Question

A linear control system is described by the equations

ẋ(t) = (1.2 -2 0) x(t) + (1) u(t), y(t) = (0 0 -1) x(t),
( 1 a 0) (3)
( 0 1 0) (0)

where a varies from -[infinity] to [infinity]

Find the characteristic polynomial

Answer 1

To find the characteristic polynomial of the linear control system, we compute the determinant of the matrix \(\lambda I - A\), where \(\lambda\) is an eigenvalue and I is the identity matrix. The characteristic polynomial is this determinant expressed as a function of \(\lambda\).

Step-by-step explanation:

The student's question pertains to finding the characteristic polynomial of a given linear control system. The system is described by differential equations for the state and output vectors. To find the characteristic polynomial, we need to write down the system matrix A and then compute the determinant of \(\lambda I - A\), which is the characteristic equation of the system. Here, \(\lambda\) represents an eigenvalue of the matrix A, and I is the identity matrix of the same dimension as A. The characteristic polynomial is the determinant expressed in terms of \(\lambda\).

In this case, if we represent the system matrix A as A = \begin{pmatrix} 1.2 & -2 & 0 \\ 1 & a & 0 \\ 0 & 3 & 0 \end{pmatrix}, then the characteristic polynomial can be found by calculating det(\lambda I - A), which results in a polynomial equation in \(\lambda\). The coefficients of the polynomial will depend on the specific values in A.