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∫∞ (eT(t)CT 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.