EE221A Linear System Theory Final Exam

Transcription

1 EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall /16/16, 8-11am Your answers must be supported by analysis, proof, or counterexample. You may quote results that we derived in class in your solutions. There are 9 questions: Please make sure your exam paper has all 9 questions. Approximate points for each question are indicated. You are allowed to use crib sheet (both sides). 1

2 Problem 1: Linear Quadratic Regulator (8 points). Consider the following discrete linear time-invariant dynamical system: and the cost function: J(U,x 0 ) = N 1 τ=0 x t+1 = Ax t + Bu t, t {0,1,...,N 1} (1) x 0 = x init, ( (xτ x ) T Q(x τ x ) + (u τ u ) T R(u τ u ) ) + (x N x ) T Q f (x N x ), (2) where Q,R,Q f 0 are positive definite matrices and U := (u 0,u 1,...,u N 1 ). We say a reference point (x,u ) is dynamically feasible if it satisfies the system dynamics, that is: x = Ax + Bu, otherwise the reference point is called dynamically infeasible. 1. Suppose that the reference point (x,u ) is dynamically feasible. Derive the optimal LQR control policy and cost-to-go function that minimize (2) subject to the system dynamics in (1). 2. Now suppose that the reference point is not dynamically feasible. Also, let m = Ax +Bu x. Derive the optimal LQR control policy and cost-to-go function. 2

3 Problem 2: Cayley-Hamilton Theorem (4 points). Consider the following matrix: 1. Find the characteristic polynomial of A. A = [ ] Express A 4 in terms of the lowest order polynomial in A. 3. Using Cayley-Hamilton theorem, show that e At = α 0 (t)i + α 1 (t)a for some scalar functions α 0 ( ) and α 1 ( ). Explicitly compute these scalar functions. 3

4 Problem 3: Minimal polynomials (3 points). Given a matrix A with minimal polynomial s 3 (s + 1) 2 (s + 2), is the system ẋ = Ax stable? 4

5 Problem 4: Exponential Stability and Controllability (6 points). Suppose that the LTI system (A, B) is completely controllable, and that there exists a symmetric, positive definite matrix P such that AP + PA T = BB T Show that all eigenvalues of A have negative real parts. HINT: If λ is an eigenvalue of A, and e is a corresponding left eigenvector, then e θ and Also, e is a left eigenvector for the eigenvalue λ. e T A = λe T 5

6 Problem 5: Controllability (4 points). For each of the following, provide either a proof or a counterexample: 1. Suppose (A,B) is controllable. Is the system (A 2,B) controllable? 2. Suppose (A 2,B) is controllable. Is the system (A,B) controllable? 6

7 Problem 6: Controllability and Observability of Augmented Systems (8 points). Suppose that the single input single output systems L 1 = (A 1,b 1,c 1 ) and L 2 = (A 2,b 2,c 2 ) are each completely controllable and completely observable. Discuss the controllability and observability of the systems: ([ ] [ ] A1 0 b1 L 3 =,, [ ] ) c 0 A 2 b 1 c 2 2 ([ ] [ ] [ ]) A1 0 b1 0 c L 4 =,, A 2 0 b 2 0 c. 2 in the two cases (a) when A 1 and A 2 have no common eigenvalues; (b) when A 1 and A 2 have at least one eigenvalue in common. 7

8 Problem 7: Designing B for Controllability (6 points). Consider the following system matrix A: A = (3) Find the minimum number of inputs needed for the system to be controllable. Now, find a corresponding system matrix B that ensures that the system is controllable. 8

9 Problem 8: Integral control (6 points). Consider the following simple plant input-output transfer function: G(s) = 1 s + 1 (a) (2 points) Derive the state space representation of this plant in controllable canonical form. (b) (2 points) Consider this plant in state feedback with state feedback gain F and reference input R: derive an expression for the plant input u(t) to ensure that Y (t) tracks constant reference inputs R(t) = R with zero steady state error. (c) (2 points) Now, suppose that the actual plant is subject to modeling errors G(s) = 1 s ǫ where ǫ is an unknown, but fixed, deviation from the known model. Discuss the effect of this modeling error on the steady state error computed in part (b) above. 9

10 Problem 9: Observer design (10 points). Figure 1 shows a velocity observation system where x 1 is the input u velocity 1 s x 1 observed variable 2 s x 2 2+s observer z 1 observer output Figure 1: Velocity Observation System for Problem 8. velocity to be observed. An observer is to be constructed to track x 1, using u and x 2 as inputs. The variable x 2 is obtained from x 1 through a sensor having the known transfer function 2 s 2 + s (4) as shown in Figure 1. (a) (3 points) Derive a set of state-space equations for the system with state variables x 1 and x 2, input u and output x 2. (b) (2 points) Design an observer with states z 1 and z 2 to track x 1 and x 2 respectively. Choose both observer eigenvalues to be at 4. Write out the state space equations for the observer. (c) (3 points) Derive the combined state equation for the system plus observer. Take as state variables x 1, x 2, e 1 = x 1 z 1, and e 2 = x 2 z 2. Take u as input and z 1 as the output. Is this system controllable and/or observable? Give physical reasons for any states being uncontrollable or unobservable. (d) (2 points) What is the transfer function relating u to z 1? Explain your result. 10