Stability analysis of systems with aperiodic sample-and-hold devices

doi:10.1016/j.automatica.2008.10.017

Automatica

Volume 45, Issue 3, March 2009, Pages 771-775

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Abstract

Motivated by the widespread use of networked and embedded control systems, improved stability conditions are derived for sampled-data feedback control systems with uncertainly time-varying sampling intervals. The results are derived by exploiting the passivity-type property of the operator arising in the input-delay approach to the system in addition to the gain of the operator, and are hence less conservative than existing ones.

Introduction

The sampled-data control theory has been well-developed in the last two decades. See Chen and Francis (1995) and references therein. Now one can, for example, design a robust digital controller taking the intersample behavior of the system into account by applying the results in the theory. In the development of the sampled-data control theory, one of the crucial properties is the periodicity of the closed-loop systems which comes from periodic sampling. It is indeed reasonable to assume periodic sampling in the conventional implementation of sampled-data systems.

In some recent applications, however, it is hard to perform periodic sampling. For example, resources for measurement and control are restricted in networked and/or embedded control systems (Hristu-Varsakelis & Levine, 2005), and hence the sampling operation tends to be aperiodic and uncertain. In view of the widespread use of such types of system, it is important to study the robustness against variation of sampling intervals, although it seems that there are fewer tools for the purpose.

The so-called input delay approach (Fridman, Seuret, & Richard, 2004) was proposed to treat the systems with aperiodic sampling. The basic idea of the approach is modeling the aperiodic sample-and-hold operations by a time-varying delay at control input, and hence one can apply methodologies developed for time delay systems to the aperiodic sampled-data systems. Several analysis and synthesis problems have been solved based on this approach (Fridman et al., 2004, Fridman et al., 2005, Naghshtabrizi and Hespanha, 2005, Suplin et al., 2007).

Recently it was pointed out in Mirkin (2007) that the stability condition from the input delay approach (Fridman et al., 2004) can be interpreted as an application of the small gain theorem, and a less conservative stability condition was derived based on the exact evaluation of the gain of the operator arising in the small gain argument. The derived condition is still conservative as pointed out in Mirkin (2007), even though it is based on the exact gain evaluation.

The purpose of this paper is to provide an improved stability criteria for the aperiodic sampled-data systems. Our main idea is to exploit extra information of the operator arising in the small gain argument in addition to the gain, based on the development of robust control theory in, e.g., Megretski and Rantzer (1997), Iwasaki and Hara (1998) and Scherer (2000). In particular, we will show a passivity-type property of the operator, and the stability criteria will be improved by combining the property and the gain condition.

This paper is organized as follows: The problem is formulated in Section 2 and existing results are reviewed in Section 3. Section 4 provides improved stability criteria and the validity is demonstrated in Section 5.

Section snippets

Problem setup

Let the following state–space system be given $\dot{x} (t) = A_{0} x (t) + B u (t)$ where $x$ and $u$ respectively denote the state and the input taking values in $R^{n}$ and $R^{m}$ . $A_{0}$ and $B$ are matrices of compatible dimensions.

We consider the following scenario of the feedback control of (1):

•
We measure the state of (1) when $t = τ_{k}$ ( $k = 0, 1, \dots$ ) where ${τ_{k}}$ is an uncertain set of discrete time instances satisfying $τ_{0} = 0,$ $lim_{k \to \infty} τ_{k} = \infty,$ and $τ_{k + 1} - τ_{k} \in (0, \bar{h}]$ for a given $\bar{h} > 0$ .
•
The control input $u$ is determined from the sampled-data $x (τ_{k})$

Review of existing results

In this section we review the input delay approach to stability analysis for our general scenario.

The basic idea of the pioneering paper (Fridman et al., 2004) for deriving stability criteria for $T$ is modeling the aperiodic sample-and-hold operation by a time-varying input delay, and is referred as the input delay approach.

In the input delay approach we rewrite (6) as $\dot{x} (t) = A_{0} x (t) + B F x (t - h (t))$ to introduce the time-varying delay term, where $h (t) ≔ t - τ_{k}, \forall t \in [τ_{k}, τ_{k + 1}) .$ Then we can apply methodologies

Main results

The main idea of this paper to improve stability criteria is to exploit an extra information of $Δ$ than the $L_{2}$ -induced norm. This is motivated by the use of general quadratic inequalities to characterize operators in the recent robust control theory. See, e.g., Megretski and Rantzer (1997).

In this section we first derive a non-gain type input–output property of $Δ$ , and then derive stability criteria.

Demonstrative example

Consider the following simple parameters: $A_{0} = - 2, B = 1, F = 1 (n = m = 1) .$ For the parameters $G (s)$ in (7) is given by $G (s) = \frac{- 1}{s + 1} + 1 = \frac{s}{s + 1}$ and thus ${‖ G ‖}_{\infty} = 1 .$ Consequently results in Fridman et al. (2004) and Mirkin (2007) guarantee the stability for $\bar{h} < 1, \bar{h} < \frac{π}{2}$ respectively.

Remark 1

Recently a stability condition in terms of LMI was derived based on a hybrid system model of $T$ with a Lyapunov function in Naghshtabrizi, Hespanha, and Teel (2006), and it was shown that the derived condition is less conservative than that

Concluding remarks

We have considered sampled-data feedback control systems, where the state is sampled aperiodically in general. Applications can be found in networked and/or embedded control systems.

We have taken the framework in the input delay approach (Fridman et al., 2004) and derived less conservative stability conditions for the systems, based on the combination of the existing results and the passivity-type property of $Δ$ .

In this paper we have considered a simple sampled state feedback scenario. However

Acknowledgments

The author would like to express his appreciations to Prof. Mirkin who introduced the author to this subject with fruitful discussions. The author is also thankful to Prof. Fridman for constructive comments including alternative proof of the stability of the system in Section 5 by showing the existence of a Lyapunov–Razumikhin functional with input-delay modeling. Thanks also go to Profs. Kao, Jönsson, Nešić, and Cantoni for valuable comments.

Hisaya Fujioka received the B.E., M.E., and Ph.D. degrees all from Tokyo Institute of Technology, Japan in 1990, 1992, and 1995, respectively. During 1995–1997, he was an Assistant Professor at Osaka University. In 1998 he joined Kyoto University, where he is currently an Associate Professor. His research interests include theory and application of robust and sampled-data control.

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^☆: This paper was presented at 7th IFAC Workshop on Time-Delay Systems, Nantes, France, September 17–19, 2007. This paper was recommended for publication in revised form by Associate Editor Mario Sznaier under the direction of Editor Roberto Tempo.

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Brief paperStability analysis of systems with aperiodic sample-and-hold devices☆

Abstract

Introduction

Section snippets

Problem setup

Review of existing results

Main results

Demonstrative example

Concluding remarks

Acknowledgments

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Systems & Control Letters

Automatica

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Systems & Control Letters

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Optimal sampled-data control systems

Brief paper
Stability analysis of systems with aperiodic sample-and-hold devices☆