By Francesco Borrelli
Many sensible keep an eye on difficulties are ruled by means of features resembling nation, enter and operational constraints, alternations among varied working regimes, and the interplay of continuous-time and discrete occasion structures. at the moment no method is offered to layout controllers in a scientific demeanour for such structures. This publication introduces a brand new layout conception for controllers for such limited and switching dynamical platforms and ends up in algorithms that systematically resolve keep watch over synthesis difficulties. the 1st half is a self-contained creation to multiparametric programming, that's the most process used to check and compute kingdom suggestions optimum keep watch over legislation. The book's major goal is to derive homes of the country suggestions answer, in addition to to acquire algorithms to compute it successfully. the focal point is on restricted linear platforms and restricted linear hybrid platforms. The applicability of the idea is tested via experimental case reports: a mechanical laboratory technique and a traction keep watch over method constructed together with the Ford Motor corporation in Michigan.
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Additional info for Constrained Optimal Control of Linear and Hybrid Systems
4 Multiparametric Quadratic Programming 37 Proof: We first prove convexity of K ∗ and J ∗ (x). Take a generic x1 , x2 ∈ K ∗ , and let J ∗ (x1 ), J ∗ (x2 ) and z1 , z2 the corresponding optimal values and minimizers. 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 ∗ .
Some parameter x could belong to the boundary of several regions. Differently form the LP and QP case, the value function may be discontinuous and therefore such case has to be treated carefully. If a point x belong to different critical regions, the expressions of the value function associated to such the regions have to be compared in order to assign to x the right optimizer. Such procedure can be avoided by keeping track which facet belongs to a certain critical region and which not. Moreover, if the value function associated to the regions containing the same parameter x coincide this implies the presence of multiple optimizers.
As the number of combim! nations of constraints out of a set of m is ( m ) = (m− )! , the number of possible combinations of active constraints at the solution of a QP is at most m m m =0 ( ) = 2 . 1, Nn ≥ Nr . 1 generates critical regions. The first critical region CR0 is defined by the constraints λ(x) ≥ 0 (m constraints) and Gz(x) ≤ W + Sx (m constraints). If the strict complementary slackness condition holds, only m constraints can be active, and hence every CR is defined by m constraints.