Chapter 7 2 Chapter 7 State Feedback and State Estimator Objectives: State feedback and design of feedback matrix .Design of state estimator Feedback control with estimated states Theory
Chapter 7 State Feedback and State Estimator Objectives: • State feedback and design of feedback matrix • Design of state estimator • Feedback control with estimated states Chapter 7 2
Chapter 7 3 7.1 State Feedback Output feedback control system Desired output response Comparison Controller Process ◆Oulput Measurement Desired (O加lpl oulpul Controller Process variables D Mersurement
7.1 State Feedback • Output feedback control system Chapter 7 3
Chapter 7 4 Output feedback system 文=Ax+Bu Desired oulput response Comparison Controller Process Oulput y=Cx+Du Measurement let u=V-Hy =Ax+B(V-Hy)=[A-BH(I+DH)C]x+[B-BH(I+DH)DWV y=(I+DH)Cx+(I+DH)DV H
Output feedback system 4 let u V Hy y Cx Du x Ax Bu x Ax B(V Hy) [A BH(I DH) C]x [B BH(I DH) D]V 1 1 y I DH Cx I DH DV 1 1 ( ) ( ) Chapter 7
Chapter 7 ◇ State feedback system A linear system: =Ax+Bu y=Cx+Du Kx-State feedback Let:u=V-Kc→ -input of system u-output of controller/input of plant Then =Ax+B(V-Kx)=(A-BK)x+BV y=(C-DK)x+DV
State feedback system • A linear system: • Then 5 y Cx Du x Ax Bu Let: u V Kx y (C DK)x DV x Ax B(V Kx) (A BK)x BV Kx - State feedback V – input of system u – output of controller/input of plant Chapter 7
Chapter 7 6 7.2 Controllability of Feedback system Linear system: 文=Ax+BM y=Cx State feedback Feedback system=(A-BKx+Br y=Cx Theorem:The system(A-BK,B)is controllable IFF system (A,B)is controllable. Note:The observability may be changed
7.2 Controllability of Feedback system • Theorem: The system (A-BK, B) is controllable IFF system (A,B) is controllable. • Note: The observability may be changed. 6 Linear system: y Cx x Ax Bu State feedback n i n ki xi u r Kx r- k k x r 1 1 y Cx Feedback system x (A BK)x Br Chapter 7