Modern Control System Theory and Design, 2nd Edition

6.2.  DETERMINING THE CHARACTERISTIC EQUATION USING CONVENTIONAL AND STATE-VARIABLE METHODS

The characteristic equation can be defined in terms of the state-variable equation of the control system. We have stated in the previous section that stability of linear systems is independent of the input. Therefore, the condition x(t) = 0, where x(t) is the state vector, can be viewed as the equilibrium state of the system. Let us assume that a linear system is subjected to a disturbance at t = 0, resulting in an initial state x(0) that is finite. If it returns to its equilibrium state as t approaches infinity, the system is considered to be stable. If it does not, in terms of our definition, it is considered to be unstable. These concepts of stability can be generalized. In the state-variable approach [1, 2], a linear system is considered to be stable if, for a finite initial state x(0), there is a positive number A that depends on x(0), where

The value x(t) denotes the norm of the state vector x(t). It is defined as

Equation (6.8) can be interpreted to mean that the transition of state for ...