Pole placement in optimal regulator by continuous pole-shifting

Abstract

This paper presents a method of pole placement in the linear quadratic regulator and its application to the flight control system design. There are three features in the proposed method. First, a weighting matrix that gives desired closed-loop pole locations is obtained by solving a set of differential equations. They are derived from the characteristic equation of the Hamilton matrix. Second, poles can be placed exactly at the desired positions and arbitrarily except the symmetry of complex conjugate poles with respect to the real axis. Third, it is a diagonal weighting matrix that is obtained by the proposed method. The third feature makes output regulation with desired pole locations possible. After demonstrating the effectiveness of the method through a simple literature example, it is then applied to the F-4 aircraft's lateral dynamic model. HE optimal regulator, or linear quadratic regulator (LQR), is a typical state-space design method. It has several features. For example, by imposing penalty on state variables and control inputs in a quadratic performance index, one can investigate tradeoff between system performance and control efforts. Besides, it is well known that the LQR has infinite gain margin and 60-deg phase margin for single-input systems. However, by this technique one cannot assign closed-loop pole locations. Since pole locations have a large effect on time-response characteristics, such as overshoot, rise time, etc., it is desirable to assign pole positions in the LQR design. Pole placement in the LQR is a problem associated with the selection of appropriate weighting matrices, when a set of desired closed-loop eigenvalues (CLEs) are known. This problem has been studied by many researchers since early 1970s, and many papers113 have been published in this research area. Among them, the study of Hayase 1 provides a relation between the weighting matrix and the feedback gains for a single-input system in controllable canonical form. Recently, Ohta et al. 2 showed a simple derivation of the relation. Meanwhile, for multi-input systems, Saif3 established a simple method using aggregation to compute a weighting matrix by which arbitrary and exact pole placement can be attained in a general form. He integrated the previously proposed three approaches.46 All of the aforementioned methods for multi-input systems follow Solheim's approach for placing imaginary parts as well as for placing real parts using the mirror image property.7 Another approach, which is completely different from the preceding approaches, is that of Graupe. In his method, first, differential equations of a weighting matrix with respect to the CLEs are derived, and then a weighting matrix is found as a solution of an optimization problem, wherein the difference between CLEs and assigned eigenvalues is made as small as possible.8 However, it is complicated and does not seem practical. Besides the preceding approaches, there are many approaches such as assigning eigenvalues with prescribed degree of stability, 9 placing eigenvalues into a specified region,10'11 and achieving pole placement asymptotically.12'13 In this paper, we propose a method of pole placement that is very different from the previously proposed methods. Whereas Graupe's differential equations are derived from the Riccati equation, we