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(Solved): please help 1. For the transfer function Pis2+5s+61 a) obtain the state space model in the phase ...
please help
1. For the transfer function a) obtain the state space model in the phase variable form;(5\%) b) Determine if the system is controllable. c) Design a state-feedback controller that gives overshoot of and settling time of 0.5 seconds. Please note you need to show all detailed calculation procedures. You are NOT allowed to use the MATLAB function acker() here.; d) Evaluate the steady-state error for a unit step input; (10\%) e) design an integral controller for the system to eliminate the steady state error. Place the third pole at -100 .
Expert Answer
Step 1 of a) State Space Model:First, we need to rewrite the transfer function into a standard form.G(s) = P(s)/Q(s) = 1/(s^2 + 5s + 6)Q(s) = s^2 + 5s + 6 = (s + 2)(s + 3)The state space model in the phase variable form is:x_dot = Ax + y = Cx + Duwhere x = [x1, x2]' is the state vector, u is the input, and y is the output.We can rewrite the transfer function in the following form:G(s) = Y(s)/U(s) = C(sI - A)^-1 B + Dwhere I is the identity matrix, C = [1 0], and D = 0.We can find A and B using the following equations:A = [0 1; -6 -5]B = [0; 1]b) Controllability:To determine if the system is controllable, we need to check if the controllability matrix has full rank:C = [B AB] = [0 0; 1 0; -5 -6]rank(C) = 3, which means the system is controllable.Answer has been explained above