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Mathematical Problems in Engineering
Volume 2016, Article ID 2416193, 11 pages
http://dx.doi.org/10.1155/2016/2416193
Research Article

Controller Design for Asynchronous Hybrid Systems with Multiple Delays

Institute of Cyber-Systems and Control, Zhejiang University, Yuquan Campus, Hangzhou, Zhejiang 310027, China

Received 17 December 2015; Revised 27 June 2016; Accepted 11 July 2016

Academic Editor: Masoud Hajarian

Copyright © 2016 Xiaoqiang Sun and Weijie Mao. 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

Solutions for the synthesis problems of asynchronous hybrid systems with input-output delays are proposed. The continuous-time lifting approach of sampled-data systems is extended to a hybrid system with multiple delays, and some feasible formulas to calculate the operators of the equivalent discrete-time (DT) system are given. Different from the existing methods derived from symplectic pair theory or by state augmentation, a Lyapunov-Krasovskii functional to solve the synthesis problem is explicitly constructed. The delay-dependent stability conditions we obtained can be described in terms of nonstrict linear matrix inequalities (LMIs), which are much more convenient to be solved by LMI tools.

1. Introduction

Networked Control Systems (NCSs) as shown in Figure 1 are controlled systems containing several distributed plants which are connected through a communication network. For NCSs, one has to check the robustness of a control law with respect to the additional dynamics introduced by the communication networks [15].

Figure 1: An NCS with input and output delays.

NCSs are widely applied, such as mobile sensor networks, remote surgery, automated highway systems, and unmanned aerial vehicles [6, 7]. Time delay is encountered in many dynamic systems such as chemical or process control systems and NCSs and often results in poor performance which even can lead to instability [8]. This paper focuses on the influence of transmission delay and asynchronous samplings of NCSs.

In the last decades, input-delay approach became popular in NCSs, which is based on time-independent Lyapunov-Krasovskii functionals or Lyapunov-Razumikhin functions to analyze systems and design controller under uncertain sampling delay with known upper bound on the sampling intervals [911].

Meanwhile, robust stability for sampled-data systems with delays has attracted extensive attention during recent years [1214]. Among them, most works are devoted to discuss continuous-time systems with delay based on algebraic Riccati equation (ARE) such as [15] or the LMI approach derived via Lyapunov-Krasovskii functionals [16, 17]. Many control problems cannot be dealt with easily when adopting these methods. Lyapunov-Krasovskii functionals are also adopted to analyze switched systems with delay and asynchronous switching [18, 19]. However, in these papers, input delay and switching delay are considered separately.

With regard to sampled-data systems, lifting technique is a crucial method to transform a continuous-time system without delay to a discrete-time one. Most papers adopt this technique to transform multirate sampled-data linear continuous-time systems to corresponding discrete-time-invariant systems [20, 21]. Then, the DT system is analyzed according to symplectic pair theory. However, symplectic pair theory is very complicated to analyze and is difficult to acquire a controller via guaranteed cost control method, such as control, , and LQG, because, according to the theory, it needs to compute controllers and disturbance attenuation alternately under the condition that eigenvalues of corresponding matrices and must be on . And the system we discuss in this paper is with time delay; some examples have shown that although eigenvalues of corresponding matrix are less than 1, the DT system with delays still cannot be stable even with very small delays [22].

There are several papers to tackle the problems satisfying synchronous condition [23, 24] or solve the asynchronous problems with ARE constraints; this added structure leads to significant technique challenge [25, 26]. Most papers adoping lifting methods also require that all signals should be sampled and held at a same time slot. However, in most actual situations, this condition is very strict, and there is a class of multirate systems that each output signal has its own frequency of measurement and each input signal may have its own frequency of updating. The traditional lifting technique is not applicable under the situation when the sampling and holding elements are incommensurate. A system with this sampling mechanism is referred to as asynchronous system in this paper as that in [27].

