Abstract
This paper is concerned with the problem of robust filter design for switched linear discrete-time systems with polytopic uncertainties. The condition of being robustly asymptotically stable for uncertain switched system and less conservative noise-attenuation 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.
1. 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 [1–15]. 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 [16–19]. 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 [21–25]. 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 parameter-dependent 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 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytropic domains. Reference [25] introduced Gram-tight forms, that is, forms whose minimum coincides with the lower bound provided by LMI optimizations based on SOS (sum of squares of polynomials) relaxations.
Thus, in order to suppress the influence of uncertainties on the system’s robust control, the homogeneous parameter-dependent quadratic Lyapunov functions are used to design robust filter in this paper. We first consider the system’s anti-interference of external disturbance. Along this direction, the robust filters for switched linear discrete-time systems are designed. Lastly, through the comparison, we have the fact that homogeneous parameter-dependent quadratic Lyapunov functions can suppress conservativeness which the uncertainties bring.
The rest of this paper is organised as follows. We state the problem formulation in Section 2. The main results are presented in Section 3. Section 4 illustrates the obtained result by numerical examples, which is followed by the conclusion in Section 5.
Notation. denotes the -dimension Euclidean space and is the real matrices with dimension ; means ; 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 with ; means with ; denotes the Kronecker product of vectors and . denotes Euclidean norm for vector or the spectral norm of matrices.
2. Problem Statement
Consider a class of uncertain switched linear discrete-time systems which were given in [1]: where is state vector, is disturbance input which belongs to , is the measurement output, is objective signal to be attenuated, and is switching rule, which takes its value in the finite set . 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 . The vector represents the time-invariant parametric uncertainty which affects linearly the system dynamics. The vector can take any value in , 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 , ; , , 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 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 , 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 is a form of degree in scalar variables if where and and is coefficient of the monomial .
The set of forms of degree in scalar variables is defined as
Definition 2. The function is a polynomial of degree less than or equal to , in scalar variables, if where and , .
Definition 3. Consider the vector , . 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 ; then, with ,
otherwise
For example, , , , and
Definition 4. The function is a homogeneous parameter-dependent matrix of degree in scalar variables if
We denote the set of homogeneous parameter-dependent matrices of degree in scalar variables as
and the set of symmetric matrix forms as
Definition 5. Let and such that where . Then (18) is called a square matricial representation (SMR) of with respect to . Moreover, is called a SMR matrix of with respect to .
Lemma 6 (Chesi et al. [26]). Let . Then (18) is an affine space. Moreover, where is linear space: whose dimension is given by
Lemma 7 (Chesi et al. [26]). Let . Then holds if and only if
Lemma 8 (Boyd et al. [27]). The linear matrix inequality where and , is equivalent to
3. 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 . Consider system (1); if there exist matrices and matrices such that with where is a matrix which is made up of , , , , , and the set 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 .
Before and after multiplying inequality (26) by and , we haveFrom Lemma 7, one haswhich implies
Under the disturbance , we get
If
then . It implies system (1) is robust asymptotic stable.
In terms of Lemma 8, condition (34) is equivalent to
Since (26), it follows that
which implies the matrices are nonsingular for each . Then, we have
which is equivalent to
Hence, it can be readily established that (32) is equivalent toBefore and after multiplying the above inequality by , we obtain
which implies that (35) holds. Then, the stability of system (1) can be deduced.
On the other hand, let
be performance index to establish the performance of system (1). When the initial condition , we have which implies
where
In terms of Lemma 8, we have which means that . Then, the proof is completed.
Remark 10. Within robustly performance scheme, the basic idea is to get an upper bound of noise-attenuation level. Since model uncertainties may result in significant changes in noise-attenuation level, we should effectively reduce this effect. For this purpose, the homogeneous parameter-dependent quadratic Lyapunov function is exploited in Theorem 9.
Remark 11. In Theorem 9, is a matrix which is made up of , , , , , . For example, when and , condition (26) can be written as
Remark 12. About the calculation of matrices , the following algorithm is presented in [26]: (1)choose and as in Definition 3 where (2)set and and define the variable (3)for and (4)set (5)for and (6)set and (7)set and and (8)if (9)set and (10)else(11)set and and (12)set and and (13)set (14)endif(15)endfor(16)endfor(17)set .
Next, the condition for robust filter is formulated. For convenience, we define , , and .
Theorem 13. Given a constant , if there exist matrices , , , , , , , and positive definite matrices and scalar such that with then there exists a robust switched linear filter in form of (4) such that, for all admissible uncertainties, the filter error system (5) is robustly asymptotically stable and performance index holds for any nonzero , where , , , and .
Proof. Let
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 . Then, the proof is completed.
4. 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: and two groups of vertex matrices in subsystem 2: 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. Table 1 lists the different calculation results. From Table 1, it can be clearly seen that the results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better.
In addition, for given and initial condition , Figures 1 and 2 show system (1) is robustly asymptotically stable with an noise-attenuation level bound .
Next, we consider the problem of robust filtering. In order to get noise-attenuation level bound , we define . By solving the corresponding convex optimization problems in Theorem 13, we obtain the minimum noise-attenuation level bounds . Table 2 lists our results and [1]’s results. Moreover, when , the admissible filter parameters can be obtained according to Theorem 13 as By comparison, using homogeneous parameter-dependent quadratic Lyapunov function for the existence of a robust switched linear filter in Theorem 13 is better.
By giving noise-attenuation level bound and initial condition , Figure 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.
5. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This project is supported by Major Program of National Natural Science Foundation of China under Grant no. 11190015, National Natural Science Foundation of China under Grant no. 61374006, and Graduate Student Innovation Foundation of Jiangsu province under Grant no. 3208004904.