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Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay. Exponential stability is a form of asymptotic stability, valid for more general dynamical systems.
Exponential Stability: The origin of x˙ = f(x) is exponentially stable if and only if the linearization of f(x) at the origin is Hurwitz Theorem: Let f(x) be a locally Lipschitz function defined over a domain D ⊂ Rn; 0 ∈ D. Let V (x) be a continuously differentiable function such that k1kxka ≤ V (x) ≤ k2kxka V˙ (x) ≤ −k3kxka
- 128KB
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- Energy and dissipation functions
- V (x(t)) when z = x(t), ̇x = f (x) dt
- Lyapunov theory
- A Lyapunov global asymptotic stability theorem
- • if
- Finding Lyapunov functions
- Other sources of Lyapunov functions
consider nonlinear system ̇x = f (x), and function V : Rn → R we define V ̇ : R n → R as V ̇ (z) = ∇V (z)T f (z) V ̇ (z) gives d
we can think of V as generalized energy function, and − V ̇ associated generalized dissipation function as the
Lyapunov theory is used to make conclusions about trajectories of a system ̇x = f (x) (e.g., G.A.S.) without finding the trajectories (i.e., solving the differential equation) typical Lyapunov theorem has the form: if n there exists a function V V and V ̇ R : → R that satisfies some conditions on • then, trajectories of system satisfy some proper...
suppose there is a function V such that • V is positive definite
the trajectories of system satisfy some property then there exists a Lyapunov function that proves it sharper converse Lyapunov theorem is more specific about the form of the Lyapunov function example: if the linear system ̇x = Ax is G.A.S., then there is a quadratic Lyapunov function that proves it (we’ll prove this later)
there are many different types of Lyapunov theorems the key in all cases is to find the required properties Lyapunov function and verify that it has there are several approaches to finding Lyapunov functions and verifying the properties one common approach: decide form of Lyapunov function (e.g., quadratic), parametrized by some parameters (called ...
value function of a related optimal control problem linear-quadratic Lyapunov theory (next lecture) computational methods converse Lyapunov theorems graphical methods (really!) (as you might guess, these are all somewhat related)
An equilibrium point x∗ is exponentially stable if there exists α > 0 and λ > 0 such that for all . As with stability of linear systems, these definitions are increasing in strength. That is, if an equilibrium point is exponentially stable, then it is stable and asymptotically stable.
Definition 4.1. Stability in the sense of Lyapunov. The equilibrium point x∗ = 0 of (4.31) is stable (in the sense of Lyapunov) at t = t0 if for any > 0 there exists a δ(t0, ) > 0 such that. x(t0) < δ = ⇒. x(t) <, ∀t ≥ t0. (4.32) Lyapunov stability is a very mild requirement on equilibrium points.
Stable. The origin is asymptotically stable if and only if xf(x) < 0 in some neighborhood of the origin. Let the origin be an asymptotically stable equilibrium point of the system ̇x = f(x), where f is a locally Lipschitz function defined over a domain D ⊂ Rn ( 0 ∈ D)
Example: Consider the LTV system x˙(t) = " −1 t 0 −1 # x(t) •Use the definition of asymptotic stability to show the origin is asymptotically stable. •Use the definition of uniform asymptotic stability to show the origin is not UAS. We take the initial time to be t 0 and let x 10 = x 1(t 0) and x 20 = x 2(t 0). Note that x 2(t) = e − ...
