Final answer:
To calculate the state-feedback control matrix K ensuring closed-loop poles at p = –1, –2, –3, we follow the Ackermann's formula that involves determining the desired characteristic polynomial, ensuring system controllability, computing the Ackermann's matrix, and applying the formula to find K.
Step-by-step explanation:
To calculate a state-feedback control matrix K such that the closed-loop poles are located at p = –1, –2, –3, we can use Ackermann's formula. First, we need the system's state-space representation with matrix A describing the system dynamics, matrix B describing the input to the system, and state matrix x. Assume a state-space representation is given as part of the problem statement. The characteristic polynomial for the desired closed-loop system is given by det(sI - A + BK) = (s+1)(s+2)(s+3).
Following the Ackermann's formula, we first need to calculate the desired characteristic polynomial from the specified pole locations. This characteristic polynomial is (s+1)(s+2)(s+3) = s^3 + 6s^2 + 11s + 6. Next, we construct the controllability matrix and determine if the system is controllable. If it is, we then calculate the Ackermann's matrix, which involves raising the system matrix A to the power of the system's state dimension and doing matrix operations involving the controllability matrix. Finally, apply the Ackermann's formula to obtain the state-feedback matrix K.