X y=[k飞2.kk, .kn kou Xr+l (1.80) : Xn wherek,j=l,.,r,k,i=r+l.,n can be computed by 1 di- 0-0gs-)广Gs] -lim k。=limG(s)=b S→00 k,=lim[(s-2,)G(s)] Note that,k0=0,if m<n or b0=0
k j r j , 1, , where , can be computed by 1 = ki ,i = r +1, ,n lim [( ) ( )] ( 1)! 1 1 1 1 1 1 s G s ds d j k r j j s j − − = − − → k lim[(s )G(s)] i s i i = − → 0 0 k lim G(s) b s = = → k u x x x x x y k k k k k n r r r n r 0 1 2 1 1 1 1 2 1 1 + = + + (1.80) Note that, k0=0, if m<n or b0=0
结论2（对角线规范型） if乙is an eigenvalue(特征根)of multiplicity1,in other words, the characteristic roots of systemi=1,.,n 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: T17 (1.81) 元2 X2 1 0 Xn 1 X1 (1.82) y=kk2.k】 X2 kou X n
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 i i = 1, , n u x x x x x x n n n + = 1 1 1 0 0 0 0 0 0 2 1 2 1 2 1 k u x x x y k k k n n 0 2 1 1 2 + = (1.81) (1.82) 结论2（对角线规范型）
由方框图得出状态空间描述 ·根据表1.1，将要求系统换化为其等价系统 ·将每一个积分器的输出选为系统的状态变量 ·根据等价系统中状态变量x,.,xm与其微分元，.，元 的关系写出微分方程组，并进而整理成为状态方程 的形式X(t)=AX(t)+Bu(t) ·根据等价系统列写输出方程y(t)=CX(t)+Du(t) ·最后得到状态空间描述： X(t)=AX(t)+Bu(t) y(t)=CX(t)+Du(t) Where A=[1,B=0,C=[1,D=[l,X=[x1,.,XnT
• 根据表1.1，将要求系统换化为其等价系统 • 将每一个积分器的输出选为系统的状态变量 • 根据等价系统中状态变量 与其微分 的关系写出微分方程组，并进而整理成为状态方程 的形式 • 根据等价系统列写输出方程 • 最后得到状态空间描述： 由方框图得出状态空间描述 1 , , n x x 1 , , n x x X(t) = AX(t) +Bu(t) y(t) = CX(t) + Du(t) Where A=[],B=[],C=[],D=[],X=[x1,.,xn]T X(t) = AX(t) +Bu(t) y(t) = CX(t) + Du(t)
1.5 Definitions Consider the LTI system described by the state equation X=AX+Bu (I一A)为A的特征矩阵(characteristic matrix) The polynomial about is called the characteristic polynomial (特征多项式) n- a)=M-A="+∑a,X (1.101) i=0 Q(2)=2I-A=0 is called the characteristic equation(特征方程) If the polynomial (1)can be written in factored form as Q()=detI-A)=Π(2-) (1.102) The roots(i=1,2,.,n)of the characteristic equation are called the eigenvalues(特征值/特征根)ofA
The polynomial about is called the characteristic polynomial （特征多项式） − = = − = + 1 0 ( ) n i i i n Q I A (1.101) Q I A ( ) 0 = − = is called the characteristic equation（特征方程） If the polynomial can be written in factored form as Q() = = − = − n i Q i 1 () det(I A) ( ) (1.102) The roots of the characteristic equation are called the eigenvalues （特征值/特征根）of A (i 1,2, ,n) i = 1.5 Definitions Consider the LTI system described by the state equation X = AX + Bu ( ) I A − 为A的特征矩阵（characteristic matrix）
Any nonzero vector D which satisfies the matrix equation (2I-A)V,=0 (1.104) is called the eigenvector(特征向量)of A associated with eigenvalues (i=1,2,.,n).If A has distinct eigenvalues,the eigenvalues can be solved by (1.104). 当A有重根时，广义特征向量(generalized eigenvectors)的计算 is the characteristic root of multiplicity m (21-A)'1=0 (I-A)W2=-V1 (1-A)W3=-V2 (1.105) (41A)Vim=-VI(m-1) 特征向量主要应用于由状态变换求对角线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 11 1 12 11 1 13 12 1 1 1( 1) ( ) 0 ( ) ( ) ( ) m m I A V I A V V I A V V I A V V − − = − = − − = − − = − (1.105) Any nonzero vector which satisfies the matrix equation Vi (i I − A)Vi = 0 (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) . (i 1,2, ,n) i = 特征向量主要应用于由状态变换求对角线规范型或约当规范型 1 is the characteristic root of multiplicity m