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Problem 1. (15 pts)  A nonlinear system is given by\begin{al…

Problem 1. (15 pts)  A nonlinear system is given by\begin{align*}  \dot x_1&=-x_1^3+x_2-1 \\  \dot x_2&=-x_1.\end{align*} 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.