Unformatted text preview: MEEN 651 Control System Design Homework 11: LQR and the Symmetric Root Locus Solutions
Assigned: Tuesday, 11 Nov. 2008 Due: Tuesday, 18 Nov. 2008, 5:00 pm Textbook Problems Do the following problem from "Feedback Control of Dynamic Systems" 5th ed., Franklin and Powell. Problem 1) Problem 7.29 Problem 2) Problem 7.33 Problem 3) Problem 7.47 Do parts a)  h) and also use Simulink to simulate the response of the controlled helicopter to: i) a nonzero initial condition ii) a step change in reference horizontal velocity (how will you include this in the state feedback/estimator design?) Problem 4) For the double integrator , what is the minimal value of the cost: over all possible controls , when 0 1 and 0 0? Problem 5) Consider the circuit below, and determine: a) For what values of R, L, and C is the system stable? b) For what values of R, L, and C is the system completely controllable? c) For what values of R, L, and C is the system completely observable? 2 1 1 0 1 1 Therefore the system is stable positive values of R, C, and L. 0 if Therefore the system is completely controllable if 0 if Therefore the system is completely observable if C is not infinite. Problem 6) Given the system 1 1 0 7 0 1 1 0 Use the symmetric root locus to determine the value of r that results in the fastest response in y, but without overshoot. The transfer function is given by: 1 The symmetric root locus is given by 1 Thus 1 1 7 1 0 0 1 1 1 7 7 0 1 The root locus appears as: 1 Root Locus 15 10 5 Imaginary Axis 0 5 10 15 15 10 5 0 Real Axis 5 10 15 With the poles going towards infinity along asymptotes centered at 0, with angles of 45, 135, 225, and 315. The breakaway points are given as 2. Thus the fastest closed loop poles without oscillations are at the breakaway point where Problem 7) Consider the system described by with 0, , and the cost function 0, and 0. a) Let and , and rewrite the cost function as . b) Define appropriate state space matrices and for the Algebraic Ricatti Equation 0 corresponding to the new cost function . c) Define the optimal feedback control, ? d) For 1, calculate the optimal control and show that the closed loop poles lie to the left of . e) Prove that, for the general case, all closed loop poles will lie in the region to the left of , or in other words, . Thus and Where For is the solution to 1 1 2 1 2 1 4 1 2 1 1 1 1 0 4 1 0 0 1 1 1 1 1 1 1 Let be a Lyapunov function. 0 0 From ARE 0 0 0 Problem 8) LQR theory can also be extended to problems where the derivative of the control variable also appears in the performance index (i.e. we have a soft constraint not only on the control variable but also on its derivative). Specifically, the LQR problem with cost subject to variable, , and treating can be solved by introducing the additional state as the control variable. a) By following this procedure, assuming 0, 0, 0, and 0 , 0 and are specified, obtain the general solution of the above by applying LQR theory. b) Can you formulate a generalization of the symmetric Root Locus for this more general problem? c) After obtaining the general solution, apply it to the specific problem: 0 1 0 , 0 0 1, 0 1, 0 0 1 Provide a root locus plot for 0 , and compute the optimal control for 2. ...
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 Fall '08
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 symmetric root locus, Locus Solutions

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