Instructiоns: On а sepаrаte sheet оf paper, answer each оf the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (30 pts) Consider the system, shown in the figure below. The spring is linear, that is, it obeys Hooke's law, where K {"version":"1.1","math":"(K)"}, in N/m {"version":"1.1","math":"( mbox{N/m})"}, is the spring coefficient, and B {"version":"1.1","math":"( B)"} , in N ⋅ sec/m {"version":"1.1","math":"(mbox{N$cdot$sec/m} )"}, is the damping coefficient. (5 pts) Form the Lagrangian for the system. (5 pts) Write the Lagrange equation of motion for the system. (5 pts) Represent the system model in state space format. (5 pts) Assume the input u = 0 {"version":"1.1","math":"(u=0)"}. Find the total energy of the system; denote it as V {"version":"1.1","math":"(V)"}. (5 pts) Compute the Lyapunov derivative V ˙ {"version":"1.1","math":"(dot{V})"}. (5 pts) Is the Lyapunov derivative V ˙ {"version":"1.1","math":"(dot{V})"} negative, negative semi-definite, positive, positive semi-definite, or neither? Based on the above, what can you say about the stability or instability in the sense of Lyapunov of the equilibrium point x = 0 {"version":"1.1","math":"(x=0)"} of the uncontrolled system? Problem 2. (10 pts) Sketch a phase-plane portrait for the dynamical system model, [ x ˙ 1 x ˙ 2 ] = [ 0 − 1 0 − 1 ] [ x 1 x 2 ] . {"version":"1.1","math":"left[ begin{array}{c} dot{x}_1\ dot{x}_2 end{array} right]=left[ begin{array}{cc} 0 & -1\ 0 & -1 end{array} right] left[ begin{array}{c} x_1\ x_2 end{array} right]."} Problem 3. (20 pts) Consider a dynamical system model, [ x ˙ 1 x ˙ 2 ] = [ x 2 x 2 sin x 1 − x 1 ] . {"version":"1.1","math":"left[ begin{array}{c} dot{x}_1\ dot{x}_2 end{array}right]=left[ begin{array}{c} x_2\ x_2 sin x_1 - x_1 end{array} right]."} (6 pts) Find an equilibrium state for this system. (6 pts) Linearize the system about the found equilibrium state. (8 pts) Determine if the linearized system is stable, asymptotically stable, or unstable. Problem 4. (10 pts) Is the following quadratic form, f = x ⊤ Q x = x ⊤ [ 2 2 2 0 0 2 0 2 0 0 1 0 0 0 0 1 ] x , {"version":"1.1","math":"[ f=x^{top} Q x=x^{top}left[begin{array}{cccc} 2 & 2 & 2 & 0\ 0 & 2 & 0 & 2\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 end{array}right] x, ] "}positive definite, positive semi-definite, negative definite, negative semi-definite, or indefinite? Carefully justify your answer. Problem 5. (15 pts) Consider an uncertain system model, x ˙ = A x + ζ ( t , x ) , {"version":"1.1","math":"dot{x}={A x} + zeta (t,{x}),"} where A = [ − 2 0 0 − 1 ] and ζ ( t , x ) ‖ 2 ≤ k ‖ x ‖ 2 . {"version":"1.1","math":"{A}=left[begin{array}{cc} -2 & 0\ 0 & -1 end{array}right]quadmbox{and}quad zeta(t, {x})|_2 le k|{x}|_2."} Find the largest k ∗ > 0 {"version":"1.1","math":"(k^* > 0)"} such that for any k 0 {"version":"1.1","math":"(k>0)"}, the system model is globally uniformly asymptotically stable. Problem 6. (15 pts) Consider a dynamical system model, { x ˙ 1 ( t ) = A x 1 ( t ) + b x 2 ( t ) x ˙ 2 ( t ) = − b ⊤ P x 1 ( t ) , {"version":"1.1","math":"[left{begin{array}{lll} dot{x}_1(t)&=&Ax_1(t)+ bx_2(t)\ dot{x}_2(t) &=& - b^top P x_1(t), end{array}right. ] "}where x 1 ∈ R n , {"version":"1.1","math":"( x_1in mathbb{R}^{n},)"} x 2 ∈ R , {"version":"1.1","math":"(x_2in mathbb{R},)"} and P = P ⊤ ≻ 0 {"version":"1.1","math":"(P= P^top succ 0)"} is the solution of the Lyapunov matrix equation, A ⊤ P + P A = − Q {"version":"1.1","math":"( A^top P + P A=- Q)"} for some Q = Q ⊤ ≻ 0 {"version":"1.1","math":"( Q=Q^top succ 0)"}. Use the Lyapunov-like lemma to show that lim t → ∞ x 1 ( t ) = 0. {"version":"1.1","math":"[ lim_{tto infty} x_1(t)= 0. ]"} *** Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Examity 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 1 Click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Examity session.