On Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Robust H ∞ Filtering Design in Switched Linear Discrete-Time Systems with Polytopic Uncertainties

This paper is concerned with the problem of robust H ∞ filter design for switched linear discrete-time systems with polytopic uncertainties.The condition of being robustly asymptotically stable for uncertain switched system and less conservativeH ∞ noiseattenuation level bounds are obtained by homogeneous parameter-dependent quadratic Lyapunov function. Moreover, a more feasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the given arbitrary switching signal. Lastly, simulation results are used to illustrate the effectiveness of our method.


Introduction
Switched systems are a class of hybrid systems that consist of a finite number of subsystems and a logical rule orchestrating switching between the subsystems.Since this class of systems has numerous applications in the control of mechanical systems, the automotive industry, aircraft and air traffic control, switching power converters, and many other fields, the problems of stability analysis and control design for switched systems have received wide attention during the past two decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].Reference [2] proposed the  ∞ weight learning law for switched Hopfield neural networks with time-delay under parametric uncertainty.Reference [8] dealt with the delay-dependent exponentially convergent state estimation problem for delayed switched neural networks.Some criteria for exponential stability and asymptotic stability of a class of nonlinear hybrid impulsive and switching systems have been established using switched Lyapunov functions in [9].Reference [14] investigated the problem of designing a switching compensator for a plant switching amongst a family of given configurations.
On the other hand, within robust control theory scheme, the  ∞ noise-attenuation level is an important index for the influence of external disturbance on system stability [16][17][18][19].However, the uncertainties which generally exist in many practical plants and environments may result in significant changes in robust  ∞ noise-attenuation level.In order to suppress the conservativeness, many new methods have been considered.Among these methods, homogeneous polynomial parameter-dependent quadratic Lyapunov function is one of the most effective methods.The main feature of these functions is that they are quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form.It is firstly introduced to study robust stability of polynomial systems in [20].Most results have been presented in [21][22][23][24][25].In [19], homogeneous parameter-dependent quadratic Lyapunov functions were used to establish tightness in robust  ∞ analysis.Reference [21] presented some general results concerning the existence of homogeneous polynomial solutions to parameterdependent linear matrix inequalities whose coefficients are continuous functions of parameters lying in the unit simplex.Reference [23] investigated the problems of checking robust stability and evaluating robust  2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytropic domains.Reference [25] introduced Notation.R  denotes the -dimension Euclidean space and R × is the real matrices with dimension  × ; R  0 means R  /{0}; the notation  ≥  (resp.,  > ), where  and  are symmetric matrices, represents the fact that the matrix  −  is positive semidefinite (resp., positive definite);   denotes the transposed matrix of ; () represents ( 2  1 , . . .,  2  ) with  ∈ R  ; ℎ() means +  with  ∈ R × ; ⊗ denotes the Kronecker product of vectors  and .‖ ⋅ ‖ denotes Euclidean norm for vector or the spectral norm of matrices.

Problem Statement
Consider a class of uncertain switched linear discrete-time systems which were given in [1]: where () ∈ R  is state vector, () ∈ R  is disturbance input which belongs to  2 [0, +∞), () is the measurement output, () is objective signal to be attenuated, and  is switching rule, which takes its value in the finite set Π := {1, . . ., }.
As in [1], the switching signal  is unknown a priori, but its instantaneous value is available in real time.As an arbitrary discrete time , the switching signal  is dependent on  or () or both or other switching rules. is an uncertain parameter vector supposed to satisfy  ∈ Λ = {  ≥ 0, ∑  =1 = 1}.The vector  represents the time-invariant parametric uncertainty which affects linearly the system dynamics.The vector  can take any value in Λ, but it is known to be constant in time.
The matrices of each subsystem have appropriate dimensions and are assumed to belong to a given convex-bounded polyhedral domain described by  vertices in the th subsystem, that is, where Hence, we are interested in designing an estimator or filter of the form where   () ∈ R  ,   () ∈ R  ;   ,   ,   and   are the parameterized filter matrices to be determined.The filter with the above structure may be called switched linear filter, in which the switching signal  is also assumed unknown a priori but available in real-time and homogeneous with the switching signal in system (1).
Augmenting the model of (1) to include the state of filter (4), we obtain the filtering error system: where Based on [1], the robust  ∞ filtering problem addressed in this paper can be formulated as follows: finding a prescribed level of noise attention  > 0 and determining a robust switched linear filter (4) such that the filtering error system is robustly asymptotically stable and This problem has received a great deal of attention.In order to get a better level of noise attention  > 0, we will use homogeneous polynomial functions which have demonstrated nonconservative result for serval problems.In this paper, we extend these methods to design robust switched linear filter.
The following preliminaries are given, which are essential for later developments.We firstly recall the homogeneous polynomial function from Chesi et al. [26].
Definition 1.The function ℎ : R  → R is a form of degree  in  scalar variables if where  , = { ∈   : ∑  =1   = } and  ∈ R  and   ∈ R is coefficient of the monomial   .
The set of forms of degree  in  scalar variables is defined as Definition 2. The function  : R  → R is a polynomial of degree less than or equal to , in  scalar variables, if where  ∈ R  and ℎ  ∈ Ξ , ,  = 1, . . ., .
The power transformation of degree  is a nonlinear change of coordinates that forms a new vector   of all integer powered monomials of degree  that can be made from the original  vector: Usually we take   = 1; then, with  > 0, otherwise For example,  = 2,  = 2, ⇒  (,) = 3, and Definition 4. The function  : R  → R × is a homogeneous parameter-dependent matrix of degree  in  scalar variables if We denote the set of  ×  homogeneous parameterdependent matrices of degree  in  scalar variables as ,, = { : R  → R × : (15) holds} ( 16) and the set of symmetric matrix forms as Definition 5. Let  ∈ Ξ ,2, and  ∈ S  (,) such that where Φ(,   , ) = (  ⊗   )  (  ⊗   ).Then ( 18) is called a square matricial representation (SMR) of () with respect to   ⊗   .Moreover,  is called a SMR matrix of () with respect to   ⊗   .

