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.
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 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 aswhere and are the state vectors, and are the control input vectors, and and are the output vectors. and are nonlinear functions of the states of . , , , , , and are known as constant system matrices of appropriate dimensions and is a singular matrix. State time delays and , interconnection time delays and , input time delays and , and output time delays and are assumed to be known. The time delays of interconnected state vectors and are induced from multiple sensors at different rates to accurately produce a reliable navigation solution.
The subsystem is the time-delay singular system and subsystem 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 through this paper is a time lag operator; for example, .
It is very difficult to directly design the tracker and observer for and 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 and 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:where the parameters , , , , and and input 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 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:where , , and are the identified parameters by OKID method. The corresponding continuous-time system of (4a) and (4b) is described byNotably, , , and 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 . By [21], is the sampling period. Let the state delay time be given by , where and is an integer, and let the input delay time be given by , where and is an integer. The time-delay singular subsystems (3a) and (3b), by both the Newton extrapolation method and the Chebyshev quadrature method [25, 26], becomewherein whichSome terms in (6) may be combined because of the same delay, so (6) can be reduced toThe time-delay state for must be evaluated as follows:wherein whichSome terms in (10) may be combined as in (9), and (10) can be rewritten asThen, the output (3b) can be rewritten aswhere , , and .
Similarly, some terms in (14) can be combined so (14) can be rewritten aswhere , , , and 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:wheremeans 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 aswhere is the estimate of the extended state in (17) andin whichThe 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.
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 aswhereThe first subsystem of the large-scale system is given as follows:Let the sampling period sec and the initial condition is .
The second subsystem of the large-scale system is given by two-link robot [27, 28], which is described as shown in Figure 3.
The dynamic equation of the two-link robot system can be expressed as follows:whereand , , are the angular positions, is the moment of inertia, includes Ceoriolis and centripetal forces, is the gravitational force, and is the applied torque vector. Here, we use the short hand notations and . The nominal parameters of the system are given as follows: the link masses , , the length , and the gravitational acceleration . Rewrite (25) in the following form:
Let and represent the state of the system and the nonlinear function of the state , respectively. And the notation is shown as follows:where , , and . Also, let , in which .
Calculate the inverse of the matrix , and then we can have such thatTherefore, the dynamic equation of the two-link robot system can be reformulated as follows:where , the sampling period sec, and the initial condition .
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 and are given as and , respectively. The time delays of the nonlinear interconnected terms are and , where sec and sec.
Based on Section 3.1 [20], the time-delay singular subsystem can be transformed into the equivalent time-delay regular system as follows:where By OKID [21, 22] in Figure 1, the identified subsystem is given aswherein which the input time-delay and output time-delay .
Following the proposed method in this paper, let the reference inputs and apply them to subsystem and subsystem , respectively. We obtain the observer gain matrix for and as follows: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.
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In order to confirm the independence of the control for the two subsystems, the time-varying optimal digital controller of the subsystem 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 is reduced, the tracking performance of the subsystem will not be affected by this condition and the results are shown in Figures 6 and 7.
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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 . Following [20, 21], let the subsystem be excited by the control force with white noise having zero mean and covariance . Then, the comparisons between the actual outputs and the OKID method for subsystem are shown in Figure 8, and the comparisons between the actual outputs and the proposed method for subsystem are shown in Figure 9.
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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.
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 :The decentralized stabilization problem for is to find controllers , , 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:
Definition A.1. Consider system ; is called a decentralized fixed mode if for all block diagonal matrices one has .
Lemma A.2. Necessary and sufficient condition for the existence of a decentralized feedback control law for the system 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 together stabilize then for any there exists a such that is invertible and the closed loop system replacing with is still asymptotically stable. This choice is possible because if is invertible obviously we can choose . If is not invertible, by small enough choice of we can make sure that is invertible and the closed loop system replacing with is still asymptotically stable. But the closed loop system when is in the loop is asymptotically stable. In particular, it cannot have a pole in . SoHence the block diagonal matrix has the property that Thus is not a fixed mode. Since this argument is true for any 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 . Since is not a fixed mode there exists such thatno longer has an eigenvalue in . Let be the smallest integer such that an unstable eigenvalue of is no longer an eigenvalue ofwhile can be chosen small enough not to introduce additional unstable eigenvalues. Then for the systeman 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.