Final: Signal, Systems and Control (BME )

Transcription

1 Final: Signal, Systems and Control (BME 580.) Instructor: René Vidal May 0th 007 HONOR SYSTEM: This examination is strictly individual. You are not allowed to talk, discuss, exchange solutions, etc., with other fellow students. Furthermore, you are not allowed to use the book or your class notes. You may only ask questions to the class teaching assistant. Any violation of the honor system, or any of the ethic regulations, will be immediately reported according to JHU regulations. NAME: Signature: CHEAT SHEET:. For a first order system with pole at s = p, the settling time is t s = log(0) p.. For a second order system with poles at s = ω n ξ±jω n ξ, the overshoot is M p = e πξ/ ξ, and the settling time is t s = log(0) ξω n. 3. Transfer function G(s) = C(sI A) B + D 4. Transfer function poles p : det(pi A) = 0 ( [ ] zi A B ) 5. Transfer function zeros z : det = 0 C D 6. Controllability matrix C = [ B AB A B A n B ] and observability matrix O = C CA CA. CA n 7. Ackermann s formula K = [ 0 0 ] [ B AB A B A n B ] ( A n +α A n +... α n I ) 8. Cayley-Hamilton theorem: p(a) = 0 R n n where p(s) = det(si A) = s n + α s n + + α 0. Question /0 Question /40 Question 3 /50 Question 4 /80 TOTAL /90

2 . (0 Points) One of the simplest and most fundamental of all physiological control systems is the muscle stretch reflex. The most notable example of this kind of reflex is the knee jerk, which is used in routine medical examinations as an assessment of the state of the nervous system. A sharp tap to the patellar tendon in the knee leads to an abrupt stretching of the extensor muscle in the thigh to which the tendon is attached. This activates the muscle spindles, which are stretch receptors. Neural impulses, which encode information about the magnitude of the stretch, are sent along afferent nerve fibers to the spinal cord. Since each afferent nerve is synaptically connected with one motorneuron in the spinal cord, the motorneurons get activated and, in turn, send efferent neural impulses back to the same thigh muscle. These produce a contraction of the muscle, which acts to straighten the lower leg. Figure (a) shows the basic components of this reflex. Construct a block diagram similar to the one shown in Figure (b) to represent the major control mechanisms involved in the muscle stretch reflex. Clearly identify the physiological correlates of the controller, the plant, and the feedback element, as well as the controlling, controlled, and feedback variables. (a) Schematic illustration of the muscle stretch reflex (b) Closed-loop control system Figure : Michael C.K. Khoo, Physiological Control Systems - Analysis, Simulation, and Estimation, Wiley-IEEE Press, 000.

3 . (40 Points) Find if each one of the following statements is TRUE or FALSE. If true, provide a proof. If false, provide a counter example. No credit will be given if you only answer true or false. (a) (0 points) If λ is an eigenvalue of A, then λ is a pole of G(s) = C(sI A) B. (b) (0 points) If (A, B, C, D) is observable, then (A, B, C, D) is observable. (c) (0 points) (A, B, C, D) is controllable and observable if and only if (A, C, B, D) is observable and controllable. (d) (0 points) State feedback does not affect the zeros of a system. 3

4 3. (50 Points) Let us define the exponential of a matrix A R n n as exp(a) = k=0 A k k! R n n. () Let {p i C n } n i= be the eigenvectors of A associated with the eigenvalues {λ i C} n i=, that is, for all i =,..., n, we have Ap i = λ i p i. Let λ 0 λ Λ =... Cn n 0 λ n be the matrix of eigenvalues and P = [ p p p n ] C n n be the matrix of eigenvectors. Assume that P is invertible, that is, there exists a matrix P such that P P = P P = I. (a) (5 Points) Show that for all k the matrix A k R n n has the same eigenvectors {p i } n i= as A, but with eigenvalues {λ k i }n i=. Hint: Show it first for A then for A 3 and so on. (b) (5 Points) Show that the matrix exp(at) R n n has the same eigenvectors {p i } n i= as A, but with eigenvalues {exp(λ i t)} n i= (c) (5 Points) Show that AP = P Λ. (d) (5 Points) Show that for all k 0, A k = P Λ k P. (e) (5 Points) Show that exp(at) = P exp(λt)p. [ ] 0 (f) (5 Points) Let A =. Show that exp(at) = 0 et +e t I + et e t A using the following three methods: i. From the inverse Laplace transform of (si A). ii. From the matrices of eigenvalues Λ and eigenvectors P of A as exp(at) = P exp(λt)p. iii. Using the Cayley-Hamilton theorem as follows. Let p(s) = det(si A) = s + a s + a be the characteristic polynomial of A. Then using the Cayley-Hamilton theorem we have A = a A a I. Multiplying this by A we obtain an expression for A 3 as a function of A and I. Repeating this, find an expression for A k as a function of A and I. Given A k, compute exp(at) = k=0 (At)k /k!. 4

5 4. (80 points) Consider the system u(t) s+3 x 3(t) s+ x (t) s+ x (t) s y(t) (a) (0 Points) Find the transfer function from u(t) to y(t), the zeros and the poles. system stable? (b) (0 Points) Consider the design of a PI controller with gains k p and k i. Is the i. Find the critical gain k c > 0 of a proportional controller such that the closed loop system is about to lose stability. Use this gain to design a PI controller that achieves zero steady state error to a unit step. Recall that the Ziegler-Nichols formulae for designing a PI controller are k p = 0.45k c and k i =.k p /t c, where k c and t c are, respectively, the critical gain and the critical time, respectively. ii. Use the Routh Hurwitz criterion to show that a necessary condition for the stability of the closed loop system is that 7k p k i < 60. (c) (0 Points) Find a state space representation of the system ẋ(t) = Ax(t) + Bu(t), and y(t) = Cx(t) + Du(t). Is the system controllable? Is the system observable? (d) (30 Points) Find the gains (k, k, k 3 ) of the state feedback u = k x k x k 3 x 3 so that the poles of the closed-loop system are located at 3 using three different methods. i. By finding the denominator of the closed-loop transfer function without using the statespace representation. ii. By finding the denominator of the closed loop system using the state-space representation. iii. By using Ackermann s formula. Hint: you don t need to compute the inverse of the controllability matrix, but instead solve for its last row by solving a linear system. Don t forget to check that your answers agree. (e) (0 Points) Find the gains of an observer for x of the form ˆx = Aˆx + L(y ŷ), ŷ = C ˆx, so that the observer poles are located at 5. 5