The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.

The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded output for a bounded input, which occurs when all the roots of the characteristic equation are located in the left-half of the s-plane. If any root lies in the right-half of the s-plane, the system becomes unstable. Absolute stability indicates whether a system is stable or unstable, while relative stability measures the degree of stability.

An illustrative example of these concepts is a pendulum. When undisturbed, a pendulum rests in a state of static equilibrium and exhibits stability for small motions around this equilibrium position. This stability is maintained when the pendulum is displaced and is allowed to swing. In this case, external or frictional forces are present. These forces introduce damping, causing the transient response of the system to gradually decrease over time. Eventually, the pendulum achieves stable motion around the equilibrium position, demonstrating the characteristics of a stable system.

In contrast, if the pendulum is displaced from an unstable equilibrium position, it begins to swing with its time response showing exponential growth. This unrestricted motion signifies instability, as the system fails to return to equilibrium and the amplitude of oscillations continues to increase.

Understanding the distinction between transient and steady-state responses, along with the conditions for stability, is essential for analyzing and designing LTI systems. By ensuring that all poles of the system are in the left-half s-plane, engineers can guarantee stability and predict the system's behavior in response to various inputs, leading to more robust and reliable system performance.