Investigating the jump process and the controller design, hybrid system is actually considered as a synchronous one with regard to its time slot of the input and output. To eliminate casual constraint of the lifting method, an auxiliary system is introduced; then the combined system is transformed to a hybrid system with jump. After applying the lifting technique to continuous-time systems, the hybrid system can be transformed to an equivalent DT linear time-variant (LTV) one. As is known to all, the traditional control methods cannot be used in LTV systems directly [2830]. In most papers, there are means to expand the dimension of the system through unilateral shift; then the augmented system is transformed to an equivalent infinite-dimensional input-output DT one according to the theory of symplectic pair actually [31]. If input and output signals are periodic, then the system belongs to a finite-dimensional space. However, this process greatly enlarges the dimension of the system, especially when the jump time changes with high frequency; moreover, since the delays of input and output are mixed together, there are no rules to choose an appropriate period for the whole system.

Until now, there are rare papers to extend the continuous-time lifting method to hybrid asynchronous systems with multiple delays, and there hardly exist appropriate ways to synthesize the equivalent DT systems conveniently or less conservatively. Motivated by these, in this paper we propose a solution for the synthesis problem of asynchronous multirate sampled-data DT systems with random delays, which generalizes some previous results on control of linear systems with random delays in input and measurement output. Standard diagram of such system is shown in Figure 2, where input and output delays exist separately but are analyzed comprehensively in controller design. We first propose a time schedule to transform the continuous-time system to an equivalent DT one by the lifting method. In this procedure, we extend the lifting method of continuous-time system to a hybrid one with multiple delays. Inspired by [3133], we give exact formulas to calculate the equivalent DT system. Then, different from any current methods, we turn to apply Lyapunov-Krasovskii functional to analyze the system with calculable operators [34, 35] and derive delay-dependent stability conditions in terms of LMIs. Comparing with the current methods, output-feedback controllers based on this method can be acquired easily by solving inequalities in the LMI framework. Finally, examples in two cases are presented to demonstrate the validity of these methods.

Figure 2: A dual-rate sampled-data system with sampling and holding delays.

Notation. Throughout the paper, superscript “” in the top-right corner stands for matrix transposition. “0”s in matrices are zero matrices with compatible dimensions.

2. Preliminary

Consider a hybrid system shown as Figure 2.

Its state-space realization is as follows:where , and is state of the continuous system described as follows:where is stabilizable, is detectable, , , , , and is output of the following jump system, which is defined as the input-holding mechanism:where is a strictly increasing sequence of numbers, presenting the instants when the control input is updated; then,

Sampler with period is supposed to be an ideal sampling process with bounded time delays; sampling delay is denoted as at every sampling instant; then,whereThe sampling delay is supposed to satisfyand element of is, is an integer, and is the time skew of sampling.

Similarly, holder with period is a zero-order hold with time delay, and input delay is at every updating instant; that is,whereThe transmission delay is supposed to satisfyand elements of satisfy, is an integer, and denotes the transmission skew of actuator.

The whole delay of system satisfies

The following lemmas will be used in this paper.

Lemma 1 (see [31]). Assume . Then is an eigenvalue of iff ( will be defined in the following section).

Lemma 2 (Schur complement). Given constant matrices , , and where and then if and only if

Lemma 3 (see [34]). Assume that , , and . Then, for any matrices , , and satisfying the following holds:

3. Model Transformation

In this paper, is scheduled to be uniform distribution after optimization to make a trade-off between discretization time for better performance and smaller delay bounds. Due to the fact that the sampler and holder are both with delays, thus input time and time slot for discretization need not be simultaneous, and sampling and holding need not be synchronous either.

Defining , if , , are equally separated, that is, , , consider the lifting operator , and the hybrid system can be transformed to the following DT system , shown in Figure 3.

Figure 3: The process of lifting system .

Let and ; then,where , , , , , , andHere . Since is an isometric isomorphism, with any controller in place, the induced norm of the lifted system is the same as that of the original system.

The lifted system belongs to an infinite input-output space, so traditional methods cannot be applied. The following lemma gives a proper way to transform it to an equivalent DT system; then the traditional methods can be applied.

