Elsevier

Systems & Control Letters

Volume 89, March 2016, Pages 55-60
Systems & Control Letters

Observability of singular time-delay systems with unknown inputs

https://doi.org/10.1016/j.sysconle.2015.12.002Get rights and content

Abstract

In this manuscript we give sufficient conditions guaranteeing the observability of singular linear systems with commensurable delays affected by unknown inputs appearing in both the state equation and the output equation. These conditions allow for the reconstruction of the entire state vector using past and actual values of the system output. The obtained conditions coincide with known necessary and sufficient conditions of singular linear systems without delays.

Introduction

The description of a variety of practical systems by means of singular systems, also called descriptive, implicit, or differential algebraic systems, has been shown to be useful since several decades ago as it is well explained in  [1]. Such systems, as many others, may contain time delay terms in the state, input, and/or system output, a compendium of new researching results for singular systems with time delays has been recently published, Gu et al.  [2]. Despite the increasing research on problems such as solvability, stability and controllability, up to the authors knowledge, there is quite a few works dedicated to the study of the observability of singular systems with time delays, with or without inputs. For singular systems with a time-delay in the state (without inputs), a condition is found in  [3] that ensures the observability of the system (interpreted as the reconstruction of the initial condition). However, such a condition seems to be difficult to check since it involves an integral on time that depends on a parameter within an infinite set. More papers, not too many, can be found addressing the observer design problem for singular linear time-delay systems with unknown inputs (SLSUI). In  [4], a Luenberger-like observer design is proposed by using a virtual discrete time system with the matrices of the originally considered continuous time system. In  [5] a Luenberger-like observer is proposed for singular delayed linear systems with unknown inputs not affected by time-delays.

Considering singular linear systems with unknown inputs (without delays), in  [6], conditions under which the trajectory of the state vector can be reconstructed were given, as well as a formula to reconstruct the state in terms of the system output and a finite number of its derivatives, provided any state trajectory is smooth. Regarding linear systems with commensurate delays and affected by unknown inputs, sufficient conditions allowing for the reconstruction of the state vector were obtained in  [7]. Hence, this paper takes advantage of those both results, to tackle the observability problem of a general class of singular linear time-delay systems with unknown inputs. The main contribution of this paper is the obtaining of sufficient (checkable) conditions allowing for the reconstruction of the trajectory of the state vector.

The remainder of the paper is organized as follows. In Section  2, the class of singular systems considered along the manuscript is presented, as well as the problem we aim to address. The main result is given in Section  3, which in turn is divided into 4 subsections. In Section  3.1, the studied system is transformed, and the state vector is split into two parts so that previous results may be used. Observability conditions for the first part of the transformed state vector are obtained straightforwardly in Section  3.2. Observability conditions are deduced for the second part of the transformed state vector in Section  3.3. Finally, with the results of Sections  3.2 Algebraic condition for the reconstruction of, 3.3 Algebraic condition for the reconstruction of, observability conditions for the original system are given in Section  3.4. In Section  4, a formula for the reconstruction of the state vector is obtained. Finally, an academic example that illustrates the theoretical obtained results is given in Section  5.

Notation. R is the field of real numbers. R[δ] is the polynomial ring over the real field R. In is the identity matrix of dimension n by n. A square matrix A(δ) with terms in R[δ] is called unimodular if its determinant is a nonzero constant. A matrix A(δ) of n by m dimension is called left invertible if there exists a matrix, denoted by A+(δ), such that A+(δ)A(δ)=Im. For a matrix F(δ) (over R[δ]), rankF(δ) denotes the rank of F(δ) over R[δ]. The degree of a polynomial p(δ)R[δ] is denoted by degp(δ). For a matrix M(δ), degM(δ) (the degree of M(δ)) is defined as the maximum degree of all the entries mij(δ) of M(δ). The limit from below of a time valued function is denoted as f(t).

Section snippets

System description and problem formulation

The system considered along the paper belongs to the class of delay systems whose dynamics is governed by the following equations Eẋ(t)=i=0kaAix(tih)+i=0kbBiw(tih)y(t)=i=0kcCix(tih)+i=0kdDiw(tih) where h is a positive real number. At a time t, x(t)Rn, y(t)Rp, and w(t)Rm. The initial condition φ(t) is a piecewise continuous function φ(t):[kh,0]Rn (k=max{ka,kb,kc,kd}), hence x(t)=φ(t) on [kh,0]. We also consider that w()Dm, which is the set of admissible vector functions mapping

Transformed system

By using a simple matrix decomposition, the matrix E can be transformed into the following form [Iq000] that is, there exist two invertible matrices (over R), denoted as R and S, such that RES is equal to the matrix in (3). Then, (2) can be transformed to the following form: ż1(t)=A11(δ)z1(t)+A12(δ)z2(t)+B1(δ)w(t)0=A21(δ)z1(t)+A22(δ)z2(t)+B2(δ)w(t)y(t)=C1(δ)z1(t)+C2(δ)z2(t)+D(δ)w(t) where [z1(t)z2(t)]=S1x(t), z1Rq, z2Rnq. In order to study the observability of the whole system, we will

State vector reconstruction

Now we show that if the system is BUIO then it is possible to express the state vector as a function of the system output and some of its derivatives, provided that y(t) is smooth. As in the proof of Theorem 1, let us define the vectors ξ11(t) and ξ12(t) (for thr0) as [ξ11(t)ξ12(t)]T0(δ)[0ȳ(t)]. By, , (7), we obtain the identities ξ11(t)=Γ11(δ)y(t)=F1(δ)z1(t)+G1(δ)v(t)ξ12(t)=Γ12(δ)y(t)=Δ1(δ)z1(t) where Γ11(δ) and Γ12(δ) are implicitly defined by the matrix T0(δ). Again, we define the vectors

Example

Let us consider the following academic example, where the matrices of system (2) are E=[1101010100010100]A=[1δ1020δ0110δ31δδ0],B=[00δ101δ0]C=[δ01δδ0100],D=[0100]. To put the matrix E into the form (3), we use the following matrices R and S, R=[1121212012121201212120121212],S=[1010010000010010].Hence, it is easy to verify that in this case q is equal to 3. In view of the transformation [z1(t)z2(t)]=S1x(t), the matrices of the transformed system, in its compact form given in (6),

Conclusions

The reconstruction of the entire state vector may be carried out by means of the past and actual values of the system output, provided the obtained conditions are fulfilled by the system. We have extended the results of Bejarano et al.  [6] for the observability of linear systems with unknown inputs, for the case when h=0. We notice that when h=0 the observability conditions given in this work are also necessary.

Acknowledgments

F.J. Bejarano acknowledges the SyNeR team at École Centrale de Lille and the Non-A team at INRIA Lille where he has stayed one month in 2014 as an invited researcher. He also acknowledges the financial support of the Proyecto   SIP 20151040.

References (9)

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