Robust ${\mathcal{H}}_2$ switched filter design for discrete-time polytopic linear parameter-varying systems

Highlights

• It is proposed an ${\mathcal{H}}_2$ switched filter to deal with discrete-time LPV systems.

• The filter is robust and does not require the online measurement of the time-varying parameter.

• The conditions do not require any assumption on the time evolution of the parameter.

• It is the first time that this kind of problem is treated in this general framework.

• Our design outperforms the ones available to date for time-invariant polytopic and LPV systems.

Abstract

This paper deals with a robust ${\mathcal{H}}_2$ filter design problem for discrete-time polytopic linear parameter-varying systems. The novelty is to design a set of full order filters and a switching rule to orchestrate them in order to assure an ${\mathcal{H}}_2$ performance index of the estimation error. The switched linear filter does not require the online measurement of the uncertain time-varying parameter which, in a great part of practical applications, is not available. The conditions are based on Lyapunov–Metzler inequalities with a special subclass of Metzler matrices. The proposed switched filter outperforms the ones available in the literature to date as far as the ${\mathcal{H}}_2$ norm of the estimation error is taken into account. The main theoretical implications of the switched filtering methodology in the framework of linear time-invariant uncertain systems and linear parameter-varying systems are discussed and illustrated by academical examples.

Introduction

In the last decades the topic of filter design of discrete-time uncertain systems has attracted a great interest of the scientific community. Roughly speaking two different filtering problems are in evidence. The first one is characterized by the fact that the uncertain parameter is time invariant, being the efforts focused on the design of robust filters. As for instance, using a technique based on a quadratic Lyapunov function, References [3], [31] propose conditions for the design of robust filters assuring minimum ${\mathcal{H}}_2$ or ${\mathcal{H}}_{\infty }$ guaranteed costs of the estimation error. In order to reduce the conservatism inherited by the quadratic conditions, References [13], [14], [30], [33] provide more general ones for robust filtering based on parameter dependent Lyapunov functions. All these references treat ${\mathcal{H}}_2$ filtering design, except the last one that treats the ${\mathcal{H}}_{\infty }$ design. The second filtering problem is characterized by the fact that the uncertain parameter is time-varying. For polytopic Linear Parameter-Varying (LPV) systems, see [1], [27] for details, the filtering problem is more involved and the difficulty stems from the fact that using less conservative techniques, as for instance, those based on parameter dependent Lyapunov functions, the knowledge of some information about the uncertain parameter, as for instance, its bounds or/and the bounds on its time variation rate is essential. This important issue was extensively studied in the last decades and the literature presents various contributions, as for instance [18], [19], [24], [26], [28], [34]. Basically they differ one from the other by the level of conservatism and the required computational burden. In all these references, except [18], [28], the goal is to design a gain scheduling filter where the parameter is supposed to be measured in real time (see [21], [25] for more details on gain scheduling techniques). However, in a great part of practical applications, the time-varying parameter is not available, or the measurement cannot be obtained with good precision, see [2]. Hence, robust filters as those provided in [18], [28] and also in [26], although more conservative, are of great interest.

Another research topic of particular importance developed in the last decades is switched linear systems analysis and control design. Basically, these systems are characterized by presenting a finite number of subsystems and a switching rule to orchestrate them. The switching rule can act in two different ways, as an arbitrary perturbation assuming values only on the vertices of a convex hull which characterizes a special subclass of LPV systems, or as a control variable to be designed in order to assure to the closed-loop system, global asymptotic stability and good performance. The books [23], [29] and the papers [10], [22], [35] are excellent and useful references on these topics. For stability analysis of uncertain switched systems, we can cite Reference [32], which proposes a state dependent switching function able to stabilize a polytopic switched system composed of two disjoint polytopes. As examples of filtering for switched linear systems with arbitrary switching, we can cite References [4], [11], [18], [34], which besides the arbitrary switching, the system of the last two references also presents time-varying uncertain parameters assuming values in a pre-specified polytope.

Motivated by recent results on consistency analysis for switched linear systems provided in [17] for continuous-time and [8] for discrete-time, and by the efficiency of the state feedback switched control as an alternative for the state feedback LPV control in both time domains, [7] for continuous-time and [9] for discrete-time, our main goal in this paper is to generalize the available results on switched systems to cope with robust ${\mathcal{H}}_2$ switched filter design for discrete-time LPV systems. The great advantage in this filtering technique is to avoid the online measurement of the uncertain parameter and to obtain conditions without requiring any assumption on the time evolution of the parametric uncertainty. More specifically, our main purpose is to design a full order switched linear filter and a switching rule depending only on the measured output to be connected to an LPV system assuring an ${\mathcal{H}}_2$ performance level. In the framework of switched linear systems, the literature presents some results regarding output feedback design, as for instance, [6], [16], [20]. However, the standard switched filter design problem has not been treated even for pure switched linear systems. In summary, our contribution is to generalize the Lyapunov–Metzler inequalities, defined in [15], with a key subclass of Metzler matrices in order to determine a switched filter where the switching function can be interpreted as an estimator of the time-varying parameter avoiding its online measurement. Moreover, the obtained conditions outperform the ones available in the literature for two cases: time-invariant uncertain systems and LPV systems. The efficiency and validity of the proposed theory is illustrated by means of academical examples.

For real matrices or vectors ($\prime$) indicates transpose and $(•)$ denotes generically each of its symmetric blocks. The set composed of the first N positive integers, namely $\{1,\dots ,N\}$ is denoted by $\mathbb{K}$. The set $\mathcal{M}$ of Metzler matrices consists of all square matrices $\Pi \in {\mathbb{R}}^{N\times N}$ with nonnegative elements satisfying the normalization constraints ${\sum }_{j\in \mathbb{K}}{\pi }_{\mathit{ji}}=1$ for all $i\in \mathbb{K}$. The convex combination of matrices is denoted by $J_{\lambda }={\sum }_{j\in \mathbb{K}}{\lambda }_j{J}_j$ where $\lambda$ belongs to the unitary simplex $\Lambda$ composed of all nonnegative vectors $\lambda \in {\mathbb{R}}^N$ such that ${\sum }_{j=1}^N{\lambda }_j=1$. Finally, the squared norm of a trajectory $\xi \left(k\right)$, defined for all $k\in \mathbb{N}$, denoted by $\parallel \xi {\parallel }_2^2$, is equal to $\parallel \xi {\parallel }_2^2={\sum }_{k=0}^{\infty }\xi \left(k\right)\prime \xi \left(k\right)$. For simplicity, the Cartesian product $\mathcal{A}\times \mathcal{A}\times \cdots \times \mathcal{A}$ for $n$ times is denoted by ${\mathcal{A}}^n$. For given scalars ${\pi }_{\mathit{ji}}\in \mathbb{R}$ the linear combinations of positive definite matrices is denoted as $P_{\pi i}={\sum }_{j\in \mathbb{K}}{\pi }_{\mathit{ji}}{P}_j$ for each $i\in \mathbb{K}$.