Answer:
To design a state feedback controller using the pole placement technique, we need to determine the control input u such that the closed-loop poles of the system are located at the desired values of −3 and −2±3j.
Given the state-space representation of the system:
x˙1=x1+2x3
x˙2=3x1+x2+7x3
x˙3=3x2+x3+u
We can start by defining the desired characteristic equation for the closed-loop poles:
s^3 + as^2 + bs + c = (s + 3)(s + 2 + 3j)(s + 2 - 3j)
Expanding the equation on the right-hand side:
s^3 + as^2 + bs + c = (s^2 + 5s + 6)(s + 2)
Comparing the coefficients of both sides, we get:
a = 5
b = 6
c = 12
Next, we can determine the desired control input u using the state feedback controller:
u = -Kx
where K is the gain matrix to be determined.
To find the gain matrix K, we need to compute the controller gain such that the eigenvalues of the closed-loop system are equal to the desired poles. Since we have a third-order system, we can place the poles using the Ackermann's formula.
The desired characteristic equation can be written as:
s^3 + 5s^2 + 6s + 12 = 0
Using the coefficients, we can define the following matrices:
A = [[0, 1, 0], [3, 0, 7], [0, 3, 1]]
B = [[0], [0], [1]]
C = [[1, 0, 0]]
To apply the Ackermann's formula, we need to calculate the controllability matrix Qc:
Qc = [B, AB, A^2B]
Calculating the controllability matrix:
Qc = [[0, 0, 1], [0, 1, 7], [1, 7, 22]]
Since the controllability matrix Qc has full rank, we can proceed to calculate the gain matrix K using the formula:
K = R^(-1)Qc^T
where R is a nonzero scalar, and we can choose R = 1.
Calculating the gain matrix:
K = [[-5, 11, -45]]
Finally, we can derive the control input u by multiplying the gain matrix K with the state vector x:
u = -Kx
Therefore, the control input u is given by:
u = -[-5, 11, -45] * [x1, x2, x3]^T
To summarize:
1. Determine the desired characteristic equation based on the desired closed-loop poles.
2. Calculate the coefficients a, b, and c from the desired characteristic equation.
3. Construct the state-space representation of the system.
4. Compute the controllability matrix Qc.
5. Calculate the gain matrix K using the Ackermann's formula.
6. Derive the control input u using the gain matrix and the state vector x.
Please note that the given question does not specify the values of x1, x2, and x3, so we cannot provide a specific numerical solution for the control input u.
Step-by-step explanation: