Reliable decentralized PID controller synthesis for two-channel MIMO processes

doi:10.1016/j.automatica.2008.08.014

Automatica

Volume 45, Issue 2, February 2009, Pages 353-363

https://doi.org/10.1016/j.automatica.2008.08.014 Get rights and content

Abstract

Reliable stabilization and regulation of two-channel decentralized multi-input multi-output (MIMO) control systems is considered. The system has integral-action due to using proportional + integral + derivative (PID) controllers. Closed-loop stability and asymptotic tracking of step-input references are achieved at each output channel when all controllers are operational. Stability is maintained when one of the controllers fails completely and is set to zero. Controller synthesis procedures are proposed for stable MIMO plants and for several unstable MIMO plant classes that admit PID controllers. These synthesis procedures are applied to various examples of process systems to illustrate the design methodology.

Introduction

A stabilizing controller synthesis method is developed for linear, time-invariant (LTI), multi-input multi-output (MIMO) systems using a two-channel decentralized controller structure, with the objectives of decentralized closed-loop stabilization, reliable stability in case of complete failure of one of the two channels, and integral-action achieved with low-order simple PID controllers.

The decentralized controller structure has advantages although it restricts the stabilizing controller class. Fully decentralized control designs can be difficult also because of the interaction between the control loops. This introduces the problem of input–output pairing, to be decided in the first stage of the design before controller tuning. A method used to measure interactions and assess appropriate pairing is the relative gain array (RGA) (Campo & Morari, 1994). It is assumed here that the input–output pairing of the decentralized structure is already completed and the given plant model is partitioned into two MIMO channels a priori. An important control design requirement is reliability of the system’s closed-loop stability against complete failure of certain channels. Reliable designs were considered under full-feedback and decentralized controller structures in Braatz, Morari, and Skogestad (1994), Gündeş and Kabuli (2001), Siljak (1980) and Tan, Siljak, and Ikeda (1992). In reliable control systems, when sensor and/or actuator failures occur and controllers in failed channels are taken out of service, the remaining controllers maintain closed-loop stability of the entire system. The completely different approach of fault tolerant control, based on first defining and storing all feasible controllers, guarantees stability by using a switching strategy among these controllers depending on failures (Seron, Zhuo, De Dona, & Martinez, 2008). Reliable stabilization requires no switching or re-tuning of controllers. An important performance objective is asymptotic tracking of constant reference inputs with zero steady-state error, achieved by designing controllers with integral-action. The simplest integral-action controllers are in proportional + integral + derivative (PID) form (Goodwin, Graebe, & Salgado, 2001). Although PID controllers are desirable due to easy implementation and tuning, their simplicity presents a major restriction that they can control only certain plants.

The problem studied here has several layers of difficulty due to the restricted decentralized structure of the controller, the requirement of closed-loop stability when one controller is taken out, and the restrictions in the class of (unstable) processes that can be stabilized using PID controllers. Achieving reliable closed-loop stability with either one of the controllers subject to failure is more demanding on the design than expecting that stability is maintained when a pre-specified one of the two controllers may fail. It is assumed that the failure of the controller $C_{j}$ , $j = 1, 2$ , is recognized and the failed controller is taken out of service (with its states reset to zero). This catastrophic failure is modeled by setting $C_{j} = 0$ . A design that is reliable against either failure is called fully reliable; if only one specific controller may fail, then the control design is called partially reliable. These and other definitions are provided in Section 2, where the problem considered is formally stated. Section 3 is devoted to MIMO processes that are open-loop stable. Although both partially and fully reliable decentralized controllers exist for stable plants, the requirement of PID controllers and integral-action imposes additional conditions on the plants. These conditions and a reliable decentralized PID controller synthesis procedure are given in Section 3.1. For the stable plant case in Section 3, any other PID tuning method can be used to design $C_{2}$ that stabilizes one sub-block of the plant; for a fully reliable design, $C_{1}$ simultaneously stabilizes two systems related to the plant. Each of these blocks to be stabilized are MIMO systems if each channel has multiple inputs and outputs. There are no established PID tuning methods achieving simultaneous stabilization of two systems applicable in the MIMO setting. The synthesis methods proposed in this section are illustrated by examples. For certain PID stabilizable unstable plant classes (Gündeş & Özgüler, 2007), Section 4 investigates existence conditions for reliable decentralized controllers and proposes controller synthesis procedures. Partially reliable PID controller designs are illustrated for two examples.

