Robust Guaranteed Cost Observer Design for Singular Markovian Jump Time-Delay Systems with Generally Incomplete Transition Probability

and Applied Analysis 3 U i k , then generally uncertain TR matrix (4) reduces to partly uncertain TR matrix (6) as follows:


Introduction
Descriptor systems are also referred to as singular systems, implicit systems, generalized state-space systems, or semistate systems and provide convenient and natural representations in the description of economic systems, power systems, robotics, network theory, and circuits systems [1].The stability for singular system is more complicated than that for nonsingular systems because not only the asymptotic stability but also the system regularity and impulse elimination are needed to be addressed [2][3][4][5].
On the other hand, state estimation plays an important role in systems and control theory, signal processing, and information fusion [50,51].Certainly, the most widely used estimation method is the well-known Kalman filtering [52,53].A common feature in the Kalman filtering is that an accurate model is available.In some applications, however, when the system is subject to parameter uncertainties, the accurate system model is hard to obtain.To overcome this difficulty, the guaranteed cost filtering approach has been proposed to ensure the upper bound of guaranteed cost function [54].Robust  ∞ filtering for uncertain Markovian jump systems with mode-dependent time delays was proposed in [55].In [56], guaranteed cost and  ∞ filtering for timedelay systems were presented in terms of LMIs.However, to the best of our knowledge, there are few considering the robust guaranteed cost observer for a class of linear singular Markovian jump time-delay systems with generally incomplete transition probability, which is still an open problem.
In this paper, based on LMI method, we address the design problem of the robust guaranteed cost observer for a class of uncertain descriptor time-delay systems with Markovian jumping parameters and generally uncertain transition rates.The design problem proposed here is to design a memoryless observer such that for all uncertainties, including generally uncertain transition rates, the resulting augmented system is regular, impulse-free, and robust stochastically stable, and satisfies the proposed guaranteed cost performance.

Problem Formulation
Consider the following descriptor time-delay systems with Markovian jumping parameters: where () ∈   and () ∈   are the state vector and the controlled output, respectively. represents the state time delay.For convenience, the input terms in system (1) have been omitted.() ∈  2 [−, 0] is a continuous vector-valued initial function.The random parameter () represents a continuous-time discrete-state Markov process taking values in a finite set S = {1, 2, . . ., } and having the transition probability matrix Π = [  ], ,  ∈ .The transition probability from mode  to mode  is defined by where Δ > 0 satisfies lim Δ → 0 ((Δ)/Δ) = 0,   ≥ 0 is the transition probability from mode  to mode  and satisfies In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements in matrix Λ have been exactly known, some have been merely known with lower and upper bounds, and others may have no information to use.For instance, for system (1) with four operation modes, the transition rate matrix might be described by where π −Δ  ≥ 0 (for all  ∈ S,  ̸ = ), π = − ∑  =1, ̸ =  π ≤ 0, and Δ  = ∑  =1, ̸ =  Δ  ; if   = 0, for all  ∈ S, for all  ∈    , then generally uncertain TR matrix (4) reduces to partly uncertain TR matrix (6) Our results in this paper can be applicable to the general Markovian jump systems with bounded uncertain or partly uncertain TR matrix.((), ),   ((), ), ((), ), and   ((), ) are matrix functions of the random jumping process ().To simplify the notion, the notation   () represents ((), ) when () = .For example,   ((), ) is denoted by   () and so on.Further, for each () =  ∈ , it is assumed that the matrices   (),   (),   (), and   () can be described by the following form: where   ,   ,   are   known real coefficient matrices with appropriate dimensions.Time-varying matrices Δ  (), Δ  (), Δ  (), and Δ  () represent normbounded uncertainties and satisfy where  1 ,  2 ,  1 , and  2 are known constant real matrices of appropriate dimensions, which represent the structure of uncertainties, and   () is an unknown matrix function with Lebesgue measurable elements and satisfies   ()   () ≤ .
Further, for convenience, we assume that the system has the same dimension at each mode and the Markov process is irreducible.Consider the following nominal unforced descriptor time-delay system: Let  0 ,  0 , and (, ,  0 ) be the initial state, initial mode, and the corresponding solution of the system (9) at time , respectively.Definition 5. System ( 9) is said to be stochastically stable if, for all () ∈  2 [−, 0] and initial mode  0 ∈ , there exists a matrix  > 0 such that The following definition can be regarded as an extension of the definition in [2].Definition 6. (1) System ( 9) is said to be regular if det(E −   ),  = 1, 2, . . .,  are not identically zero.
(3) System ( 9) is said to be admissible if it is regular, impulse free, and stochastically stable.
The linear memoryless observer under consideration is as follows: where x() ∈   is the observer state, and the constant matrices  1 and  2 are observer parameters to be designed.Denote the error state () = () − x(), and the augmented state vector represent the output of the error states, where  is a known constant matrix.Define and combine (1) and ( 11); then we derive the augmented systems as follows: Similar to [5], it is also assumed in this paper that, for all  ∈ [−, 0], there exists a scalar ℎ > 0 such that ‖  ( + )‖ ≤ ℎ‖  ()‖.
Associated with system (13) is the cost function Definition 7. Consider the augmented system (13), if there exist the observer parameters  1 ,  2 and a positive scalar J * , for all uncertainties, such that the augmented system (13) is robust, stochastically stable and the value of the cost function ( 14) satisfies J ≤ J * , then J * is said to be a robust guaranteed cost and observer ( 11) is said to be a robust guaranteed cost observer for system (1) with (4).
Problem 8 (robust guaranteed cost observer problem for a class of linear singular Markovian jump time-delay systems with generally incomplete transition probability).Given system (1) with GUTR Matrix (4), can we determine an observer (11) with parameters  1 and  2 such that the observer is a robust guaranteed cost observer for system (1) with GUTR Matrix (4)?
Remark 12.The solution of LMIs ( 39)-( 45) parameterizes the set of the proposed robust guaranteed cost observers.This parameterized representation can be used to design the guaranteed cost observer with some additional performance constraints.By applying the methods in [14], the suboptimal guaranteed cost observer can be determined by solving a certain optimization problem.This is the following theorem.
Theorem 13.Consider system (13) with GUTR Matrix (4) and the cost function (14) it follows that the suboptimal guaranteed cost observer problem is turned into the minimization problem (55).
Remark 14. Theorem 13 gives the suboptimal guaranteed cost observer conditions of a class of linear Markovian jumping time-delay systems with generally incomplete transition probability and LMI constraints, which can be easily solved by the LMI toolbox in MATLAB.

Numerical Example
In this section, a numerical example is presented to demonstrate the effectiveness of the method mentioned in Theorem 11.Consider a 2-dimensional system (1) with 3 Markovian switching modes.In this numerical example, the singular system matrix is set as  = [  (60) Therefore, we can design a linear memoryless observer as (11) Finally, the observer (11) with the above parameter matrices for this numerical example is a suboptimal guaranteed cost observer by Theorems 11 and 13.

Conclusions
In this paper, the robust guaranteed cost observer problem for a class of uncertain descriptor time-delay systems with Markovian jumping parameters and generally uncertain transition rates is studied by using LMI method.In this GUTR singular model, each transition rate can be completely unknown or only its estimate value is known.The parameter's uncertainty is time varying and is assumed to be norm-bounded.Memoryless guaranteed cost observers are designed in terms of a set of linear coupled matrix inequalities.The suboptimal guaranteed cost observer is designed by solving a certain optimization problem.Our results can be applicable to the general Markovian jump systems with bounded uncertain or partly uncertain TR matrix.