Given a 2-DOF system with the following nonlinear equations of motion 3 x_{1}-2 x_{2}+12 \dot{x}_{1} \dot{x}_{2}-11 \dot{x}_{1}+20 \ddot{x}_{1}=5 f_{1}+3 f_{2} 2 x_{1}+0.9 x_{2}^{2}+12 \dot{x}_{2}+11 \ddot{x}_{2}=f_{1}+5 f_{2} a. (25%) Put the

nonlinear equations in a vector form i = f(x, t, u). b. (25%) Simulate the natural response of the nonlinear system with the following initial condi- \text { tions } x_{0}=\left[\begin{array}{llll} 0.0 & 0.05 & 0.05 & 0.0 \end{array}\right]^{T} \text { for } t=0: 15 s c. (25%) Linearize the system (ignore the square and coupling terms) and put the system into the state space form x = Ax + Bu d. (25%) Design a full-state feedback controller using pole placement method, to stabilize the system. Test your controller on the linearized system first using Isim, then simulate the closed-loop system response using the linear and nonlinear model in a numerical integration setup.Assume you can control both inputs (What is the dimension of the gain matrix?) Your controller is a regulator, so observe the closed-loop response with r = 0 and initial conditions xo = | 15.0 0.7 0.510.0 Plot the response of all the 4 states, for both the linear and non-linear system in one subplot.Can you control the non-linear (actual) system with the same controller? Explain why or why not.

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