EE C128 / ME C134 Final Exam Fall 2014 December 19, 2014 Your PRINTED FULL NAME Your STUDENT ID NUMBER Number of additional sheets 1. No computers, no tablets, no connected device (phone etc.) 2. Pocket calculator allowed. 3. Closed book, closed notes, closed internet. 4. Allowed: 2 sheets (each double sided) Chi Chi. 5. Additional sheets are available and may be submitted (e.g. for graphs). 6. Write your name below, and your SID on the top right corner of every page (including this one). 7. If you turn in additional sheets: ˆ Write your name and/or SID on every sheet, and ˆ Write the number of additional sheets you are turning in above where indicated. 8. Do not write on the back of any page. Part 1 2 3 4 5 6 7 8 Score 1
1. Laplace transform, controllable canonical form (a) Derive the following Laplace property for convolution integrals L {g 1 (t)} = G 1 (s) L {g 2 (t)} = G 2 (s) { t } L g 1 (τ)g 2 (t τ)dτ = G 1 (s)g 2 (s) 0 (b) Find the solution to the following ODE with the given initial conditions: d 2 x dt 2 dx dt + 2x = t ẋ(0) = 2, x(0) = 1 0 δ(τ) sin (t τ)dτ UC Berkeley, 12/19/2014 2 of 22
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2. Bode Plotting and Nyquist Stability Consider the following transfer function: G(s) = 100(s + 4) (s 2 + 12s + 20)(s + 2) (a) Sketch a Bode plot of the system (magnitude and phase). Label all slopes and points on the graph. UC Berkeley, 12/19/2014 4 of 22
(b) Using the Bode plots you created in Part (a), calculate the phase margin and gain margin for G(s). (c) Draw a Nyquist plot of the system and use the Nyquist stability criterion to determine if the closed loop system under unity feedback is stable. UC Berkeley, 12/19/2014 5 of 22
(d) Assume we have a closed loop system below where G(s) is given in Part (a) and C(s) = K. For what values of K is the system stable? UC Berkeley, 12/19/2014 6 of 22
3. Solution in time domain, stability (a) For the following system, explicitly determine the time-domain solution x(t) ẋ = where u(t) is a unit step function. [ 4 1 0 10 ] x + [ 1 0 ] [ y = 1 0 u, x(0) = ] x [ 1 1 ] UC Berkeley, 12/19/2014 7 of 22
(b) Determine the transfer function G(s) for the zero initial state response, given the system in Part (a). (c) BONUS: Does the degree of your state space model and transfer function match? Why or why not? UC Berkeley, 12/19/2014 8 of 22
4. Electrical Circuit and Root Locus (a) For the above circuit derive the transfer function C(s) = V out V in. UC Berkeley, 12/19/2014 9 of 22
(b) Assume the system below, G(s), is in unity negative feedback. Determine the value of K such that the steady state error to a step response is 1 11. Also determine the percent overshoot and settling time of the feedback system at this K. G(s) = K (s + 5)(s + 15) (c) Now assume that the system is described by the figure below, where C(s) and G(s) are obtained from parts (a) and (b) respectively. Draw the root locus given R 1 = 125 MΩ, C 1 = 15 µf, R 2 = 625 MΩ and C 2 = 0.1 µf. Watch your signs! UC Berkeley, 12/19/2014 10 of 22
(d) Label on the root locus a suitable region if the goal is to achieve a settling time (T s ) 0.4 sec and percent overshoot (%OS) 20%. UC Berkeley, 12/19/2014 11 of 22
5. Controller design using State-feedback Consider the mechanical system shown above. Here, V denotes the voltage applied to the motor (control input) and x(t) is the position of the mass. You may assume the back emf from the motor is negligible (EMF = 0) and the torque supplied by the motor is equal to T = IK m J m θ where, K m : Constant relating T and I J m : Inertia of the motor (a) Show that G(s) is the transfer function from V to x. To do this, you MUST derive the governing equations for the mechanical/electrical system. [( G(s) = X(s) V (s) = N ( ) ) ] 2 K m r 2 1 N2 J 1 + J 2 + J m s 2 + r 2 (Ms 2 + f v s + k) N 1 R + Ls N 1 UC Berkeley, 12/19/2014 12 of 22
