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# lecture_24 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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+ - R ( s ) C ( s ) G ( s ) G ( s ) c o n t r o l l e r p l a n t c p + - R ( s ) C ( s ) G ( s ) G ( s ) c p Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 24 1 Reading: Nise: Chapter 7 1 The Poles and Zeros of Closed-loop systems: Consider the unity feedback system shown below with a controller G c ( s ) and plant G p ( s ): Combine the two cascaded blocks to form a single forward transfer function G f ( s ) = G c ( s ) G p ( s ) and write N f ( s ) G f ( s ) = D f ( s ) in terms of the numerator polynomial N f ( s ) and denominator polynomial D f ( s ). The closed- loop transfer function is G f ( s ) N f ( s ) G cl ( s ) = = 1 + G f ( s ) D f ( s ) + N f ( s ) from which we see that 1 copyright c D.Rowell 2008 24–1
0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Step Response Time (sec) Amplitude The closed-loop poles are the roots of the characteristic equation N f ( s ) + D f ( s ) = 0. The closed-loop zeros are the same as the zeros of the forward transfer function. Example 1 Find the closed-loop transfer function of the plant G p ( s ) = 3 / ( s + 3) under P-D control where G c = 10 + 2 s . The forward transfer function is 6(5 + s ) G f ( s ) = G c ( s ) G p = s + 3 The closed-loop transfer function is: N f ( s ) 6(5 + s ) 6( s + 5) 6 ± s + 5 G cl ( s ) = = = = D f ( s ) + N f ( s ) ( s + 3) + 6(5 + s ) (7 s + 33) 7 s + 33 / 7 so that the closed -loop pole is at s = 33 / 7 = 4 . 7143 and the closed-loop zero is at s = 5 (the same as the open loop zero deﬁned by the P-D controller). Aside: The system can be analyzed using the following MATLAB commands: forward_system = zpk(-5, -3, 6) closed_loop =feedback(forward_system,1) pole(closed_loop) %Find the closed-loop system poles zero(closed_loop) %Find the closed-loop system system zeros pzmap(closed_loop) %Make a pole-zero plot step(closed_loop) %Plot the closed-loop step response. 24–2

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lecture_24 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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