## Servicios Personalizados

## Articulo

## Indicadores

- Citado por SciELO

## Links relacionados

- Similares en SciELO

## Compartir

## 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. García ^{1} and S. M. Hernández^{2}**

*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 - T*_{0}, *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 *U* ⊂ ^{m}, 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 *x* ∈ ^{n} , *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 *L _{f}* and

*L*such that the functions

_{h}*f*and

*h*verify for all ξ, ξ' ∈ and all ν, ν' ∈

*U*

|*f*(ξ, ν) - *f*(ξ',ν')| ≤ *L _{f}* (|ξ - ξ| + |ν - ν'|) (2)

|*h*(ξ) - *h*(ξ')|≤ *L _{h}*|ξ - ξ'|. (3)

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

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

with *z* ∈ ^{p}, 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 *T _{u}* that may be made arbitrarily small such that

|*x*(*t*, τ, ξ, *u*) - (*t*, τ, ζ, *u*)| < ε ∀*t* ∈ [τ + *T _{u}*,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 *y*_{ν} (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 Δ*y*_{ν} (τ, ξ, ξ)= *y*_{ν} (τ, ξ) - *y*_{ν} (τ, ξ'). System (1) is *finite-time observable* if there exists α_{T} ∈ such that

Δ*y*_{ν}(·, ξ, ξ')[*-T*,0] ≥ α_{T} (|ξ - ξ'|), (5)

for any ξ, ξ' ∈ and any ν ∈ *U*.

Observe that finite-time observability characterizes the variations Δ*y*_{ν}(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*, Δ*y*_{ν}(·, ξ, ξ') = 0 for all ξ = ξ' ∈ .

In fact, let *d* = diam(). Due to (2)-(3) *y*_{ν}(τ, ξ) is continuous in (ξ, ν), and hence we may take α_{T} ∈ defined by

**Definition II..4 :** We say that system (1) is *strongly finite-time observable* if for every *T* > 0 there exists α_{T} 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 (*A*_{ν}, *b*_{ν}) is controllable, *A*_{ν} ∈ , *b*_{ν} ∈ , and *A*_{ν} is diagonal and Hurwitz of prescribed eigenvalues, and |*b*_{ν}|_{∞} ≤ 1.

The mapping *q*_{T,ν} : → assigns to each ξ ∈ a point *q*_{T,ν}(ξ) ∈ via the output trajectory *y*(*s*, *t*, ξ, ν), *s* ≤ *t* of system (1).

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

= *f*_{ν}(*x*) *y* = *h*(*x*), (7)

with *f*_{ν}(·)= *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*)= *z*_{0}(*t*), *t* ∈ [τ - *T*,τ] and with *Q*_{T,ν} : → ^{n}, which ideally satisfies *Q*_{T,ν}(*q*_{T,ν}(ξ)) = ξ for all ξ ∈ , and is an *extended inverse* of *q*_{T,ν}.

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

*q*_{T,ν}: → is injective;*Q*_{T,ν}: →^{n}satisfies*Q*_{T,ν}(*q*_{T,ν}(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* , *q*_{T,ν}(*x*(*t*)) = *q*_{ν}(*x*(*t*)) - *q*_{ν}(*x*(*t - T*)) and

and since η(*t*)= *q*_{T,ν}(*x*(*t*)) + [*z*(*t*) - *q*_{ν}(*x*(*t*))] - [*z*(*t - T*) - *q*_{T,ν}(*x*(*t - T*))], it follows that η(*t*)= *q*_{T,ν}(*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 *q*_{T,ν} is *uniformly injective* if there exists β ∈ such that

|*q*_{T,ν}(ξ) - *q*_{T,ν}(ξ)| ≥ β(|ξ - ξ'|)

for all ξ, ξ' ∈ .

The next property characterizes the variations Δ*y*_{ν}(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 (*A*_{ν}, *b*_{ν}) which renders the observation map *q*_{T,ν}uniformly injective in (see Theorem II..10 below). On the other hand, the uniform injectivity of the map *q*_{T,ν}guarantees the existence of an extended inverse *Q*_{T,ν}for 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 *p*_{ν} ∈ and a controllable pair (*A*_{ν}, *b*_{ν}) with *A*_{ν} ∈ Hurwitz, which can be taken diagonal and of prescribed eigenvalues, and *b*_{ν} ∈ with |*b*_{ν}|_{∞} ≤ 1, such that the observation map *q*_{T,ν}given 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 *q*_{T,ν}is uniformly injective for some dimension *p*_{ν}, it will also be so for every integer *m* > *p*_{ν} . 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 *Q*_{T,ν} for *q*_{T,ν}. Moreover *Q*_{T,ν}(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; = {λ_{i}, *i* ∈ } a (from now on fixed) strictly decreasing sequence of negative real numbers, and consider that is an extended inverse, when the eigenvalues of *A*_{ν} are the first *p*_{ν} 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 Λ_{k} ⊂ λ given by: Λ_{k} = {λ_{}, λ_{k+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 *p*_{ν} , 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*)= {*t*_{0} = *a* < *t*_{1} < *t _{N}* =