The designs proposed here achieve closed-loop stability and asymptotic tracking of step-input references with zero steady-state error when all channels are operational, and maintain closed-loop stability of the overall system when either channel fails, with integral-action still present in the channel that remains active. The proposed designs also achieve asymptotic rejection of output disturbances since this is mathematically equivalent to the tracking problem. The proposed controllers also achieve robust closed-loop stability under sufficiently small additive or multiplicative plant uncertainty. The synthesis procedure for each plant class considered here allows freedom in choosing many of the design parameters. These parameters may be chosen to optimize the response in case of other performance specifications. Since the only goal here is reliable regulation, only stability and asymptotic tracking of constant inputs are emphasized and other performance objectives are not specified.

Notation

Let $C$ , $R$ , $R_{+}$ denote complex, real, positive real numbers; $U = {s \in C ∣ R e (s) \geq 0} \cup {\infty}$ is the extended closed right-half plane; $I_{n}$ is the $n \times n$ identity matrix; $R_{p}$ denotes real proper rational functions of $s$ ; $S$ is the stable subset with no $U$ -poles; $M (S)$ is the set of matrices with entries in $S$ . A square matrix $M \in M (S)$ is called unimodular iff $M^{- 1} \in M (S)$ . The $H_{\infty}$ -norm of $M (s) \in M (S)$ is $‖ M ‖ ≔ {sup}_{s \in \partial U} \bar{σ} (M (s))$ ; $\bar{σ}$ is the maximum singular value and $\partial U$ is the boundary of $U$ . Wherever this causes no confusion, $(s)$ in transfer functions such as $G (s)$ is dropped. We use coprime factorizations over $S$ . We abbreviate right-coprime (RC) and left-coprime (LC).

Section snippets

Problem statement and preliminaries

Consider the LTI decentralized feedback system $S y s (G, C_{D})$ with two MIMO channels as in Fig. 1. The feedback system is well-posed; the plant and controller have no hidden-modes associated with eigenvalues in the unstable region $U$ . The plant $G \in {R_{p}}^{r \times m}$ and the decentralized controller $C_{D} \in {R_{p}}^{m \times r}$ are partitioned as: $G = [\begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix}], G_{i j} \in {R_{p}}^{r_{i} \times m_{j}}, i, j = 1, 2,$ $C_{D} = [\begin{matrix} C_{1} & 0 \\ 0 & C_{2} \end{matrix}] = diag [C_{1}, C_{2}], C_{j} \in {R_{p}}^{m_{j} \times r_{j}} .$ Let $m ≔ m_{1} + m_{2}$ , $r ≔ r_{1} + r_{2}$ . We assume throughout that $rank G (s) = r$ and $rank G_{j j} (s) = r_{j}$ for $j = 1, 2$ . Let $u ≔ [\begin{matrix} u_{1} \\ u_{2} \end{matrix}], v ≔ [\begin{matrix} v_{1} \\ v_{2} \end{matrix}], e ≔ [\begin{matrix} e_{1} \\ e_{2} \end{matrix}]$

Reliable stabilization of stable plants

Let the plant $G$ partitioned as in (1) be stable, i.e., $G_{i j} \in S^{r_{i} \times m_{j}}, i, j = 1, 2$ . Then $G$ can be factorized as $G = Y^{- 1} X = I^{- 1} G$ . Lemma 2 gives necessary and sufficient conditions for decentralized stabilization.

Lemma 2

Let $G \in S^{r \times m}$ and $C_{D} \in {R_{p}}^{m \times r}$ be as in(1), (2), respectively. Let $N_{c j} D_{c j}^{- 1}$ be an RCF of $C_{j}$ , $j = 1, 2$ .

