An engineering approach to the simultaneous stabilisation and strong simultaneous stabilisation with D stability

The simultaneous stabilisation problem (SSP) and strong simultaneous stabilisation problem (SSSP) with the simple Hurwitz stability requirement are open problems in the control community. In this paper, an engineering approach for controller synthesis is presented, with an extension to the general D stability requirement rather than simply the Hurwitz stability requirement. The method developed permits the designer to specify a set of desired stable regions, joint or disjoint, in the complex root plane. A numerical algorithm is used to find the controller parameters such that all the roots of the closed-loop system are within the specified regions. The desired dynamic performance of the closed-loop system is controlled by the specification of the desired D-stable region. The advantages of this approach include: (1) The desired D stability region can be of any form. It can 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 SSP and the SSSP can be dealt with in a unified way, although the SSSP is more complicated than the SSP. (4) The traditional PID controller tuning problem can be treated within this general framework. Examples which are harder than the previous results are provided to show the merits of this synthesis approach