Brief paperSet membership state and parameter estimation for systems described by nonlinear differential equations☆
Introduction
The purpose of this paper is to study the state and parameter estimation for nonlinear continuous-time systems in a bounded-error context.
In the literature, state or parameter estimation problems are usually solved by probabilistic methods, which are relevant when an explicit characterization of the measurement noise is available. This, however, is not always the case in practice, and it is often more natural to assume that the perturbations belong to a known set. In this case, bounded-error approaches allow the characterization of the whole set of state or parameter vectors that are compatible with the measured data, a model structure and some prior error bounds.
The bounded-error approach to estimation problems (state and parameters) has attracted the attention of many researchers (Milanese, Norton, Piet-Lahanier, & Walter, 1996; and the references therein). For the linear case, the solution set is a convex polyhedron for which an exact characterization can be attempted if the dimension of the state or parameter vector is not too large; in practice however, this set can be very complicated and other simple-shaped forms, such as ellipsoids, parallelotopes, zonotopes or boxes have been used to give an enclosure of the exact solution set (Durieu, Walter, & Polyak, 2001; Maksarov & Norton, 2002). When the model is nonlinear, the previous algorithms are no longer relevant and other methods based on interval analysis and constraint propagation have been developed (Jaulin, Kieffer, Didrit, & Walter, 2001; Kieffer, Jaulin, & Walter, 2002). Interval observers for a limited class of nonlinear systems have also been proposed in (Gouzé, Rapaport, & Hadj-Sadok, 2000). An alternative but not guaranteed technique has been proposed as an extension of Kalman filtering to intervals by Becerra, Roberts, and Griffths (2001).
The techniques cited above are meant for discrete-time systems only. In fact, actual systems are often described by differential equations, for which no bounded-error method has been studied until Jaulin (2002), where a guaranteed state estimator is proposed for a continuous-time system where the state equation involves no state perturbation. Jaulin (2002) uses interval analysis and a first-order enclosure of the solution of the ordinary differential equation (ODE) describing the system. A well-known disadvantage of first-order enclosures is the large wrapping effect which leads to very pessimistic results, thus making the use of bisections necessary to reduce the pessimism. Such a procedure is known to be time-consuming and is not practical when the dimension of the state vector is large.
The first problem investigated in our paper is the state estimation for nonlinear continuous-time systems in a bounded error context. Our contribution is the technical improvement of the prediction part within the scheme published by Jaulin (2002) by using a more accurate interval computation of the solution of the ODE. We will show that this makes it possible to build a reliable state estimator without resorting to bisections in the prediction part of the estimator.
The second contribution deals with the guaranteed parameter estimation for systems described by nonlinear ODEs in a bounded-error context. To our knowledge, this problem has not been investigated before.
Our approach to solve the estimation problems relies on the use of validated methods for solving the initial value problem (IVP) for ODE. These methods use high-order interval Taylor models to derive intervals which are guaranteed to contain the solution of the IVP for ODE; in the applied mathematics community, they are called validated as they make it possible to validate existence and uniqueness of the latter.
The validated methods present two important advantages comparatively to the standard numerical methods. The latter compute only approximations satisfying a user-specified tolerance, and in some cases these approximations will constitute in fact inaccurate solutions. However, if the validated methods return a solution to the problem, then the problem is guaranteed to have a unique solution, and a guaranteed enclosure of this solution is derived (Nedialkov, Jackson, & Corliss, 1999).
When all initial data and parameters are exactly known, the validated methods present some disadvantages as they usually require considerably more computing time than the standard ones. However, this is no longer the case when the state equations contain parameters and/or initial state which cannot be known exactly, but are known to belong to a given set. To compute solutions for such problems with standard methods, the same process has to be executed many times with different values of the parameters and/or different values of the initial state, which is clearly numerically expensive and also is no more guaranteed.
The paper is structured as follows: the problem statement is given in Section 2. Some properties of interval analysis are presented in Section 3. The principle of the validated methods to solve ordinary differential equations is given in Section 4. Section 5 deals with set inversion. These techniques are used in Section 6 to build a guaranteed state estimator for systems represented by nonlinear ordinary differential equations. This estimator will be extended in Section 7 to solve parameter estimation problems.
