Unification of independent and sequential procedures for decentralized controller design

Abstract

This paper is concerned with decentralized control problems. There are two typical procedures to design decentralized controllers, ‘independent’ and ‘sequential’ design procedures. As the concepts and the techniques in these two approaches are too different from each other, there have been no attempts to unify these approaches. This paper proposes an iterative independent design procedure for decentralized control systems, which is a unified approach to these conventional approaches.

Introduction

Decentralized control systems are the systems with the constraints on the controller structure. This paper focuses on the decentralized control systems with block diagonal controllers. To design decentralized controllers, there are two typical procedures called ‘independent’ and ‘sequential’ design procedures. Although there are some attempts to design decentralized controller based on matrix inequality approaches (D’Andrea and Dullerud, 2003, Ebihara et al., 2004, Zhai et al., 2001), these two design procedures are still very useful for theoretical analysis and practical design for decentralized control systems. In this paper, a unified approach of these two design procedures is proposed.

The independent design procedure constructs local controllers for corresponding local subsystems or diagonal approximation of the given plant. Each local controller is designed independently to the other local controllers (Araki and Nwokah, 1975, Grosdidier and Morari, 1986, Rosenbrock, 1969, Skogestad and Morari, 1989). Generally, the procedure only uses local loop information, thus designed controllers might be simple and easy to tune. Furthermore, the complexity of controllers is proportional to that of local subsystems. On the other hand, it is not easy to attain the best performance.

In the sequential design procedure, local loops are closed step by step (Bernstein, 1987, Chiu and Arkun, 1992, Davison and Gesing, 1979). Each local controller is designed with the information of previously designed local controllers. Thus, the sequential design would attain better performance than independent design does. The drawback is complexities of local controllers. The later the local controller is designed, the more complex it becomes.

As the concepts and techniques in these two approaches are too different from each other, there have been no attempts to unify the approaches. In this paper, an iterative independent design procedure for decentralized control systems is proposed. The proposed procedure has parameters to balance the difficulty and significance of local loops. With particular selections of these parameters, both the independent and sequential design procedures can be implemented. Furthermore, a modification to $H_{\infty }$ controller design is also discussed in this paper.

This paper is organized as follows. In Section 2, a parametrization of plants and controllers are stated. In Section 3, the stability condition with balancing weights is derived. Section 4 contains the proposed design procedures with balancing weights. Design examples are given in Section 5.

Notations: $M_{\mathrm{d}}$ denotes the block diagonal part of a matrix M, where the block diagonal structure is compatible with a decentralized controller. $\overline{\sigma }(M)$ denotes the maximum singular value of a matrix M. ${\mu }_{\Delta }(M)$ denotes the structured singular value of a matrix M with respect to the block diagonal structure $\Delta$ (Doyle, 1982). Throughout this paper, the diagonal structure $\Delta$ is same as that of a decentralized controller, and $\Delta$ is dropped in many cases for the sake of simplicity. For matrices M and N partitioned as $M_{\mathit{ij}}$, $N_{\mathit{ij}}$$(i,j=1,2)$, the upper linear fractional transformation (LFT), the lower LFT, and the star product of M and N are defined as

$${\mathcal{F}}_u(N,M_{22})=N_{22}+N_{21}{M}_{22}(I-N_{11}{M}_{22}{)}^{-1}{N}_{12}\text{,}{\mathcal{F}}_l(M,N_{11})=M_{11}+M_{12}{N}_{11}(I-M_{22}{N}_{11}{)}^{-1}{M}_{21}\text{,}M\star N=\left[\begin{array}{cc}\hfill {\mathcal{F}}_l(M,N_{11})\hfill & \hfill M_{12}(I-N_{11}{M}_{22}{)}^{-1}{N}_{12}\hfill \\ \hfill N_{21}(I-M_{22}{N}_{11}{)}^{-1}{M}_{21}\hfill & \hfill {\mathcal{F}}_u(N,M_{22})\hfill \end{array}\right]\text{,}$$

respectively. The inverse LFT of M is also defined as

$$M^{-}=\left[\begin{array}{cc}\hfill O\hfill & \hfill I\hfill \\ \hfill I\hfill & \hfill O\hfill \end{array}\right]M^{-1}\left[\begin{array}{cc}\hfill O\hfill & \hfill I\hfill \\ \hfill I\hfill & \hfill O\hfill \end{array}\right]\text{.}$$

Note that

$$M\star M^{-}=\left[\begin{array}{cc}\hfill O\hfill & \hfill I\hfill \\ \hfill I\hfill & \hfill O\hfill \end{array}\right],M^{-}\star M=\left[\begin{array}{cc}\hfill O\hfill & \hfill I\hfill \\ \hfill I\hfill & \hfill O\hfill \end{array}\right]\text{.}$$

${\mathcal{RH}}_{\infty }$ denotes the set of stable transfer function matrices, and $\parallel G(s){\parallel }_{\infty }$ denotes the $H_{\infty }$ norm of $G(s)$.