Download Set-Theoretic Methods in Control by Franco Blanchini PDF

By Franco Blanchini

This self-contained monograph describes uncomplicated set-theoretic equipment for keep watch over. It offers a dialogue in their hyperlinks to primary difficulties in Lyapunov balance research and stabilization, optimum regulate, keep an eye on below constraints, continual disturbance rejection, and unsure platforms research and synthesis. The paintings offers a number of verified and in all likelihood new purposes, besides numerical examples and case experiences. A key subject is the trade-off among unique (but computationally extensive) and approximate (but conservative) options to difficulties. Mathematical language is saved to the minimal necessary.

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X)/∂xn ]T ). , locally bounded), possibly nondifferentiable convex function defined in an open domain. 10 (Subgradient). The vector z ∈ Rn is a subgradient of Ψ , at x if (see Fig. 26) The set of all the subgradients at x is the subdifferential ∂Ψ (x). 12 10 Ψ(x) 8 6 4 T 2 Ψ(x1)+z (x−x1) 0 x1 −2 −2 −1 0 1 2 Fig. 3. Subgradient of Ψ at point x. Note that for a differentiable (convex) function ∂Ψ (x) is a singleton including the gradient at x: ∂Ψ (x) = {∇Ψ (x)}. 3 For a convex (possibly nondifferentiable) Lyapunov function Ψ (x) the above definition allows to compute the Lyapunov derivative as D+ Ψ (x, w) = sup z T f (x, w).

The theory of differential inclusions, by exploiting the set valued maps theory, and especially the concept of semicontinuity (or continuity) of set-valued maps, is a framework to prove the existence of solutions in the general case (see Chapter 2 of [AC84] for details). 2 Model absorbing An idea that sometimes is extremely useful is the approximation achieved by absorbing a nonlinear uncertain systems in a linear (controlled) differential inclusion. 18) for all x and u. 18). 18) (the opposite is clearly not true in general).

The corresponding differential inclusion is ⎧ if x > x ¯ ⎪ ⎨ −λx x, −λ¯ x+u ¯] if x = x ¯ x˙ ∈ F (t) := [−λ¯ ⎪ ⎩ −λx + u¯ if x < x ¯ Let us assume the desired temperature is x ¯ ≥ 0 (reasonable for both Celsius and Fahrenheit scale) and that (a desirable assumption) λ¯ x+u ¯ > 0. Then, it is not difficult to see that the (unique in this case) absolutely continuous function which satisfies the inclusion for x(0) ≤ x ¯ is x(t) = min{(x(0) − u ¯/λ)e−λt + u¯/λ, x ¯} which becomes, by definition, the solution of the relay system.

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