SENSITIVITY MINIMIZATION OF MIMO SYSTEMS WITH A ST ... 14th World Congress ofIFAC Copyright © 1999 IF AC C-2a-Ol-2 14th Triennial World Com!ress. Beiiinl!. P.R. China IVllnlmlzatlon of MIMO Systems with A Stable Controller Abdul- Wahid A. Saif King Fahd University of Petroleum & Minerals Systems Engineering Department P.O.Box 1626 Da-Wei Gu University of Leicester Department of Engineering Leicester LE 7RH U.K. Dhahran 31261 Saudi Arabia E-mail: awsaif@dpc.kfupm.edu.sa Abstract In this paper, the sensitivity minimization prob- lem of multivariable linear time-invariant, bi- proper or strictly proper, systems using a stable controller is considered. The problem is reduced to a I-block or 2-block 1too - optimization problem in the case of square or nons quare systems, respec- tively. The solution of this optimization problem, if satisfying an upper bound constraint which de- pends on an initial choice of a unimodular transfer function in 'iJni oo , would be the Youla pararne- ter and hence a stable stabilizing controller to the original system. A search procedure for an ini- tial choice of an unimodular function is discussed. Copyright© 1999 [FAC Keywords:Strong Stabilization, Sensitivity mini- mization, l-Block/2-Block problems. 1 Introduction The strong stabilization problem eStSP) is defined as that of finding a st.able controller t.hat stabi- lizes a given plant. A study of strong stabiliz.- ability is mainly of practical interest for control engineers. If a plant is stabilized using a stable compensator then the resulting closed-loop sta- ble transfer function matrix from the input dis- turbance to the output, or equivalently, the con- trol signal to the output, would have the same number of right half plane zeros as the original plant, (Vidyasagar, 1985). On the other hand, sta- bilization using an unstable compensator always introduces additional right half plane zeros into this closed-loop transfer function matrix beyond those of the original plant. As it is known that the right half plane zeros of a system affect its ability to track reference signals and/or to reject disturbances, it is preferable to use a stable sta- bilizing compensator whenever possible. In addi- tion, several other problems in reliable stabiliza- Copyright 1999 IF AC E-mail: dwg@sun.engg.le.ac.uk tion are related to strong stabilizability, such as the simultaneous stabilization problem. The 8tSP of linear time-invariant systems was first addressed and solved for SI80 systems in (Youla, et al.,1974), who presented a tractable condition, known as the parity interlacing property (PIP), to check whether a given plant is strongly stabilizable or not. While the PIP is an elegant result, it does not indicate how to find such a controller if it ex- ists. Some attempts to provide such a stable con- troller were given in (Youla , et al., 1974; Ganesh and Person, 1989; Dorato, et al., 1989; Ita, et aL, 1993) for S180 systems using the Nevanlinna-Pick interpolation technique. References (Jacobus, et al., 1990; Halevi, et al., 1991; William and Bern- stein, 1993; Halevi, 1994) suggested some modifi- cations in the Riccati equations that would result in a stable controller. While the modifications are different, all these methods result in nonstandard Riccati equation(s) with solutions that are difficult to obtain and, furthermore, not even guaranteed to exist. A greater restriction on the zeros of the sought stable controller was considered in (Wei, 1990). In (Sideris and Safonov, 1985), the prob- lem was addressed for MIMO systems where a fi- nite dimensional optimization problem was formu- lated to satisfy the Pick interpolation condition. In (Saif, et al., 1997), the St8P was formulated as an 'Hoo optimization problem, while the matrix Nevanlinna-Pick interpolation technique was used to solve the StSP in (Saif, et al., 1998a). In practice, stability alone is never sufficient and a procedure just for finding a stable stabilizing controller is not of much use by itself. In this di- rection, the sensitivity minimization problem us- ing stable controller was addressed in (Ito, et al., 1993) for SI80 systems using the Nevanlinna-Pick interpolation technique and in (Nakayama, 1995, Saif,A., 1998b,c) by tuning the free parameter of the H.oc> control problem to get a stable controller. ISBN: 008 0432484 1025