Fixed, low-order controller design with time response specifications using non-convex optimization

In this paper, we present a new algorithm for designing a fixed, low-order controller with time response specifications for a linear time invariant (LTI), single input single output (SISO) plant. For a two-parameter feedback configuration, the problem of finding a fixed or low-order controller to meet the desired time response specification is reduced to the least square estimation (LSE) in the sense of partial model matching (PMM), which minimizes a quadratic cost function. The closed-loop stability condition imposed on the controller parameters is formulated by the polynomial matrix inequality (PMI) constraint associated with the cost function. When the cascade feedback structure is considered, the zeros of the controller may be a substantial obstacle when designing a controller that has a good time response. This problem can also be formulated using polynomial constraints. Consequently, it is shown that the total problem here can be formulated as an optimization problem with a quadratic objective function and several polynomial constraints in the controller parameter space. We show that the SeDuMi with YALMIP interface [Löfberg J. YALMIP: A toolbox for modeling and optimization in MATLAB, in: Proceedings of the IEEE symposium on computer aided control systems design 2004. p. 284–9. http://control.ee.ethz.ch/~joloef/yalmip.php] can be used for solving this problem. Finally, several illustrative examples are given.