Research Article | Open Access
Finite Frequency Filtering for Time-Delayed Singularly Perturbed Systems
This paper investigates the problem of finite frequency (FF) filtering for time-delayed singularly perturbed systems. Our attention is focused on designing filters guaranteeing asymptotic stability and FF disturbance attenuation level of the filtering error system. By the generalized Kalman-Yakubovich-Popov (KYP) lemma, the existence conditions of FF filters are obtained in terms of solving an optimization problem, which is delay-independent. The main contribution of this paper is that systematic methods are proposed for designing filters for delayed singularly perturbed systems.
Several physical processes are on one hand of high order and on the other hand complex, what returns their analysis and especially their control, with the aim of certain objectives, very delicate. However, knowing that these systems possess variables evolving in various speeds (temperature, pressure, intensity, voltage…), it could be possible to model these systems by singularly perturbed technique [1, 2]. They arise in many physical systems such as electrical power systems and electrical machines (e.g., an asynchronous generator, a DC motor, and electrical converters), electronic systems (e.g., oscillators), mechanical systems (e.g., fighter aircrafts), biological systems (e.g., bacterial-yeast-virus cultures, heart), and also economic systems with various competing sectors. This class of systems has two time scales, namely, “fast” and “slow” dynamics. This makes their analysis and control more complicated than regular systems. Nevertheless, they have been studied extensively [3–5].
As the dual of control problem, the filtering problems of dynamic systems are of great theoretical and practical meaning in the field of control and signal processing and the filtering problem has always been a concern in the control theory [6–8]. The state estimation of singularly perturbed systems also has attracted considerable attention over the past decades and a great number of results have been proposed in various schemes, such as Kalman filtering [9, 10] and filtering .
Like all kinds of systems which can contain a time-delay in their dynamic or in their control, the singularly perturbed systems can also contain a delay, which has been studied in many references such as [12–15]. For example, Fridman  has considered the effect of small delay on stability of the singularly perturbed systems. In [13, 14], the controllability problem of nonstandard singularly perturbed systems with small state delay and stabilization problem of nonstandard singularly perturbed systems with small delays both in state and control have been studied, respectively. In  a composite control law for singularly perturbed bilinear systems via successive Galerkin approximation was presented. However, there is seldom literature dealing with the synthesis design for the delayed singularly perturbed systems, which is the main motivation of this paper.
With the fundamental theory-generalized KYP lemma proposed by Iwasaki and Hara , the applications using the GKYP lemma have been sprung up in recent years [17–23]. Actually, if noise belongs to a finite frequency (FF) range, more accurately, low/middle/high frequency (LF/MF/HF) range, design methods for the entire range will be much conservative due to overdesign. Consequently, the FF approach has a wide application range. In future work, we can use this approach to Markovain jump systems [24, 25].
Lately, using FF approach to analyze and design control problems becomes a new interesting in singularly perturbed systems [19–21]. In  the author has studied the control problem for singularly perturbed systems within the finite frequency. In  the positive control problem for singularly perturbed systems has been studied based on the generalized KYP lemma. The idea is that design in the finite frequency is critical to singularly perturbed systems since the transfer function of singularly perturbed systems has two different frequencies, that is, the high frequency and low frequency, which are corresponding to the fast subsystem and the low subsystem separately. Hence the idea based on the generalized KYP lemma actually is constructing the design problem in separate time scale and also separate frequency scale. Obviously, it could be seen that the conservativeness is much less than existing results. As far as the author’s knowledge, there is seldom literature referring to the filtering design problem within the separate frequencies for delayed singularly perturbed systems, which is also the main motivation of this paper.
In this paper we are concerned with FF filtering for singularly perturbed systems with time-delay. The frequencies of the exogenous noises are assumed to reside in a known rectangular region, which is the most remarkable difference of our results compared with existing ones. Based on the generalized KYP lemma, we first obtained an FF bounded real lemma in the parameter-independent sense. Filter design methods will be derived by a simple procedure. The main contribution of this paper is summarized as follows: the standard filtering for singularly perturbed systems has been extended to the FF filtering for singularly perturbed systems with time-delay, and systematic filter design methods have been proposed.
