By Francesco Borrelli

Many useful keep watch over difficulties are ruled by way of features resembling kingdom, enter and operational constraints, alternations among diverse working regimes, and the interplay of continuous-time and discrete occasion platforms. at the present no method is obtainable to layout controllers in a scientific demeanour for such structures. This e-book introduces a brand new layout conception for controllers for such restricted and switching dynamical platforms and ends up in algorithms that systematically resolve regulate synthesis difficulties. the 1st half is a self-contained creation to multiparametric programming, that's the most strategy used to review and compute kingdom suggestions optimum keep an eye on legislation. The book's major goal is to derive houses of the country suggestions answer, in addition to to acquire algorithms to compute it successfully. the focal point is on restricted linear structures and limited linear hybrid platforms. The applicability of the speculation is verified via experimental case reviews: a mechanical laboratory method and a traction keep watch over process built together with the Ford Motor corporation in Michigan.

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**Extra resources for Constrained Optimal Control of Linear and Hybrid Systems**

**Example text**

Let α ∈ [0, 1], and define zα αz1 + (1 − α)z2 , xα αx1 + (1 − α)x2 . By feasibility, z1 , z2 satisfy the constraints Gz1 ≤ W + Sx1 , Gz2 ≤ W + Sx2 . 34) where x(t) = xα . This proves that z(xα ) exists, and therefore convexity of K ∗ = ∗ ∗ i CRi . In particular, K is connected. e. J ∗ (αx1 + (1 − α)x2 ) ≤ αJ ∗ (x1 ) + (1 − α)J ∗ (x2 ), ∀x1 , x2 ∈ K, ∀α ∈ [0, 1], which proves the convexity of J ∗ (x) on K ∗ . 42). The boundary between two regions belongs to both closed regions. Since H 0, the optimum is unique, and hence the solution must be continuous across the boundary.

3 generates regions Ri to explore the set of parameters K. The following analysis does not take into account that (i) redundant constraints are removed, and that (ii) possible empty sets are not further partitioned. The first region R0 = CRA(x0 ) is defined by the constraints Gz(x) ≤ W +Sx (m constraints). If there is no dual degeneracy and no primal degeneracy only s constraints can be active, and hence CRA(x0 ) is defined by q = m − s constraints. 9, Rrest consists of q convex polyhedra {Ri }, each one defined by at most q inequalities.

To the authors’ knowledge, there does not exist an efficient method for solving general mp-MIQPs. In Chapter 8 we will present an algorithm that efficiently solves mp-MIQPs derived from optimal control problems for discrete time hybrid systems. 2 Constrained Finite Time Optimal Control F. Borelli: Constrained Optimal Control of Linear and Hybrid Systems, LNCIS 290, pp. 51--69, 2003 Springer-Verlag Berlin Heidelberg 2003 52 2 Constrained Finite Time Optimal Control For discrete time linear systems we prove that the solution to constrained finite time optimal control problems is a time varying piecewise affine feedback control law.