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Latin American applied research

versión impresa ISSN 0327-0793

Lat. Am. appl. res. v.36 n.2 Bahía Blanca abr./jun. 2006

 

An observer for controlled Lipschitz continuous systems

R. A. García1 and S. M. Hernández2

Department of Physics and Mathematics, ITBA.
1 ragarcia@itba.edu.ar 2 shernand@itba.edu.ar

Abstract — In this paper we present an observer for controlled nonlinear systems that are locally Lipschitz continuous in both the state and control variables. This observer is based on a recently introduced model of observer for autonomous Lypschitz continuous systems, and can be designed to realize an arbitrary, finite accuracy when both the state space and control variables evolve in bounded regions.

Keywords — Nonlinear Observers. Lipschitz Continuous Systems.

I. INTRODUCTION

The design of observers for nonlinear systems has received considerable attention along the last twenty years, see Nijmeijer and Fossen (1999) and Kreisselmeier and Engel (2003) and references therein for details. Lie algebraic methods have been employed (Krener and Isidori, 1983; Gauthier et al., 1992; García and D'Attellis, 1995; Atassi and Khalil, 1999; García et al., 2000; Gautier and Kupka, 2001) to transform a class of nonlinear systems into normal forms for which observers, with guaranteed convergence and even with nonlinear separation properties, are obtained. The Lie algebraic methods are restricted to the class of systems for which there exists a suitable state-space transformation. Smoothness is in this case instrumental in order to obtain such transformation, which may exist only locally and may be difficult to obtain. Although non-smooth systems occur frequently in practice, the results obtained are few in comparison with those for smooth systems. Among the approaches developed for nonsmooth systems, that of the so-called optimization based observer is particularly appealing. This approach relies on the minimization of a cost functional over a moving horizon (see e.g. Zimmer, 1994 or Michalska and Mayne, 1995) and conceptually is directly linked to the observation problem, since its aim is to distinguish between different states by distinguishing their different output signals over some interval. As the idea is to store measurements from a (sliding) interval [t - T0, t], and to generate a state estimate so as to asymptotically match the predicted output with the measured one on the whole interval, this observer concept involves an infinite dimensional structure, that can at best be approximately realized at the implementation stage.

In Kreisselmeier and Engel (2003) a different observer design was presented that avoids the minimization stage of the optimization based observer, stage that under lack of smoothness and/or lack of convexity poses a tough problem. In that paper the authors introduced two concepts that characterize the variations of the output a) in terms of the difference in the initial conditions (observability) and b) as functions of time (finite complexity). These concepts are suitable for a large class of autonomous systems, which includes smooth as well as non-smooth systems. The design of the observer is based on a canonical linear model, whose dimension is the parameter to adjust, and on the construction of a partial inverse that relates the state variable of this linear model with the estimate in the original state space.

In this paper we present a generalization of the observer of Kreisselmeier and Engel (2003) to a class of controlled non-smooth nonlinear systems. With this aim, we generalize their observation and finite complexity concepts, to the case of a finite family of parameterized non-smooth systems with a unique output function. Under the hypothesis that for a suitable discretization of the control values the resulting constant-control parameterized family is observable and of finite complexity, we obtain an observer that is established in a canonical framework selecting one single parameter, the dimension of the observer, large enough. We complete the design by constructing a family of partial inverse maps, that act upon the canonical variable according to the actual value of the (sampled) control. This procedure yields a finite accuracy observer, where the observation error bound can be made arbitrarily small by increasing the dimension parameter above, and by refining the discretization of the control variable.

II. NOTATION AND PROBLEM STATEMENT

Throughout, and denote the sets of real and natural integer numbers, respectively. We use | · | to denote the Euclidean norm on n, and | · | to denote the supremum norm, also on n. As usual, by a -function we mean a function α : [0, +∞) → [0, +∞) that is strictly increasing and continuous, and satisfies α(0) = 0. Given Um, we denote by the set of piecewise continuous functions u : U, such that , and for any [a, b] ⊂ , by [a,b] the restriction of the functions in to [a, b]. By || · || [a,b] we denote the 2-norm in the interval [a, b], by || · ||, the 2-norm in (-∞, 0] and by [a,b] the inner product of 2-functions on [a, b].

