Elsevier

Automatica

Volume 64, February 2016, Pages 208-216
Automatica

Design of stable parallel feedforward compensator and its application to synchronization problem

https://doi.org/10.1016/j.automatica.2015.11.020Get rights and content

Abstract

This paper addresses the design problem of a stable parallel feedforward compensator V for a given SISO LTI plant P (possibly being of non-minimum phase and/or having relative degree greater than one). The objective of the problem is that their parallel interconnection P+V becomes minimum phase having relative degree one. Based on the classical results of simultaneous stabilization, a necessary and sufficient condition for solving the problem is presented as well as a design procedure for constructing such a compensator. The proposed feedforward compensator allows the control system to have the three useful features: (1) the ability that assigns the zeros of P+V to a region of complex numbers having arbitrary negative real parts, (2) infinite gain margin property of P+V controlled by a static output feedback, and (3) block diagonal structure of P+V. These features are extensively exploited in the synchronization problem of multi-agent systems to achieve arbitrary fast convergence rate and to have the synchronized trajectory independent of the initial conditions and parameters of the involved dynamic controllers.

Introduction

A class of systems being of minimum phase and having relative degree one is frequently encountered, and is dealt with in depth in the literature because it is closely related with passive systems (Byrnes et al., 1991, Sepulchre et al., 1997) and a system belonging to this class can be stabilized through a simple static high-gain output feedback (Byrnes et al., 1991, Isidori, 1995). Although the high-gain stabilization schemes can be extended to systems having higher relative degree as long as they remain as minimum phase (Teel & Praly, 1995), the output feedback control of non-minimum phase systems still suffers from its intrinsic limitations.

A possible approach to overcome these difficulties is to find a parallel feedforward compensator (PFC) V such that the parallel interconnection P+V of the plant P and the compensator V, shown in the shaded region of Fig. 1(a), has desired properties (e.g., passivity, minimum phaseness, and/or relative degree 1) when the signal y=yp+yv is viewed as a new output. (Therefore, this idea belongs to the category of output redefinition methods.) If this task is successfully done, then, relying on the obtained properties of P+V, one may design an output feedback controller Q for P+V to achieve the objectives of the original problem with relative ease. These feedforward and feedback controllers are actually implemented in a feedback form like in Fig. 1(b), i.e., V becomes a part of a feedback controller.

In this direction, Bar-Kana (1987) has used a PFC to make P+V almost strictly positive real (ASPR), which results in an implementable simple adaptive controller (Sobel, Kaufman, & Mabius, 1982) when the plant itself does not satisfy the positivity condition. In particular, he showed in Bar-Kana (1986) that if P(s), the transfer function of the system P, can be stabilized by a biproper output feedback controller V1(s), then P(s)+V(s) becomes ASPR. Following the approach and results of Bar-Kana, 1986, Bar-Kana, 1987, a robust design problem of such PFCs for uncertain plants has been addressed in, e.g., Deng, Iwai, and Mizumoto (1999), Iwai and Mizumoto, 1992, Iwai and Mizumoto, 1994 and Iwai, Mizumoto, and Deng (1994) with constructive design methods given. A common restriction of them is P(s) has to be of minimum phase uniformly in the uncertainties. Several passification methods of non-passive systems via suitable compensation, including PFC, were also introduced by Kelkar and Joshi (1997).

On the other hand, state space design approaches of PFC have been reported in, e.g., Isidori and Marconi (2008), Misra and Patel (1988), Patel and Misra (1992) and Son et al., 2003, Son et al., 2002. In Misra and Patel (1988) and Patel and Misra (1992), the design methods of PFC V were provided in order that P+V is of minimum phase and has relative degree zero. Son et al., 2003, Son et al., 2002 have considered the problem of designing a PFC that achieves relative degree 1 and minimum phaseness of the interconnected system. In particular, they showed the problem is solvable if there exists a static output feedback controller that stabilizes a certain system derived from the plant P. Requiring the existence of a static stabilizer is a consequence of the PFC to be input-dimensional (i.e., the dimension of the PFC is the same as that of the plant input). In Isidori and Marconi (2008), the design problem of a PFC for a class of nonlinear plants was addressed and it was shown that the output feedback stabilization problem of a nonlinear plant P is solved by means of the proposed PFC V and the static high-gain output feedback of P+V. The PFC V in Isidori and Marconi (2008) was constructed from the prior knowledge of a controller that possesses a certain structural property and stabilizes the auxiliary system (again derived from P).

