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

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


Introduction
The singular system model is a natural presentation of dynamic systems, such as power systems [1] and largescale 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][5][6] for timedelay singular systems.In [7], the problem of decentralized stabilization has been discussed for nonlinear singular largescale 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][11][12].However, a survey of the literature indicates that the singular system issue has seldom been studied in such systems.Many research [13][14][15][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][18][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.

Mathematical Problems in Engineering
The equivalent time-delay linear nonsingular system The equivalent time-delay linear system .
x . x Figure 1: The schematic design methodology for the interconnected time-delay singular/nonlinear system.

System and Problem Description
Consider the time-delay system consisting of two interconnected MIMO subsystems shown as where  The subsystem 1 is the time-delay singular system and subsystem 2 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 1 and 2 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 1 and 2 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.

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 timedelay 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   , Â ,   ,  11 , and  1 and input V  () 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  1 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  2 ,  2 , and  2 are the identified parameters by OKID method.The corresponding continuous-time system of (4a) and (4b) is described by Notably,  2 ,  2 , and  2 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.[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).

The Observer-Based Suboptimal Digital Tracker Design
Consider the continuous time-delay singular subsystems (3a) and (3b) without the time delay of interconnected state vector ℎ  12  2 ( −  2 −  1 ).By [21],  1 is the sampling period.Let the state delay time be given by  1 =  1  1 + Γ 1 , where 0 ≤ Γ 1 <  1 and  1 ≥ 0 is an integer, and let the input delay time be given by  1 =  1  1 +  1 , where 0 ≤  1 <  1 and  1 ≥ 0 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 where in which Some terms in ( 6) may be combined because of the same delay, so (6) can be reduced to The time-delay state  1 ( −  1 ) for  1 ≤  −  1 < ( + 1) 1 must be evaluated as follows: where in which Some terms in (10) may be combined as in (9), and ( 10) can be rewritten as Then, the output (3b) can be rewritten as where 1 , and Similarly, some terms in ( 14) can be combined so ( 14) can be rewritten as where  * 0 ,  *  ,  0 , and   are the summation of multiple inputdelay terms.
In the following work, we use ( 13) and ( 15) to derive the equivalent extended delay-free system as follows: where means the extended virtual state vector.By the previous method [21], we derive the observerbased suboptimal tracker for the time-delay singular system with unavailable states using the equivalent extended delayfree system.The extended observer-based suboptimal digital tracker can be represented as where X ( 1 ) is the estimate of the extended state   ( 1 ) in ( 17) and in which   ( 1 ) = [ (0)  ( 1 )  (1)   ( 1 ) ⋅ ⋅ ⋅ ( 1 ) = [ (1)   ( 1 ) ⋅ ⋅ ⋅ 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.
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.
The second subsystem 2 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: where and  = [ 1  2 ]  ,  1 ,  2 are the angular positions, () is the moment of inertia, (, q ) includes Ceoriolis and centripetal forces, () is the gravitational force, and Γ is the applied torque vector.Here, we use the short hand notations   = sin(  ) and   = cos(  ).The nominal parameters of the system are given as follows: the link masses  1 = 5 kg,  2 = 2.5 kg, the length  1 =  2 = 0.5 m, and the gravitational acceleration   = 9.81 ms −2 .Rewrite (25) in the following form: Let  1 and  1 ( 1 ) represent the state of the system and the nonlinear function of the state  1 , respectively.And the notation is shown as follows: where , and Calculate the inverse of the matrix , and then we can have Therefore, the dynamic equation of the two-link robot system can be reformulated as follows: where  2 = [ 1 0 0 0 0 0 1 0 ], the sampling period  2 = 0.02 sec, and the initial condition  2 (0) = [0 0 0 0]  .
Based on Section 3.1 [20], the time-delay singular subsystem 1 can be transformed into the equivalent time-delay regular system as follows: where By OKID [21,22] in Figure 1, the identified subsystem 2 is given as where in which the input time-delay  2 = 0.5 ×  2 and output timedelay  2 = 0.5 ×  2 .
Following the proposed method in this paper, let the reference inputs () = [0.5 sin() 0.5 cos()] and apply them to subsystem 1 and subsystem 2, respectively.We obtain the observer gain matrix   for 1 and 2 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.In order to confirm the independence of the control for the two subsystems, the time-varying optimal digital controller of the subsystem 2 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 2 is reduced, the tracking performance of the subsystem 1 will not be affected by this condition and the results are shown in Figures 6 and 7.
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 2.
Following [20,21], let the subsystem 2 be excited by the control force () with white noise () = [ 1 ()  2 ()] having zero mean and covariance diag [cov( 1 ()) cov( 2 ())] = diag [0.2 0.2].Then, the comparisons between the actual outputs and the OKID method for subsystem 2 are shown in Figure 8, and the comparisons between the actual outputs and the proposed method for subsystem 2 are shown in Figure 9. From the comparison between Figures 8 and 9, the effectiveness of the proposed method is better than OKID method in the tracking error performance.

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.