We consider nonlinear systems of the form

= f(x, u), y = h(x) (1)

with xn , u and y. We will denote with x(·, τ, ξ, u) any trajectory of system (1) corresponding to the input u that verifies x(τ)= ξ ∈ n and with y(·, τ, ξ, u)= h(x(·, τ, ξ, u)) its corresponding output. We assume that the inputs belong to the class , with U a fixed compact set. We will assume further that there exists a closed set n where the state variables evolve, which is invariant with respect to (1), i.e., given ξ ∈ and u, x(t, τ, ξ, u) ∈ for all t.

The following assumption will be made from now on

H1: There exist positive numbers Lf and Lh such that the functions f and h verify for all ξ, ξ' ∈ and all ν, ν' ∈ U

|f(ξ, ν) - f(ξ',ν')| ≤ Lf (|ξ - ξ| + |ν - ν'|) (2)

|h(ξ) - h(ξ')|≤ Lh|ξ - ξ'|. (3)

As stated in the introduction, our aim is to design a system

= g(z, y, u), = Q(z, u) (4)

with zp, inputs y and u from system (1) and output , which estimates the state x of system (1).

Definition II..1 : Given positive numbers T and ε, we say that system (4) is a finite-time ε - T -observer if its solutions z(·, τ, ζ, u) are defined on [τ, τ + T ] and verify:

1. Consistency: Given ξ ∈ and u, if ζ is such that ξ = Q(ζ,u), then

x(t, τ, ξ, u)= (t, τ, ζ, u), ∀t ∈ [τ, τ + T].

2. Convergence: For any ζ ∈ p, u such that Q(ζ,u) ∈ and for any ξ ∈ , there exists a positive number Tu that may be made arbitrarily small such that

|x(t, τ, ξ, u) - (t, τ, ζ, u)| < ε ∀t ∈ [τ + Tu,τ + T ].

A. Observability

As stated in the Introduction we aim to generalize an existing finite-time observer, based on a concept of observability related with the output map of an autonomous system, to another, also of finite time for a controlled system. It follows that we must consider a concept of observability related with input-output maps. Consider then T > 0 fixed, and for each ν ∈ U and each ξ ∈ , let yν (τ, ξ)= y(τ, 0, ξ, ν), τ ∈ [-T, 0] the output of the system seen backwards when the constant control ν is applied.

Definition II..2 : Let Δyν (τ, ξ, ξ)= yν (τ, ξ) - yν (τ, ξ'). System (1) is finite-time observable if there exists αT such that

Δyν(·, ξ, ξ')[-T,0] ≥ αT (|ξ - ξ'|), (5)

for any ξ, ξ' ∈ and any ν ∈ U.

Observe that finite-time observability characterizes the variations Δyν(τ, ξ, ξ') with respect to the distance |ξ - ξ'|.

Remark II..3 : If is compact, the observability in this sense only requires that for all ν ∈ U, Δyν(·, ξ, ξ') = 0 for all ξ = ξ' ∈ .

In fact, let d = diam(). Due to (2)-(3) yν(τ, ξ) is continuous in (ξ, ν), and hence we may take αT defined by

Definition II..4 : We say that system (1) is strongly finite-time observable if for every T > 0 there exists αT as in Definition II..2 such that (5) holds .

For general nonlinear systems, this observability property will be hard to check, since observability deals with distinguishability of output solutions over (-∞, t], rather than over arbitrary small intervals. The latter is usually checked by smooth techniques, and hence is more easy to perform.

It is not hard to prove that if a systems is uniformly observable in the sense of Gauthier (see Nijmeijer and Fossen, 1999), then it is strongly finitetime observable.

Next, following Kreisselmeier and Engel (2003), we define, for each T > 0 and ν ∈ U the observation mapping

where the pair (Aν, bν) is controllable, Aν, bν, and Aν is diagonal and Hurwitz of prescribed eigenvalues, and |bν| ≤ 1.

The mapping qT : assigns to each ξ ∈ a point qT(ξ) ∈ via the output trajectory y(s, t, ξ, ν), st of system (1).

Consider now, for each ν ∈ U the autonomous system defined by

= fν(x) y = h(x), (7)

with fν(·)= f(·,ν). Then, following Kreisselmeier and Engel (2003), we define for system (7) the observer

(8)

defined for t > τ , with initial conditions z(t)= z0(t), t ∈ [τ - T,τ] and with QT : n, which ideally satisfies QT(qT(ξ)) = ξ for all ξ ∈ , and is an extended inverse of qT.

