Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 515232, 16 pages

http://dx.doi.org/10.1155/2015/515232

## Periodic Switched Control of Dual-Rate Sampled-Data Systems

School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

Received 21 October 2014; Revised 9 April 2015; Accepted 15 April 2015

Academic Editor: Dan Simon

Copyright © 2015 Dawei Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the periodic switched control of a linear dual-rate sampled-data system. The state variables of the continuous-time plant are sampled by two types of sensors. The ratio of two sampling rates is assumed to be a rational number. Depending on whether the sampled-data of state variables at two sampling rates is available simultaneously or separately, a periodic switched controller is constructed. Applying an input delay approach, the closed-loop system is modeled as a switched system with subsystems having different input delays. Some delay-dependent criteria for the performance of the switched system and the existence of the switched controller are derived by employing a Lyapunov-Krasovskii functional that includes information about two sampling periods. The dual-rate sampled-data control of a vehicle dynamic system is given to show that the proposed method is effective and it can achieve a better control performance than the single-rate design method.

#### 1. Introduction

Sampled-data control of continuous-time practical systems, especially complex industrial systems, offers several advantages such as flexibility, low cost, and increased reliability [1, 2]. In a sampled-data control system, where a continuous-time plant is controlled with a digital controller, the sampling rate is a critical design parameter. The choice of the sampling rate mainly depends on some factors like bandwidth and response time of closed-loop systems, physical limitations of sensors and actuators, and the effect of the noise. The sampling rate must be chosen as fast as possible to ensure the high precision and the fast response time of the system but as slow as possible to satisfy the hardware limits and to eliminate the effect of the noise on the control input. Taking these factors into account, the effective range of the sampling rate is determined.

In many industrial applications, it is impractical to sample all physical signals uniformly at one single-rate, which demands a multirate sampling scheme. For instance, for an industrial vehicle, laser sensors are used to measure the heave position and the heave velocity, and gyrometers are chosen to measure the angular velocity and the heading angle [3]. Due to the sensor restrictions and the control performance requirement, it is often necessary to sample the signals for different types of sensors at different sampling rates. The multirate sampling technique has received much attention since the early 1950s. Compared with the single-rate sampling scheme, the use of the multirate sampling technique is of two main benefits: (i) it may improve the performance such as improving the transient system behavior and enhancing the disturbance rejection property and (ii) it can provide a better tradeoff between the system performance and the implementation cost, which can be achieved by using analog-to-digital converters and digital-to-analog converters at different rates. Motivated by these benefits, much work has been done to deal with system modeling and identification, stability analysis, and controller synthesis of multirate sampled-data systems in the past few decades [3–12]. For example, in [4], a general framework of a multirate sampled-data control system is presented using nest operators and nest algebras, and an suboptimal controller satisfying causality constraint is designed by the lifting technique, in which the outputs and the control inputs are paired with sampling periods and holding periods , respectively, where , , is the set of positive integers, and is the base sampling period. Based on this framework, some particular cases of the multirate sampled-data systems are considered in [5–15]. More specifically, when the multirate sampled-data system involves a fast sampling rate and a slow control input rate (i.e., ), a new multirate sampling method for acceleration control is proposed in [9]; on the other hand, when the multirate sampled-data system involves a slow sampling rate and a fast control input rate (i.e., ), some estimation and/or control problems for several practical systems such as polymer reactors [10], visual servo control systems [11, 12], read-write arm of the hard disk drive [13, 14], and a pilot plant [15] are addressed. It should be mentioned that for the multirate sampling scheme with , in [9], the control signal is calculated by output measurement at each sampling rate , but only the one produced at the input rate is implemented. Although such a multirate scheme is effective in the realization of acceleration control in wide bandwidth, some of the output measurements may not be used to update the control actuation in time; for the multirate sampling scheme with in [3, 5–8, 10–15], the control signal is calculated recursively at each input rate only using the available data at each sampling rate . Moreover, using the lifting technique, the multirate sampled-data system is converted into an equivalent discrete-time single-rate time-invariant system in [3, 5–8, 10–15]. However, the equivalent conversion is not readily applicable to the continuous-time systems with polytopic uncertainties [16–18]. In [16], an input delay approach is proposed to investigate the sampled-data stabilization of linear systems, which can be extended to deal with the sampled-data control for systems with polytopic uncertainties and the networked control systems [19–24]. Most of the existing results developed by using the input delay approach, such as [16–18], have been largely focused on the sampled-data control of single-rate sampled-data systems. However, there are few results available on the dual-rate or multirate sampled-data control of a continuous-time system using the input delay approach except [20, 25], which provides the main motivation of the current study. In [20], exponential stability and the induced -gain of networked control systems are investigated, in which the sampled-data via dual-rate samplings are transmitted one after another by introducing a Round-Robin scheduling protocol. In [25], multirate sampled-data systems are modeled as systems with multiple input delays by reordering the updating instants, and some stability and stabilization conditions are established in terms of linear matrix inequalities. Without reordering sensor instants [20] or updating instants [25], this paper attempts to apply the input delay approach for a dual-rate sampled-data control system.

In this paper, we apply the input delay approach to investigate the periodic switched control of a linear continuous-time system system with two different sampling rates and , where the sampling periods and satisfy and , with being two positive integers, being the unique basic time period, and having no common factors greater than unity. Once the sampled-data of the state variable or is available, the control input is computed to update the system. Depending on whether the sampled-data of state variables at two sampling rates are available simultaneously or separately, a periodic switched controller with three switching modes is constructed to implement the sampled-data control. Using such a controller and the input delay approach, the resulting closed-loop system is modeled as a switched system with subsystems that have different input delays. A Lyapunov-Krasovskii functional that involves information about two sampling periods is constructed to derive some delay-dependent criteria for the performance of the switched system and the existence of the periodic switched controller. Comparing with the existing results for multirate sampled-data systems based on lifting technique [3–15], the proposed results can be trivially extended to handle the multirate sampled-data systems or networked control systems with polytopic uncertainties. The effectiveness of the proposed method and its advantage over a single-rate sampled-data control is shown by performing the dual-rate sampled-data control of a vehicle system.

*Notation*. The superscript “” stands for the transposition of a vector or a matrix. is the dimensional Euclidean space. is the set of nonnegative integers and is the set of positive integers. For symmetric matrices and , (resp., ) means that is negative semidefinite matrix (resp., negative definite matrix). is the maximum eigenvalue of a symmetric matrix . We use an asterisk “” to denote a term induced by symmetry and to denote the block-diagonal matrix. The space of square-integrable vector functions over is denoted by .

#### 2. Modeling of a Dual-Rate Sampled-Data System with a Switched Controller

Consider the linear system described bywhere , , and are the state, the control input, and the controlled output, respectively; is the external disturbance acting on system (1) and ; is the initial state; , , , , and are constant matrices of appropriate dimensions. It is assumed that all state variables of system (1) are sampled by two different types of sensors. Let , where , , and . Without loss of generality, we assume that and are paired with two sampling periods and , respectively, and , where and . Then the sequences of sampled-data of the state variables and are and .

In the proposed dual-rate sampling scheme, once or is available, the control signal is computed immediately for input update. By taking full advantage of and () in a real-time way, we construct the following switched controller, which is shown in Figure 1:where are the control gain matrices to be determined and is the switching signal. Notice that there are three cases of available sampled-data of the state variables for control computation of the controller (2): (i) both and are available, (ii) only is available, and (iii) only is available. Define the switching rules as follows. Set when and are available simultaneously, when only is available, and when only is available, respectively.