Section snippets
Problem statement
Consider a nonlinear system represented by the following equations:where t∈[t0,T], , is an open set, is a set containing the parameters; n, m and np are respectively the dimension of the state vector , the dimension of the output vector and the dimension of the parameter vector . The functions and are possibly nonlinear. The initial state is assumed to belong to a prior known box . Assume in addition that
Interval analysis
Interval analysis was initially developed to account for the quantification errors introduced by the rational representation of real numbers with computers and was extended to validated numerics (Moore, 1966). A real interval is a connected and closed subset of . The set of all real intervals of is denoted by . If , then [a] is a point interval, if , then [a] is nonnegative, and if then [a] is symmetric. Two intervals [a] and [b] are equal if and only if
Validated integration of ordinary differential equations using Taylor expansions
Consider the following equation:where the function is assumed to be at least k-times continuously differentiable in a domain . Interval arithmetic is used to compute guaranteed bounds for the solution of (5) at the sampling times {t1,t2,…,tN}. The most effective methods to solve such a problem are based on Taylor expansions (for more details see Moore, 1966; Nedialkov et al., 1999; Rihm, 1994). These methods consist in two parts: they first verify existence and
Set inversion
Consider the problem of determining a solution set for the unknown quantities defined bywhere is known a priori, is an a priori search set for and a nonlinear function not necessarily invertible in the classical sense. Eq. (11) involves computing the reciprocal image of , it is a set inversion problem which can be solved using set inversion via interval analysis (SIVIA). SIVIA (Jaulin & Walter, 1993) is a recursive algorithm which explores all the search
State estimation
Consider a system represented by Eqs. (1) with the parameters assumed to be known. The aim of this section is to estimate the state vector at the sampling times {t1,t2,…,tN} corresponding to the measurement times of the outputs. Remark 1 In this paper, state equation is assumed with no perturbations. However, if a state perturbation ought to be introduced, the predictable effect will be to increase the width of the interval state vector solution of the ODE and hence the pessimism in the prediction
Parameter estimation
In this section, the problem of the guaranteed estimation of the unknown parameter vector in (1) is investigated in a bounded-error context.
When both the state and parameter vectors ought to be estimated (or when the parameter vector is not constant), the state estimator defined in the previous section can be extended to estimate both of the state and the parameter vectors. The extended state is obtained by the original state supplemented with the parameters to be estimated. Such an approach
Conclusion
In this paper, high-order Taylor models and interval analysis have been applied to study the guaranteed state and parameter estimation for nonlinear continuous-time systems in a bounded-error context.
The first goal of the paper was to derive a reliable state estimator for nonlinear continuous-time systems in a bounded-error context. It was shown that using high-order Taylor expansion to solve the state equations allied with centred forms and matrices pre-conditioning makes the use of bisections
Acknowledgements
The authors wish to thank the anonymous reviewers for their constructive comments and suggestions.
Tarek Raı̈ssi was born in Kasserine, Tunisia, in 1976.
He obtained his DEA in automatic control from the “Ecole Centrale de Lille” in 2001.
He is currently a Ph.D. candidate at the University of Paris XII.
His research interests include nonlinear systems identification, interval analysis and global optimization.
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Tarek Raı̈ssi was born in Kasserine, Tunisia, in 1976.
He obtained his DEA in automatic control from the “Ecole Centrale de Lille” in 2001.
He is currently a Ph.D. candidate at the University of Paris XII.
His research interests include nonlinear systems identification, interval analysis and global optimization.
Nacim Ramdani was born in Algiers, Algeria, in 1967.
He graduated from Ecole Centrale de Paris, France in 1990, and obtained his Ph.D. in automatic control from the University of Paris XII, France in 1994.
He is currently Assistant Professor at the University of Paris XII, France since 1996.
His research interests revolve around robust model identification and its applications to robotics, thermal and electrical engineering.
Yves Candau received the M.E. degree from the Polytechnical School in 1980 and the Ph.D. in Energetics Sciences.
Prof. Yves Candau became the Head of the Thermal Sciences Research Laboratory (CERTES—Centre d'Etudes et Recherche Thermique et Systèmes) at the University Paris XII of Créteil, France.
His centres of interests include signal processing and inverse methods applied to thermal measurements engineering.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Ian Petersen.