Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 515232, 16 pages
http://dx.doi.org/10.1155/2015/515232

Research Article

Periodic Switched Control of Dual-Rate Sampled-Data Systems

Dawei Zhang, Wei Bai, and Xinchun Jia

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 by

where , , 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.

Figure 1: The multirate sampled-data system architecture using a digital switched controller.

Due to the fact that , the interval of the control input is or equally . Define and . Based on the switching rules of controller (2), we propose the following interval partition:

where

To clearly show the partition process, a special case for the interval partition of with and is depicted by Figure 2.

Figure 2: A special case for the interval partition of with and .

Considering interval partition (3) and the availability of and , one can see that switching controller (2) is periodically activated on the time instants . Then the control input can be described by

Define the following input delays on the subintervals in (3):

where . It follows that

Using (1), (5), and (6), the resulting closed-loop system can be described by

where and and denote and identity matrices, respectively.

The purpose of this paper is to investigate the periodic switched control such that system (8) is exponentially stable with a prescribed performance, which means that

(i)system (8) with is exponentially stable; that is, there exist constants and such that for , where [23, 26];(ii) can be ensured for all nonzero under the zero initial condition , where and are positive constants.

To end this section, we introduce the following lemma.

Lemma 1. For any constant matrices , (), , , and scalars , () satisfying and a vector function such that the integration concerned is well defined, the following inequality holds:

where

Proof. If (resp., ), inequality (9) reduces to be the one in Lemma 1 [27]. If , using Jensen inequality to obtain

where

for any , the following inequalities hold:

It follows from (13) that

Combining (14) and (15), we obtain inequality (9). This completes the proof.

3. Performance Analysis and Periodic Switched Controller Design

In this section, we first derive a delay-dependent criterion such that system (8) is exponentially stable with prescribed performance by using the Lyapunov-Krasovskii functional method. Based on the derived performance criterion, we establish the delay-dependent criterion on the existence of a switched controller for system (8).

Proposition 2. Given , , , , satisfying , and the gain matrices , system (8) is exponentially stable with a given performance for any switching signal with , if there exist matrices , , , and and symmetric matrices , , and such that , , , , , , , , , , , , and

where

Moreover, the parameters in the exponential stability and performance are given by

with

Proof. See Appendix A.

We now state and establish the delay-dependent criterion for the switched controller design.

Proposition 3. Given , , , , , and satisfying , system (8) is exponentially stable with a given performance for any switching signal with , if there exist matrices , , , , , , , , and and symmetric matrices , , and such that , , , , , , , , , , , , and

where

Moreover, the control gains of controller (2) are obtained by , , and , where and . The parameters in the exponential stability and performance are given by

with

Proof. See Appendix B.

Remark 4. Suppose that the system matrices are not exactly known and they reside in the uncertain polytope [16, 17, 26]. Consider

where the vertices of the polytope are described by . For the performance analysis and the periodic switched controller design of the dual-rate system with polytopic uncertainties, one can solve the LMIs in Propositions 2 and 3 for all the vertices by applying the same decision matrices, respectively.

For comparison purpose, a criterion for the existence of a sampled-data controller for system (1) via single-rate sampling is developed and given by the following corollary.

Corollary 5. Given , and , system (1) with a single-rate sampled-data controller is exponentially stable with a given performance , where the exponential decay rate , if there exist a matrix and symmetric matrices , and such that

where

Moreover, the control gain is obtained as .

Remark 6. For the case ( and ), periodic switched controller (2) reduces to a switched controller with switching modes and . Correspondingly, Proposition 3 with , , , and () reduces to a sampled-data control design result for system (1) via two sampling rates and . Particularly, when , the design result can be reduced to the one in [28].

Remark 7. In this paper, we obtain some results on periodic switched control of dual-rate sampled-data systems. It should be pointed out that these results can be extended to multirate sampled-data systems although the process is very tedious. The corresponding results are omitted.

Remark 8. As shown in [20, 29], analysis and synthesis of research on networked control systems with multiple samplings are promising and significant issues. Compared with [20], this paper is distinguished based on the following evidences: (i) The dual-rate sampling schemes in this paper and [20] are different, resulting in two essentially different switched system models. In [20], by introducing a Round-Robin scheduling protocol, the sampled-data via two different sampling rates is available one after another. In this paper, however, the sampled-data is available in three cases (both and , only , and/or only ). In other words, under the sampling relationship in this paper, the case that and are available simultaneously must occur, but this surely does not occur in [20]. (ii) The problems addressed in this paper and [20] are different. Stability and -gain analysis carried out for dual-rate networked control systems is considered in [20], while this paper focuses on controller design for a dual-rate sampled-data system without considering the effect of network-induced delays. It should be mentioned that the constant constraint on network-induced delays in [20] does not significantly change the nature of the proposed results. (iii) The methodologies of performance analysis in this paper and [20] are different. In [20], a common discontinuous Lyapunov-Krasovskii functional is constructed to derive some delay-dependent stability and -gain criteria. However, in this paper, a switched continuous Lyapunov-Krasovskii functional is proposed to establish delay-dependent criteria for the performance and switched controller design by fully utilizing the relation of multiple input delays and the information about two sampling periods.

