JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III LINEAR SYSTEMS ANALYSIS Previous Question Paper (March - 2024 - R16)

Phase variables are a specific set of state variables defined as the output of a system and its successive derivatives. For an n-th order linear differential equation, the state vector is composed of the variable y(t) and its first (n-1) derivatives, represented as x1 = y, x2 = dy/dt, ..., xn = d^(n-1)y/dt^(n-1). This representation is commonly used to express a system in the standard phase-variable canonical form, which simplifies the construction of state-space models and controller design.

The transfer function of a linear time-invariant (LTI) system can be derived from its state-space representation, defined by the state equation ẋ = Ax + Bu and the output equation y = Cx + Du. Taking the Laplace transform of these equations with zero initial conditions yields sX(s) = AX(s) + BU(s) and Y(s) = CX(s) + DU(s).

Solving the state equation for X(s) gives X(s) = (sI - A)^-1BU(s). Substituting this into the output equation results in Y(s) = [C(sI - A)^-1B + D]U(s). The transfer function matrix H(s) is defined as the ratio of the output to the input, given by the expression:

Here, (sI - A)^-1 is the resolvent matrix, and the poles of the transfer function are the eigenvalues of the system matrix A.