Robust H∞ control for switched systems with input delays: A sojourn-probability-dependent method

doi:10.1016/j.ins.2014.05.017

Information Sciences

Volume 283, 1 November 2014, Pages 22-35

https://doi.org/10.1016/j.ins.2014.05.017 Get rights and content

Abstract

In this paper, a sojourn-probability-dependent method is proposed to investigate the robust $H_{\infty}$ control for a class of switched systems with input delays. The considered system has the following characteristics: (1) it is a switched system consisting of a set of subsystems; (2) sojourn probabilities (i.e. the probability of switched systems staying in each subsystem) are assumed to be known (or partly known) a prior; and (3) there are input delays and parameter uncertainties in each subsystem. By using the sojourn probability information, a new type of switched system model is built. By using the Lyapunov functional method, the robust mean square stability criteria are obtained for switched systems under two conditions: (A) all sojourn probabilities of the subsystems are known; (B) only partly sojourn probabilities are known. Then the robust $H_{\infty}$ controller feedback gains are derived by using the cone complement linearization method. An inverted pendulum system and a numerical example are given to demonstrate the effectiveness and applicability of the proposed method.

Introduction

Switched systems have many applications in nature science, engineering and social sciences because various real-world systems cannot be described simply by a single model, and those systems exhibit switching behavior between several subsystems [28], [24], [15], [25], [23]. The switching law $r (k)$ (also called logical rule) orchestrates switching between these subsystems and generates switching signals, which are usually described as classes of piecewise constant maps, $r (k) : Z^{+} \to Ω$ , and $r (k) = i \in Ω$ is called the active subsystem at time k. There are various switching laws in real switched systems, and the following types have been widely investigated [10], [7]: (1) time-dependent switching law, wherein switching depends on the time k; (2) state-dependent switching law, wherein switching depends on variation of the state; (3) memory-dependent switching law [27], wherein switching depends on the history of active subsystems with Markov switching law as an example; and (4) arbitrary switching law [2], wherein switching between subsystems has no restrictions.

In most switched systems, elements of switching law should be known a prior and will be used in system analysis and synthesis. For example, the minimum dwell time of dwell-time switched systems and the transmission probability matrix (TPM) of Markovian jump systems (MJSs) should be known a prior. However, in some practical systems, all or parts of those elements are probably hard or costly to obtain [21]. Consequently, analysis and control methods based on existing switching laws will fail to work, or switched systems have to be regarded as arbitrary switched systems [27], which brings about great conservatism. Therefore, alternative switching rules should be sought for switched systems.

The motivation of the proposed switched system with KSP comes from two aspects, the first is the difficulty to obtain full transition probabilities in MJSs, which motivates us to find an alternative switching rules in the stochastic switched systems. The KSP information is more easier to be measured than the transition probabilities; the second is that there have been some systems with various problems, such as systems with missing measurement [19], random time delay [20], random packet loss [22], probabilistic sensor/actuator failure [5] or randomly occurring nonlinearities [18], which have been modeled as switched-like systems with a few sojourn probability information. This motivates us to consider that is there a general modeling and analysis method to cover those stochastic problems. The proposed switched system model with KSP can be seen as a generalization of the systems with the above mentioned system models.

In this paper, a sojourn-probability-dependent switching law is proposed and investigated for system analysis and control synthesis of switched linear discrete systems. This is based on the fact that over a long time horizon, the probability of the system staying in a particular subsystem can be easily obtained. For each subsystem, we call this probability sojourn probability and assume that this is known (or partly known) a prior. It should be noted that the sojourn-probability-dependent switching law does not change the rules of original switched systems, but utilize some existing information (sojourn probabilities) which has not been fully concerned before. To the best of the authors’ knowledge, such an approach appears not to have been reported in related literature.

The main contributions of this paper are highlighted as follows: (1) by using sojourn probability information, a new type of switching law is proposed for switched systems, and a statistical approach on how to measure the sojourn probability is proposed for linear discrete switched systems; (2) based on the sojourn-probability-dependent switching law, a new type of switched system model is constructed, which can be seen as a generalization of many practical systems with a class of problems; (3) some new analysis and control methods are proposed for the developed system model, sufficient conditions for the robust mean square stability (RMSS) of switched systems are obtained and the corresponding controller for each subsystem can also be obtained; (4) when only partly sojourn probabilities are known, the system modeling, analysis and control are also considered in this paper.

