23.5 : 安定性

The time response of a linear time-invariant system is divided into transient and steady-state responses.

The transient response diminishes to zero over time. The steady-state response persists after the transient response has faded.

The stability of such a system is directly linked to the roots of the system's characteristic equation or its poles.

A system maintaining bounded output for a bounded input, with roots of the characteristic equation in the left-half s-plane, is stable. However, if any root lies in the right-half s-plane, it becomes unstable.

Absolute stability confirms system stability, while relative stability measures its degree.

Consider a pendulum. When undisturbed, it rests in a state of static equilibrium, maintaining stability for minor motions around the equilibrium.

When external or frictional forces are added, the transient response of the system gradually decreases over time due to the influence of the damping. The pendulum achieves stable motion around the equilibrium position.

In contrast, an inverted pendulum is inherently unstable and requires active control to prevent it from toppling. When disturbed by an external force, it does not return to its original position, illustrating instability.