Question

A LTIC system is specified by the equation (D2 + 5D + 6)y(t)-(D + 1)x(t) a) Find the characteristic polynomial, characteristic equation, characteristic roots, and b) Find the zero-input response ya(t) for t 〉 0 if the initial conditions are ya(0-) = 2 characteristic modes corresponding to each characteristic root.

Answer 1

The characteristic polynomial of the LTIC system is obtained by setting the output-related part of the differential equation to zero. The characteristic roots are -2 and -3, and the zero-input response ya(t) for initial condition ya(0-) = 2 is found using a combination of e^{-2t} and e^{-3t}, which are the system's characteristic modes.

Step-by-step explanation:

The question pertains to a Linear Time-Invariant Control (LTIC) system represented by a differential equation. To address part (a), we find the characteristic polynomial by equating the part of the equation with the highest order derivatives that involve the output y(t) to zero (D2 + 5D + 6)y(t) = 0. The characteristic equation is therefore D2 + 5D + 6 = 0. Solving for D, we find the characteristic roots, which are D = -2 and D = -3.

For part (b), we need to find the zero-input response ya(t) given the initial condition ya(0-) = 2. The zero-input response of the system is the solution of the homogeneous differential equation obtained by setting the input x(t) to zero. Since the initial condition is given and we have the characteristic roots, we can write ya(t) as a combination of the system's characteristic modes. These modes are e-2t and e-3t, considering the roots of the characteristic equation. The general solution will be a linear combination of these modes, with constants determined by the initial conditions.