(i) $S y s (G, C_{1}, C_{2})$ is stable if and only if $M_{c l} ≔ D_{c} + G N_{c} = [\begin{matrix} D_{c 1} + G_{11} N_{c 1} & G_{12} N_{c 2} \\ G_{21} N_{c 1} & D_{c 2} + G_{22} N_{c 2} \end{matrix}] is unimodular.$

(ii) $S y s (G, 0, C_{2})$ is stable if and only if $C_{2}$ stabilizes $G_{22} \in M (S)$ ; equivalently, $M_{2} ≔ D_{c 2} + G_{22} N_{c 2} is unimodular.$

(iii) $S y s (G, C_{1},$

Reliable stabilization of unstable plants

Let the plant $G \in {R_{p}}^{r \times m}$ be partitioned as in (1). Let $G = Y^{- 1} X$ be an LCF of $G$ and $G_{22} = Y_{22}^{- 1} X_{22}$ be an LCF of $G_{22}$ . It is assumed that the denominator matrix $Y$ is in upper-block-triangular form as in (23) (Gündeş and Desoer, 1990, Vidyasagar, 1985): $G = [\begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix}] = Y^{- 1} X = {[\begin{matrix} Y_{11} & Y_{12} \\ 0 & Y_{22} \end{matrix}]}^{- 1} [\begin{matrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix}] .$ Lemma 4 gives necessary and sufficient conditions for the stability of the two channel decentralized system for normal and failure modes.

Lemma 4

Let $G \in {R_{p}}^{r \times m}$ and $C_{D} \in {R_{p}}^{m \times r}$ be as in(23), (2), respectively. Let $N_{c j} D_{c j}^{- 1}$ be

Conclusions

We proposed systematic synthesis of fully reliable or partially reliable decentralized PID controllers that achieve closed-loop stability and asymptotic tracking of step-input references at each output channel when both channels are operational, and maintain closed-loop stability even when one of the controllers is turned off. The proposed PID controllers provide robust stability for small plant uncertainty. For stable processes, we gave illustrative examples for a fully reliable PID design and

A.N. Gündeş received the B.S., M.S., Ph.D. (1988) degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley. She has been with the Electrical and Computer Engineering Department, University of California at Davis since 1988.

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A.N. Mete received the B.S. degree in Electrical Engineering from Gaziantep University, Turkey, in 2000, and the M.S. degree from the University of California, Davis, in 2004. He is currently a Ph.D. candidate at UC Davis. His research interests include reliable control and decentralized control.

A. Palazoğlu received his B.S. and M.S. degrees in chemical engineering from Middle East Technical University, Turkey and Bogazici University, Turkey, in 1978 and 1980, respectively. He received his Ph.D. in Chemical Engineering in 1984 from Rensselaer Polytechnic University in Troy, NY, focusing on robust control design. He joined the Chemical Engineering and Materials Science Department at University of California, Davis in 1984, and has been there since that time. He has over 100 publications and two co-authored books, Introduction to Process Control (Boca Raton, FL: Taylor & Francis, 2006) and Chemical Process Performance Evaluation (Boca Raton, FL: Taylor & Francis, 2007). His research areas include process control and monitoring, nonlinear systems, as well as statistical modeling in biological and environmental systems. Dr. Palazoglu received the Tarik Somer Award from Middle East Technical University, Turkey in 2005.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo.

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Reliable decentralized PID controller synthesis for two-channel MIMO processes☆

Abstract

Introduction

Section snippets

Problem statement and preliminaries

Reliable stabilization of stable plants

Reliable stabilization of unstable plants

Conclusions

Automatica

Process control modeling, design and simulation

Achievable closed-loop properties of systems under decentralized control: Conditions involving the steady-state gain

IEEE Transactions on Automatic Control

Hybrid predicative control of nonlinear systems: Method and applications to chemical processes

International Journal on Robust Nonlinear Control

Control system design