che446_10_hw7

# che446_10_hw7 - G c s using a ﬁrst-order ﬁlter f s = 1...

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Homework #7 ChE 446 Fall 2010 Problem 1. Consider the following transfer function model: G ( s ) = s - 1 ( s + 1) 2 ( s + 2) 1. Use direct substitution to determine the range of controller gains ( K c ) that yield a stable closed-loop system. 2. Use the Ziegler-Nichols rules for a 1/4 decay ratio to determine PID controller parameters. Explain the sign of the controller gain ( K c ). 3. Design the IMC controller G * c ( s ). The ﬁlter f ( s ) should by chosen as a ﬁrst-order system with unity gain and time constant τ c . Is this a PID controller? Does the controller have integral action? 4. Use G * c ( s ) to ﬁnd the equivalent feedback controller G c ( s ). Is this a PID controller? Does the controller have integral action? Problem 2. Consider the following transfer function: Y ( s ) U ( s ) = G ( s ) = s - 1 ( s + 2)( s + 3) e - 2 s 1. Consider a proportional controller G c ( s ) = K c . Use the Routh test to determine the range of controller gains which yield a stable closed-loop system. 2. Derive the IMC controller

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Unformatted text preview: G * c ( s ) using a ﬁrst-order ﬁlter f ( s ) = 1 τ c s +1 . Specify the closed-loop transfer function Y ( s ) R ( s ) . Derive the closed-loop response y ( t ) for a unit step change in the setpoint r ( t ) when τ c = 2. Problem 3. Consider the following state-space system: dx dt = ±-1 4 1-1 ² x + ± 1-1 ² u = Ax + Bu y = ( 1 0 ) u = Cx 1. Compute the eigenvalues of the system and determine if the system is stable. Use the controllability and observability matrices to determine if the system is controllable and/or observable. 1 2. Compute the state feedback controller gains k 1 and k 2 such that the closed-loop character-istic equation is equal to ( λ + 3) 2 = 0. 3. Compute the observer gains l 1 and l 2 such that the characteristic equation of the observer error dynamics is equal to ( λ + 10) 2 = 0. 2...
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che446_10_hw7 - G c s using a ﬁrst-order ﬁlter f s = 1...

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