Nonlinear Systems¶
Khalil, H. K. (2002). Nonlinear Systems. Prentice Hall.
Cited by¶
7 citations across 7 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Continuity
- For a real-valued function
f : ℝ → ℝ, both are the real line with standard metric; for a control systemf : 𝒰 → 𝒴, the input space might be square-integrable signals and the output space a state trajectory inℝⁿ, in the framework systematized by Khalil (2002)This sourceStandard graduate-level reference for continuous-time nonlinear control systems; develops input-output and state-space frameworks, BIBO and input-to-state stability, and Lyapunov-based stability analysis for continuous feedback systems.
- For a real-valued function
- Saddle Point
- Ecology: coexistence equilibria stable along total-biomass but unstable along the species-ratio axis, persisting until ratio-perturbations grow. International relations: balance-of-power equilibria stable against bilateral perturbations but unstable against multilateral defection. Control engineering: closed-loop systems with one well-damped and one unstable mode; inverted-pendulum stabilization is the canonical case, with active control supplying damping on the unstable mode.
This sourceStandard reference on saddle equilibria, stable/unstable manifolds, and feedback stabilization of unstable modes (e.g., the inverted pendulum).
- Ecology: coexistence equilibria stable along total-biomass but unstable along the species-ratio axis, persisting until ratio-perturbations grow. International relations: balance-of-power equilibria stable against bilateral perturbations but unstable against multilateral defection. Control engineering: closed-loop systems with one well-damped and one unstable mode; inverted-pendulum stabilization is the canonical case, with active control supplying damping on the unstable mode.
- Stability
- A Lyapunov function — total mechanical energy \(E = \frac{1}{2}\dot\theta^2 + \frac{g}{L}(1-\cos\theta)\) — proves stability without solving the equation: \(\dot E = -b\dot\theta^2 \le 0\), so energy monotonically decreases inside the basin.
This sourceCanonical treatment of Lyapunov's direct method, including energy functions that certify stability without solving the dynamics.
- A Lyapunov function — total mechanical energy \(E = \frac{1}{2}\dot\theta^2 + \frac{g}{L}(1-\cos\theta)\) — proves stability without solving the equation: \(\dot E = -b\dot\theta^2 \le 0\), so energy monotonically decreases inside the basin.
- Stock Disabled Control
- Model a system whose output \(y\) responds to a flow input \(u\) with a gain that depends on a stock state \(s\): \(\dot{y} = g(s)\,u\), where the propagation gain \(g(s)\) is positive and roughly constant for \(s\) above a threshold \(s^*\) but collapses toward zero for \(s < s^*\).
This sourceStandard reference for gain-scheduled and state-dependent control laws of the form modeled here, where loop gain depends on a system state and collapses outside a valid regime.
- Model a system whose output \(y\) responds to a flow input \(u\) with a gain that depends on a stock state \(s\): \(\dot{y} = g(s)\,u\), where the propagation gain \(g(s)\) is positive and roughly constant for \(s\) above a threshold \(s^*\) but collapses toward zero for \(s < s^*\).
Domain-specific¶
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