Lemma 4 (see [32]). Suppose that holds for all ; then can be transformed to system as follows:where is stabilizable and is detectable; , , , and ; is sampling delay and is input delay at the th time instant, and, , and satisfyand then, the following are equivalent:(i)The controller stabilizes the DT system , and the closed-loop system satisfies .(ii)The controller stabilizes the equivalent system , and the closed-loop system satisfies .

Finally, we obtain system , which can be analyzed by traditional method. And there is an example in [31] that shows that direct sampled-data design requires a sampling rate which is 10 times slower than digital implementation via step-invariant transformation and thus can let us schedule sampling time easily.

4. Static Output-Feedback Robust Control

In this section, a sufficient condition for the existence of a delay-dependent output-feedback controller which guarantees the stability of system (1) and meets performance is discussed.

The static output-feedback stabilization problem is to design the following controller, for discrete-time system , where is an appropriately dimensioned matrix to be determined. If and , the output-feedback controller turns to be a static state-feedback one; the proposed method in this paper is also valid and the corresponding results will be clean and concise. However, this is not what we care about in this paper as the state-feedback problem can be solved more easily than output-feedback case.

By connecting the controller (22) with system , we can obtain the following closed-loop system of :where , and

Denote ; the bounds of delays also satisfy (13).

The following theorem presents a sufficient condition for the existence of such controller that can stabilize system .

Theorem 5. For a given disturbance attenuation , system with a static output-feedback controller (22) can be stabilized if there exist appropriate matrices , , , , , and , satisfying the following inequalities:where , , and are appropriate identity matrices:where , , , and , respectively, satisfy the following:and the other matrices in the operators can be obtained by the following matrices exponential computing or partitioning some blocks on the left-hand side of the following matrices into block matrices:where

Proof. To calculate the operators , , and , we can extend methods in [12, 32]. The derivation is standard on the ground of matrix exponential computing and adjoint operators; as the process is very trivial, we omit it here and only give the results we have derived.
The final DT system is obtained by symmetric factorization of two expressions of (29) and (30); all that remains is to synthesize a controller for .
Since we havewhere , then the closed system of can be transformed toChoose a Lyapunov-Krasovskii functional candidate as follows:where , , and are positive definite matrices to be determined.
Define ; then along the solution of (19) we getAccording to Lemma 3, the following inequalities can be obtained:In addition, we haveNote thattherefore, we haveSimilarly, we can haveFinally,Then, combining (39)–(41) and (43)–(45), we haveIn order to obtain robustly stable system with a disturbance attenuation , it is required that the associated Hamiltonian function satisfiesThen, from (19) and (47), we can obtainwhere , andBy Schur complement, the following inequality is obtained:where as (27) shows that , , and satisfy inequality (26). A congruence transformation to (50) by pre- and postmultiplying together with the substitution of the matrices defined as (24) leads to (25). Therefore, we can conclude from the stability theory, to a given disturbance attenuation , that conditions (25)–(35) guarantee the closed system of to be robustly stable if all time-varying delays satisfy . Due to the fact that the lifting process is an isometric isomorphism and the transformation from to is equivalent, the hybrid system can equivalently be stabilized by controller (22) under the same disturbance attenuation level.

Remark 6. The obtained condition in Theorem 5 is less conservative due to the proper Lyapunov functional we choose. First, more detailed information (the minimum and the maximum delay bounds) is included in the Lyapunov-Krasovskii functional candidate; that is, the result is delay-dependent. Second, the constructed Lyapunov-Krasovskii functional candidate is sampling-dependent; that is, the tuning matrices , , and are time-varying, which enlarges the feasible domain and thus leads to less conservative results.

Remark 7. Due to the existence of inverse matrices of and , (25) is not a strict LMI and cannot be directly solved by the LMI tools. However, notice that the matrices and their inverse matrices appear in pairs; we can solve the inequality by iteration on LMI according to cone complementarity linearization algorithm (CCLA) [22].

If the delay of the system is constant, the minimum and the maximum delay bounds become identical; that is, , and Theorem 5 is specialised to the following corollary.

