Lesson+13

# Lesson+13 - 1 Lesson 13 Lesson 13 Poles and Zeros Ch 12.8-9...

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Lesson 13 1

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Poles and Zeros Ch 12.8-9 Lesson 13 Statement: This lesson will be an expansion of the author’s limited study of poles and zeros. Material not found in the textbook will not appear on an examination. Lesson 13 Fast and Furious 2 Lesson 13
Challenge 12 Find x(t) if X(s)=(s+2)/(s(s+1) 2 ) (strictly proper) Heaviside expansion: X(s) = A/s + B/(s+1) + C/(s+1) 2 2 2 1 ) ( 1 1 2 ) ( 1 2 1 2 ) ( 1 2 1 2 1 1 2 0 2 0 s s s s s s s s s ds s X s d B s s s X s C s s s sX A Old School 3 Lesson 13

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» num=[1,2]; den=[1,2,1,0]; » [R,P,K]=residue(num,den) R = -2 (B: p = -1) -1 (C: p = -1) 2 (A: p = 0) P = -1 -1 0 K = [] K absent because X(s) is strictly proper. 4 "Lesson 13
Interpret the Heaviside expansion: X(s) = 2/s -2/(s+1) -1/(s+1) 2 ) ( 1 2 2 t u te e t x t t Initial value x(t) = 0 and sX(s)| s  =s(s+2)/(s(s+1) 2 ) | s  =0 Final value x( ) = 2 and sX(s)| s 0 =s(s+2)/(s(s+1) 2 ) | s 0 = (s+2)/((s+1) 2 ) | s 0 = 2 5 Lesson 13

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Lesson 13 Linear System ODEs x(t) X(s) y(t) Y(s) Linear System (system at-rest) Y(s)/X(s)=H(s) N i i N i i N i i i N i i i p s z s K s a s b s H 1 1 0 0 ) ( z i called zeros p i called poled H(s) is called monic if a 0 = 1. ICs=0 System-level view. ICs=0 6 Lesson 13 or
Feedback Structures Historically, continuous-time systems have been extensively used in circuit, automatic control and communication applications ranging from disk head controllers, to manned space vehicles, to cell phones. Types of controllers. G(s) Open loop G(s) Closed loop - 7 Lesson 13

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General forms of closed loop systems. ) ( ) ( 1 ) ( ) ( 2 1 1 s G s G s G s H G 1 (s) - G 2 (s) KG 1 (s) - ) ( 1 ) ( ) ( 1 1 s KG s KG s H G 1 (s) - ) ( 1 ) ( ) ( 1 1 s KG s G s H K 8 Lesson 13
Example: K = feedforward gain H 1 (s)=1/(s 2 +2s+1) E(s) = error in s-domain ) ( ) ( ) ( ) ( 1 2 1 2 1 1 2 1 2 2 2 1 1 K s s K s s K s s K s KH s KH s H MATLAB’s rlocus routine computes the poles locations of H(s) for K [0, ) H 1 (s) K Y(s) E(s) - X(s) 9 Lesson 13

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1 2 1 2 1 s s s H ) ( » sys=tf([1],[1 2 1]); » sys Transfer function: 1 ------------- (H 1 (s)) s^2 + 2 s + 1 » [R,K]=rlocus(sys); (Closed loop poles) R = - 1.0000 -1.0000 + 0.2236i -1.0000 + 0.3303i -1.0000 + 0.4880i -1.0000 -1.0000 - 0.2236i -1.0000 - 0.3303i -1.0000 - 0.4880i -1.0000 + 0.7209i - 1.0000 + 1.0649i -1.0000 + 1.5732i Inf -1.0000 - 0.7209i -1.0000 - 1.0649i -1.0000 - 1.5732i Inf K = 0 0.0500 0.1091 0.2381 0.5197 1.1341 2.4750 Inf Poles K H 1 (s) K Y(s) E(s) - X(s) 10 Lesson 13 (open loop gain)
) ( ) ( 1 2 2 K s s K s H Display pole zero locations » a1=real(R(1,:)); » a2=real(R(2,:)); » a=[a1,a2]; » b1=imag(R(1,:)); » b2=imag(R(2,:)); » b=[b1,b2]; » stem(a,b) 11 Lesson 13 (closed loop gain)

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-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 j -j K=0 K= 2.5 1.1 0.5 0.2 0.1 0.05 [2] [0,0] Step response K=0.5. Time (sec.) Amplitude Step Response 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 From: U(1) To: Y(1) 12 Lesson 13
PID Controllers Positional, integral, differential ( PID ) controllers The performance of the system is shaped by the gains K i .

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• Fall '07
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