2.1 Time Response of the LTI Homogeneous System（齐次系统The stateequation of the LTI system canbe described by(2.1)X(t) = AX(t)+ Bu(t)u(t) is the input vector of the system. The first term onthe right hand of the state equation (2.1) is known as thehomogenous part of the stateequation.If u(t) = 0 the system is called a LTI homogeneous systemor,a freemotionsystem.In this section, we focus on the response of the LTlhomogeneous system(2.2)X(t) = AX(t)
The state equation of the LTI system can be described by X(t) = AX(t) + Bu(t) (2.1) u(t) is the input vector of the system. The first term on the right hand of the state equation (2.1) is known as the homogenous part of the state equation. If the system is called a LTI homogeneous system, or, a free motion system. u(t) = 0 X(t) = AX(t) In this section, we focus on the response of the LTI homogeneous system (2.2) 2.1 Time Response of the LTI Homogeneous System （齐次系统）
求解线性时不变齐次系统（homogeneous system）(2.2)X(t) = AX(t)Considerthefirst-order differential equation(2.3)x(t) = a · x(t)Where x(t) is a time function, taking Laplace transform of (2.3)1(2.5)x(0)sx(s) - x(O) = ax(s)x(s) =s-aThe inverse Laplace transform of Eq.(2.5) results in the solution ofEq.(2.3)(2.6)x(t) = et x(0)
X(t) = AX(t) 求解线性时不变齐次系统（ homogeneous system） (2.2) The inverse Laplace transform of Eq.(2.5) results in the solution of Eq.(2.3) x(t) e x(0) at = (2.6) Consider the first-order differential equation x (t) = a x(t) (2.3) Where x(t) is a time function, taking Laplace transform of (2.3) sx(s) − x(0) = ax(s) (0) 1 ( ) x s a x s − = (2.5)
x(t) = eat x(0)oatAccording to the power series(幂级数) of the exponential functioneα?f?α't3eat =1+ at +(2.7)2!3!（矩阵指数函数）Define matrix exponentialfunctionA?f?A't32AktkeAt = I+ At +(2.8)T2!3!k=o k!Obviously,the state vectorZero-inputresponseX(t) = eAt X(O)is the solution of the LTI homogenous system (2.2)
According to the power series(幂级数) of the exponential function at e = + + + + 2! 3! 1 2 2 3 3 a t a t e at at (2.7) Define matrix exponential function （矩阵指数函数） 2 2 3 3 0 1 2! 3! ! t k k k t t e t A t k = = + + + + = A A A I A (2.8) Obviously, the state vector X( ) X(0) At t = e is the solution of the LTI homogenous system (2.2). x(t) e x(0) at = Zero-input response
结论「零输入响应Zero-input Response]线性时不变齐次系统X(t) = AX(t), X(to)= X(O)(2.2)的零输入响应，具有如下形式：(2.9)AtX(0)X(t) =eto不等于0如果是更一般的初始条件X(t.)，系统(2.2)的零输入响应为eA(r-1o)X(t) =X(to)(2.10)
结论 [零输入响应Zero-input Response] 线性时不变齐次系统 0 (2.2) X AX ( ) ( ), ( ) (0) t t X t X = = 的零输入响应，具有如下形式： X( ) X(0) At t = e (2.9) 如果是更一般的初始条件 , 系统(2.2) 的零输入响应为： ( ) ( ) 0 ( ) 0 t e t t t X X A − = (2.10) 0 X t( ) t0不等于0
矩阵指数函数的性质lim eAt = It-0eA(t+t) =eAt =色.e4t(eAt)-1 = e-AtA,F为同维方阵(A+F)tFHtAtAfe且可交换d4F=eAAdtA')"=e4(mt)Om=0,1,2...,PAP-1= Pe^ p-1e
➢矩阵指数函数的性质 1 PAP A 1 e Pe P − − = A,F为同维方阵 且可交换