The transfer-function of the closed-loop system is G(s)=(c-d)(sI-A+bK)b+d···(5.5) ↓ If.G(s)is strictly proper,i.e.d=0,we have the closed-loop system as. X(t)=(A-bK)X(t)+br(t) 6 y(t)=cX(t) the transfer function of(5.6)is. Gx(S)=c(sI-A+bK)b·. (5.7) In the following discussion,we mainly consider the case,d=0
5.1.2 Controllability and Observability of the Closed-Loop System 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. In other words,the controllability can be preserved by introducing the state feedback control into the open-loop system.However,the observability may be changed by introducing the state feedback control into the open-loop system. 状态反馈控制的引入，不改变系统的能控性。 值得注意的是，它可能改变系统的能观测性
5.1.2 Controllability and Observability of the Closed-Loop System 状态反馈控制的引入，不改变系统的能控性。 值得注意的是，它可能改变系统的能观测性。 Theorem 5.1 The closed-loop system (5.4) and (5.6) with the state feedback control is completely controllable iff the original openloop system (5.1) is completely controllable. In other words, the controllability can be preserved by introducing the state feedback control into the open-loop system. However, the observability may be changed by introducing the state feedback control into the open-loop system
Proof.The controllability matrix of the open-loop system(5.1)is. Q=「b Ab Aib.Am-b Furthermore,the controllability matrix of the closed-loop system(5.4)and(5.6) is =b (A-bK)b (A-bK)2b .(A-bK)"67 [1-Kb-K(A-bK)b.* 0 1 -Kb =[b Ab A'b.A-6] 0 0 1 0 0 Obviousyrak。=ramk2,”状态反愤控制的引入，不改变系统的能控性。 It means that the controllability can be preserved by introducing the state feedback control into the open-loop system
状态反馈控制的引入，不改变系统的能控性
Suppose the eigenvalues of the. closed-loop system are (4-bK),i=1,2,.,n.They are different fromthe eigenvalues(4)of the original open-loop system.So,introducing the state feedback control into the open-loop system may yield the pole-zeros cancellation(零极点对消). 定理4.19SS0LT1系统是完全能控完全能观的充分必要条件是它 的传递函数不存在零极点对消。 Itmeansthat,iftherialopen-loopsystemisompletelycontrollableand completelyobservable,theclosed-loopsystemwith the"state feedbackcontrol must be completely controllable,but may beunobservable
定理4.19 SISO LTI系统是完全能控完全能观的充分必要条件是它 的传递函数不存在零极点对消
5.l.3 Poles Placement(极点配置)by State Feedback Control 问题的提法 已知：x=Ax+bu,x(0)=xo,t≥0 y=Cx 性能指标：期望闭环极点，入，.，入 要求：构造u=-Kx+r,（即求K),使满足 2(A-BK)=2,i=1,2,.,n 任务：什么条件下可任意配置闭环极点，如何配置？
已知: 性能指标: 期望闭环极点 要求: 构造u=-Kx+r,(即求K),使满足 任务:什么条件下可任意配置闭环极点，如何配置？ 0 x Ax bu x x t , (0) , 0 y cx = + = = ➢ 问题的提法 * * * 1 2 , , , n * ( ) , 1,2, , i i A BK i n − = = 5.1.3 Poles Placement （极点配置）by State Feedback Control