Main Result
In this section, the sufficient condition for existence of robust  ∞ filter for uncertain switched systems is formulated.For this purpose, we firstly consider the anti-interference of system (1) for disturbance.
Theorem 9.For a given scalar  > 0. with where Ψ(⋅) is a matrix which is made up of   ,   ,   ,   ,   ,   and the set L ,, is defined in Lemma 6, then system ( 1) is robustly asymptotically stable with  ∞ performance  for any switching signal.
Proof.Consider the following homogeneous parameter-dependent quadratic Lyapunov function: Then, along the trajectory of system (1), we have When  = , the switched system is described by the th mode.When  ̸ = , it represents the switched system being at the switching times from mode  to mode .
In terms of Lemma 8, condition (34) is equivalent to Since (26), it follows that which implies the matrices   (  ) are nonsingular for each .

Theorem 13. Given a constant
) , ) , ) , ) , ) , ) , then there exists a robust switched linear filter in form of ( 4) such that, for all admissible uncertainties, the filter error system (5) where From Theorem 9, the filter error system is robustly asymptotically stable with a prescribed  ∞ noise-attenuation level bound  if the following matrix inequality holds: By (48) and Lemma 7, one can obtain that inequality (45) is equivalent to (49).Thus, if (45) holds, the filter error system is robustly asymptotically stable with an  ∞ noise-attenuation level bound  > 0.Then, the proof is completed.

Examples
The following example exhibits the effectiveness and applicability of the proposed method for robust  ∞ filtering problems with polytopic uncertainties.Consider the following uncertain discrete-time switched linear system (1) consisting of two uncertain subsystems which is given in [1].There are two groups of vertex matrices in subsystem 1: Moreover, we define the disturbance: Firstly, consider the problem of being robustly asymptotically stable with an  ∞ noise-attenuation for the uncertain switched system which was given in Theorem 9.The different minimum  ∞ noise-attenuation level bounds  can be obtained by different methods.In addition, for given  = 1.2 and initial condition (0) = [0.1,−0.3]  , Figures 1 and 2 show system (1) is robustly asymptotically stable with an  ∞ noise-attenuation level bound  = 4.0988.
Next, we consider the problem of robust  ∞ filtering.In order to get  ∞ noise-attenuation level bound , we define   3 shows the filtering error system is robustly asymptotically stable and Figure 4 shows the error response of the resulting filtering error system by applying above filter.It is clear that the method in Theorem 13 is feasible and effective against the variations of uncertain parameter.

Conclusions
In this paper, the problems of being robustly asymptotically stable with an  ∞ noise-attenuation level bounds  and switched linear filter design for uncertain switched linear system are studied by homogeneous parameter-dependent quadratic Lyapunov functions.By using this method, the less conservative  ∞ noise-attenuation level bounds are obtained.Moreover, we also get a more feasible and effective method against the variations of uncertain parameter under the given arbitrary switching signal.Numerical examples illustrate the effectiveness of our method.

Table 1 :
Different minimum  for uncertain switched system.

Table 2 :
Different minimum  for robust switched linear filter.