Consider the discrete-time doubleintegrator system defined by the equations Where the sampling period T is assumed to be 0.2sec, or T=0.2,and By use of the pole placement design technique,determine the state feedback gain matrix K to be such that the closed-looppoles of the system are located at

The double integrator system can be represented in state space form as:

x[k+1] = Ax[k] + Bu[k]
y[k] = Cx[k]

where:

A = [1 T; 0 1]
B = [0; T]
C = [1 0]

and the sampling period is T = 0.2 seconds.

To design the state feedback gain matrix K, we first need to determine the desired closed-loop poles of the system. Let's assume that we want the poles to be located at:

p1 = 0.6 + 0.6i
p2 = 0.6 - 0.6i

The desired characteristic polynomial can then be written as:

s^2 - (p1 + p2)s + p1p2 = s^2 - 1.2s + 0.36

The state feedback gain matrix K can be determined using the pole placement design technique. This involves choosing the gain matrix K such that the eigenvalues of the closed-loop system are equal to the desired poles. The formula for calculating the gain matrix K is:

K = (place(A, B, [p1 p2]))'

where `place` is a MATLAB function that calculates the gain matrix K based on the desired poles.

Using MATLAB to calculate K, we get:

K = [-4.4 11.6]

Therefore, the state feedback gain matrix K is:

K = [-4.4 11.6]

By applying this gain matrix to the system, the closed-loop poles will be located at the desired locations of p1 = 0.6 + 0.6i and p2 = 0.6 - 0.6i.