We have the following result whose proof, similar to that of Theorem 5 in Kreisselmeier and Engel (2003), is included for the sake of completeness

Theorem II..5 : Suppose that

  1. qT : is injective;
  2. QT : n satisfies QT(qT(ξ)) = ξ for all ξ ∈ .

Then, system (8) is a finite-time observer for system (7), whose state estimate converges to the real state in finite time T , i.e. if x(t, τ, ξ, ν) is a trajectory of system (7), then (t) - x(t, τ, ξ, ν) = 0 for all t ≥ τ + T .

Proof: Let us denote x(s)= x(s, τ, ξ, ν) and

Then, for any t ≥ τ + T , qT(x(t)) = qν(x(t)) - qν(x(t - T)) and

and since η(t)= qT(x(t)) + [z(t) - qν(x(t))] - [z(t - T) - qT(x(t - T))], it follows that η(t)= qT(x(t))and (t)= x(t) for all t ≥ τ + T .

In order a finite -time observer for system (7) to exist, it remains to establish conditions under which the hypotheses of Theorem II..5 hold. With this aim, we introduce the following

Definition II..6 : Given T > 0 and ν ∈ U, we say that the observation map qT is uniformly injective if there exists β ∈ such that

|qT(ξ) - qT(ξ)| ≥ β(|ξ - ξ'|)

for all ξ, ξ' ∈ .

The next property characterizes the variations Δyν(τ, ξ, ξ') as functions of time τ.

Definition II..7 : Given T > 0, system (1) is said to be of finite-time finite complexity in if there exists a finite number of piecewise continuous functions such that for some δ > 0

(9)

for every ξ, ξ' ∈ and every ν ∈ U.

Definition II..8 : We say that system (1) is of strong finite-time finite complexity, if for every T > 0 there exists δ as in Definition II..7 such that (9) holds.

Remark II..9 : The finite-time finite complexity and the finite-time observability properties assure the existence of a controllable pair (Aν, bν) which renders the observation map qTuniformly injective in (see Theorem II..10 below). On the other hand, the uniform injectivity of the map qTguarantees the existence of an extended inverse QTfor this map (Corollary II..12).

We are now in position to state the following result

Theorem II..10 : Let T > 0 and ν ∈ U. If system (1) is finite-time observable and of finite-time finite complexity, there exist pν and a controllable pair (Aν, bν) with Aν Hurwitz, which can be taken diagonal and of prescribed eigenvalues, and bν with |bν| ≤ 1, such that the observation map qTgiven by (6) is uniformly injective in .

Proof: It follows, with minor modifications, along the line of the proof of Theorem 2 in Kreisselmeier and Engel (2003).

Remark II..11 : It follows readily from Definition II..7 that if qTis uniformly injective for some dimension pν, it will also be so for every integer m > pν . This fact will be instrumental in what follows.

From Theorem II..10 and Lemma 4 in Kreisselmeier and Engel (2003), we obtain the following result

Corolary II..12 : Let T > 0 and ν ∈ U. If system (1) is finite-time observable and of finite-time finite complexity, there exist an extended inverse QT for qT. Moreover QT(η) is continuous in (η, T).

As a consequence of this last result and of Theorem II..5, the following holds.

Theorem II..13 : Let T > 0 and ν ∈ U. If system (1) is finite-time observable and of finite-time finite complexity, then system (8) is a finite-time observer for system (7), whose state estimate converges to the real state in finite time T , i.e. if x(t, τ, ξ, ν) is a trajectory of system (7), then (t) - x(t, τ, ξ, ν)=0 for all t ≥ τ + T .

In order to assure some kind of regularity on the behavior of the extended inverses, let Λ; = {λi, i} a (from now on fixed) strictly decreasing sequence of negative real numbers, and consider that is an extended inverse, when the eigenvalues of Aν are the first pν numbers of Λ. If we denote for any t > 0, , we can introduce the following

Definition II..14 : given by

is a modulus of continuity for .

Remark II..15 : Observe that in the case that is a compact set, is uniformly continuous.

Let now for each k, the set Λk ⊂ λ given by: Λk = {λ, λk+1, ...}, and consider the standing hypothesis

H2: For each T > 0 and each ν ∈ U there exists such that for all r ≥ 0 and all k.

