Techniques for Reduced-Order Control of Stochastic Discrete-Time Weakly Coupled Large Scale Systems

Xuemin Shen    Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Zijad Aganovic    Cylex Systems Inc., 6001 Broken Sound Parkway, Boca Raton, FL 33487

Zoran Gajic    Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08855–0909

I INTRODUCTION

The weakly coupled systems were introduced to the control audience by (Kokotovic et al., 1969). Since then many control aspects for the linear weakly coupled systems have been studied (Medanic and Avramovic, 1975; Ishimatsu et al., 1975; Ozguner and Perkins, 1977; Delacour et al., 1978; Mahmoud, 1978; Petkovski and Rakic, 1979; Washburn and Mendel, 1980; Arabacioglu et al., 1986; Petrovic and Gajic, 1988; Gajic and Shen, 1989, 1993; Harkara et al., 1989; Gajic et al., 1990; Shen, 1990; Shen and Gajic, 1990a, 1990b, 1990c; Su, 1990; Su and Gajic, 1991, 1992; Qureshi, 1992).

The general weakly coupled systems, in different set ups, have been studied by Siljak, Basar and their coworkers (Ikeda and Siljak, 1980; Ohta and Siljak, 1985, Sezer and Siljak, 1986, 1991; Kaszkurewicz et al., 1990; Siljak, 1991; Srikant and Basar, 1989, 1991, 1992a, 1992b; Skataric, 1993; Skataric et al., 1991, 1993; Riedel, 1993). The weak coupling has been also considered in the concept of multimodeling (Khalil and Kokotovic, 1978; Ozguner, 1979; Khalil, 1980; Saksena and Cruz, 1981a, 1981b; Saksena and Basar, 1982; Gajic and Khalil, 1986; Gajic, 1988; Zhuang and Gajic, 1991) and for nearly completely decomposable Markov chains (Delebecque and Quadrat, 1981; Srikant and Basar, 1989; Aldhaheri and Khalil, 1991). The nonlinear weakly coupled control systems have been studied in only a few papers (Kokotovic and Singh, 1971, Srikant and Basar, 1991, 1992b; Aganovic, 1993; Aganovic and Gajic, 1993).

The discrete-time linear control systems have been the subject of recent research (Shen and Gajic, 1990a, 1990b). In this chapter we first give an overview of the obtained results on filtering and control of discrete-time stochastic systems and then present some new results. For the reason of completeness, we study the main algebraic equations of the linear control theory, that is, the Lyapunov and Riccati equations. Corresponding parallel reduced-order algorithm for solving discrete Lyapunov and Riccati equation of weakly coupled systems are derived and demonstrated on the models of real control systems. Algorithms for both the Lyapunov and Riccati equations are implemented as synchronous ones. Their implementation as the asynchronous parallel algorithms is under investigation.

II RECURSIVE METHODS FOR WEAKLY COUPLED DISCRETE-TIME SYSTEMS

In this section parallel reduced-order algorithms for solving discrete algebraic Lyapunov and Riccati equations of weakly coupled systems and the corresponding linear-quadratic optimal control problem are presented.

A PARALLEL ALGORITHM FOR SOLVING DISCRETE ALGEBRAIC LYAPUNOV EQUATION

Consider the algebraic discrete Lyapunov equation

TPA−P=Q,A<0,Q≥0

In the case of a weakly coupled linear discrete system the corresponding matrices are partitioned as

=A1ϵA2ϵA3A4,Q=Q1ϵQ2ϵQ2TQ3,P=P1ϵP2ϵP2TP3

where Ai, i = 1, 2, 3, 4, and Qj, j - 1, 2, 3, are assumed to be continuous functions of e. Matrices Pi and P3 are of dimensions n × n and m × m, respectively. Remaining matrices are of compatible dimensions.

The partitioned form of (1) subject to (2) is

1TP1A1−P1+Q1+ϵ2A1TP2A3+A3TP2TA1+A3TP3A3=0

1TP1A2−P2+Q2+A1TP2A4+ϵ2A3TP2TA2=0

4TP3A4−P3+Q3+ϵ2A2TP1A2+A2TP2A4+A4TP2TA2=0

Define, O(ε2) perturbations of (3)-(5) by

1TP¯1A1−P¯1+Q1=0

1TP¯1A2+A1TP¯2A4+A3TP¯3A4−P¯2+Q2=0

4TP¯3A4−P¯3+Q3=0

Note that we did not set ε = 0 in Ai's and Qj's. Under the assumption made in (1), A < 0, it follows that for sufficiently small ε the matrices A1 and A4 are stable. Then the unique solutions of (6)-(8) exist.

Define errors as

1=P¯1+ϵE1P2=P¯2+ϵE2P3=P¯3+ϵE3

Subtracting (6)-(8) from (3)-(5), the following error equations are obtained

1TE1A1−E1=−A1TP2A3−A3TP2TA1−A3TP3A3A4TE3A4−E3=−A2TP1A2−A2TP2A4−A4TP2TA2A1TE2A4−E2=−A1TE1A2−A3TP2TA2−A3TE3A4

The proposed parallel synchronous algorithm for the numerical solution of (10) is as follows (Shen, et al., 1991).

1 CASE STUDY: DISCRETE CATALYTIC CRACKER

A fifth-order model of a catalytic cracker (Kando et al., 1988), demonstrates the efficiency of the proposed method. The problem matrix A (after performing discretization with the sampling period T = 1) is given by

d=0.0117710.0469030.0966790.071586−0.0191780.0140960.0564110.1150700.085194−0.0228060.0663950.2522600.5808800.430570−0.116280.0275570.1049400.2404000.178190−0.0481040.0005640.0026440.0034790.002561−0.000656

The small weak coupling parameter is ε = 0.21 and the state penalty matrix is chosen as Q = I.

The simulation results are presented in Table 1.