compensator4 - From the necessary and sufficient conditions...

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Compensators Q UESTION 4 For the closed–loop system in Figure 4 with and , complete the following: a. Sketch the Root Locus Diagram. Qualitatively describe the closed–loop system stability and transient characteristics for 0 < K < ∞. b. Calculate K such that the steady–state error for a unit step input is 0.15. c. For the value of K calculated in part b, simulate the closed–loop system for r ( t ) = 1 and r ( t ) = t . For both simulations plot y ( t ), e ( t ), and u ( t ). d. For the value of K calculated in part b, calculate the closed–loop pole locations. For each complex conjugate pair, determine its natural frequency and damping ratio. For each real pole, determine its time constant. Turn in your Matlab code. G(s) KG C (s) U(s) R(s) + - Y(s) E(s) Figure 4 There is m = 1 finite open–loop zero at –5 There are n = 2 finite open–loop poles located at There are n = 2 branches and n m = 1 branch that goes to infinity The Root Locus is on the real axis between –5 and –∞ The closed–loop characteristic equation is
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Unformatted text preview: From the necessary and sufficient conditions for second order systems, all of the coefficients must be positive, which is true for K &gt; 0.2. Since only positive values for K are considered, the closedloop system is stable for K &gt; 0. Compensators The closedloop system is underdamped for small values of K and becomes overdamped when the two branches break into the real axis. The 2% settling time decreases as K increases until the branches break into the real axis, and then increases and approaches 4/5 s as K approaches . For r ( t ) = 1, The closedloop poles are located at The damping ratio and natural frequency, respectively, of the poles at are and The Root Locus Diagram and simulation plots are given below. 5 10 0.5 1 time (s) r y 5 10 5 10 time (s) r y 5 10 0.5 1 time (s) e(t) 5 10 1 2 Compensators-15-10-5-4-3-2-1 1 2 3 4 Real Axis Imaginary Axis...
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compensator4 - From the necessary and sufficient conditions...

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