control for singular systems with interval time-varying delays via dynamic feedback controller
Introduction
The integration and analysis of singular systems (SSs) (also called generalized state–space systems, differential–algebraic systems, semi–state systems, or descriptor systems, and so on) have attracted long attention from a large number of scholars, especially in the control or mathematical circles [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. These types of systems have unique characteristics. For example, they contain algebraic equations in addition to differential equations, which means their states can not be completely determined by their dynamic [11], [12], [13], [14], [15]. Therefore, the research on SSs is much more intricate than that of regular ones. In addition to studying the stability of the systems, we also need to think about their regularity and non–impulsiveness. Furthermore, the effect of time-delay in the practical use of control systems is unavoidable. It may lead to the decline of system performance and even the instability of the systems in serious cases. As a result, for SSs with time delays, in the light of the challenge of their research, numerous rich research results and applications have been surging out [16], [17], [18], [19], [20].
In recent decades, the application of feedback control has been one of the hot directions in the analysis and synthesis of SSs. It is controlled according to deviation. Meanwhile, it has the ability to suppress the influence of any internal and external disturbance on the controlled quantity, and has high control accuracy. Many valuable research results have been developed [21], [22], [23], [24], [25], [26], [27], [28]. For example, Chen et al. [21] studied the mixed and passive control problem based on the static out feedback for SSs with time delays. The dissipative fault–tolerant control with slow state feedback was discussed for nonlinear singular perturbed systems in [22]. The stabilization of neutral singular Markovian jump systems (SMJSs) was explored under state feedback control in [23], [24]. The output feedback control problem for SMJSs with uncertain transition rates was studied in [25]. We find that the feedback control discussed in the above literature is either state feedback control or static output feedback. However, the performance of dynamic feedback (DF) control is much better than that of static feedback control, mainly because its control response and control gain have greater degrees of freedom. So, the DF control issue has attracted more attention [29], [30], [31], [32], [33], [34], [35], [36]. Zhang et al. [29] considered the dynamic output–feedback stabilization of singular LPV systems and acquired the necessary and sufficient conditions for stability. Lin et al. [30] studied the resilient dynamic output–feedback controller design problem. Park et al. [31], by using the linear matrix inequality (LMI) approach, concerned the dynamic output–feedback control for singular interval–valued fuzzy system. Long and zhong in [32] designed a DF controller for a class of SSs and explored the control issue simultaneously. To sum up, we find that to obtain the desired feedback controller parameters, the usual processing method is to convert the obtained conditions containing nonlinear matrix inequality conditions into LMI conditions through some matrix transformation techniques or other technical means. Nevertheless, the linearization of some matrix terms including singular matrix or system coefficient matrix increases the computational complexity. Naturally, there will be a problem is how to better reduce the computational complexity and conservatism in such a processing process? Recently, a state decomposition recombination (SDR) approach in [37] to deal with the filtering for SSs has aroused our great interest. The main idea is to decompose and reorganize the closed–loop systems with a full-order filter through the characteristics of a singular matrix, obtain the equivalent transformation systems, and then analyze their admissibility. One of the main superiority of this method is that it can deal with decision variables more flexibly and remove the number of redundant decision variables to abate the math-based complex difficulty.
Inspired by the above statement, the DF control issue for SSs in this paper is revisited. First, by constructing a novel state decomposition augmented Lyapunov-Krasovskii functional (LKF), the admissibility criteria with performance index for unforced nominal SSs are derived. Second, with the help of the established admissibility criteria, some delay dependence sufficient conditions for the closed–loop systems with a DF controller are obtained by employing an SDR method. Finally, through three numerical examples, the advantages and feasibility of the presented results will be reflected. But here is what we know, few literature have tried to use the SDR method to study the DF issue for SSs. Thus, this article fills this gap. The main innovations are :
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In this paper, one of the differences from previous literature work on DF control in [29], [30], [31], [32], [33], [34], [35], [36] is the construction of a new state decomposition augmented LKF. More specifically, we construct LKF from the perspective of state decomposition components rather than from the overall state. According to the characteristics of the singular matrix and combined with the tighter integral inequality technique, the established LKF can decrease the conservatism and computational complexity.