The paper is organized as follows. Section 2 gives the problem formulation and preliminaries used in the next sections. The main results are given in Section 3. An illustrative example is given in Section 4. And conclusions are given in the last section.
Notation. For a matrix , its transpose is denoted by ; is an arbitrary matrix whose column forms a basis of the null space of . indicates . denotes maximum singular value of transfer function. stands for a block-diagonal matrix.
2. Problem Formulation and Preliminaries
The following time-delayed singularly perturbed systems will be considered in this paper: where is “slow” state and is “fast” state. Denote ; is the state vector; is the measured output signal, is the signal to be estimated, and is the noise input signal in the functional space domain. Time-delay is known and time-invariant is the known initial condition in the domain ; let , , , , , , , , , , , , , , , , be appropriate constant matrices. Then the above system can be regulated as First, we give the following assumption on noise signal .
Assumption 1. Noise signal is only defined in the low, medium, and high frequency domains
Remark 2. By the generalized KYP lemma in  and by an appropriate choice and , the set can be specialized to define a certain range of the frequency variable . For the continuous time setting, we have , , where is defined in (4); in this situation, can be chosen as
The main objective of this paper is to design the following full-order linear filtering: where is the state vector, is the measured output signal, is the output for the filtering systems, and , , , are the filtering matrices to be solved. Combine (2) and (6) and let ; the following filtering error system is obtained: where is the filtering error and ; the filtering system matrices are The transfer function of the filtering error system in (7) from to is given by where “” is the Laplace operator.
Due to the asymptotic stability of the filtering error system (7) depends on system (2), while delayed system (2) does not include input channel for the input signal, so the following assumption is given.
Assumption 3. Time-delayed singularly perturbed system in (2) is asymptotically stable.
Now the problems to be solved can be summarized as follows.
Problem 4. For the continuous time-delay system in (2), find a full-order linear filtering (6), such that the filtering error system in (7) satisfies the following conditions.(1)The filtering error system in (7) is asymptotically stable.(2)Given the appropriate positive real , under the zero initial condition, the following finite frequency index is satisfied:
To conclude this section, we give the following technical lemma that plays an instrumental role in deriving our results.
Lemma 5 ((projection lemma) ). Let , , and be given. There exists a matrix satisfying if and only if the following projection inequalities are satisfied:
3. Main Results
3.1. FF Filtering Performance Analysis
To ensure the asymptotic stability and FF specification in (11) for the filtering error system, we need to resort to the generalized KYP lemma in . Based on this, the following lemma can be obtained.
Lemma 6. (i) Given system in (2) and scalars , , , the filtering error system in (7) is asymptotically stable for all and satisfies the specifications in (11) if there exist matrices with the form of (16), , , , , with the form of (17), , , , such that , , and matrices , , and that satisfy
(ii) Given , if there exist matrices with the form of (16) when , , , , , with the form of (17) when , , , , , , , and such that (14), (15) are feasible for then filtering error system (7) is asymptotic stable and satisfies the specifications in (11) for all small enough and .
Proof. (i) Let .
Give the following Lyapunov-Krasovskii functional where Then
Define the following performance index:
Using the zero initial conditions, we could get Let and use the Parseval equality to get Considering frequency and combining (23) and (24), we know that is a sufficient condition of , for all . Additionally, due to , we know that is a zero space of .
By the zero space theory, we know that Then by generalized KYP lemma in , the sufficient condition for is existing , , such that the following inequality is satisfied: By redefining as , we obtain inequality (14) that completes the first part of (i).
As for the asymptotic stability, the Lyapunov-Krasovskii functional can be reselected as follows: where and .
Then similar to the proof of the first part of (i), it could be shown that inequality (15) can guarantee the asymptotic stability of (7).
(ii) If (14), (15) are feasible for , then they are feasible for all small enough and thus, due to (i), filtering error system in (7) is asymptotic stable and satisfies the specifications in (11) for these values . Furthermore, linear matrix inequalities (14), (15) are convex with respect to ; hence they are feasible for some ; then they are feasible for all .
