vakauskriteerien

Vakauskriteerien are mathematical conditions used to determine whether a system or process is stable under a particular notion of stability. They appear across fields such as mathematics, control engineering, physics and economics, and range from criteria that are necessary, sufficient, or both. A common theme is to relate stability to the location of poles of a system’s transfer function or to the behavior of an energy-like function.

One well-known example is Lyapunov stability criterion. If there exists a scalar function V(x) that is positive definite and whose derivative along system trajectories is nonpositive, the equilibrium is stable; if the derivative is negative definite, the equilibrium is asymptotically stable. This approach is powerful because it can apply to nonlinear and high-dimensional systems without requiring explicit solutions.

For linear time-invariant systems, several criteria focus on polynomial roots. The Routh–Hurwitz criterion provides necessary and sufficient conditions for all roots of a characteristic polynomial to have negative real parts, guaranteeing continuous-time stability. In the discrete-time setting, the Jury criterion serves a similar purpose for roots inside the unit circle. The Nyquist stability criterion connects stability to the frequency-domain plot of the open-loop system and encirclements of a critical point, given certain pole conditions.

A broader criterion is BIBO stability (bounded-input, bounded-output), which holds when a system’s impulse response is absolutely summable (continuous-time: poles lie in the left half-plane; discrete-time: inside the unit circle). Vakauskriteerien thus function as practical tests to assess and assure stability across modeling contexts, while recognizing that many criteria are either sufficient, necessary, or depend on modeling assumptions.