[ T (b) Choosing x = x ẋ] as the state vector and x as the output, derive a state space model (matrices A, B, C and D) for the above system. Use the following parameters: R = 1, K m = 0.1, L = 0, N 2 /N 1 = 10, r = 1, J 1 = J 2 = 1, J m = 0, M = 1, k = 1, f v = 1. Note your input to the system should be V. (c) Explicitly write the observability and controllability matrices. Is the system controllable? Is it observable? (Use parameters from Part (b)) UC Berkeley, 12/19/2014 13 of 22
(d) Determine the eigenvalues of matrix A you found in Part (b). (e) We will now control the system[ using ] a state feedback controller as shown in the diagram below ] r 1 where K = [k 1, k 2 and r =. r 2 Write the dynamics of the closed-loop system as ẋ = Ãx + Br. That is, find both à and B in terms of the system parameters given in Part (b) and the elements of the controller gain matrix K. UC Berkeley, 12/19/2014 14 of 22
6. State-feedback and observer design Consider the following system: ẋ = Ax + Bu, y = Cx with A = [ 7 1 2 4 (a) Compute the eigenvalues and eigenvectors for A. ], B = [ 0 1 ] [, C = 0 1 ] (b) Use state-feedback of the form of u = Kx. Determine the gain K = [k 1 k 2 ] such that the poles of the closed loop system are located at s 1,2 = 2 ± 5j. UC Berkeley, 12/19/2014 15 of 22
(c) Unfortunately for this system we are unable to measure all the states. In order to do state feedback we must use a Luenberger observer of the form: ˆx = Aˆx + Bu + L(y ŷ) ŷ = C ˆx and the system is controlled using state feedback, given by u = K ˆx Determine the error dynamics of the system, ė, where e = ˆx x. The result must be in terms of e only. (d) Determine the observer matrix L = [l 1 l 2 ] T such that the error dynamics have poles at s 1,2 = 2 ± 5j. UC Berkeley, 12/19/2014 16 of 22
(e) Comment on the performance of the state observer given the previously placed poles. What are we interested in when designing an observer and how could we improve the observer? (f) Complete the following block diagram of the system described in part(c). Plant Controller u y Observer ˆx UC Berkeley, 12/19/2014 17 of 22
7. Linear Quadratic Regulator Consider the LTI system ẋ = Ax + Bu x = [x 1, x 2 ] T where A = [ ] [ ] 1 1 1, B = 0 1 0 We would like to solve an LQR problem for the system. control u (t) that minimizes the cost functional That is, we want to find the optimal J = t=0 (x 2 1(t) + u 2 (t)) dt (a) Solve the Algebraic Riccati Equation for the infinite horizon LQR. Hint: the solution of the Algebraic Riccati equation is a positive semi-definite matrix. A (2 2) matrix P is positive semi-definite everywhere when: [ ] p 11 0 p 22 0 p 2 p 11 p 12 12 p 11 p 22 for P = p 12 p 22 UC Berkeley, 12/19/2014 18 of 22
(b) Determine the optimal feedback matrix K 1 such that the optimal control is u (t) = K 1 x(t). UC Berkeley, 12/19/2014 19 of 22
8. Linear Quadratic Regulator Consider the system where the dynamics are scalar: ẋ = ax + bu x is a scalar We want to create a finite horizon optimal controller given the cost function: J = tf t=0 (qx 2 (t) + ru 2 (t))dt (a) Write the Ricatti Equation for this system as well as the terminal condition. (b) Find P for the static case where t f =. Your answer should be in terms of q, r, a, and b. Note the P should be positive semi-definite everywhere. UC Berkeley, 12/19/2014 20 of 22
(c) For this scalar system, given any t f, the Riccati equation can be analytically solved: P (τ) = (ap (t f ) + q) sinh(βτ) + βp (t f ) cosh(βτ) ) a sinh(βτ) + β cosh(βτ) ( b 2 P (t f ) r where τ = t f t, β = a 2 + b2 q and sinh(.) and cosh(.) are the hyperbolic trigonometric r functions. Taking the limit as t f, the solution becomes: P (τ) = q a + β Show that this is equivalent to your solution from Part (b). UC Berkeley, 12/19/2014 21 of 22
f(t) F (s) δ(t) 1 1 u(t) s 1 tu(t) s 2 t n n! u(t) s n+1 ω sin(ωt)u(t) cos(ωt)u(t) e αt sin(ωt)u(t) e αt cos(ωt)u(t) s 2 +ω 2 s s 2 +ω 2 ω (s+α) 2 +ω 2 s+α (s+α) 2 +ω 2 Table 1: Laplace transforms of common functions sinh(θ) cosh(θ) tanh(θ) e θ e θ 2 e θ + e θ 2 sinh(θ) cosh θ = 1 e 2θ 1 + e 2θ Table 2: Trigonometric functions UC Berkeley, 12/19/2014 22 of 22