Problem 1- Consider a dynamical system characterized by the state-space equation

x? (t) = Ax (t) + Bu (t)

y (t) = Cx (t) .

The goal is to design a state feedback of the  form

u (t) = -Kx (t) + Gr (t)

such that the scalar output y (t) closely tracks the scalar reference input r (t). For this purpose, the matrix G is obtained from

G = - (C (A - BK)^-1 B)^-1

for any given K. To determine an appropriate value for the gain matrix K, the error dynamics

e? (t) = Ae (t) + Bv (t)

is considered, and K  is obtained in such a manner that under the state    feedback

v (t) = -Ke (t),

the closed-loop system demonstrates a rapid response, i.e., e (t) tends to 0 quickly. For the numerical values

determine K using two methods of pole placement and Linear Quadratic Regulator (LQR) as instructed below.

(a) Determine the value of K to place the poles of the closed-loop system at -0.2 ± j0.2 and -5. Compute the value of G associated with this value of K. Simulate the closed-loop system by applying a step function as r (t).

(b) For different choices of β > 0, determine K to minimize the cost function

J = ₀∫^∞ (e^T(t)C^T Ce (t) + βv²(t))dt.

For each resulting value of K, compute G and simulate the system similar to part (b).

Problem 2- In Problem 1, assume that instead of the entire state, only the output y (t) is measured.  Thus, to implement the state feedback, the state must be replaced with its estimate xˆ(t), so that

u (t) = -Kxˆ(t) + Gr (t).

The procedure for computation of K and G is unchanged, but a new gain matrix L must be designed to implement a state estimator of the form

xˆ? (t) = (A - LC) xˆ (t) + Bu (t) + Ly (t)

u (t) = -Kxˆ (t) + Gr (t) .

(a) Determine the value of L to place all the eigenvalues of A - LC at -10.

(b) For the values of K and G determined in part (a) of Problem 1., simulate the closed-loop system and determine its step response. Compare this response with the one in part (a) of Problem 1.

Problem 3- Consider the state-space equation in Problem 1, and determine a transfer function to equivalently represent the system. For this transfer function, use the root locus method to design a

(a) Proportional (P) controller,

(b) Integral (I) controller,

(c) Proportional-integral (PI) controller,

(d) Proportional-integral-differential (PID) controller.

For each controller, simulate the closed-loop system to determine its step response. Compare these responses in terms of transient and steady state behaviors.

Problem 4- Repeat Problem 3, using the frequency response method.