1.5 State Transformation of the LTI system（特征值）andEigenvector1.5. 1 Eigenvalue（特征向量）Consider the LTI system described bythe state equationX = AX + BuWhere XeR"is the state vector, AeRnxnis the systemmatrixandplays animportant rolein system properties.When u = 0 , the system given by (1.99) is calledafreesystem（自由系统）(1.99)X = AX当X和X方向相同但大小不同的时候，引入标量因子几，得到AX = ^X(-A)X =0有非零解iff-A=0（I一A）为A的特征矩阵（characteristicmatrix)
1.5 State Transformation of the LTI system 1.5.1 Eigenvalue （特征值）and Eigenvector （特征向量） Consider the LTI system described by the state equation X = AX + Bu Where is the state vector, is the system matrix and plays an important role in system properties. When , the system given by (1.99) is called a free system（自由系统） n X R nn AR u = 0 X = AX (1.99) AX = X 当 X 和 X 方向相同但大小不同的时候，引入标量因子 ，得到 (I − A)X = 0 有非零解iff I A − = 0 ( ) I A − 为A的特征矩阵（characteristic matrix）
DefinitionsThe polynomial about △ is called the characteristic polynomial（特征多项式）0(2) =|ul - A| = " +Zα,2(1.101)i=0Q(a)=aI -A|=0 is called the characteristic equation （特征方程)If the polynomial Q(a) can be written in factored form as(1.102)Q(a) = det(al - A) = IT(a - a,)i=1The roots 2, (i = l,2,..:,n) of the characteristic equationarecalledtheeigenvalues（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 = Definitions
Propertiesabouteigenvalue1. If the elements of A are real, then its eigenvalues are eitherrealorincomplexconjugatepairs.2. If 2, (i = l,2,.".,n) are the eigenvalues of A, thentr(A)=≥,(1.103)i-l3. If , (i=1,2,...,n) are the eigenvalues of A, then they are theeigenvalues of AT .4. If A is nonsingular, with eigenvalues a, (i = l,2,..,n) , then,1aretheeigenvalues of A-l元
Properties about eigenvalue 1. If the elements of A are real, then its eigenvalues are either real or in complex conjugate pairs. 2. If are the eigenvalues of A, then (i 1,2, ,n) i = = = n i i 1 tr(A) (1.103) 4. If A is nonsingular, with eigenvalues , then, (i 1,2, ,n) i = 1 i are the eigenvalues of 1 A − 3. If are the eigenvalues of A, then they are the eigenvalues of . (i 1,2, ,n) i = T A
Any nonzero vector V which satisfies the matrix equation(1.104)(2,I - A)V, = 0iscalledtheeigenvector（特征向量）ofAassociatedwitheigenvalues,(i =1,2,.,n)注：特征向量常用于状态变换（statetransformation）的计算。自学：广义特征向量的计算如果A只含有单根，可直接由Eq.(1.104)求出对应的特征向量如果A有重根且非对称，则由Eq.(1.105)求出所有的广义特征向量(AI-A)V1 = 0(a, I - A)Vi2 = -Vi1( I - A)Vi3 = -Vi2(1.105).(21 - A)Vim = -V(m-)
Any nonzero vector Vi which satisfies the matrix equation (i I − A)Vi = 0 (1.104) is called the eigenvector (特征向量) of A associated with eigenvalues (i 1,2, ,n) i = 注：特征向量常用于状态变换（ state transformation ）的计算。 如果 A只含有单根，可直接由Eq.(1.104)求出对应的特征向量. 如果 A有重根且非对称，则由Eq.(1.105)求出所有的广义特征向量. 自学：广义特征向量的计算 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)
1.5.2StateTransformation（状态变换）Consider theLTl system described by (1.107)X(t) = AX(t) + Bu(t)(1.107)y(t) = CX(t) + Du(t)注：由于状态X(t)的选取不唯一（not unique），那么它所对应的状态空间描述也不唯一。linearstatetransformationX→XDefineanXnnonsingular（非奇异）matrixP，andletX(t) = PX(t)(1.108)X(t) = P-IX(t)(1.109)Eq.(1.108) or Eq.(1.109) is called the linear nonsingular statetransformation（线性非奇异状态变换）：
1.5.2 State Transformation（状态变换） Consider the LTI system described by (1.107) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t y CX Du X AX Bu = + = + (1.107) 注：由于状态X(t)的选取不唯一（not unique），那么它所对应的状态空间 描述也不唯一。 X X → linear state transformation Define a n×n nonsingular （非奇异） matrix P, and let X(t) = PX(t) ( ) ( ) 1 X t P X t − = (1.108) (1.109) Eq.(1.108) or Eq.(1.109) is called the linear nonsingular state transformation （线性非奇异状态变换）