Introduction to Stability Analysis

Definition of Stability and Transient/Steady State
📌 The steady-state is the final condition of the system, and the transient state is the time difference between the process starting and reaching the steady state.
📐 Stability concepts are clarified through examples involving a body in a container.

Examples of System Stability
🎱 Stable System (Example 1): A body placed inside a container that returns to its original position regardless of the force applied; the system is not disturbed.
💥 Unstable System (Example 2): A body placed on top of a system where even a small external force pushes the body down, causing the system to become completely unbalanced.
⚖️ Conditionally Stable System (Example 3): A body placed inside a container where stability depends on the magnitude of the applied force; it returns to the original position for small disturbances but not for large ones.
〰️ Marginally Stable/Oscillatory System: Systems where the output continuously oscillates between two extreme values, like a pendulum.

Effect of Closed-Loop Poles on Stability
📍 The stability of a linear closed-loop system is determined by the location of its closed-loop poles in the s-plane (left half-plane, right half-plane, or imaginary axis).
📉 Left Half-Plane Poles (Example 1: poles at $s=-2$ and $s=-4$): Result in a stable system with a constant, finite steady-state output (e.g., 1.25 for the given transfer function).
📈 Pole in the Right Half-Plane (Example 2: pole at $s=2$): Results in an unstable system due to the presence of a positive exponential term ($e^{+2t}$), causing the output to tend towards infinity as time approaches infinity.
🪞 Poles on the Imaginary Axis (Example 3: poles at $s=\pm j2$): Result in a marginally stable system, where the output oscillates continuously between two fixed extreme values (e.g., $\sin(2t)$ for the given transfer function).

Characteristic Equation and Poles
📝 The Characteristic Equation is derived by setting the denominator of the closed-loop transfer function to zero.
🔗 The closed-loop poles of the system are precisely the roots of the Characteristic Equation.
🛠️ To determine stability when only given the characteristic equation, one must find its roots to locate the poles and assess their positions in the s-plane.

Key Points & Insights
➡️ Stability is dictated by the location of closed-loop poles in the s-plane (Left Plane = Stable, Right Plane = Unstable, Imaginary Axis = Marginally Stable).
➡️ The presence of even a single pole in the right half-plane guarantees the system will be unstable.
➡️ The roots of the Characteristic Equation are equivalent to the closed-loop poles, which are essential for stability analysis.

📸 Video summarized with SummaryTube.com on Nov 12, 2025, 13:37 UTC