Interval Observers for Linear Systems with Time-Varying Delays

Авторы: Mustapha Ait Rami, Jens Jordan and Michael Schoenlein

Источник: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary .

    Abstract – This paper considers linear observed systems with time-varying delays, where the state as well as the observation of the state is subject to delays. It is assumed that the delays are unknown but stay below a certain bound. Similar to the case of uncertainties in the systems parameters we aim to derive upper and lower estimates for the state of the system under consideration. A pair of estimators providing such bounds is called an interval observer. In particular, the case where the estimators converge asymptotically is of notable interest. In this case the interval observer is said to be convergent. In this paper we derive necessary and sufficient conditions for the existence of a convergent interval observer for linear observed systems with time-varying bounded delays.

    I. INTRODUCTION
    Differential delay systems represent a class of infinitedimensional systems which may ba used to model population dynamics and many physical and biological dynamical systems. As a matter of fact, the reaction of real world systems to exogenous signals is often infected by certain time delays, e.g. in logistics networks the transportation of products between different locations is subject to traffic jams etc. In practice, these delays vary over time and are frequently unknown. Further, a direct measurement of certain state variables is also subject to delays. Such phenomena can be described by a mathematical model in which the behavior of the system is described by an equation that includes information on the past evolution of the system.
    A common and frequently used technique to obtain information on the unknown state of the system is to use state estimation that bounds the state from below and above. This technique, called interval observer, was introduced by [1] to obtain state estimates for biological systems that are subject to parameter uncertainties. Later the framework of interval observers was used and extended in many works, e.g. [2]. Moreover, it is of interest whether the difference between the upper and lower bound on the state of the
system, called the interval error, converges. If this is the case, the interval observer is said to be convergent. In recent years the framework of interval observers is also used to derive state estimates for linear systems that are subject to time delays, see e.g. [3], [4]. In [3] the interval observers for positive linear systems with constant time-delay is based on an observer of extended Luneberger type [5]. On the other hand, [4] considers input-free linear systems that are subject to disturbances. But both works are not concerned with timevarying delays.

    In this paper we regard the problem of the existence of convergent interval observers for linear observed systems with time-varying bounded delays. We prove necessary and sufficient conditions for the existence of a convergent interval observer. The design of interval observers relies on the theory of positive systems (see [6] for general references). In fact, enforcing the positivity of the error estimation will necessarily lead to a guaranteed bounds on the estimated states, once we start with a priori bounds on the unknown initial condition of the observed system. Here, we provide a simple and efficient way of designing observers that ensures guaranteed bounds on the estimated states.

    II. PRELIMINARIES
    This section is composed of two subsections. In the first subsection we introduce the objective under consideration and state the main result of this paper. The second subsection provides necessary preliminary statements about positive systems and technical keys that are primordial for the characterization and the treatment of positive systems satisfying a differential delayed equation. The introduced facts and results will be essentially used in the verification of our main result.
    A. Problem formulation
    In this paper we deal with linear observed systems with bounded time-varying delays 550

to be Lebesgue measurable. Throughout the paper we use the notation

The systems (not necessarily positive) under consideration are of the following form

where 553 for all 554 The initial condition for the system (1) is given by

Here 556 is a given continuous function that is defined on 557 . The aim of this paper is to derive necessary and sufficient conditions for the existence of estimator functions that bound the state of the system from below and above. This is formalized in the following definition.
Definition 1: For any delays such that 558 , an interval observer for system (1) is a pair of lower and upper estimator functions 559 such that

561 is said to be aconvergent interval observer for the system (1) if the interval error 562 converges to zero.
The objective is to design convergent interval observers for systems of the form (1), in the case when the delays are not known but bounded. In fact, the main idea behind the proposed interval observers machinery is to reconstruct the error dynamics of the estimators in such way that it can be governed by a differential dynamical equation of the form

which enforced to be inherently positive by choosing an adequate matrix gain L. That is, we have to ensure that if the initial error is nonnegative, i.e. e(0) >0, the error will remain nonnegative over time, i.e. e(t) >0 for all t >0. Consequently, by using this positivity concept we will show the following necessary and sufficient condition for a convergent interval observer that ensures an estimation with guaranteed lower and upper bounds on the observed states.

Note that the existence of matrix L which satisfies conditions (i) and (ii) in Theorem 1 can be equivalently described by a LMI formulation, see [3], [7].

B. Positive systems
This section provides necessary notations and useful results from the theory of positive systems. As mentioned in the previous subsection the structure of the error dynamics
will be described by a linear differential delayed equation of the following form

where the given matrices 566 are timeinvariant and 567 are time-varying delays such that 568 is the instantaneous system state at time t. The whole state at time t of system (2) is infinite dimensional which is given by the set

Following [8], it can be shown that the solution to the system’s equation (2) exists, is unique and totally determined by any given initial locally Lebesgue integrable vector function 5556 such that

For any nonnegative initial condition 571 such that 572 system (2) is said to be positive if the corresponding trajectory is nonnegative, that is 573 We recall intrinsic properties of the delayed system’s positivity behavior are related to Metzlerian matrices and positive matrices. A real matrix M is called a Metzler matrix if its off-diagonal elements are nonnegative. A real matrix M is called a positive matrix if all its elements are nonnegative, that is 574 Note that the following result from [9] shows how Metzlerian matrices are intrinsically connected to positivity. For a more detailed presentation of positive matrices and their properties see e.g. [10].

Further, we cite from [11] the following characterization of the positivity of a linear system of the form (2).

Moreover, Theorem 2.1 in [3] contains a useful characterization for the stability of a positive system of the form (2) in terms of the matrices that define the system.

    III. CONCLUSIONS
    This paper presents a necessary and sufficient condition for the existence of convergent interval observers for linear systems with time-varying bounded delays. Our approach is based on observers of extended Luenberger type. This framework yields to positive systems with subject to the error between the state and the observation of the system. Thus the analysis of the evolution of the error uses techniques from positive systems.

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