Delay-dependent sampled-data control based on delay estimates

doi:10.1016/j.sysconle.2010.12.001

Systems & Control Letters

Volume 60, Issue 2, February 2011, Pages 146-150

https://doi.org/10.1016/j.sysconle.2010.12.001 Get rights and content

Abstract

In this paper the sampled-data stabilization of linear time-invariant systems with feedback delay is considered. We assume that the delay is time-varying and that its value is approximatively known. We investigate how to use the available information about the evolution of delays for adapting the control law in real time. Numerical methods for the design of a delay-dependent controller are presented. This allows for providing a control for some cases in which the stabilization cannot be ensured using a controller with a fixed structure.

Introduction

Since the middle of nineties, the sampled-data control theory has known a considerable development [1] which has shown its effectiveness in conventional engineering applications. However, the development of embedded and networked control algorithms brought up new challenges [2]. In such control configurations, the resources are limited and time-varying delays have to be dealt with. In the literature, a great effort has been devoted to robust stability analysis and controller design methods [3], [4], [5], [6], [7], [8], [9]. Since the desired control specification cannot always be ensured with a fixed structure controller, the current trend is to adapt the control law according to the delay evolution (see e.g. [10] or [11]). A notable delay-dependent control in the context of sampled-data systems is the compensator proposed by Zhang et al. [2] which uses the exact value of the delay. However, it is generally very difficult to know or identify the exact delay value and to use it in real time. For this reason, such a controller may be a little bit too sensitive in configurations with delay variation or uncertain delay knowledge. One can notice from the literature that formal methods for deriving a robust delay-dependent controller are needed [2].

The goal of this paper is to present an LMI approach for the design of a delay-dependent stabilizing state feedback controller in the context of linear time invariant (LTI) systems with sampled-data control. The robustness of the method with respect to the approximative knowledge of delay value is explicitly considered. The method is based on a discrete-time linear time variant (LTV) model of the closed-loop system which enables one to reformulate the control design problem in a more tractable framework [12].

This paper is structured as follows. In Section 2 we provide motivations for our work and formalize the problem under study. Section 3 presents a discrete-time model that is appropriate for control design. A generic control design method is presented as a set of parametric LMI. In Section 4, we show how this parametric set of LMI can be addressed numerically. A numerical example is presented in Section 5.

Notations: $‖ M ‖$ denotes the induced matrix norm of a square matrix $M$ . By $M > 0$ or $M < 0$ we mean that the symmetric matrix $M$ is positive or negative definite respectively. We denote the transpose of a matrix $M$ by $M^{'}$ . By $I$ (or $0$ ) we denote the identity (or the null) matrix with the appropriate dimension. $λ_{\max} (M)$ denotes the maximum eigenvalue of a symmetric matrix. For a given set $S, co (S)$ denotes the convex hull of $S$ . The symbol $*$ denotes a block that can be inferred by symmetry.

Section snippets

Motivations and problem formulation

Consider the following continuous-time system $\dot{x} (t) = A x (t) + B u (t), \forall t \in R^{+} .$ Here $x (t) \in R^{n}$ represents the system state and $u (t) \in R^{m}$ represents the input. $x (t)$ is sampled with the period $T$ and $x_{k}$ denotes its discrete-time version at time $k T, k \in N$ . The implementation of the discrete control $u_{k}$ is affected by a time-varying delay $τ_{k}, 0 \leq τ_{k} \leq T$ , i.e. $u (t) = {\begin{matrix} u_{k - 1}, & \forall t \in [k T, k T + τ_{k}) \\ u_{k}, & \forall t \in [k T + τ_{k}, (k + 1) T) . \end{matrix}$ The exact value of the delay $τ_{k}$ is unknown. We consider that an estimated value ${\hat{τ}}_{k} \in T \subseteq [0, T]$ is available and that the

Control design

This section presents a delay-dependent control design procedure for the uncertain system (3). First, we rewrite the system (3) in a form that is more suitable to LMI control design. Next, generic conditions are presented as a parametric set of LMI.

Numerical issues

This section illustrates how the set of LMIs (16) can be used in practice. Two cases are being considered. First, we consider a simplified case in which the domain $T$ represents a quantization of the domain $[0, T]$ . Next, we consider the more complex case in which the estimation of the delay may take any value in $[0, T]$ .

Numerical example— Example 1 (revisited)

As follows we illustrate the polytopic delay-dependent feedback control design for the motivating example given in (9). We consider $T = 0.005, T = [0, 0.005]$ and $δ τ_{k}$ bounded in $[- 0.0015, 0.0015]$ . For this example it was not possible to obtain a (delay independent) state feedback controller based on the recent LMI in [8]. However a delay-dependent controller can be obtained using the method proposed in this paper (the LMI (26)). It can be shown that $‖ Ω (δ τ_{k}) ‖ \leq 1.6 \times 1 0^{- 3}$ . In order to derive an accurate

Conclusion

This paper presents a method for the stabilization of linear time-invariant systems with feedback delay. We present numerical methods for the design of a delay-dependent sampled-data state feedback. The controller is adapted in real time according to an estimate of the delay value. Robustness with respect to the delay uncertainty is explicitly taken into account. Numerical examples illustrate the approach.

Acknowledgements

The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no 257462: HYCON2 Network of Excellence “Highly-Complex and Networked Control Systems”.

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Delay-dependent sampled-data control based on delay estimates

Abstract

Introduction

Section snippets

Motivations and problem formulation

Control design

Numerical issues

Numerical example— Example 1 (revisited)

Conclusion

Acknowledgements

Automatica

Systems and Control Letters

Automatica

Systems & Control Letters

Systems & Control Letters

Computer-Controlled Systems

Stability of networked control systems

IEEE Control Systems Magazine