回 顾 结论1：状态反馈控制性质定理 Theorem 5.1 The closed-loop system (5.4)and (5.6)with the state feedback control is completely controllable iff the original open-loop system (5.1)is completely controllable. 状态反馈控制的引入，不改变系统的能控性。 但是，状态反馈控制的引入可能改变系统的能观测性。 结论2：极点配置定理-极点可配置条件 Theorem5.2The polesorienvaluesof theclosed-loopsystem by-introducing the statefeedbackcontrolcanbe placed arbitrarily iff theoriginal-open-loopsystemis completely controllable. 结论：线性定常闭环系统可通过状态反馈任意配置其全部极 点的充分必要条件是：原开环系统为完全能控的
Theorem 5.1 The closed-loop system (5.4) and (5.6) with the state feedback control is completely controllable iff the original open-loop system (5.1) is completely controllable. 状态反馈控制的引入，不改变系统的能控性。 但是，状态反馈控制的引入可能改变系统的能观测性。 结论 ：线性定常闭环系统可通过状态反馈任意配置其全部极 点的充分必要条件是：原开环系统为完全能控的。 结论1：状态反馈控制性质定理 结论2：极点配置定理-极点可配置条件 回 顾
5.2 Design of the State Observer 引言 Q:在什么条件下，才可以使用状态反馈控制改善系统的性能？ A: 系统状态是可观测或可量测的条件下。 事实上，不是所有的系统状 用已知的输入输出构造所 态都是可观测或可量测的。 有状态的观测器： 状态观测器、状态估计器 状态反馈控制不可用 得到全部的系统状态 ↓ 状态反馈控制可用
5.2 Design of the State Observer 引 言 Q：在什么条件下，才可以使用状态反馈控制改善系统的性能？ A：系统状态是可观测或可量测的条件下。 事实上，不是所有的系统状 态都是可观测或可量测的。 状态反馈控制不可用 用已知的输入输出构造所 有状态的观测器： 状态观测器、状态估计器 得到全部的系统状态 状态反馈控制可用
5.2.1Full-Order State Observer全阶状态观测器 0可80□學日9 X=AX+bu y=cX 复制 交=AN+bu =c8 open-loop observer Figure 5.4 Open-Loop Observer is the estimate of the actual state X Define the error:=-
5.2.1 Full-Order State Observer 全阶状态观测器 Figure 5.4 Open-Loop Observer ˆ ˆ ˆ ˆ u y = + = X AX b cX Define the error： X X X = − ˆ 复制
Define the error:=X When lim=lim=0 the estimates can be used in place of the actual state variables. ↓两边微分 京=水-氵=A(X-X)=A成 ↓求解得零输入响应 =e"x)=e“[Xt)-X(i】 ↓目标：误差为零 X(t)-X(t)=0 However,the information about X(t)is unknown. 又 Hence,the open-loop observer is not practical
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 = − = 求解得零输入响应 0 0 0 ˆ ( ) [ ( ) ( )] t t = = − e t e t t A A X X X X ˆ ˆ X X X A X X AX = − = − = ( ) 两边微分 目标：误差为零 =0 Hence, the open-loop observer is not practical
The closed-loop form observer u(t) !反馈 (t) !复制 A closed-loop observer 量测输出y(t)和估计输出()之间的修正量当做反馈量
The closed-loop form observer 量测输出y(t)和估计输出 之间的修正量当做反馈量 反馈 复制
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
Furthermore,the closed-loop system by introducing the state feedback control- (5.38)into(5.37)can be described by.引入状态反馈后，对偶系统的闭环表达式 Z(t)=(A"-c"K)Z(t)+c"n(t) ·.··.(5.39) w(t)=b"Z(t) It means the eigenvalues of(-K)can-be placed arbitrarily at the expected locationss,S2,.,Sn Moreover,since. 令L=[l4.1]-KT sI-(AT-c"K)=sI-(A-K"c). then,itcan beobtained that the eigenvalues of (A-Le)can also be placed arbitrarily at theexpectedlocations The procedure above iscalled-the poles placement forthe full-orderobserver, andis the expectedpoles of the observer
T 1 2 T n 令 L K = = l l l 引入状态反馈后，对偶系统的闭环表达式
The poles placement for the full-order observer: The original SISO LTI system is completely observable The dual system of it is completely controllable introducing the state feedback control 7(t)=-KZ(t)+v(t) Z(t)=(AT-c"K)Z(t)+c"n(t) w(t)=b'Z(t) 该闭环系统是完全能控的 It means the eigenvalues of (Kcan be placed arbitrarily at the expected locations 因为：sI-(AT-cTK)=sI-(A-KTc) ↓令L=[l.]=KT The eigenvalues of (A-Lc)can also be placed arbitrarily at the expected locations
The original SISO LTI system is completely observable The dual system of it is completely controllable introducing the state feedback control ( ) ( ) ( ) t t v t = − + KZ T T T T ( ) ( ) ( ) ( ) ( ) ( ) t t t w t t = − + = Z A c K Z c b Z 该闭环系统是完全能控的 T T T s s I A c K I A K c − − = − − ( ) ( ) T 1 2 T n 令： L K = = l l l The eigenvalues of can also be placed arbitrarily at the expected locations ( ) A Lc − The poles placement for the full-order observer： 因为：