Decentralized Control for the Interconnected Time-Delay Singular/Nonlinear Subsystems with Closed-Loop Decoupling Property

Abstract

This paper presents the decentralized trackers using the observer-based suboptimal method for the interconnected time-delay singular/nonlinear subsystems with closed-loop decoupling property. The observer-based suboptimal method is used to guarantee the high-performance trajectory tracker for two different subsystems. Then, due to the high gain that resulted from the decentralized tracker, the closed-loop system will have the decoupling property. An illustrative example is given to demonstrate the effectiveness of the proposed control structure.

Nien-Tsu Hu 1 and Ter-Feng Wu 2 and Sendren Sheng-Dong Xu 3 and Pu-Sheng Tsai 4

Academic Editor:Jun Hu

1, Integrated Logistical Support Center, National Chung-Shan Institute of Science and Technology, Taoyuan 32599, Taiwan
2, Department of Electrical Engineering, National Ilan University, Ilan 26047, Taiwan
3, Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, Taipei 10607, Taiwan
4, Department of Electronic Engineering, China University of Science and Technology, Taipei 11581, Taiwan

Received 26 September 2014; Accepted 4 December 2014; 1 April 2015

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.

1. Introduction

The singular system model is a natural presentation of dynamic systems, such as power systems [1] and large-scale systems [2, 3]. In general, an interconnection of state variable subsystems is conveniently described as a singular system, even though an overall state space representation may not even exist. Over the past decades, much attention has been focused on the decentralized control [4-6] for time-delay singular systems. In [7], the problem of decentralized stabilization has been discussed for nonlinear singular large-scale time-delay control systems with impulsive solutions. The [figure omitted; refer to PDF] control for singular systems with state delay has been presented in [8]. And the decentralized output feedback control problem [9] is considered for a class of large-scale systems with unknown time-varying delays.

In the recent years, a large number of control systems are characterized by interconnected large-scale subsystems, and many practical examples have been applied to decentralized control systems. The decentralized control of interconnected large-scale systems has commonly appeared in our modern technologies, such as transportation systems, power systems, and communication systems [10-12]. However, a survey of the literature indicates that the singular system issue has seldom been studied in such systems. Many research [13-16] results concerning the singular/nonlinear system have successfully solved lots of complex problems. For the above reasons, we will discuss the decentralized control of the interconnected large-scale time-delay singular subsystem and nonlinear subsystem.

In this paper, we consider the time-delay effect. In practical applications, the time-delay effect [17-19] may result in an unexpected and unsatisfactory system performance, even including the serious instability, if it is ignored in the design of control systems. In order to overcome this problem, the controller design method [20, 21] is necessary to be further explored in this paper. Sequentially, the decentralized tracker with the high-gain property will make the closed-loop system own the decoupling property.

This paper is organized as follows. Section 2 describes the problem of interest. Section 3 presents the observer-based suboptimal digital tracker. Section 4 presents the simulation results of interconnected time-delay singular/nonlinear subsystems. Finally, Section 5 draws conclusions.

2. System and Problem Description

Consider the time-delay system consisting of two interconnected MIMO subsystems shown as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the state vectors, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the control input vectors, and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the output vectors. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are nonlinear functions of the states [figure omitted; refer to PDF] of [figure omitted; refer to PDF] . [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are known as constant system matrices of appropriate dimensions and [figure omitted; refer to PDF] is a singular matrix. State time delays [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , interconnection time delays [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , input time delays [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and output time delays [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are assumed to be known. The time delays of interconnected state vectors [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are induced from multiple sensors at different rates to accurately produce a reliable navigation solution.

The subsystem [figure omitted; refer to PDF] is the time-delay singular system and subsystem [figure omitted; refer to PDF] is the time-delay nonlinear subsystem. Before designing the controller, the decentralized modeling of the interconnected time-delay system is proposed in Figure 1. The notation [figure omitted; refer to PDF] through this paper is a time lag operator; for example, [figure omitted; refer to PDF] .

Figure 1: The schematic design methodology for the interconnected time-delay singular/nonlinear system.

[figure omitted; refer to PDF]

It is very difficult to directly design the tracker and observer for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] because their system models are not nonsingular and linear models. To solve this problem, the previously proposed method in [21] and the OKID (observer/Kalman filter identification) method in [22] are appropriately utilized to make [figure omitted; refer to PDF] and [figure omitted; refer to PDF] become the equivalent linear time-delay nonsingular subsystems. As a result, the process becomes quite simple. Besides, as long as the designed tracker for each subsystem has the high-gain property, the designed global system will have the closed-loop decoupling property.

