Stability of Control Systems refers to the ability of a control system to produce a bounded and predictable output when subjected to a bounded input or disturbance. A stable system returns to its equilibrium condition or settles to a finite value after a disturbance, whereas an unstable system may exhibit continuously increasing oscillations or an unbounded output. For a linear time-invariant system, stability is commonly determined by the locations of the closed-loop poles in the complex plane: a continuous-time system is stable when all poles have negative real parts. Stability analysis is essential for ensuring safe, reliable, and satisfactory performance of control systems.
Stability of Control Systems
Explore how the location of closed-loop poles affects the stability
and time response of a continuous-time control system.
1. Pole Location in the s-Plane
2. System Time Response
Stable: poles have negative real parts.
Closed-loop poles: s = σ ± jω
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Stable when Re(s) < 0
How the simulation works:
A continuous-time control system is stable when all of its closed-loop poles have negative real parts. In the s-plane, this means the poles must lie in the left half-plane. When the real part is negative, the system response decays with time and approaches equilibrium.
When the poles are located approximately on the imaginary axis (σ = 0), the response can exhibit sustained oscillations, representing the idealized marginally stable case.
When the poles move into the right half-plane (σ > 0), the response grows with time and the system becomes unstable.
A continuous-time control system is stable when all of its closed-loop poles have negative real parts. In the s-plane, this means the poles must lie in the left half-plane. When the real part is negative, the system response decays with time and approaches equilibrium.
When the poles are located approximately on the imaginary axis (σ = 0), the response can exhibit sustained oscillations, representing the idealized marginally stable case.
When the poles move into the right half-plane (σ > 0), the response grows with time and the system becomes unstable.