examII_s04 - EE 530 EXAM 2 Spring 2004 Name: ID#: DO NOT...

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Unformatted text preview: EE 530 EXAM 2 Spring 2004 Name: ID#: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 30 1 35 2 35 3 100 Total This test consists of three problems. Answer each problem on the exam itself; if you use additional paper, repeat the identifying information above, and staple it to the rest of your exam when you hand it in. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. 1 Problem 1: (30 points) Consider the rst-order plant where a is an unknown constant, and the adaptive control law dy =ay+u dt u = 0ky dk = y 2 dt where > 0. 1. (5 points) Using x1 = y and x2 = k, nd a state-space representation of the adaptive closedloop system. 2. (25 points) Consider the candidate Lyapunov function V (x1; x2) = where 1 2 1 x + (x 0 k3 )2 2 1 2 2 V (x1; x2) = 0, if (x1; x2) = (0; k3) V (x1; x2) > 0, if (x1; x2) = (0; k3). 6 Under what restriction of k3 is the adaptive system stable ? 2 Problem 2: (35 points) A nonlinear system with input u(t) and output y (t) is described by the ODE y = 02y + ky2 + u: _ If the parameter k is known, then the control law u = ay (t) 0 ky2 will stabilize the system for an appropriate choice of the constant parameter a. This method of control design, known as feedback linearization, is not robust. If there is any uncertainty in the value of k, it can be shown that the resulting closed-loop system will not be stable. In order to work around this problem, we can use an adaptive control law u = ay (t) 0 y2 where is an adjustable control parameter. Using the candidate Lyapunov function V = y 2 + ( 0 k)2 nd an update law for so that the closed-loop adaptive control system is stable. What restrictions, if any, must be imposed on a ? 1 2 1 2 3 4 Problem 3: (35 points) 1. (15 points) Show that if H1(s) an H2(S ) are PR (SPR), then H1 (s) + H2 (s) is PR (SPR) for ; > 0. 5 2. (20 points) Show that the dierence in the degrees of the numerator and denominator of a PR transfer function cannot dier in magnitude by more than 1. 6 ...
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This note was uploaded on 07/23/2008 for the course EE 530 taught by Professor Schiano during the Fall '06 term at Pennsylvania State University, University Park.

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examII_s04 - EE 530 EXAM 2 Spring 2004 Name: ID#: DO NOT...

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