Simultaneous stabilisation and strong simultaneous stabilisation of r-tuple plants with D stability: a constrained optimisation approach

A systematic synthesis approach to the simultaneous stabilisation problem (SSP) and the strong simultaneous stabilisation problem (SSSP) with D stability is presented in this paper. The SSPs and SSSPs are respectively converted into a standard, computationally tractable feasibility problem for a set of equalities and inequalities (SEI) which can be solved using an available numerical algorithm in a finite number of iterations, with guarantee of convergence to a solution. This sort of problem is further considered as a non-linear constrained optimisation problem, for which a commercial software package has been developed in the Matlab optimisation toolbox, in order to find an optimal solution. A distinguishing feature of this approach is that the local minima problem existing with the non-linear constrained optimisation, which is serious for certain applications, is only a mild restriction for the SSP and SSSP, because the satisfaction of both the stability and transient performance of the closedloop system is guaranteed by the formulation of the problem as a set of equalities and inequalities and the possibility of finding a solution. We will explain the principle underlying the approach and the steps taken to cast the problem into the standard optimisation problem. The advantages of this approach include: 1) the desired D stability region can be of any form and be connected or disjoint. This leads to a unified treatment of continuous and discrete systems, and consequently encompasses the Hurwitz and Schur stability regions as special cases; 2) the size of the family of plants may be finite and more than three, which is the upper bound with the available approaches; 3) both the SSPs and the SSSPs can be dealt with in a unified way, although a SSSP is more complicated than a SSP. 4) the traditional PID controller tuning problem can be treated within this general framework. Examples are provided to show the merits of this synthesis approach.