We will use the proposed schematic design in Figure 1 to construct the methodology of the decentralized control for the interconnected time-delay singular/nonlinear subsystems with the closed-loop decoupling property.

3. Main Results

In this section, we construct the methodology of the decentralized control by using the design concept of the observer-based suboptimal digital tracker to control time-delay singular subsystem and time-delay nonlinear subsystem, respectively. Before designing the controller, we need to obtain the equivalent time-delay linear nonsingular subsystem and the equivalent time-delay linear subsystem. The problem of decentralized stabilization is discussed in the appendix.

3.1. The Equivalent Time-Delay Linear Nonsingular Subsystems for the Time-Delay Singular/Nonlinear Subsystems

From the schematic design methodology of Figure 1, and by using the previous method in [20], we can make the time-delay singular subsystems (1a) and (1b) become the equivalent time-delay regular system as follows: [figure omitted; refer to PDF] where the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and input [figure omitted; refer to PDF] can be referred to in [20].

Remark A.0.

Notably, definitions of the regular pencil [23] and the standard pencil [24] are satisfied on no state delay term in systems (1a) and (1b). If [figure omitted; refer to PDF] exists, then definitions of the regular pencil and the standard pencil do not guarantee that systems (1a) and (1b) can be decomposed into the equivalent time-delay regular system.

Similarly, the time-delay nonlinear subsystems (2a) and (2b) can transform the equivalent time-delay linear subsystem by OKID method [21, 22] as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the identified parameters by OKID method. The corresponding continuous-time system of (4a) and (4b) is described by [figure omitted; refer to PDF] Notably, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are known as constant system matrices of appropriate dimensions.

The equivalent subsystems (3a), (3b), and (5a) and (5b) will be applied to the observer-based suboptimal digital tracker [21] for the singular/nonlinear subsystem in the next subsection and finally we proposed the schematic design methodology of decentralized control for the interconnected time-delay singular/nonlinear subsystems with closed-loop decoupling property.

3.2. The Observer-Based Suboptimal Digital Tracker Design [21]

Consider the continuous time-delay singular subsystems (3a) and (3b) or the time-delay subsystems (5a) and (5b). Here, we take the time-delay singular subsystems (3a) and (3b) to design the observer-based suboptimal digital tracker and the design results are similar to the time-delay subsystems (5a) and (5b).

Consider the continuous time-delay singular subsystems (3a) and (3b) without the time delay of interconnected state vector [figure omitted; refer to PDF] . By [21], [figure omitted; refer to PDF] is the sampling period. Let the state delay time be given by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an integer, and let the input delay time be given by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an integer. The time-delay singular subsystems (3a) and (3b), by both the Newton extrapolation method and the Chebyshev quadrature method [25, 26], become [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] Some terms in (6) may be combined because of the same delay, so (6) can be reduced to [figure omitted; refer to PDF] The time-delay state [figure omitted; refer to PDF] for [figure omitted; refer to PDF] must be evaluated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] Some terms in (10) may be combined as in (9), and (10) can be rewritten as [figure omitted; refer to PDF] Then, the output (3b) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Similarly, some terms in (14) can be combined so (14) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the summation of multiple input-delay terms.

In the following work, we use (13) and (15) to derive the equivalent extended delay-free system as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] means the extended virtual state vector.

By the previous method [21], we derive the observer-based suboptimal tracker for the time-delay singular system with unavailable states using the equivalent extended delay-free system. The extended observer-based suboptimal digital tracker can be represented as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the estimate of the extended state [figure omitted; refer to PDF] in (17) and [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] The details of the parameters can be referred to in [21]. Here, the observer-based suboptimal tracker has been completely obtained. Figure 2 presents the realization of decentralized control for the interconnected time-delay singular/nonlinear subsystems.

Figure 2: The decentralized control for the interconnected time-delay singular/nonlinear subsystems.

[figure omitted; refer to PDF]

From Figures 1 and 2, the design procedure can be summarized as the following steps.

Step 1.

Perform the previously proposed method [21] and the OKID method [22] to determine the equivalent time-delay linear subsystems in Figure 1.

Step 2.

Design the observer-based suboptimal digital trackers from the equivalent time-delay linear subsystems obtained in Step 1.

Step 3.

Perform the observer-based suboptimal digital trackers obtained in Step 2. The decentralized control for the interconnected time-delay singular/nonlinear subsystems is shown in Figure 2.

