（约当）结论2【对角规范型ifl ,is an eigenvalue（特征根） of multiplicity l，inother words, the characteristic rbotsf,Eystemare all distinct（n个两两互异的特征根），thestatespace description (1.79) and (1.80) can be reduced to adiagonal canonical form:L0élO uéx ué&uélu(1. 81)eu<euLieeu&01.0<Xie2e21Uher2u+e'ue Mi0éM0O ue Mieneeuué,uuee&lL0e0elatexnaéx,u(1.82)éuk,F2u+kouy=[kk2Le Miéü实现问题exnu
if is an eigenvalue (特征根) of multiplicity 1, in other words, the characteristic roots of system are all distinct（n个两两互异的特征根）, the state space description (1.79) and (1.80) can be reduced to a diagonal canonical form： (1.81) (1.82) 结论2【对角（约当）规范型】 实现问题
状态变换1. 5 DefinitionsConsider the LTI system described by the state equation&= AX + Bu（lI－ A）为A的特征矩阵（characteristic matrix）The polynomial aboutlis called the characteristic polynomia（特征多项式）n-1Q(l)=[I- A=1"+a a,li(1. 101)Q(l)=l I- A=O is called the characteristic equation （特征方程）If the polynomia(1)can be written in factored form as2(1. 102)Q(l)=det(II- A)=O(l - l,)i-1The rootsl, (i=l,2,L ,n)of the characteristicequation（特征值/特征根）ofAare called the eigenvalues
The polynomial about is called the characteristic polynomial （特征多项式） (1.101) is called the characteristic equation（特征方 程） If the polynomial can be written in factored form as (1.102) The roots of the characteristic equation are called the eigenvalues （特征值/特征根）of A 1.5 Definitions Consider the LTI system described by the state equation 为A的特征矩阵（characteristic matrix） 状态变换
状态变换Any nonzero vectorVwhich satisfies the matrix equation(1. 104)(l ,I- A)V, = 0is calledtheeigenvector（特征向量）ofAassociated witheigerIf A has distinct eigenvalues, the él(i=1,2,L ,n)solved by (1. 104) :当A有重根时，广义特征向量（generalizedeigenvectors）的计算l , is the characteristic root of multiplicity m(I,I- A)V=0(l I - A)V2 = - V(1. 105)(1 ,I - A)Vi3 = - Vi2M(l,I- A)Vm=- V(m-1特征向量主要应用于由状态变换求对角规范型或约当规范型
当A有重根时，广义特征向量(generalized eigenvectors)的计算 (1.105) Any nonzero vector which satisfies the matrix equation (1.104) is called the eigenvector (特征向量) of A associated with eigenvalues . If A has distinct eigenvalues, the eigenvalues can be solved by (1.104) . 特征向量主要应用于由状态变换求对角规范型或约当规范型 is the characteristic root of multiplicity m 状态变换
状态变换&= AX + Bu（约当）规范型由状态变换得对角(1. 117)y=CX+DuIf a system described by(1. 117)and A has distincteigenvalues I,/,L't,there is a nonsingulartransformationX(t) = Px(t)which transforms the general state description (1.117) intothe diagonal canonical form (1.120)&= AX+Bu(1. 120)y=cX+Du&ouéuA=P-IAP=0Whereis aa diagonal matrixeu@0I,tB=P-B=CPD=D
If a system described by (1.117) and A has distinct eigenvalues , there is a nonsingular transformation which transforms the general state description (1.117) into the diagonal canonical form (1.120) (1.120) Where is a diagonal matrix 由状态变换得对角（约当）规范型 (1.117) 状态变换
状态变换约当规范型1.5.4--转化为对角1、通过状态变换 X=PX，将一般状态空间描述转化为对角规范形。前提：系统矩阵A的n个特征根两两互异。Step:）求出n个互异特征根/,,L，l,，1）由特征方程,I-A=0求出特征根L，l,2）根据方程 1V=AV或(,I-A)V=0对应的特征向量V,L,Vn和p-13）得出状态转移矩阵P=[VV,L V,]4）代入状态变换后状态空间描述{A,B,C,D)，得到对角线规范形：B=P-'Belo u=AX+Bu福euu=CPA-P-AP-Oey =CX+Du0[ ,ED5=D
1.5.4-转化为对角（约当）规范型 1、通过状态变换，将一般状态空间描述转化为 对角规范形。 前提：系统矩阵A的n个特征根两两互异。 Step： 1）由特征方程求出n个互异特征根 2）根据方程求出特征根 对应的特征向量 3）得出状态转移矩阵 和 4）代入状态变换后状态空间描述，得到对角 线规范形： 状态变换