By Francesco Borrelli

ISBN-10: 354000257X

ISBN-13: 9783540002574

ISBN-10: 3540362258

ISBN-13: 9783540362258

Many functional keep an eye on difficulties are ruled by means of features resembling nation, enter and operational constraints, alternations among various working regimes, and the interplay of continuous-time and discrete occasion platforms. at the present no method is out there to layout controllers in a scientific demeanour for such platforms. This e-book introduces a brand new layout idea for controllers for such limited and switching dynamical structures and results in algorithms that systematically clear up regulate synthesis difficulties. the 1st half is a self-contained creation to multiparametric programming, that is the most approach used to review and compute kingdom suggestions optimum keep an eye on legislation. The book's major goal is to derive houses of the kingdom suggestions answer, in addition to to procure algorithms to compute it successfully. the point of interest is on restricted linear platforms and limited linear hybrid structures. The applicability of the speculation is validated via experimental case stories: a mechanical laboratory procedure and a traction keep an eye on method built together with the Ford Motor corporation in Michigan.

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**Additional info for Constrained Optimal Control of Linear and Hybrid Systems**

**Sample text**

11. Consider two set of active constraints Aj and Aj and let CRi and CRj be the corresponding critical region, respectively. 34) is non-degenerate and that CR ¯ i and CR ¯ j denote the closure of the sets CRi and CRj , respectively. where CR Then, Ai ⊂ Aj and #Ai = #Aj − 1 or Aj ⊂ Ai and #Ai = #Aj + 1. 2. 10. This fact, together with the convexity of the set of feasible parameters K ∗ ⊆ K and the piecewise linearity of the solution z ∗ (x) is proved in the next Theorem. 12. 34) and let H 0. Then, the set of feasible parameters K ∗ ⊆ K is convex.

The function J ∗ : K ∗ → R will denote the function which expresses the dependence on x of the minimum value of the objective function over K ∗ , J ∗ (·) will be called value function. The set-valued function sc s Z ∗ : K ∗ → 2R × 2{0,1} d will describe for any fixed x ∈ K ∗ the set of ∗ optimizers z (x) related to J ∗ (x). We aim at determining the region K ∗ ⊆ K of feasible parameters x and at finding the expression of the value function J ∗ (x) and the expression an optimizer function z ∗ (x) ∈ Z ∗ (x).

38) where nT is the number of rows Ti of the matrix T . 34) is feasible for such an x0 . 34) is infeasible for all x in the interior of K. 34), in order to obtain the corresponding optimal solution z0 . 34). 10. Let H 0. Consider a combination of active constraints A0 , and assume that LICQ holds. Then, the optimal z ∗ and the associated vector of Lagrange multipliers λ∗ are uniquely defined affine functions of x over the critical region CR0 . 39a) λi (Gi z − Wi − Si x) = 0, i = 1, . . 39b) to obtain the complementary slackness condition λ∗ ( −G H −1 G λ∗ − W − Sx) = 0.

### Constrained Optimal Control of Linear and Hybrid Systems by Francesco Borrelli

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