4. An Illustrative Example

Consider the time-delay system consisting of two interconnected MIMO subsystems shown as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] The first subsystem [figure omitted; refer to PDF] of the large-scale system is given as follows: [figure omitted; refer to PDF] Let the sampling period [figure omitted; refer to PDF] sec and the initial condition is [figure omitted; refer to PDF] .

The second subsystem [figure omitted; refer to PDF] of the large-scale system is given by two-link robot [27, 28], which is described as shown in Figure 3.

Figure 3: Two-link robot.

[figure omitted; refer to PDF]

The dynamic equation of the two-link robot system can be expressed as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are the angular positions, [figure omitted; refer to PDF] is the moment of inertia, [figure omitted; refer to PDF] includes Ceoriolis and centripetal forces, [figure omitted; refer to PDF] is the gravitational force, and [figure omitted; refer to PDF] is the applied torque vector. Here, we use the short hand notations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The nominal parameters of the system are given as follows: the link masses [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the length [figure omitted; refer to PDF] , and the gravitational acceleration [figure omitted; refer to PDF] . Rewrite (25) in the following form: [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the state of the system and the nonlinear function of the state [figure omitted; refer to PDF] , respectively. And the notation is shown as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Also, let [figure omitted; refer to PDF] , in which [figure omitted; refer to PDF] .

Calculate the inverse of the matrix [figure omitted; refer to PDF] , and then we can have [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Therefore, the dynamic equation of the two-link robot system can be reformulated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , the sampling period [figure omitted; refer to PDF] sec, and the initial condition [figure omitted; refer to PDF] .

Combining the above systems with the nonlinear interconnected terms, the large-scale system can then be shown in Figures 1 and 2, where the nonlinear interconnected terms [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are given as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. The time delays of the nonlinear interconnected terms are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] sec and [figure omitted; refer to PDF] sec.

Based on Section 3.1 [20], the time-delay singular subsystem [figure omitted; refer to PDF] can be transformed into the equivalent time-delay regular system as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By OKID [21, 22] in Figure 1, the identified subsystem [figure omitted; refer to PDF] is given as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which the input time-delay [figure omitted; refer to PDF] and output time-delay [figure omitted; refer to PDF] .

Following the proposed method in this paper, let the reference inputs [figure omitted; refer to PDF] and apply them to subsystem [figure omitted; refer to PDF] and subsystem [figure omitted; refer to PDF] , respectively. We obtain the observer gain matrix [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Finally, the scheme of Figure 2 is implemented. For simplification, the numerical analysis is not presented and Figures 4 and 5 show the results of the simulation.

Figure 4: (a) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] . (b) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 5: (a) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] . (b) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

In order to confirm the independence of the control for the two subsystems, the time-varying optimal digital controller of the subsystem [figure omitted; refer to PDF] is reduced by multiplying a scalar 0.97 during 4 sec to 6 sec in this simulation. Although the time-varying optimal digital controller of the subsystem [figure omitted; refer to PDF] is reduced, the tracking performance of the subsystem [figure omitted; refer to PDF] will not be affected by this condition and the results are shown in Figures 6 and 7.

Figure 6: The unanticipated failure occurs without fault-tolerant control during [figure omitted; refer to PDF] = 4~6 sec. (a) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] . (b) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 7: The unanticipated failure occurs without fault-tolerant control during [figure omitted; refer to PDF] = 4~6 sec. (a) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] . (b) Output responses of the subsystem [figure omitted; refer to PDF] : output [figure omitted; refer to PDF] and reference [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

To show the effectiveness of the proposed method, we compare it with the observer/Kalman filter identification (OKID) method in the simulation for the subsystem [figure omitted; refer to PDF] . Following [20, 21], let the subsystem [figure omitted; refer to PDF] be excited by the control force [figure omitted; refer to PDF] with white noise [figure omitted; refer to PDF] having zero mean and covariance [figure omitted; refer to PDF] . Then, the comparisons between the actual outputs and the OKID method for subsystem [figure omitted; refer to PDF] are shown in Figure 8, and the comparisons between the actual outputs and the proposed method for subsystem [figure omitted; refer to PDF] are shown in Figure 9.

