The reconstructed system can be described by =(A-Le)X+Ly+bu j=cx where, is the estimate of the actual state X, and the feedback gain vector L=will be chosen to achieve satisfactory error characteristics. Obviously,the dimension of the closed-loop observer(5.33)is equal to it of the estimatedoriginal system(5.9),so the observer(5.33)is calleda full-order observer
The reconstructed system can be described by ˆ ˆ ( ) ˆ ˆ y u y = − + + = X A Lc X L b cX
Define the error:= When lim=lim=0 the estimates can be used in place of the actual state variables. ↓两边微分 及=疗-=(A-Lc)8 ↓求解 X=e(A-L(t)=e(A-L[X(to)-X(to)] ↓目标：误差为零 choose L so that A-LC has asymptotically stable and reasonably fast eigenvalues independent onthe input(and the initial-conditionX(. ↓闭环观测器存在性条件 Under what-conditioncantheigenvalues of A-Lebe placedarbitrarily?
Define the error： X X X = − ˆ When the estimates can be used in place of the actual state variables. ˆ lim lim 0 t t → → X X X = − = 求解 ( ) ( ) 0 0 0 ˆ ( ) [ ( ) ( )] t t e t e t t − − = = − A Lc A Lc X X X X 两边微分 ˆ X X X A Lc X = − = − ( ) 目标：误差为零 choose L so that A-LC has asymptotically stable and reasonably fast eigenvalues 闭环观测器存在性条件
5.1.2 Design of the Full-Order State Observer Theorem5.3The eigenvalues of (A-Le)can be placed arbitrarily iff the estimated originalsystem(5.9)isobservable. Proof.Consider a SISO-LTI-system,whichis-completelyobservable,such as-(5.9) The dual -system of it can be described by. Z(0=AZ(0+cv() (5.37) w(t)=b'Z(t) Since the system(5.9)is completelyobservable,then its'dual system(5.37)is completely controllable,and the eigenvalues of it can be placed arbitrarily by introducing the state feedback control. 7t)=-KZ(t)+v(t) (5.38)
5.1.2 Design of the Full-Order State Observer