1.7 LINEARIZATION OF THE STATE EQUATION OF A GENERAL TIME-INVARIANT CONTROL SYSTEM

The state equation of a general time-invariant system is given by

For simplicity, we are just writing Equation (1.13) as

Let ( x ₀, u ₀) be an equilibrium point. This means the system is in equilibrium under the conditions x ₀ and u ₀, i.e.,

Equation (1.15) shows that the derivatives of all the state-variables are zero at the equilibrium point. Thus, the system continues to lie at the equilibrium point unless otherwise disturbed.

The state-Equation (1.14) of a general time-invariant system can be linearized about the equilibrium point ( x ₀, u ₀) by expanding it into Taylor's series and neglecting second-and higher-order terms.

Thus, the i ^th state equation is given by

Define the variations about the equilibrium point ( x ₀, u ₀) as

Also, from (1.16 A) we have

Now, for f _i ( x ₀, u ₀) = 0; the linearized i ^th state-equation is given by

When writing the above linearized state equations in vector-matrix form

The matrices A and B are called the Jacobian Matrices and all the partial derivatives in matrices A and B are evaluated at the equilibrium point ( x ₀, u ₀).