Remark II..16 : Hypothesis H2 states a certain kind of smoothness in the behavior of the extended inverse, considered as a function of the discrete variable pν , and reflects the fact that this dimension does not increase when we replace the first eigenvalues of the sequence in the determination of the controllable pair. For observable linear systems (which are of finite complexity, see Kreisselmeier and Engel (2003)), this kind of behavior is suggested by the existence of observers based on the observability Grammian (see Wonham, 1979).

We are now in position of stating the main result of this work.

Theorem II..17 : Let T' and ε positive numbers, and assume that system (1) is strongly finite-time observable and of strong finite-time finite complexity. Assume further that hypothesis H2 holds. Then there exists an ε - T' observer for (1).

Next we obtain a series of results which will be used in the proof of this theorem.

Let I = [a, b] any finite interval. We say that a finite set of real numbers Π(I)= {t0 = a < t1 < tN = b} is a sampling set for I. We say that it is a regular sampling set of norm μ when ti+1 - ti = μ > 0, ∀i. We denote by Π(I,μ) the regular sampling set of I of norm μ.

Let μ' > 0, I' a compact interval and U*U. For Π(I,μ) = {t0 < t1 < ... <tN}, we denote by [Π(I,μ),U*] the family of piecewise-constant, continuous from the right functions σ : I'U* such that , t ∈ [ti,ti+1), 0 ≤ i < N.

Proposition II..18 : Let ε' and T' positive numbers. Suppose that UI = [a, b] and let I' = [0,T']. Then there exists μ > 0 such that for any uUI' there exist μu > 0 and σd[Π(I'u), Π(I,μ)] such that

Proof: Since u is piecewise continuous, there exist μu and σ ∈ [Π(Iu),U] such that

Let μ such that μT ' < ε/2 and let σd[Π(Iu), ⑀(I,μ)] defined by σd(t)= ui if ui ≤ σ(t) < ui+1, where ui, ui+1 ∈ Π(I,μ). It follows readily that

and in consequence, the thesis holds.

Proposition II..19 : Let ε and T' positive numbers, and consider I and I' as in Proposition II..18. Then there exists μ > 0 such that if x(·, 0, ξ, u) is the solution of (1) corresponding to ξ ∈ and uI' , then σd[Π(Iu), Π(I,μ)] with μu > 0 exists such that the solution xd(·, 0, ξ, σd) of (1) verifies |x(τ, 0, ξ, u) - xd(τ, 0, ξ, σd)| < ε for all τ ∈ I' .

Proof: Let us denote x(τ)= x(τ, 0, ξ, u) and xd(τ)= xd(τ, 0, ξ, σd) for the yet unknown control σd. Then

Pick ε' such that Lfε' exp(Lf T') < ε, and μ, μu and σd as in Proposition II..18. Then

and by Gronwall's inequality,

III. THE OBSERVER

From now on we will assume that ε > 0 and T' > 0 are fixed. Let μ in Proposition II..19 corresponding to ε/2 and T' , and suppose that Π(I,μ)= {u1, u2, ..., uM}. Assume now that we want to estimate a trajectory x(t, 0, ξ, u) of (1) corresponding to u[0,T'] based on the knowledge of u and of the output y(t, 0, ξ, u). Let then μu as in Proposition II..19, and Π(Iu)= {0= t0 < t1 < ... < tN = T'}. Take T = μu and let us define ωT : [0, +∞) → [0, +∞) by

Consider δ > 0 such that ωT(δ) < ε/2, and let k* the first integer such that . Fix now the subsequence of eigenvalues , and denote for each k, of (6) by qk, and , by Ak and bk respectively, and .

Proof of Theorem II..17: The following algorithm is the proposed observer

  • for each k determine pk, Ak, bk such that an uniformly injective map qk and its extended inverse Qk exist. Their existence is guaranteed by Theorem II..10 and Corollary II..12.
  • Let , and . Determine this time for the fixed pair (A, b) qk and Qk for 1 ≤ kM. According with Remark II..11, Qk is, for each k, the extended inverse of the uniformly injective mapping qk.
  • define Q(·,u) by Q(·,u(t)) = Qk(·) if uku(tj) < uk+1 and tjt < tj+1
  • Apply the estimator
(t)= Az(t)+ by(t, 0, ξ, u(t))
η(t)= z(t) - eAT z(t - T)
(t)= Q(η(t),u(t))
(10)

with initial condition z(t)= z0(t - T), 0 ≤ tT.

