tailieunhanh - CS 205 Mathematical Methods for Robotics and Vision - Chapter 6

Ordinary Differential Systems Trong chương này chúng tôi sử dụng các lý thuyết phát triển trong chương 5 để giải quyết hệ thống phương trình vi phân tuyến tính bậc nhất với hệ số không đổi. Các hệ thống này có các hình thức sau đây: | Chapter 6 Ordinary Differential Systems In this chapter we use the theory developed in chapter 5 in order to solve systems of first-order linear differential equations with constant coefficients. These systems have the following form x Ax bt x x0 where x xt is an n-dimensional vector function of time t the dot denotes differentiation the coefficients a jj inthe n X n matrix A are constant and the vector function b t is a function of time. The equation in which x0 is a known vector defines the initial value of the solution. First we show that scalar differential equations of order greater than one can be reduced to systems of first-order differential equations. Then in section we recall a general result for the solution of first-order differential systems from the elementary theory of differential equations. In section we make this result more specific by showing that the solution to a homogeneous system is a linear combination of exponentials multiplied by polynomials in t This result is based on the Schur decomposition introduced in chapter 5 which is numerically preferable to the more commonly used Jordan canonical form. Finally in sections and we set up and solve a particular differential system as an illustrative example. Scalar Differential Equations of Order Higher than One The first-order system subsumes also the case of a scalar differential equation of order n possibly greater than 1 dny dn iy dy cn-l ci T7 coy bt dtn dtn 1 dt In fact such an equation can be reduced to a first-order system of the form by introducing the n-dimensional vector -I r y X1 dy_ dt . . . . Xn J dn 1y L -I x With this definition we have dÀ dny dtn for i . n dxn dt 69 70 CHAPTER 6. ORDINARY DIFFERENTIAL SYSTEMS and x satisfies the additional n - equations Xi 1 at for i . n - . If we write the original system together with the n - differential equations we obtain the first-order system x Ax b t where . . . A . . . . . Co Cl Ỡ2 .