MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems--Fall 2003 PROBLEM SET 10 Issued: November 25. 2003 Due: December 5. 2003 REMINDER: Computer Lab 3 is also due on December 5 Reading Assignments Lectures #21-22 PS#10: Chapters 10 11(through Subsection 11. 3.4)of O& Lectures #23-24 PS#11: Chapters 10 11(through Subsection 11.3.4)of O&w Exercise for home study (not to be turned in, although we will provide solutions): El)O&W 11.32(a)through(d) Problems to be turned in Problem 1 Consider the following feedback configuration G(s Sketch the root loci for K>0 and K <0 for each of the following (a)G(s)= )c(s)=(-5(8+8 +1 (c)G(s) For this part, clearly indicate the point at which the closed-loop system has a double-pole
y(t) MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 10 Issued: November 25, 2003 Due: December 5, 2003 REMINDER: Computer Lab 3 is also due on December 5. Reading Assignments: Lectures #21-22 & PS#10: Chapters 10 & 11 (through Subsection 11.3.4) of O&W Lectures #23-24 & PS#11: Chapters 10 & 11 (through Subsection 11.3.4) of O&W Exercise for home study (not to be turned in, although we will provide solutions): (E1) O&W 11.32 (a) through (d) Problems to be turned in: Problem 1 Consider the following feedback configuration. x(t) − + e(t) K G(s) Sketch the root loci for K > 0 and K < 0 for each of the following: 1 (a) G(s) = . s + 1 1 (b) G(s) = (s − 5)(s + 3) s + 1 (c) G(s) = 2 . For this part, clearly indicate the point at which the closed-loop system s has a double-pole. 1
Problem 2 Consider the system shown below e() K(s (s+10) y(t) (a) Compute the steady state tracking error, e(oo), due to a unit step input a(t)=u(t) when K(s)=K. Does the steady state tracking error change as K changes? (b)Compute the steady state tracking error, e(oo), due to a ramp input a)=tu(t)when K(s)=K. Does the steady state tracking error change as K changes? (c)Assume, for this part, that K(s)=1. Find systems K,()and Ks(s)in the modified system shown below such that the steady state tracking error due to the ramp input (t)=tu(t) becomes zero. Hint: One of the two systems, K,(s)and Ks(s), is a constant gain. Note that the tracking error is defined to be e(t)=a(t-y(t) K。(s) K(s s(s+10 Problem3 O&W 11.27 Problem 4 Determine the z-transform for each of the following sequences. Sketch pole-zero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists (a)rn]=26n+3-6{n-2] (b)x{n]=2u{n-1]+4"u-n]
y(t) Problem 2 Consider the system shown below: x(t) − + e(t) K(s) 1 s(s + 10) (a) Compute the steady state tracking error, e(�), due to a unit step input x(t) = u(t) when K(s) = K. Does the steady state tracking error change as K changes ? (b) Compute the steady state tracking error, e(�), due to a ramp input x(t) = tu(t) when K(s) = K. Does the steady state tracking error change as K changes ? (c) Assume, for this part, that K(s) = 1. Find systems Kf (s) and Ks(s) in the modified system shown below such that the steady state tracking error due to the ramp input x(t) = tu(t) becomes zero. Hint: One of the two systems, Kf (s) and Ks(s), is a constant gain. Note that the tracking error is defined to be e(t) = x(t) − y(t). x(t) Ks(s) + − + Kf (s) K(s) 1 s(s y(t) + 10) Problem 3 O&W 11.27 Problem 4 Determine the z-transform for each of the following sequences. Sketch pole-zero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. (a) x[n] = 2�[n + 3] − �[n − 2] (b) x[n] = 2nu[n − 1] + 4nu[−n] 2
Problem 5 For each of the following z-transforms, determine the inverse z-transform (a)X(z)=122-4-2-1+6+92-825 x(3)=1+21-=,52 Problem 6 Consider a signal y(n] which is related to two signals 21[n] and T2[n]by ym]=x1[-m-2]*x2{n+4 where un and r2[nl an Determine the z-transform Y(z)of yn], together with its ROC Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for tests
Problem 5 For each of the following z-transforms, determine the inverse z-transform 5 (a) X(z) = 12z−4 − z−1 + 6 + 9z2 − 8z 5 1 1 (b) X(z) = 1 + 1 z−1 − 1 z−2 , < |z| < 3 2 6 6 Problem 6 Consider a signal y[n] which is related to two signals x1[n] and x2[n] by y[n] = x1[−n − 2] � x2[n + 4] where � � 1 n 1 n x1[n] = − u[n] and x2[n] = u[n]. 2 4 Determine the z-transform Y (z) of y[n], together with its ROC. Reminder: The first 20 problems in each chapter of O&W have answers included at the end of the text. Consider using these for additional practice, either now or as you study for tests. 3