Corollary 8. The closed system of with is robustly stable with a given disturbance attenuation , if there exist , , , , and , satisfying (26) andhere , , and are appropriate identity matrices, and , , , , , and are defined in (28), (29), and

5. Dynamic Output-Feedback Robust Control

In this section, dynamic output-feedback controller is adopted; that is, we choose the following controller:where , , (), and , , , and are appropriately dimensioned matrices to be determined.

By defining , combining controller (53) with system , the closed-loop system of can be given bywhere , is an identity matrix with appropriate dimension, andwhere , , , , , and are defined as (28) and (29). Similar to Theorem 5, we can obtain the following theorem.

Theorem 9. For a given disturbance attenuation , system with a dynamic output-feedback controller (53) can be stabilized if there exist appropriate matrices , , , and , and , , , , , satisfying the following inequalities:where , , and are appropriate identity matrices with appropriate dimensions; , , , , , , and are defined as (55), and

Proof. From (36) we havewhere .
To the closed system of with dynamic output-feedback controller (53), because there is no delay in the state of dynamic controller, we can construct mixed descriptor Lyapunov-Krasovskii functionals as follows:Then, the proof of Theorem 9 can be carried out as that of Theorem 5 and hence omitted here.

Remark 10. From Theorems 5 and 9, we can notice that theories we proposed are different from the current methods, which are difficult to calculate and feedback gains alternatively under the condition that eigenvalues of corresponding matrices of the closed system are on the unit circle, especially to dynamic feedback-output control system; the method based on eigenvalues is more difficult and hard. However, applying methods we proposed, can be assigned before calculation, and all parameters in the LMIs can be solved clearly and conveniently.

6. Numerical Example

In this section, a numerical example is presented to illustrate the effectiveness and applicability of the proposed methods. As an example, we construct a hybrid system described by the following matrices: According to Lemma 2, we can calculate . The output-sampling matrices with period 3 are chosen to be as follows:The input-holding mechanism is assumed with period 2 and can be described as follows:The period for discretization is chosen to be 0.3 s. is a Gaussian white noise with amplitude of 0.1. Bounds of sampling and holding delays are chosen to be and , respectively, which are much larger than those of most current literatures.

Disturbance attenuation level is given as ; however, the levels obtained by current methods are all larger than 1. Then, we simulate the system with two control strategies, respectively; the results are shown below.

Case 1 (static output-feedback robust control). Initial status is specified as .
From the figures, we can see that the system can be stabilized by static output-feedback controller.

Case 2 (dynamic output-feedback robust control). Here, initial status also is . From the figures, we can see that the system can be stabilized by dynamic output-feedback controller, and comparing Figures 4 and 5 with Figures 6 and 7, respectively, we can see that the performance of the system is improved by offering more parameters of controller.

Figure 4: The states response to static output-feedback control.
Figure 5: The static output-feedback control.
Figure 6: The states response to dynamic output-feedback control.
Figure 7: The dynamic output-feedback control.

7. Conclusion

Motivated by a desire to provide a better solution for NCSs with delays, a computable solution to the asynchronous multirate hybrid synthesis problem is addressed. At the beginning, the traditional continuous-time lifting method is extended to the hybrid system with random delays. Due to asynchronism, the sampling time should be scheduled to be equal for discretization. Next, the lifted DT system that lies in infinite input-output space is transformed to an equivalent DT system, which belongs to a finite space. To solve the operators in the equivalent DT system, a series of computable formulas are provided for the first time according to adjoint operator theory. Then, different from any existing approach, a Lyapunov-Krasovskii functionals candidate is proposed to deal with the synthesis problem of the equivalent DT system. The results based on this method are delay-dependent and can be solved by LMI tools. Compared with the current methods, this method is more convenient to calculate the controller, especially in dynamic feedback case. Finally, two examples exhibit the validity of the two theories, respectively.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61473252 and 61633019) and the Zhejiang Provincial Natural Science Foundation of China (LR12F03002).

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