Q1 Find the state space description of the system shown in Figure 1. Use state variables as indicated Figure 1 Q2 Consider the following nonlinear system (a) Linearize the system around the...

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Q1 Find the state space description of the system shown in Figure 1. Use state variables as indicated Figure 1 Q2 Consider the following nonlinear system (a) Linearize the system around the equilibrium point x1 = x2 = x3 = 0. Is the linearized system controllable? (b) Linearize the system around the equilibrium point x1 = x2 = x3 = 1. Is the linearized system controllable? Q3 Q4 For the LTI system (a) Given x(0) =[1 1]T ( transpose), compute the zero-input response of y(t). ( use two methods) (b) Given u(t) =1 for t ≥0, compute the zero-state response of y(t) (c) If the system is sampled and let the sampling period be T = 1, Compute the discrete matrices A, B ( i.e Ad, Bd) for discrete system. Q5 Consider the following system Q6 (a) Obtain the Kalman controllable form (b) Can the state x(0) =[0 1 0]T be driven to zero ( zero state) in a finite amount of time? (c) Obtain the transfer function of H(s) of the system Q7 A system is described by the following dynamic equation (a) Use the Popov-Belevitch-Hautus test to determine the controllability and observability of the system. (b) Design a controller to relocate the poles of the systems so that they are relocated at -1+2j, -1- 2j,-2? (c) As the states sensors are expensive, design an observer with closed loop poles located at - 10+2j, -10-2j,-20. Use MATLAB to plot both the original states and their estimate. (on the same axis). (d)What is the transfer function of the controller based observer. Q8 (a) Is the system above stabilizable? (b) Obtain the Algebraic Riccati Equation (ARE) associated with the infinite horizon LQR problem above and calculate a positive definite solution (c) Determine the optimal control law. (d) Is the closed-loop system is asymptotically stable? verify.