Remark 7. Though the essence of the proof of Lemma 6 follows from that of Theorem 1 in , there are also some differences in Lemma 6. First, in Lemma 6, time-delayed singularly perturbed systems are considered, which are totally different from the regular systems. Since the singularly perturbed systems have perturbed parameter , the existence of can lead to the ill-conditioned numerical problems, so here comes part (ii) of Lemma 6. Furthermore, considering the special structure of singularly perturbed systems, note that, in the proof of part (i), the selected Lyapunov-Krasovskii functionals are different from the regular systems.
To facilitate and reduce the conservatism of the filter design using the projection lemma, we present an alternative of Lemma 6.
Lemma 8. Given delayed systems in (2) and scalars , , filter in (6) exists such that the filtering error system in (7) is asymptotically stable and satisfies the specifications in (11) if there exist matrices with the form of (16), with the form of (17), , , and and satisfying with the other notations are defined in (14) and (15).
Proof. Define . By the Schur complement, (28) is equivalent to
By the definition of matrices and , one can choose and ; thus the first inequality in (13) vanishes and then by Lemma 5, (31) is equivalent to
By calculation, we can obtain (32) is equivalent to (14) when .
Similarly, by introducing the following null space, Using projection lemma, (29) is equivalent to the following inequality: By calculation, (34) is equivalent to (15) when .
Thus, Lemma 8 is equivalent to Lemma 6.
3.2. Design of FF Filters
Lemma 8 does not give a solution to filter realization explicitly. Based on the result in the following section we focus on developing methods for designing FF filters. The following result can be derived via specifying the structure of the slack matrices and in Lemma 8.
Theorem 9. Given time-delayed systems in (2) and scalars , , filter in (6) exists such that the filtering error system in (7) is asymptotically stable and satisfies the specifications in (11) if there exist matrices with the form of (16), with the form of (17), , , , , , , , , such that the following matrices are satisfied for the given scalars : where, , , and are defined in (28) and (29). Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the filter in (6) is given by
Proof. First, in order to prove Theorem 9, we just need to prove that (28) and (29) in Lemma 8 can be deduced from (35) in Theorem 9. It is noted that the slack matrix has the following form:
Here, , are complex matrices with dimension . In fact, is nonsingular and can be implied from (24); then by multiplying from the left side and the right side, respectively, with and , we could get
Without loss of generality, we could restrict .
On the other hand, to overcome the difficulty of filtering design, more work should be done. Next, we will consider and to be linearly -dependent; that is, , , and are scalars, respectively. Similarly, the slack matrix has the same structure restriction; that is, Define By substituting , in (28) and (29) and of (40) into , one can obtain . Moreover, by using (41), one can also obtain . The proof is complete.
4. Numerical Example
In this section, we use an example to illustrate the effectiveness and advantages of the design methods developed in this paper. Consider the singularly perturbed system with time-invariant delay in (2) with matrices given by
Suppose the frequencies , , we calculate the achieved minimum performance by using Theorem 9 in this paper. For brevity, the scalar parameters in Theorem 9 are given by , . The obtained minimum performance is , when ; the problem becomes a standard filtering problem and the minimum performance of the nominal filtering is .
And the obtained state-space matrices of filtering are
This paper has studied the problem of FF filtering for time-delayed singularly perturbed systems. The frequencies of the exogenous noise are assumed to reside in a known rectangular region and the standard filtering for singularly perturbed systems has been extended to the FF case. The generalized KYP lemma for singularly perturbed systems has been further developed to derive conditions that are more suitable for FF performance synthesis with time-delay. Via structural restriction for the slack matrices, systematic methods have been proposed for the design of the filters that guarantee the asymptotic stability and FF disturbance attenuation level of the filtering error system.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by National Natural Science Foundation of China under Grants (no. 61304089, no. 51405243), in part by Natural Science Foundation of Jiangsu Province Universities (no. BK20130999), and in part by Natural Science Foundation of Jiangsu Province (no. BK2011826).
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