MEM255Su06-07.12.ssRealizatn_studs

MEM255Su06-07.12.ssRealizatn_studs - MEM255 Introduction 2...

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Unformatted text preview: MEM255 Introduction 2 Control (Realizations) Dr. Ajmal Yousuff Dept. MEM Drexel University overview Non-uniqueness of realizations through analog diagrams Canonical realizations Recall... Example U(s) b2 s 2 b1s b0 1s 3 a2 s 2 a1s a0 Y(s) 0 x y 0 a0 b0 1 0 a1 0 1 a2 x 0 0 u 1 b1 b2 x [0]u Yousuff MEM255: Intro2Control 3 An analog simulation a2 a1 y a0 y y y b2u b1u b0u y b2u a2 b1u a1 y b0u a0 y y y u (b2u a2 y (b1u a1 y (b0u a0 y )dt )dt )dt b0 1 x3 s b1 1 x2 s b2 1 x1 s y a2 a1 a0 Observer canonical form u b0 1 x3 s b1 1 x2 s b2 1 x1 s y a2 a1 a0 x1 x2 x3 y a2 a1 a0 1 0 0 1 x 0 0 b2 b1 u b0 1 0 0 x Another simulation diagram a2 a1 y a0 y y y b2u b1u b0u y x1 (b2 s 2 b1s b0 ) 1 s 3 a2 s 2 a1s a0 u x1 u a0 x1 a1 x1 a2 1 ; y x b2 1 b1 x1 b0 x1 x y b2 u b1 1 s b0 1 s 1 x1 s a2 a1 1 x x1 x1 a0 Controller (phase) canonical form y b2 u 1 s b1 1 s b0 1 x1 s x3 x2 a2 a1 a0 x1 x2 x3 y 0 0 a0 b0 1 0 a1 b1 b2 x 0 1 a2 x 0 0 u 1 Jordan canonical form y (s 2 1 u 3s 2) 1 s 1 u 1 u s 2 y y1 y2 u 1 y1 y x1 x2 y 1 0 1 0 x 2 1 x 1 u 1 y2 2 Example #2 x y 3 1 x 2 0 3 2 x ( A) {1, 2} y ( s) u ( s) C (sI A) 1 B 1 s 1 Pole-zero cancellation 1 u 2 Obtain a state space representation of: y( s) u ( s) s 2 s 2 3s 2 ( s 2) ( s 1)( s 2) poles of y ( s) {1} u ( s) What happened to the eigenvalue {2} ? Yousuff MEM 255 Controls 9 Example #3 x y 1 1 1 0 x 1 2 x ( A) {1, 1} y ( s) u ( s) C (sI A) 1 B 1 s 1 Pole-zero cancellation 1 u 0 Obtain a state space representation of: y( s) u ( s) s 1 s2 1 (s 1) ( s 1)(s 1) poles of y ( s) { 1} u ( s) What happened to the eigenvalue {1} ? Yousuff MEM 255 Controls 10 #2 Examples, contd. 2 #3 2 for 1 1 1 and 1 , 2 2 2 are : for 1 1 and 2 , 1 2 1 are : 0 1 [ 1 2 1 1 [ 1 2 1 Transform : x Tq; T Transform : x Tq; T q y 1 0 q 0 2 1 1q 1 u 0 q y 1 0 0 0 q 1 2 q 1/ 2 u 1/ 2 u(t) does not affect q2(t) q2(t) has no effect on y(t) MEM 255 Controls 11 Yousuff "steering" states t x(t ) e x0 Initial condition response t At e Bu ( )d 0 Input response A( t ) e A( t 0 ) Bu ( )d x(t ) e At x0 Reflects the control effort required to steer the states from x0 to x(t). Can this be achieved for arbitrary x0 and x(t) in a finite time? Controllable system The system (A,B) is controllable if and only if there exists a (piecewise continuous) control signal u(t) that will take the state from any initial state x0 to any desired final state xf in a finite time interval. Controllability Matrix The Controllability Matrix associated with (A,B) is C B AB A2 B An 1B The system is controllable if and only if Rank(C ) = n. (Proof omitted.) Controllability is a notion in state space, and not seen in s-domain approach. Yousuff MEM 255 Controls 14 The omitted "proof" set {x0 0, u 0, t dt} and consider x Ax Bu. Range of B x(dt ) x(2dt ) Ax0 dt Bu0 dt Ax(dt )dt Bu0 dt x (dt ) B x(2dt ) Range B AB AB An 1B ABu0 dtdt x(t ) B + AB ... + An 1B ...
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This note was uploaded on 06/03/2008 for the course MEM 255 taught by Professor Yousuff during the Spring '08 term at Drexel.

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