4. The Dual-Rate Sampled-Data Control of a Vehicle Dynamic System

In this section, we consider the dual-rate sampled-data control of a vehicle dynamic system described by the following state-space representation [30]:

where

, is the inertial lateral displacement of the vehicle mass center, is lateral velocity in the vehicle body axis system, is the angular velocity, and is the vehicle heading angle.

The state variables and are measured by using laser sensors at a sampling rate , and the state variables and are measured by using gyrometers at a sampling rate , where are two sampling periods.

We now show the effectiveness of the proposed dual-rate sampling design result. Set , , , , , , ,  s, and  s. Using Proposition 3, we can obtain the minimum performance , an exponential decay rate , and corresponding control gain matrices , , and , which means that switched controller (2) with the above can stabilize system (39) and achieve an performance and for system (39) with an external disturbance. Using Corollary 5, the minimum performance , the exponential decay rate , and the control gain is . Clearly, Proposition 3 can be used to search for a better control performance than Corollary 5. For comparison purpose, we choose different delay bounds of for both dual-rate sampling case and single-rate sampling case. The solvability of Proposition 3 and Corollary 5 is shown in Table 1, from which one can see that the dual-rate sampling design result can provide a larger sampling period than the single-rate design result.

Table 1: Solvability comparison between Proposition 3 with and Corollary 5 with .

0.24 s
0.27 s
0.30 s
0.36 s

Proposition 3 (dual-rate)
Feasible
Feasible
Feasible
Feasible

Corollary 5 (single-rate)
Feasible
Feasible
Infeasible
Infeasible

Then we compare the proposed dual-rate sampling design result and the single-rate design result in simulation. Choose the initial state and the external disturbance . Using the switched controller with , we can depict the state responses of system (39) by Figure 3 and the corresponding control input by Figure 4, respectively. Under the zero initial condition, -norm using the switched controller and using the single-rate sampled-data controller are shown in Figure 5. From Figures 3–5, one can conclude that the dual-rate sampled-data can stabilize the system (39) and achieve a better performance than the single-rate sampled-data controller.

Figure 3: The state responses of system (39) using the switched controller.

Figure 4: The control input of system (39) using the switched controller.

Figure 5: The comparison of under the zero initial condition by two different methods.

5. Conclusion

This paper has dealt with the sampled-data control for a multirate system with two sampling rates by using an input delay approach. A periodic switched controller has been proposed to implement the sampled-data control such that the sampled-data of state variables with different sampling rates can be used real-timely. The resulting closed-loop system has been modeled as a switched system, where the subsystems have different input delays. Some delay-dependent criteria for the performance of the switched system and the existence of the switched controller have been established by using the Lyapunov-Krasovskii functional method. By comparing with single-rate design methods, the effectiveness of the proposed design method has been illustrated by an example.

Appendices

A. Proof of Proposition 2

Proof. Construct the following Lyapunov-Krasovskii functional:

where , and .
When , taking the time derivative of , we have

For system (8) with , by using Lemma 1 and Jensen’s inequality to (A.2), it can be derived that , where

If the inequality holds, then we obtain . On the other hand, is implied by (20).
For the cases and , similar to the above process, one can obtain from (21) and (22) that and , respectively.
On different subintervals , it can be obtained that

For , , , , , , , , , , , and , one has

Defining and , we obtain

where .
For , using (A.4) and (A.6), we have

where

Using to (A.7), one can see that

For , it is clear that

For , it can be seen that

For , one has

For , one has

Denote . If is satisfied, then one obtains . Integrating all cases (A.9)–(A.13), we have , which means that system (8) is exponentially stable for system (8) with , where and are given in Proposition 2.
Now, we are in a position to show the performance for system (8) with nonzero under the zero initial condition.
Using Lemma 1 and Jensen’s inequality to (A.2), we have

where .
Similarly, we obtain

where ().
It follows from (20)–(22) and (A.14)-(A.15) that

Let . When , integrating both sides of (A.16) from to , we have

where .
Combining (A.6) and (A.17), we obtain

where

For , it can be seen that

Multiplying both sides of (A.20) by yields

Taking zero initial condition and into consideration, we have

Notice that , where . So it is easy to obtain

Integrating both sides of (A.23) from to leads to

This completes the proof.

B. Proof of Proposition 3

Proof. By Schur complement, (20) is equivalent to

where

Define . Performing a congruence transformation to (B.1) by , together with the changes of matrix variables defined by , , , , , and using , we can obtain (30).
Using a similar method to (21) and (22), we can obtain (31) and (32), respectively, where , , , , , , , , , and . Before and after multiplying the inequalities , , , and by , , , and , respectively, we have , , , and . Similarly, we obtain , , , , , , , , and the equalities (26)–(29). The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grants 61403240, 61374059, and U1334210, Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi under Grant 2013105, International S&T Cooperation Program of Shanxi Province, China, under Grant 2013081040.

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