(t) = Ξ(t) ˆΞ(t) where (Ξ(t), ˆΞ(t)) is a solution of the system ξ = Aξ, ˆξ = Aˆξ + L(L)K(C ˆξ Cx) satisfies the two inequalities, { { ε (t)pε(t) (L c 1 ) ε (t)pε(t) + c 2 L(L) 1 v(t) 2 + c 3 w(t) 2, where ε = L(L) 1 e and which gives, e(t) 2 c 4 exp( (L c 1 )t)l 2m 2 e(0) 2 + L 2m L c 1 [ max c4 w(s) 2 + c 6 L(L) 1 v(s) 2] s [0,t]

36 Increasing L makes the error going faster toward zero. This inequality highlights the fact that model uncertainties and measurement noises act in a very different manner on the quality of the estimate. 1. About the measurement noises w : As L increase, the term which multiplies w is in L 2m 1 which goes to infinity. high-gain observer behaves very badly with respect to measurement noises. 2. About the model uncertainties v : In this case, is we assume the disturbance acts only on the last component. Then it yields that the term which multiplies v is in 1 L which goes to zero.

37 Introduction and problem description Definition of observability Consequences of instantaneous observability ms

38 It is possible to asymptotically estimates the successive time derivatives of the measured output if 1. The system is differentially observable of order m in O. Hence, the evolution of the successive time derivatives of the output can be described by a system in canonical normal form. 2. The model of the successive output derivative satisfies a global Lipschitz property.

39 About differential observability: This injectivity property says that the map H m is left invertible. With the high gain observer we have H m (X (x, t)) ˆΞ(ξ, ˆξ, t) 0 With injectivity there exists τ such that τ(h m (x)) = x. To get we need τ uniformly continuous. X (x, t) τ(ˆξ(ξ, ˆξ, t)) 0

40 About the global Lipschitz property: The function ϕ m is only precisely defined on the set H m (O). ϕ m (ξ) = L f h m (τ(ξ))), ξ H m (O). Outside the set H m (O) this function has to be synthesized in order to guarantee the global Lipschitz property. If we have: L m f h(x a ) L m f h(x b ) c L H m (x a ) H m (x b ), (x a, x b ) cl(o) 2. then ϕ m can be computed as a Lipschitz extension of the function ϕ m (τ(ξ)).

41 back to the system ẋ = f (x), y = h(x) Theorem (High-gain estimation) If there exist an open set O and a positive real number m such that 1. The mapping H m is uniformly injective in O. In other words, there exists a class K function ρ such that we have the inequality: x a x b 2 ρ( H m (x a ) H m (x b ) 2 ), (x a, x b ) cl(o) 2 2. There exists a positive real number c L such that for all x a and x b in O, L m f h(x a ) L m f h(x b ) c L H m (x a ) H m (x b ), (x a, x b ) cl(o) 2.

42 Then there exist a mapping τ, a function ϕ m, a matrix K in R m, two positive real number L min > 1 and c 0 and a class K function ρ such that the solutions of the system with the observer ẋ = f (x), y = h(x), ˆx = τ(ˆξ), ˆξ = Aˆξ + Bϕ m (ˆξ) + L(L)K(ˆξ 1 y) where L(L) = Diag(L,..., L m ), L > L min satisfy for all (x, ˆξ) in O R m, L > L min, and 0 < t < σ + O (x) X (x, t) τ(ˆξ((x, ˆξ), t)) 2 ρ ( c 0 exp( (L L min )t)l 2m 2 H m (x) ˆξ 2).

43 To construct the observer we need to construct τ the left inverse. Giving an explicit formulation to this left inverse may not be an easy task. In some specific case, this crucial step may be avoided. Indeed, when m = n and if the mapping H n defines a diffeomorphism from the open set O toward the set H n (O) another representation of the high-gain observer is simply given as ( ) 1 Hn ˆx = f (ˆx) + x (ˆx) L(L)K(h(ˆx) y), in which τ don t appear anymore.

44 To get asymptotic convergence of a solution initiated from x, we need σ + O (x) = +. This is obtained for instance if O is invariant. Due to the presence of the class K function ρ it is not possible to say that the observer converges exponentially toward the state of the system. In order to obtain this further property on the estimate, we need the matrix Hm x (x) to be full rank in cl(o) and O to be bounded.

45 The high gain observer recipe Given a system ẋ = f (x), y = h(x) 1. Find an order m and a BOUNDED open set O such that H m is injective in its closure. 2. Construct and Extend a left inverse τ. 3. Construct and Extend the function ϕ m.

High gain observer for a class of implicit systems

High gain observer for a class of implicit systems Hassan HAMMOURI Laboratoire d Automatique et de Génie des Procédés, UCB-Lyon 1, 43 bd du 11 Novembre 1918, 69622 Villeurbanne, France E-mail: hammouri@lagepuniv-lyon1fr