�6.241 Spring 2011 Final Exam 5/16/2011, 9:00am — 12:00pm The test is open books/notes, but no collaboration is allowed: i.e., you should not discuss this exam or solution approaches with anybody, except for the teaching staff. Problem 1 Let (A, b, c, 0) be a state-space model of a LTI system, with A ∈ Rn×n , b, c� ∈ Rn . Assume that λi(A) + λj (A) = 0, for all i, j. Consider the equation AX + XA + bc = 0; show that there exists a non-singular matrix X that satisfies the equation if and only if (A, b) is controllable and (c, A) is observable. Problem 2 Consider a LTI system described by the following state-space model: ⎡ ⎤ ⎡ ⎤ −1 2 2 1 � � A = ⎣−3 −1 −3⎦ , B = ⎣−1⎦ , C =1 −1 0 , D = 0. 3 −2 0 1 1. Construct a Kalman decomposition for this system, and compute the transfer function of the system. Is the system controllable/stabilizable, observable/detectable? 2. Design a stabilizing model-based compensator (i.e., composed of a full-state controller and an observer). 3. What is the transfer function of the compensator? Can you give a “classical” inter-pretation of the control law? 1Problem 3 Consider a plant with transfer function G(s) = 1/(s − 1). Find all feedback compensators K(s) such that (i) the closed-loop system is stable, and (ii) the output response to a unit step disturbance at the output is asymptotically zero. (Recall that, assuming the closed-loop transfer function Tyd is stable, then limt +∞ y(t) = →lims 0 sTyd(s)D(s), where D(s) is the Laplace transform of the input signal.) →Problem 4 Consider the block diagram shown below. P is an uncertain SISO plant with transfer function P (s) = P0(s) + W1(s)Δ1(s), where W1(s) is a stable transfer function, P (s) and P0(s) have the same number of right half-plane poles, and |Re[Δ1(s)]| ≤ α, |Im[Δ1(s)]| ≤ β, ∀s ∈ C. The transfer function W2(s) is a stable frequency weight. W2 d PK yu − 1. Derive necessary and sufficient conditions for robust stability, i.e., such that the closed-loop system shown in the figure is externally stable for all admissible Δ1(s). 2. Assume P = P0. Derive necessary and sufficient conditions for nominal performance, i.e., to ensure that �y� ≤ �d�, for all square-integrable disturbance inputs d ∈ L2. 3. Derive necessary and sufficient conditions for robust stability and performance, i.e., such that the closed loop system is stable, and �y� ≤ �d�, for all square-integrable disturbance inputs d ∈ L2, and for any admissible Δ1(s). 4. Can you give a graphical interpretation of these conditions? 2MIT OpenCourseWarehttp://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control Spring 2011 For information about citing these materials or our Terms of Use, visit: