By Franco Blanchini
This self-contained monograph describes simple set-theoretic tools for regulate. It offers a dialogue in their hyperlinks to basic difficulties in Lyapunov balance research and stabilization, optimum keep watch over, keep watch over below constraints, chronic disturbance rejection, and unsure platforms research and synthesis. The paintings provides a number of demonstrated and very likely new purposes, in addition to numerical examples and case experiences. A key subject matter is the trade-off among certain (but computationally extensive) and approximate (but conservative) recommendations to difficulties. Mathematical language is saved to the minimal necessary.
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Extra resources for Set-Theoretic Methods in Control
Sample text
Then it is globally exponentially stable. 3 Lyapunov functions and stability 43 Proof. 24) and bound it as t+T Ψ (x(t + T )) = Ψ (x(t)) + D+ Ψ (x(σ), w(σ))dσ t t+T Ψ (x(σ))dσ ≤ Ψ (x(t)) − γT Ψ (x(t + T )) ≤ Ψ (x(t)) − γ t where the last inequality follows by the fact that Ψ (x(t)) is nonincreasing. 36), after simple mathematics the following inequality holds γ p β x(t) ≤ e− p t x(0) α which implies exponential convergence with convergence speed γ/p. 37) becomes the ˙ differential inequality Ψ(x(t)) ≤ −γΨ (x), implying Ψ (x(t)) ≤ e−γt Ψ (x(0)).
13) u(t) ∈ U. 26, the control constraints are also satisfied. Note also that the problem with state constraints can be addressed. If one assumes that x(t) ∈ X is a hard constraint to be satisfied, it can be immediately argued that, as long as N [Ψ, μ] ⊆ X , for some μ, and Ψ is a control Lyapunov function (either global, inside or outside P), then the constraints can be satisfied by means of a proper control action as long as x(0) ∈ N [Ψ, μ]. 1 Associating a control law with a Control Lyapunov Function: state feedback In this section, the state feedback and the full information feedback cases will be basically analyzed.
The basic idea of dynamic programming is that of determining the solution backward in time by starting from the target. There is an interesting connection between dynamic programming and the construction of Lyapunov functions for uncertain systems. Let us consider the case of a simple linear time-varying uncertain system x(k + 1) = A(w(k))x(k) where A(w), for w ∈ W, and W is a compact set. Being w time-varying, the natural way to check robust stability of the system is to seek for a Lyapunov function.
