By Michael Malisoff, Frédéric Mazenc
The building of strict Lyapunov services is a difficult challenge that's of important ongoing learn curiosity. even though speak Lyapunov functionality thought promises the lifestyles of strict Lyapunov services in lots of events, the Lyapunov features that speak concept offers are frequently summary and nonexplicit, and as a result won't lend themselves to engineering purposes. usually, even if a procedure is understood to be strong, one nonetheless wishes particular Lyapunov services; in spite of the fact that, as soon as a suitable strict Lyapunov functionality has been developed, many robustness and stabilization difficulties will be solved shortly via ordinary suggestions designs or robustness arguments. against this, non-strict Lyapunov capabilities are usually comfortably built, e.g., from passivity, backstepping, or forwarding (especially within the time various context), or through the use of the Hamiltonian in Euler–Lagrange systems.
Constructions of Strict Lyapunov Functions incorporates a vast repertoire of Lyapunov buildings for nonlinear structures, concentrating on equipment for remodeling non-strict Lyapunov services into strict ones. Many very important periods of dynamics are lined: Jurdjevic–Quinn platforms; time-varying structures pleasant LaSalle or Matrosov stipulations; slowly and swiftly time-varying platforms; adaptively managed dynamics; and hybrid platforms. The explicitness and straightforwardness of the buildings lead them to compatible for suggestions layout, and for quantifying the consequences of uncertainty. Readers will enjoy the authors’ mathematical rigor and unifying, design-oriented strategy, in addition to the various labored examples, masking a number of functions which are of compelling curiosity together with the adaptive keep watch over of chemostats and the stabilization of underactuated ships.
Researchers from applied-mathematical and engineering backgrounds operating in nonlinear and dynamical structures will locate this monograph to be most dear and for graduate scholars of keep watch over thought it is going to even be an authoritative resource of knowledge on an important subject.
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Additional info for Constructions of Strict Lyapunov Functions
Example text
W1 (x) ≤ V (t, x) ≤ W2 (x); and 2. V˙ (t, x) ≤ −W3 (x) for all t ≥ 0 and x ∈ X . 4). 4). In the special case where there exist positive constants ci and p so that the preceding assumptions hold with Wi (x) = ci |x|p and X = Rn , then the equilibrium is GES. The preceding theorem reduces the stability analysis to a search for an appropriate Lyapunov function. 4) is known to be UGAS, it is often important to be able to go in the converse direction, by constructing a strict Lyapunov function for the system.
1 for the main ideas from the proof. The challenge is to obtain explicit formulas for global strict Lyapunov functions that do not involve the flow map. 14) satisfying our standing assumptions from the previous chapter. 14) is 30 2 Review of Lyapunov Functions all of Rn . When X = Rn , we call a function V : [0, ∞) × X → [0, ∞) a storage function provided that there exist α1 , α2 ∈ K∞ so that α1 (|x|) ≤ V (t, x) ≤ α2 (|x|) everywhere; in this case, we also say that V is uniformly proper and positive definite, or of class UPPD.
44) is such that (s(t), x(t)) ∈ K0 for all t ≥ t0 . 44) in terms of the Erdmann Transformation t τ= y1 (l) dl . 45) for t → (˜ s, x˜, γ˜ )(τ −1 (t)). 45) is D = (−s∗ , ∞) × (−x∗ , ∞) × (γm − γ∗ , γM − γ∗ ). , its (˜ s, γ˜ )subsystem does not depend on x ˜ (because ν is independent of x), and the x ˜-subsystem is globally input-to-state stable with respect to γ˜ with the ISS Lyapunov function x˜2 . 46) ⎩ γ˜˙ = (γ − γ )(γ − γ)ν m M with state space X = (−s∗ , ∞) × (γm − γ∗ , γM − γ∗ ) is GAS to 0.