Pole assignment with improved control performance by means of periodic feedback

Abstract

This paper is concerned with the pole placement of continuous-time LTI systems by means of periodic feedback. It is known that any desired pole placement can be achieved for a controllable and observable LTI system by using a proper controller. The corresponding control law, however, is not unique in general. Analogously, a generalized sampled-data hold function (GSHF) which places the modes of the discrete-time equivalent model of a continuous-time system at the desired locations in the complex plane is not unique. This paper aims to present a systematic method to attain a GSHF which not only does it satisfy the pole-placement criterion, but it also results in an optimal performance for the original continuous-time system. To this end, the GSHF being sought is written as the sum of a particular GSHF and a homogeneous GSHF. The particular GSHF can be obtained using any existing methods. To characterize the homogeneous GSHF, a set of basis functions is constructed (e.g., in the form of polynomials or Fourier exponentials). Then, it is shown any linear combination of the constructed basis functions is, in fact, a homogeneous GSHF. To find the optimal homogeneous GSHF (associated with the particular GSHF obtained before), a pre-specified LQ performance index is considered. The corresponding optimization problem is then formulated in a LMI framework. The method proposed here can be simply extended to the case when a structurally constrained GSHF (such as a block-diagonal one for a decentralized system) is to be designed. The efficacy of this work is demonstrated in an illustrative example.