20075ee141_1_hw3_solutions

20075ee141_1_hw3_solutions - EE141 Principles of Feedback...

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EE141 HW 3 Principles of Feedback Control Instructor: Balakrishnan, A.V. Problem Set 3: Solutions 1. Determine the stability of the following transfer functions using the Nyquist criterion. First specify an appropiate contour (as shown in class, line segment in the imaginary axis plus a semicircle in the right half plane), then draw the contour plot (sketch or with MatLab), and finally evaluate the number of encirclements of 0 to find the number of poles in the right half plane. Ψ( s ) = 1 s + a , for a = 1 , a = - 1 , a = - 1 + i . (10 points) Ψ( s ) = 1 s 2 + bs +1 , for b = 1 and b = - 1 . (15 points) The product of the two previous cases (for a = - 1 , b = 1 ) (15 points) F ( s ) = Ψ( s ) 1+ K Ψ( s ) , with Ψ( s ) as defined in the 2 nd case. Determine whether the sta- bility of the system depends on the value of K. Verify your answer by calculating the roots of the polynomials involved.(20 points) Hint: Follow the solution to Ψ( s ) = 1 s +1 and Ψ( s ) = 1 s - 1 and their contour plots. Solution: For a, b and c it is enough to pick a semicircle of radius R = 2 to cover all the poles of our transfer function.
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