Numerical Methods for Linear Control Systems 论文

摘要

The design and analysis of linear control systems: 1.1 $$\dot x(t) = Ax(t) + Bu(t),\,\,\,\,\,\,\,\,\,\,y = Cx(t)$$ and 1.2 $$x_{k + 1} = Ax_k + Bu_k ,\,\,\,\,\,yk = Cx_k$$ give rise to many interesting linear algebra problems. Some of the important ones are: controllability and observability problems, the problem of computing the exponential matrix e At ,the matrix equations problems: (Lyapunov equations, Sylvester equations, the algebraic Riccati equations), the pole-placement problems, stability problems, and frequency response problems. These problems have been very widely studied in the literature. There exists a voluminous work both on theory and computations. Theory is very rich. Unfortunately, the same can not be remarked about computations. Many of the earlier methods were developed before the computer era, and are not based on numerically sound techniques. Fortunately, in the last twenty years or so, numerically effective techniques have been developed for most of these problems, and numerical analysis aspects of these methods (e.g. study of stability by round-off error analysis) and of the problems themselves (e.g. study of sensitivity) have been studied.