*b*} is a

*sampling set*for

*I*. We say that it is a

*regular sampling set of norm*μ when

*t*

_{i}_{+1}-

*t*= μ > 0, ∀

_{i}*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) = {*t*_{0} < *t*_{1} < ... <*t _{N}*}, we denote by [Π(

*I*,μ),

*U*

^{*}] the family of piecewise-constant, continuous from the right functions σ :

*I'*→

*U*

^{*}such that ,

*t*∈ [

*t*,

_{i}*t*

_{i}_{+1}), 0 ≤

*i*<

*N*.

**Proposition II..18 :** Let ε' and *T'* positive numbers. Suppose that *U* ⊂ *I* = [*a*, *b*] and let *I'* = [0,*T'*]. Then there exists μ > 0 such that for any *u* ∈ *U _{I'}* there exist μ

_{u}> 0 and σ

_{d}∈ [Π(

*I'*,μ

_{u}), Π(

*I*,μ)] such that

*Proof*: Since u is piecewise continuous, there exist μ_{u} and σ ∈ [Π(*I*,μ_{u}),*U*] such that

Let μ such that μ*T '* < ε/2 and let σ_{d} ∈ [Π(*I*,μ_{u}), ⑀(*I*,μ)] defined by σ_{d}(*t*)= *u _{i}* if

*u*≤ σ(

_{i}*t*) <

*u*

_{i}_{+1}, where

*u*,

_{i}*u*

_{i}_{+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 *u* ∈ _{I'} , then σ_{d} ∈ [Π(*I*,μ_{u}), Π(*I*,μ)] with μ_{u} > 0 exists such that the solution *x _{d}*(·, 0, ξ, σ

_{d}) of (1) verifies |

*x*(τ, 0, ξ,

*u*) -

*x*(τ, 0, ξ, σ

_{d}_{d})| < ε for all τ ∈

*I'*.

*Proof*: Let us denote *x*(τ)= *x*(τ, 0, ξ, *u*) and *x _{d}*(τ)=

*x*(τ, 0, ξ, σ

_{d}_{d}) for the yet unknown control σ

_{d}. Then

Pick ε' such that *L _{f}*ε' exp(

*L*) < ε, and μ, μ

_{f}T'_{u}and σ

_{d}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)= {*u*_{1}, *u*_{2}, ..., *u _{M}*}. 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 μ

_{u}as in Proposition II..19, and Π(

*I*,μ

_{u})= {0=

*t*

_{0}<

*t*

_{1}< ... <

*t*=

_{N}*T'*}. Take

*T*= μ

_{u}and let us define ω

^{T}: [0, +∞) → [0, +∞) by

Consider δ > 0 such that ω^{T}(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

*q*, and , by

_{k}*A*and

_{k}*b*respectively, and .

_{k} *Proof of Theorem II..17*: The following algorithm is the proposed observer

- for each k determine
*p*,_{k}*A*,_{k}*b*such that an uniformly injective map_{k}*q*and its extended inverse_{k}*Q*exist. Their existence is guaranteed by Theorem II..10 and Corollary II..12._{k} - Let , and . Determine this time for the fixed pair (
*A*,*b*)*q*and_{k}*Q*for 1 ≤_{k}*k*≤*M*. According with Remark II..11,*Q*is, for each_{k}*k*, the extended inverse of the uniformly injective mapping*q*._{k} - define
*Q*(·,*u*) by*Q*(·,*u*(*t*)) =*Q*(·) if_{k}*u*≤_{k}*u*(*t*) <_{j}*u*_{k}_{+1}and*t*≤_{j}*t*<*t*_{j}_{+1} - Apply the estimator

(t)= Az(t)+ by(t, 0, ξ, u(t)) η( t)= z(t) - e(^{AT} zt - T) ( t)= Q(η(t),u(t)) | (10) |

with initial condition *z*(*t*)= *z*_{0}(*t - T*), 0 ≤ *t* ≤ *T*.