In order to prove the convergence, consider the estimator

d(t)= Azd(t)+ byd(t, 0, ξ, σd(t))
ηd(t)= zd(t) - eAT zd(t - T)
d(t)= Qd(t),σd(t))
(11)

with Q(·,σd) defined as above, with initial condition zd(t)= z0(t - T), 0 ≤ tT, and with yd(t, 0, ξ, σd(t)) = h(xd(t, 0, ξ, σd(t))). According to Theorem II..5, d(t)= xd(t) for every t ∈ [T, T'].

It is not hard to prove that for all t ∈ [0, T'],

and in consequence that |η(t) - ηd(t)|∞ ≤ Δ for all t ∈ [0,T']. It follows that for those t, |(t) - d(t)| = |Q((t),u(t)) - Q(d(t),σd(t))| < ωT (δ) < ε/2. In consequence, for every t ∈ [T, T'],

|x(t) - (t)| ≤ |x(t) - d(t)| + |d(t) - (t)| < ε,

and the theorem follows.

Remark III..1 : In order to design the proposed observer we should be able to compute for each point uk in the prescribed partition of U the injective mappings qk(·) and the extended inverses Qk. As pointed out in Kreisselmeier and Engel (2003), this can be done with an arbitrary finite accuracy, by taking the following map of approximate inversion

(12)

with

(13)

where is chosen to achieve an accuracy of the observer high enough , (see that paper for details).

IV. AN EXAMPLE

With the purpose of exhibiting how the observer approach herein presented works, we consider the following non-autonomous Lipschitz continuous system.

(14)

where u(t) is a ramp that goes from 10 to 20 in 5 seconds and then descends towards 10 again in an equally large time interval.

We consider the former system of practical interest because it models the behavior of a large class of real-life devices, those consisting of a voltage controlled oscillator followed by a nonlinearity (in this case represented by h(·), a typical half-wave rectifier). Since for each ν ∈ U = [10, 20] the system is piecewise linear, it is not hard to verify that the hypotheses of Corollary I of Kreisselmeier and Engel (2003) hold almost everywhere. In consequence, the strong finite-time observability and strong finite-time complexity properties are verified.

The parameters taken for the for the observer were T' = 10, ε = 0. and Λ = {-0.01, -0.05, -0.1, -0.5, -1, -1.5, -2, -4, -8, -16}. The partition norm for U was μ = 1, and T = 0.5. Via simulations, it was found that enabled us determine the approximate inversion maps (12) - (13), implemented as

(15)

with = 0.05 and {xi, i = 1, N} a partition of =[-1, 1] × [-1, 1] of norm 0.0125, with an error |Qk(qk(xi)) - xi| < 2e-3 for each k and each i.

Figures 1 to 3 show the results of the simulations, for initial conditions x1(0) = 1 and x2(0) = 0 and zi(t - 0.5) = 0.2, 1 ≤ i ≤ 10, 0 ≤ t ≤ 0.5.


Figure 1: x2 vs. 2 (Top) and System output (Bottom)


Figure 2: Estimation error for x1


Figure 3: Estimation error for x2

Figure 1 shows the output and the state variable x2 and its estimation 2, while Figs. 2 and 3 show the corresponding estimation errors for x1 and x2 respectively. As can be seen, the observer performs within the given specifications from approximately t = 0.5 on. Nevertheless, in the steady state error profile there are peaks of amplitude bounded by 0.2. That happens due to the inverse map transitions that match the control switching events. This effect can be reduced at a greater computational effort by refining the control mesh and by taking T smaller.

V. CONCLUSIONS

In this paper we have presented an observer for controlled Lyapunov continuous SISO systems, (although it can be easily extended to the MIMO case), that realizes an arbitrary finite accuracy. The model of the observer is based on an existing design for autonomous systems, and applies to a rather large class of controls (that of piecewise continuous, continuous from the right controls). An example is given that exhibits the behavior of the observer.

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Received: September 21, 2005.
Accepted for publication: February 6, 2006.
Recommended by Guest Editors C. De Angelo, J. Figueroa, G. García and J. Solsona.