Figure 8: (a) The comparison between the system output [figure omitted; refer to PDF] and its observer-based output [figure omitted; refer to PDF] by OKID for the subsystem [figure omitted; refer to PDF] . (b) The comparison between the system output [figure omitted; refer to PDF] and its observer-based output [figure omitted; refer to PDF] by OKID for the subsystem [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 9: (a) The comparison between the system output [figure omitted; refer to PDF] and its observer-based output [figure omitted; refer to PDF] by the proposed method for the subsystem [figure omitted; refer to PDF] . (b) The comparison between the system output [figure omitted; refer to PDF] and its observer-based output [figure omitted; refer to PDF] by the proposed method for the subsystem [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

From the comparison between Figures 8 and 9, the effectiveness of the proposed method is better than OKID method in the tracking error performance.

5. Conclusion

This paper presents a systematical methodology of the decentralized control for the interconnected time-delay singular/nonlinear subsystems with closed-loop decoupling property. We use the observer-based suboptimal digital tracker with high gain property to keep the good tracking performance. Moreover, the decoupling property performs very well such that even if some unanticipated fault occurs in some of subsystems, it still will not affect the tracking performance of each subsystem. The proposed methods depend on the decentralized modeling of the interconnected sampled-data time-delay subsystems in Section 2 and the controller design is suitable to time-delay singular/nonlinear subsystems in Section 3. Thus, the proposed method can deal with the signal quantization and sensor delay but cannot deal with intermittent measurements and missing/fading measurements. In future works, we will pay more attention to fault-tolerant control, intermittent measurements, and missing/fading measurements by using the proposed methods.

Acknowledgment

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract no. NSC 101-2511-S-197-002.

Conflict of Interests

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

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Appendix

Appendix

The Decentralized Control Stabilization

The necessary and sufficient conditions for the decentralized stabilization are presented in [29]. Here, we provide the proof for the decentralized stabilization and more details can be seen [29]. The following proofs are cited from [29].

Consider the given system [figure omitted; refer to PDF] : [figure omitted; refer to PDF] The decentralized stabilization problem for [figure omitted; refer to PDF] is to find controllers [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that the poles of the closed loop system are in the desired locations in the open unit disc. In order to provide an easier bookkeeping, we define the following matrices: [figure omitted; refer to PDF]

Definition A.1.

Consider system [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is called a decentralized fixed mode if for all block diagonal matrices [figure omitted; refer to PDF] one has [figure omitted; refer to PDF] .

Lemma A.2.

Necessary and sufficient condition for the existence of a decentralized feedback control law for the system [figure omitted; refer to PDF] such that the closed loop system is asymptotically stable is that all the fixed modes of the system are asymptotically stable (in the unit disc).

Proof.

We first establish necessity. Assume local controllers [figure omitted; refer to PDF] together stabilize [figure omitted; refer to PDF] then for any [figure omitted; refer to PDF] there exists a [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] is invertible and the closed loop system replacing [figure omitted; refer to PDF] with [figure omitted; refer to PDF] is still asymptotically stable. This choice is possible because if [figure omitted; refer to PDF] is invertible obviously we can choose [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is not invertible, by small enough choice of [figure omitted; refer to PDF] we can make sure that [figure omitted; refer to PDF] is invertible and the closed loop system replacing [figure omitted; refer to PDF] with [figure omitted; refer to PDF] is still asymptotically stable. But the closed loop system when [figure omitted; refer to PDF] is in the loop is asymptotically stable. In particular, it cannot have a pole in [figure omitted; refer to PDF] . So [figure omitted; refer to PDF] Hence the block diagonal matrix [figure omitted; refer to PDF] has the property that [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] is not a fixed mode. Since this argument is true for any [figure omitted; refer to PDF] on or outside the unit disc, this implies that all the fixed modes must be inside the unit disc. This proves the necessity of the Lemma A.2.

Next, we establish sufficiency. To prove that we can actually stabilize the system, we use a recursive argument. Assume the system has an unstable eigenvalue in [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is not a fixed mode there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] no longer has an eigenvalue in [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the smallest integer such that an unstable eigenvalue of [figure omitted; refer to PDF] is no longer an eigenvalue of [figure omitted; refer to PDF] while [figure omitted; refer to PDF] can be chosen small enough not to introduce additional unstable eigenvalues. Then for the system [figure omitted; refer to PDF] an unstable eigenvalue is both observable and controllable. But this implies that there exists a dynamic controller which moves this eigenvalue in the open unit disc without introducing new unstable eigenvalues. Through a recursion, we can move all eigenvalues one-by-one in the open unit disc and in this way find a decentralized controller which stabilizes the system. This proves the sufficiency of Lemma A.2.