Problem 1. (15 pts) A nonlinear system is given by\begin{al…
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Prоblem 1. (15 pts) A nоnlineаr system is given bybegin{аlign*} dоt x_1&=-x_1^3+x_2-1 \ dot x_2&=-x_1.end{аlign*} Find all the equilibrium points of the system and determine their local stability, if possible. Problem 2. (20 pts) Find a quadratic Lyapunov function (V(x)=x^T P x) for the continuous-time LTI system x·=-130-2x{"version":"1.1","math":"x·=-130-2x"}. Problem 3. (15 pts) Consider the continuous-time LTI system x·=Ax+Bu=-133-1x+1-1u, y=Cx=-11x,{"version":"1.1","math":"x·=Ax+Bu=-133-1x+1-1u, y=Cx=-11x,"} with eAt=11-11e2t00e-4t12-121212=12(e2t+e-4t)12(e2t-e-4t)12(e2t-e-4t)12(e2t+e-4t){"version":"1.1","math":"eAt=11-11e2t00e-4t12-121212=12(e2t+e-4t)12(e2t-e-4t)12(e2t-e-4t)12(e2t+e-4t)"} Suppose x(0)=11{"version":"1.1","math":"x(0)=11"} and the input (u(t)equiv 1) is the unit step signal. Find the corresponding output (y(t)). Problem 4. (20 pts) Consider the following continuous-time LTI system x·=Ax+Bu=2-101x+11u, y=Cx=1-1x{"version":"1.1","math":"x·=Ax+Bu=2-101x+11u, y=Cx=1-1x"} (a) Is the system reachable? Find its reachable subspace. (b) Is the system observable? Find its unobservable subspace. (c) Find a state coordinate transformation (T) and the transformed system ((tilde A,tilde B)) where the reachable part is separated from the non-reachable part. Problem 5. (15 pts) A discrete-time LTI system is given by x[k+1]=1-10-1x[k]+11u[k]{"version":"1.1","math":"x[k+1]=1-10-1x[k]+11u[k]"} Suppose x[0]=01{"version":"1.1","math":"x[0]=01"} and the goal is to steer the state to x[3]=1-1{"version":"1.1","math":"x[3]=1-1"}. Find the sequence (u[0]), (u[1]), (u[2]) with the minimum energy (|u[0]|^2+|u[1]|^2+|u[2]|^2) to achieve the above goal. Problem 6. (15 pts) A discrete-time system is given as follows: x[k+1]=3-2-11x[k], y[k]=01x[k], k=0, 1, ...{"version":"1.1","math":"x[k+1]=3-2-11x[k], y[k]=01x[k], k=0, 1, ..."} Suppose (x[0]) is unknown, and the first two outputs are measured as (y[0]=2), (y[1]=0). Determine the unknown (x[0]) and use it to predict the next output (y[2]). Congratulations, you are almost done with Midterm Exam 2. 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 go to Gradescope to submit your work: Midterm Exam 2 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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