The rest of the paper is organized as follows. Section 2.1 proposes the definition of sojourn probabilities and the measurement method, and introduces the relation between the sojourn-probability-dependent switching law and the Markovian jump switching law. By using the sojourn probability information, a new type of switched system model is constructed in Section 2.2. The generalization and applicability of the proposed system model is illustrated in Section 2.3. In Section 3.1, sufficient conditions are derived for the RMSS of switched systems with completely known sojourn probabilities. When only partly sojourn probabilities can be measured, the system model and RMSS criteria are provided in Section 3.2. In both Sections 3.1 , 3.2 , the $H_{\infty}$ controller feedback gain can be obtained by using the cone complement linearization method. An inverted pendulum system and a numerical example are given to demonstrate the effectiveness and applicability of the proposed method in Section 4. Finally, Section 5 concludes the paper.

Section snippets

Problem formulation

Consider a discrete-time switched system $x (k + 1) = A_{r (k)} (k) x (k) + B_{r (k)} (k) u (k - d_{r (k)}) + L_{r (k)} ω (k),$ $z (k) = C_{r (k)} (k) x (k) + F_{r (k)} (k) u (k - d_{r (k)}) + D_{r (k)} ω (k),$ where $x (k) \in R^{m}$ is the state vector, $z (k) \in R^{p}$ is the controlled output, $u (k) \in R^{q}$ is the control input, and $ω (k) \in R^{r}$ is the disturbance input. $r (k) : Z^{+} = {0, 1, 2, \dots} \to {1, 2, \dots, n} ≜ Ω$ is the switching law. For $r (k) = i \in Ω, d_{i}$ is a constant input delay of the $i$ th subsystem, $A_{i} (k), B_{i} (k), L_{i}, C_{i} (k), F_{i} (k)$ and $D_{i}$ are matrices with compatible dimensions for the $i$ th subsystem and $\begin{matrix} A_{i} (k) = A_{i} + Δ A_{i} (k \end{matrix}$

$H_{\infty}$ Control for switched systems under Condition A

In this section, some criteria are proposed for the RMSS of system (12), (13) under Condition A, i.e., all the sojourn probabilities of the subsystems are known.

Theorem 1

For a given parameter $γ > 0$ , system (12), (13) under Condition A is RMSS with an $H_{\infty}$ norm bound $γ$ if there exist matrices $P > 0, Q_{i} > 0, R_{i} > 0$ and $K_{i}$ $(i \in Ω)$ with compatible dimensions such that the following inequality holds $[\begin{matrix} Ξ_{11} & * & * & * \\ Ξ_{21} (k) & Ξ_{22} & * & * \\ Ξ_{31} (k) & 0 & Ξ_{33} & * \\ Ξ_{41} (k) & 0 & 0 & Ξ_{44} \end{matrix}] < 0,$ where $Λ_{1 i} (k), Λ_{2 i} (k)$ and $Λ_{3 i} (k)$ are defined in (14), (15), (16) and $\begin{matrix} Ξ_{11} = [\begin{matrix} I_{11} & * & * \\ I_{21} & I_{22} & * \\ 0 & 0 \end{matrix}] \end{matrix}$

Illustrative examples

In this section, an inverted pendulum system and a numerical example are proposed to demonstrate the effectiveness of the analysis method.

Example 1

Consider an inverted pendulum system with delayed control input taken from [3]. The inverted pendulum on a cart is depicted in Fig. 1, wherein a pendulum is attached to the side of a cart by means of a pivot which allows the pendulum to swing in the plane and the system equations are $(M + m) \ddot{x} + ml \ddot{θ} \cos θ - ml {\dot{θ}}^{2} \sin θ = u ml \ddot{x} \cos θ + (4 / 3) {ml}^{2} \ddot{θ} - mgl \sin θ = 0,$ where $u (t)$ is the

Conclusion

This paper investigates robust $H_{\infty}$ control for switched systems with input delays. The probabilities of a system staying in each subsystem are assumed to be known a prior. By using these probability information, new kind of switched system model is proposed and the robust mean square stability criteria are derived for the switched system with an $H_{\infty}$ performance index $γ$ . Furthermore, when only partial sojourn probabilities are measured, the corresponding system model and analysis method are also

Acknowledgements

This work was partly supported by The Hong Kong Polytechnic University (Project code: G-YM53), the National Natural Science Foundation of China (Grant Nos. 61273115, 61374055 and 61375012) and the Natural Science Foundation of Jiangsu Province of China (Nos. BK2012847 and BK2012469).

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Robust H∞ control for switched systems with input delays: A sojourn-probability-dependent method

Abstract

Introduction

Section snippets

Problem formulation

H∞ Control for switched systems under Condition A

Illustrative examples

Conclusion

Acknowledgements

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