However, while those references present different sufficient conditions for the existence of the PFC, a complete necessary and sufficient characterization and a systematic design method are still lacking. Motivated by this fact, we study a necessary and sufficient condition for the existence of stable PFC and its constructive design procedure. It will be seen in Section  3 that, by imposing the stability of PFC itself, the problem can be converted into the classical simultaneous stabilization problem and therefore, a few off-the-shelf design methods become applicable to the PFC problem. The search for PFC within the stable systems could be a restriction, but it also provides with some benefits. One particular application, where the structure of P+V and stability of V play important roles, is the synchronization problem dealt with in Section  4. Synchronization problem has its origin at the asymptotic convergence of each first-order dynamic agent to their average of initial conditions (Jadbabaie et al., 2003, Ren et al., 2007). It has been extended to higher order dynamic agents, but instead some dynamic controller is introduced into each agent. As a result, the synchronized trajectory is not purely an average of each agent, but the initial conditions of the controllers now take part in the average (see, e.g., Kim, Shim, Back, & Seo, 2013; Li et al., 2011, Li et al., 2010; Scardovi & Sepulchre, 2009 and Seo, Shim, & Back, 2009). On the other hand, the proposed design of synchronizing controller overcomes this drawback and the problem reverts into its original philosophy. Moreover, the proposed design of PFC allows arbitrarily fast zero assignment (that has not been addressed in the previous results, e.g., Bar-Kana, 1986; Deng et al., 1999; Iwai & Mizumoto, 1994 and Son et al., 2002), which in turn yields fast synchronizing rate to the average. This feature also eliminates the intrinsic limitation of Seo et al. (2009), in which the synchronizing rate is usually very slow because of its low-gain based design. Finally, the PFC admits infinite gain margin property of P+V,2 which enables the design of fully distributed synchronizing controllers (Li, Ren, Liu, and Fu, 2013, Li, Ren, Liu, and Xie, 2013) that do not use the information on the interconnection structure of network. We mention that the topics covered in this paper have their origins in Kim (2011) and Kim, Kim, Back, Shim, and Seo (2011) in part.

Notation

For aR, Ca denotes the set of complex numbers whose real parts are greater than or equal to a, namely, Ca{sC:Re(s)a}. In addition, Ra{sR:sa} and RaRa{s=+}. The sets R>a, C<a, and Ca are defined analogously. S<a denotes the set of proper rational functions whose poles are all in C<a. A controller C(s) is said to stabilize a plant P(s) (Vidyasagar, 1985) if the closed-loop system H(P(s),C(s)) is stable in the sense that H(P(s),C(s))[1/Δ(s)P(s)/Δ(s)C(s)/Δ(s)1/Δ(s)]M(S<0), where Δ(s)1+P(s)C(s) and M(S<a) is the set of 2×2 matrices whose elements are in S<a. Two plants P0(s) and P1(s) are simultaneously stabilizable if there is a common controller C(s) that stabilizes both of the plants, i.e., H(Pi(s),C(s))M(S<0) for i=0,1. [x;y] stands for the stack of two vectors (or matrices of compatible dimensions) x and y. The symbols and denote transpose and Kronecker product, respectively.

Section snippets

Design problem of stable PFC

Consider a SISO LTI plant P given by P:{ṗ=App+Bpu,pRn,uR,yp=Cpp,ypR, where (Ap,Bp,Cp) is minimal, i.e., controllable and observable.

In this paper, whenever we call a ‘stable parallel feedforward compensator (PFC)’, we mean a dynamical system of the form V:{v̇=Avv+Bvu,vRm,yv=Cvv,yvR such that the parallel interconnection P+V (see Fig. 1) ẋ=Ax+Bu[Ap00Av]x+[BpBv]u,y=Cx[CpCv]x is of minimum phase and has relative degree one, in which x[p;v] is the state and y=yp+yv the output of P+V, and A

Main results: necessary and sufficient conditions and PFC design procedure

From the design problem stated in Section  2, one observes that the PFC V has to have relative degree 1 if P has relative degree greater than 1, for otherwise P+V would not have relative degree 1. This observation leads to a PFC having relative degree 1 as a candidate solution for the problem as in the sufficient part of Theorem 1. Conversely, the necessary part of Theorem 1 shows that, among the solutions, there always exists a PFC (possibly different from (2)) having relative degree 1. Thus,

