Hurty James C. Dayon
BSEE-3 day
Stability Control System
Stability –A measure of the tendency of a system’s response to return to zero after being
disturbed.
Stability Control System
Stability Analysis
(a). Free motion type stability small disturbance large disturbance
(b). Stability of the force motion type BIBO stability (bounded-input-bounded-output) In the
sense of Lyapunov asymptotic stability, a stable system is a dynamic system with a limited
response to bounded input stability.
Consider x t Ax t &() () = with xt x ( ) 0 0 = (1). The system is stable in the sense of Lyapunov iff
Re{ } λ I ≤ 0 , and the eigenvalue with zero real parts are distinct. (2). The system is
asymptotically stable iff Re{λi} < 0, for all t. (3). For time invariant linear systems, asymptotical
stability implies exponential stability.
*For time-varying systems, the system can be unstable even if Re{λi} < 0, for all t.
BIBO stability of the linear time-invariant system (A, B, C, D) does not always guarantee
asymptotic stability. E.g., If any canceled pole has a positive real part but all the remaining poles
have negative real parts the system is BIBO stable but not asymptotically stable.