State Space Analysis and Controller Design Lecture Overview
Explore the key concepts of controller design in state space, controllability, observability, state feedback control, Linear Quadratic Regulator (LQR), and more. Understand the principles of feedback controlled systems, state feedback models, and equations, with insights into RL circuit dynamics and pole placement methods.
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EEE3001 EEE8013 State Space Analysis and Controller Design This lecture will be recorded and you will be able to download it Dr Damian Giaouris http://www.staff.ncl.ac.uk/damian.giaouris/
Goals/Aims of Chapter 4 Controller design in state space Controllability/Observability State feedback control (pole placement) Linear Quadratic Regulator (LQR) Estimator design in state space Open loop estimator Closed loop estimator Summary
Feedback Controlled Systems Open Loop Transfer Function (OLTF): U Y G(s) Closed Loop Transfer Function (CLTF): OLTF + Feedback of the output U Y R Gc(s) G(s) U U U Y Y Y R R R R1 R1 R1 Gc(s) K K1 G(s) G(s) G(s) Gc(s) K K
State Feedback State Space Models Open Loop Transfer Function (OLTF): dt U Y u Dx x y G(s) B C A Closed Loop Transfer Function (CLTF): OLTF + Feedback of the State U r Y R Gc(s) G(s) u Dx x y dt Controller B C x A
State Feedback State Space Models U Y R R1 K1 G(s) K x y + Bu+ x r u dt K1 B C + + A Ax Plant -K
State Feedback - Equations The task of the controller is to produce the appropriate control signal u that will insure that y=r. x y + Bu+ x r u dt K1 B C + + A Ax ( ) ( ) ( ) = u t K r t 1 Kx t Plant ( ) ( ) x t ( ) ( ) ( ) ( ) = = u t K r t 1 Kx t -K + Ax t Bu t WE MUST CHECK IF THE SYSTEM ( ) ) ( ) ( ) ( ) y t ( ) ( ) IS CONTROLLABLE. ( ) = ( ) DK x t ( ) ( x t DK r t + ) ( ) ( ) = + = + x t Ax t B K r t Kx t A BK x t BK r t 1 1 ( ) ( ( ) = + Cx t Du t C 1 ( ) ( ) y t ( ) ( ) ( ) ( ) = = + + x t A x t B r t CL CL C x t D u t CL CL K? : ACL faster/stable. This method is called pole placement. = = = = A B C D A BK C DK BK CL 1 CL DK CL 1 CL
RL Circut L R 1 1 di di R ( ) = + x Ax Bu = = + V iR i V dt L dt L L I A=-R/L, x=i, B=1/L and u=V V Obviously the eigenvalue is R/L (assume here 0.5 and u=0) 1 0.8 0.6 x(t) 0.4 0.2 0 0 2 4 6 8 10 time, s
RL Circut L R 1 1 di di R ( ) = + x Ax Bu = = + V iR i V dt L dt L L I A=-R/L, x=i, B=1/L and u=-Kx=-Ki V R L K L R K = = = A A BK CL L R K R Choose to place the eigenvalue at -6R/L = = 6 5 K R L L u=V=-5Ri 0.5 Open loop Closed Loop 0.4 0.3 x 0.2 0.1 0 0 2 4 6 8 10 12 Time, s
Unstable System ( )x = 3 Hence the eigenvalue of the CL system is 3-k = 3 + = x k x x u u kx k=13 Hence the eigenvalue of the CL system is -10 (stable) 2 Open loop Closed Loop 1.5 1 x 0.5 0 0 0.2 0.4 0.6 0.8 1 Time, s
LQR 0 1 3 2 4 1 1 0 0 ( ) ( ) = = 26 = + = = + , 72 ( ) x t ( ) x t , ( ) y t ( ) x t u Kx K u t Ix x x dt 1 2 1 2 4 x1 x2 1 |x1| 2 0 0 0 0.2 0.4 0.6 0.8 1 x1, x2 time, s 2 -1 |x2| 1 -2 0 0 0.2 0.4 0.6 0.8 1 -3 time, s 0 0.2 0.4 0.6 0.8 1 time, s 1 0.9383 0.8 0.6 Ix 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 time, s
LQR ( ) 0 1 3 2 4 1 1 0 0 ( ) = + 2 2 xI x x dt = = 26 = + = , 72 ( ) x t ( ) x t , ( ) y t ( ) x t u Kx K u t 1 2 1 2 2 x1 x2 1 1 x1 2 0 0 0 0.2 0.4 0.6 0.8 1 x1, x2 time, s -1 2 1 x2 2 -2 0 0 0.2 0.4 0.6 0.8 1 -3 time, s 0 0.2 0.4 0.6 0.8 1 time, s 1.5 Ix=1.403 1 Ix 0.5 0 0 0.2 0.4 0.6 0.8 1 time, s
LQR Poles at -10, -11 2 1.5 x1 x2 Ix=1.403 1 1 0 x1, x2 Ix -1 0.5 -2 0 0 0.2 0.4 0.6 0.8 1 time, s -3 0 0.2 0.4 0.6 0.8 1 time, s Poles at -15, -20 1.5 5 Ix=1.271 0 x1 1 -5 0 0.2 0.4 0.6 0.8 1 Ix time, s 0.5 2 1 x2 0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 time, s time, s
LQR 50 0 case 1 case 2 -50 u=-Kx -100 -150 -200 0 0.2 0.4 0.6 0.8 1 time, s
LQR 2 = Iu u dt 600 500 Case 1 Case 2 400 540 300 Iu 200 230.5 100 0 0 0.2 0.4 0.6 0.8 1 time, s
LQR ( ) t Qx t ( ) ( ) t Ru t dt ( ) = + T T I x u + + = 1 T T Reduced Riccati Equation 0 A P PA PBR B P Q = 1 T K R B P
Estimating techniques y Bu+ x x u dt B C + A Ax ( ) e t Ax t ( ) x t Bu t + ( ) x t ( ) e t ( ) x t Bu t ( ) x t = = Plant ( ) ( ) ( ) ( ) A t x t ( ) ( ) e t ( ) = Ae t y Bu+ dt B C x + x Ax A Estimator
Estimating techniques x y Bu+ x u dt B C + y + y A - Ax Plant G - Bu+ y x x dt B C + Ax A Estimator ( ) ( e t ) ( ) ( ) ( ) ( ) ( ) = A GC x t A GC x t = A GC e t But the system must be observable
Estimating techniques = = + + ( ) ( ) y t ( ) ( ) ( ) ( ) x t Ax t Cx t Bu t Du t x y Bu+ x u dt B C + y + y ( ) ( ) A - = u t Kx t Ax ( ) ( = = ( ) x t ( ) Ax t BKx t Plant G ) ( ( ) ) ( ) x t + ( ) ( ) A BK x t BK x t - Bu+ y x x dt ( ) B C + ( ) A BK x t BKe t + Ax ( ) ( e t ) = A GC e A Estimator -K ( ) ( ) e t ( ) ( ) e t x t x t A BK 0 BK ( ) sI A 0 BK BK A GC ( ) = = 0 A GC = ( ) 0 sI A BK sI ( ) A GC sI This means that the pole placement design and the estimator design are independent of each other. Then they can be designed separately and combined to form the observed-state feedback control system.
Estimation and Tracking x y + Bu+ x rss u dt K1 B C + + y + y A - Ax Plant G + Bu+ y x x dt B C + Ax A Estimator -K 1 N N 0 1 A C + B 0 x = u = K u N KN Kx 1 u x = K r = 1 ss A GC e ( ) ( e t )
