Prоblem 1. (10 pts) Find the fundаmentаl mаtrix (Phi(t)) and the state transitiоn matrix (Phi(t,tau)) оf the following LTV system: x˙(t)={"version":"1.1","math":"x˙(t)="}-2t01-2t{"version":"1.1","math":"-2t01-2t"}x(t){"version":"1.1","math":"x(t)"} Problem 2. (10 pts) Consider the following system:begin{align*} dot x_1 &= -(x_1)^2+x_2\ dot x_2 &= 2-x_2+x_1.end{align*} Find all the equilibrium points of the system and determine theirlocal stability. Problem 3. (15 pts) Consider a discrete-time LTI system (x[k+1]=) 1-110{"version":"1.1","math":"1-110"}xk+{"version":"1.1","math":"xk+"}10{"version":"1.1","math":"10"}uk{"version":"1.1","math":"uk"} with (x[0]=)10{"version":"1.1","math":"10"}. Find the minimum-energy control (u[0],u[1],u[2]) to steer the system state to (x[3]=)11{"version":"1.1","math":"11"}. Problem 4. (15 pts) Suppose a matrix (A) is given by A=121-1-2-1111=00101-11-10⏟T 10000011-1 ⏟A~ 111110100⏟T-1{"version":"1.1","math":"A=121-1-2-1111=00101-11-10⏟T 10000011-1 ⏟A~ 111110100⏟T-1"} (a) Find the eigenvalues of matrix (A) (hint: (tilde A) is lower triangular). Find the corresponding eigenvectors. (b) For the continuous-time LTI system (dot x=Ax), determine its stability. Is it possible to find (x(0)neq 0) such that as (ttoinfty) we have (i)~(x(t)to 0); (ii)~(x(t)toinfty); (iii) (x(t)) neither converges to zero nor diverges to infinity. In each case, explain why if the answer is no, and find one such (x(0)neq 0) if the answer is yes. (c) For the discrete-time system (x[k+1]=Ax[k]), redo part (b), i.e., determine its stability and whether we can find (x[0]neq 0) so that (x[k]) has the properties in the above three cases. In each case, explain why if the answer is no, and find one such (x[0]neq 0) if the answer is yes. Problem 5. (15 pts) Consider the continuous-time LTI system (dot x=Ax+Bu) where A=121-1-2-1111 = [00101-11-10]⏟T [10000011-1] ⏟A~ [111110100]⏟T-1, B=01-1{"version":"1.1","math":"A=121-1-2-1111 = [00101-11-10]⏟T [10000011-1] ⏟A~ [111110100]⏟T-1, B=01-1"} Note that the (A) matrix is the same as in Problem 4. (a) Is the system controllable? Find the reachable subspace of the system (b) Is the system stabilizable, i.e., there exists some gain (Kinmathbb R^{1times 3}) such that (A-BK) is stable? Why? You may find it convenient to work with the system after a coordinate change (x=Ttilde x). (c) Find a gain (K) that places as many eigenvalues of (A-BK) at (-2) as possible. Problem 6. (10 pts) Consider the continuous-time LTI system (dot x=Ax), (y=Cx) with A=[121-1-2-1111] =[00101-11-10]⏟T [10000011-1]⏟A~ [111110100]⏟T-1 , C=[001]{"version":"1.1","math":"A=[121-1-2-1111] =[00101-11-10]⏟T [10000011-1]⏟A~ [111110100]⏟T-1 , C=[001]"} Note that the (A) matrix is the same as in Problem 4. (a) Is the system observable? What is its unobservable subspace? (b) Is the system detectable, i.e., there exists some (L in mathbb R^3) such that (A-LC) is stable? Explain why. You may find it convenient to work with the system after a coordinate change (x=Ttilde x). Problem 7. (10 pts) Consider the continuous-time LTI system (dot x=Ax+Bu), (y=Cx), with A=121-1-2-1111{"version":"1.1","math":"A=121-1-2-1111"}=[00101-11-10]⏟T [10000011-1]⏟A~ [111110100]⏟T-1 , {"version":"1.1","math":"=[00101-11-10]⏟T [10000011-1]⏟A~ [111110100]⏟T-1 , "}B=01-1, C=001{"version":"1.1","math":"B=01-1, C=001"} Note that the (A), (B), and (C) matrices are the same as appeared in the previous three problems. What are the poles of the transfer function (frac{Y(s)}{U(s)})? Is the system BIBO stable? Problem 8. (15 pts) A system is given as follows: x·=1112x+11u{"version":"1.1","math":"x·=1112x+11u"}, and y=01x{"version":"1.1","math":"y=01x"} . (a) Design a feedback controller (u=-Kx) with gain (K) such that the closed-loop system (A-BK) has the eigenvalues ({-1,-1}). (b) Design an output feedback observer gain (L) such that the estimated state (hat x) by the resulting state observer has the dynamics (A-LC) with both eigenvalues at ({-2,-2}). (c) Combining your designs in (a) and (b), we obtain an observer-based state-feedback controller with the combined state x~=xx^{"version":"1.1","math":"x~=xx^"}. Write the state space equation of the combined system (frac{d}{dt}tilde x =tilde A tilde x) for some proper (tilde A), and find all the eigenvalues of the matrix (tilde A). Congratulations, you are almost done with the Final Exam. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to submit your work to Gradescope: Final Exam Submit your exam to the assignment Final Exam. Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.
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