JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B. Tech II CONTROL THEORY Previous Question Paper (March - 2022 - R16)

Linearization is the process of approximating a non-linear system by a linear model in the vicinity of an operating point. This is essential because linear systems allow for the use of powerful analytical tools like Laplace transforms, transfer functions, and frequency response analysis.

The most common method is using the Taylor Series expansion. For a non-linear function y = f(x), we expand the function around an operating point x0:

By neglecting higher-order terms, we obtain the linear approximation:

where Δx = x - x0. For multi-variable systems, we use the Jacobian matrix consisting of partial derivatives of the system equations with respect to each state variable and input, evaluated at the equilibrium point. This results in the state-space representation:

where A and B are the Jacobian matrices of the system dynamics and input mappings, respectively.

To construct a Signal Flow Graph (SFG), we represent each variable as a node and each algebraic relationship as a directed branch with a gain. The given equations are:

1. x1 = -x2 - x3 + 3
2. x2 = -x1 + x3 + 5
3. x3 = x1 - 2x2 - x3 + 2

In an SFG, these are represented by nodes x1, x2, x3, and a source node for the constants (represented as 1). For the first equation, branches with gains -1, -1, and 3 originate from nodes x2, x3, and the constant node respectively, all terminating at node x1.

The SFG is not unique because the algebraic equations can be rearranged algebraically before drawing. For example, x3 can be expressed as x3 = (x1 - 2x2 + 2) / 2, or the equations can be solved for different variables first. Since any set of equivalent algebraic equations represents the same system, different topological arrangements of nodes and branches can be drawn to represent the same underlying mathematical relationship, demonstrating that multiple valid SFGs exist for the same system.