In order to prove the convergence, consider the estimator

d(t)= Az(_{d}t)+ by(_{d}t, 0, ξ, σ_{d}(t)) η _{d}(t)= z(_{d}t) - e(^{AT} z_{d}t - T) (_{d}t)= Q(η_{d}(t),σ_{d}(t)) | (11) |

with *Q*(·,σ_{d}) defined as above, with initial condition *z _{d}*(

*t*)=

*z*

_{0}(

*t - T*), 0 ≤

*t*≤

*T*, and with

*y*(

_{d}*t*, 0, ξ, σ

_{d}(

*t*)) =

*h*(

*x*(

_{d}*t*, 0, ξ, σ

_{d}(

*t*))). According to Theorem II..5,

*(*

_{d}*t*)=

*x*(

_{d}*t*) for every

*t*∈ [

*T*,

*T'*].

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

and in consequence that |η(*t*) - η_{d}(*t*)|∞ ≤ Δ for all *t* ∈ [0,*T'*]. It follows that for those *t*, |(*t*) - * _{d}*(

*t*)| = |

*Q*((

*t*)

*,u*(

*t*)) -

*Q*(

*(*

_{d}*t*),σ

_{d}(

*t*))| < ω

^{T}(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 *u _{k}* in the prescribed partition of U the injective mappings

*q*(·) and the extended inverses

_{k}*Q*. 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

_{k}(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 {*x _{i}*,

*i*= 1,

*N*} a partition of =[-1, 1] × [-1, 1] of norm 0.0125, with an error |

*Q*(

_{k}*q*(

_{k}*x*)) -

_{i}*x*| < 2

_{i}*e*

^{-3}for each

*k*and each

*i*.

Figures 1 to 3 show the results of the simulations, for initial conditions *x*_{1}(0) = 1 and *x*_{2}(0) = 0 and *z _{i}*(

*t*- 0.5) = 0.2, 1 ≤

*i*≤ 10, 0 ≤

*t*≤ 0.5.

Figure 1: *x*_{2} vs. _{2} (Top) and System output (Bottom)

Figure 2: Estimation error for *x*_{1}

Figure 3: Estimation error for *x*_{2}

Figure 1 shows the output and the state variable *x*_{2} and its estimation _{2}, while Figs. 2 and 3 show the corresponding estimation errors for *x*_{1} and *x*_{2} 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.

**REFERENCES**

1. Attasi, A. N. and H.K. Khalil "Separation results for the stabilization of nonlinear systems using different highgain observer designs", *Systems & Control Letters*, **39**, 183-191 (2000). [ Links ]

2. García, R. A. and C. E. D'Attellis, "Trajectory tracking in nonlinear systems via nonlinear reduced observers" , *International Journal of Control*, **62**, 685-715 (1995). [ Links ]

3. García, R. A., M. I. Troparevsky and J. L. Mancilla-Aguilar, "An observer for nonlinear noisy systems", *Latin American Applied Research*, **20**, 87-92 (2000). [ Links ]

4. Gauthier, J. P., H. Hammouri and S. Othman, "A simple observer for nonlinear systems with applications to byoreactors", *IEEE Trans. Autom. Control*, **37**, 875-880 (1992). [ Links ]

5. Gauthier, J. P. and I. Kupka, *Deterministic observation Theory and Applications*, Cambridge University Press (2001). [ Links ]

6. Kreisselmeier, G. and R. Engel, "Nonlinear observers for autonomous Lipschitz continuous systems", *IEEE Trans. Autom. Control*, **48** 451-464, (2003). [ Links ]

7. Krener, A.J. and A. Isidori, "Linearization by output injection and nonlinear observers", *Systems & Control Letters*, **3**, 47-52 (1983). [ Links ]

8. Michalska, H. and D. Q. Mayne, "Moving horizon observers and observer-based control", *IEEE Trans. Autom. Control*, **40**, 995-1006 (1995). [ Links ]

9. Nijmeijer, H. and T.I. Fossen, eds., *New Directions in Nonlinear Observer Design*, Springer Verlag, London (1999). [ Links ]

10. Wonham, W., *Linear Multivariable Control Systems, A Geometric Approach*, Springer Verlag, New York (1979). [ Links ]

11. Zimmer, G., "State observation by on-line minimization", *International Journal of Control*, **60**, 595-606 (1994). [ Links ]

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