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The delay–dependent conditions of the admissibility (according to non–impulsiveness, regularity, and stability) for the closed–loop SSs applying the DF controller are given by using the SDR method. In contrast, the results derived are less conservative than those in [7], [32], [37]. In particular, the desired DF controller parameters can be obtained more accurately and flexibly by discussing each decomposition component of the studied controller parameters.
is a positive definite matrix. The symbols and are transpose and inverse, respectively. are identity matrices and zero matrices, respectively. stands for . . . . is the determinant of . and are the set of real matrices and dimensional Euclidean space, respectively.
Section snippets
Problem statements
Consider the following SSs:where is the state satisfying , and . satisfies . , and are the disturbance input, controlled output and control input, respectively. is an initial function. and under proper dimensions are known matrices. When and , the system (1) is called the unforced nominal SS. The time
Main results
In this section, first, a less conservative admissible criterion will be given for the unforced nominal SS Eq. (1). Then, on this basis, we investigate the admissibility with performance for the closed–loop system Eq. (4) via using the SDR method. Theorem 1 System Eq. (1) satisfies admissibility with an performance for some given scalars and if there exist any matrices with appropriate dimensions and such thatwhere ,
Numerical examples
Example 1 Assume the unforced nominal system Eq. (1) has the following parameters:
Let , and with help of LMI toolbox in MATLAB, the minimum performance (MHP) for different in this example can be displayed in Table 1. We find that our results via applying Theorem 1 are better than those in [37]. Moreover, it should be pointed out that the number of decision variables (NDVs)in our results is less than that in
Conclusion
The DF control issue for SSs with interval time-varying delays has been sufficiently developed. First, via building a novel LKF with state decomposition components, some sufficient conditions including few decision variables relatively based on LMIs are acquired and the admissible criterion (namely regularity, impulse-freeness, and stability) with performance index is established for the unforced nominal SSs. Second, an SDR method is applied to the design of the DF controller and the
Data availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
CRediT authorship contribution statement
Wenbin Chen: Conceptualization, Methodology, Writing – original draft. Guangming Zhuang: Formal analysis, Writing – review & editing. Fang Gao: Software, Methodology. Wei Liu: Formal analysis, Methodology, Software. Weifeng Xia: Formal analysis, Software.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by National Natural Science Foundation of China under Grants 62103006, 61803002,62173159.
References (40)
- et al.
Noise–to–state stability criteria of switching stochastic nonlinear systems with synchronous and asynchronous impulses and its application to singular systems
Nonlinear Anal. Hybrid Syst.
(2022) - et al.
Dissipativity analysis for singular Markovian jump systems with time–varying delays via improved state decomposition technique
Inf. Sci.
(2021) - et al.
Reachable set estimation for singular systems via state decomposition method
J. Frankl. Inst.
(2020) - et al.
Stability analysis for a class of neutral type singular systems with time-varying delay
Appl. Math. Comput.
(2018) - et al.
Admissibility and stabilization of stochastic singular Markovian jump systems with time delays
Syst. Control Lett.
(2018) - et al.
A neutral system approach to stability of singular time–delay systems
J. Frankl. Inst.
(2014) - et al.
Delay-dependent hfiltering for singular Markovian jump time-delay systems
Signal Process.
(2010) - et al.
Extended passive filtering for discrete-time singular Markov jump systems with time-varying delays
Signal Process.
(2016) - et al.
On energy–to–peak filtering for semi-Markov jump singular systems with unideal measurements
Signal Process.
(2018) - et al.
New insight into reachable set estimation for uncertain singular time-delay systems
Appl. Math. Comput.
(2018)
Robust sliding mode control for nonlinear stochastic TS fuzzy singular Markovian jump systems with time-varying delays
Inf. Sci.
Mixed and passive control for singular systems with time delay via static output feedback
Appl. Math. Comput.
Dissipative fault–tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback
Appl. Math. Comput.
New results on stabilization for neutral type descriptor hybrid systems with time-varying delays
Nonlinear Anal. Hybrid Syst.
Further results on stabilization for neutral singular Markovian jump systems with mixed interval time-varying delays
Appl. Math. Comput.
Robust non–fragile proportional plus derivative state feedback control for a class of uncertain Takagi–Sugeno fuzzy singular systems
J. Frankl. Inst.
Output-feedback stabilization of singular LPV systems subject to inexact scheduling parameters
Automatica
Resilient dynamic output feedback controller design for USJSs with time-varying delays
Appl. Math. Comput.
Dynamic output–feedback control for singular interval–valued fuzzy systems: linear matrix inequality approach
Inf. Sci.
control for a class of singular systems with state time-varying delay
ISA Trans.
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