Application: synchronization problem

The synchronization problem is basically a problem of attaining an identical behavior (or group behavior) of multiple systems by interacting with others through a communication network. In this section, we present how the properties of P+V̄ (i.e., zero assignment, block diagonal structure in (3), and infinite gain margin) can be effectively used in dealing with the synchronization problem. Regarding the graph theory which is essential for describing the local interaction, the reader is referred

Conclusions

We have considered the design problem of stable PFCs for SISO LTI plants such that their parallel interconnection is of minimum phase and has relative degree one. Based on the classical simultaneous stabilizability, a complete necessary and sufficient condition for solving the problem is developed as well as a systematic design procedure for synthesizing such a PFC. The benefits of considering the proposed PFCs in the control systems are three-fold. First, the PFC allows us to assign the zeros

Acknowledgments

This work was supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2015R1A2A2A01003878). The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2013R1A6A3A03066047).

Hongkeun Kim received his B.S. degree from Hanyang University, Korea, in 2005, and Ph.D. degree from Seoul National University, Korea, in 2012. From 2014 to 2015, he was a postdoctoral researcher at University of Groningen, the Netherlands. Since 2015, he has been with Korea University of Technology and Education, Korea, where he is currently an assistant professor.

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      The use of parallel feedforward compensator (PFC) technique dates back to the work [8], where a compensator is added in parallel to render a linear system almost strictly positive real (ASPR), motivated by the need to implement an adaptive controller on positive real plants. Recently, [9] provides a necessary and sufficient condition for the existence of a stable PFC to render a single-input-single-output (SISO) LTI system minimum phase. There are also some recent applications of PFC in the literature.

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      In [15], the requirements for stability and minimum-phaseness are relaxed but the proposed scheme is merely applicable to SISO linear systems. Kim et al. investigated the method of designing stable parallel compensators for SISO linear systems and reformulated the problem as a simultaneous stabilization problem [16]. Moreover, the input-output data driven and adaptive methods have been devised to alleviate dependency on the prior knowledge about the plants [3,17].

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    Hongkeun Kim received his B.S. degree from Hanyang University, Korea, in 2005, and Ph.D. degree from Seoul National University, Korea, in 2012. From 2014 to 2015, he was a postdoctoral researcher at University of Groningen, the Netherlands. Since 2015, he has been with Korea University of Technology and Education, Korea, where he is currently an assistant professor.

    Seongjun Kim received his B.S. and M.S. degrees in the Department of Electrical Engineering and Computer Science from Seoul National University, Korea, in 2009 and 2011, respectively. Since 2011, he has worked at the Agency for Defense Development, where he is a researcher. His research interests include control system theory and design.

    Juhoon Back received the B.S. and M.S. degrees in Mechanical Design and Production Engineering from Seoul National University, in 1997 and 1999, respectively. He received the Ph.D. degree from the School of Electrical Engineering and Computer Science, Seoul National University in 2004. From 2005 to 2006, he worked as a research associate at Imperial College London, UK. Since 2008 he has been with Kwangwoon University, Korea, where he is currently an associate professor.

    Hyungbo Shim received his B.S., M.S., and Ph.D. degrees from Seoul National University, Korea, in 1993, 1995 and 2000, respectively. From 2000 to 2001 he was a post-doctoral researcher at University of California, Santa Barbara. Since 2003, he has been with Seoul National University, where he is now a professor. He has served as Associate Editor for the journals IEEE Trans. on Automatic Control and Automatica.

    Jin Heon Seo received the B.S. and M.S. degrees in Electrical Engineering from Seoul National University, Korea, in 1978 and 1980, and the Ph.D. degree in Electrical Engineering from the University of California, Los Angeles, in 1985. He served as an assistant professor from 1985 to 1989 in the Department of Electrical Engineering at Texas Tech University, Lubbock. Since 1989, he has been with the School of Electrical Engineering at Seoul National University.

    The material in this paper was partially presented at the 11th International Conference on Control, Automation and Systems, October 26–29, 2011, Gyeonggi-do, South Korea. This paper was recommended for publication in revised form by Associate Editor Tamas Keviczky under the direction of Editor Christos G. Cassandras.

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