A novel decomposition scheme to solve parametric non-convex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of alternating proximal gradient steps and a dual update per time step. Hence, the proposed approach is attractive in a real-time distributed context. Assuming that the Nonlinear Program (NLP) is semi-algebraic and that its critical points are strongly regular, contraction of the sequence of primal-dual iterates is proven, implying stability of the sub-optimality error, under some mild assumptions. Moreover, it is shown that the performance of the optimality-tracking scheme can be enhanced via a continuation technique. The efficacy of the proposed decomposition method is demonstrated by solving a centralised NMPC problem to control a DC motor and a distributed NMPC program for collaborative tracking of unicycles, both within a real-time framework. Furthermore, an analysis of the sub-optimality error as a function of the sampling period is proposed given a fixed computational power.
In-situ imaging techniques are a promising direction for monitoring the distribution of crystal sizes and shapes during a crystallization process. Nevertheless, no tractable method yet exists for estimating complex crystal shapes. In this paper, an in-situ imaging set-up is presented and a novel algorithm for crystal shape estimation from a pair of images is presented. It is shown that such a shape estimation problem can be turned into parametric polytope reconstruction from projections. Based on results in polyhedral geometry, it is demonstrated that an accurate estimate of the crystal shape can be computed by solving a nonlinear least-squares problem built from samples in images and a prior model of the crystal. Effectiveness of the approach is proven on artificial and real images. Results show that very accurate estimations of crystal shapes can be obtained from well-chosen data points sampled on images.
a b s t r a c tThe size and shape of particles crucially influences the characteristics of solid products. Until recently these quantities were evaluated using light microscopy. However, extracting the three-dimensional shape of a faceted crystal from a single image is a formidable computer vision challenge. In this work we combine stereoscopic imaging devices (e.g., commercial stereoscopic microscopes or the stereoscopic flow through cell that continuously draws samples from a crystallizer ) with a model-based approach in which parametric polytopes are used to describe faceted crystals (Hours et al., 2014). In the shape reconstruction algorithm these parametric polytopes are scaled and rotated until their projections closely match the measured stereoscopic images, which is formulated as a nonlinear optimization problem. The proposed approach is assessed using simulated images and experimental data. We also assess in which cases the proposed approach does or does not provide advantages over concepts using generic particle shapes (Schorsch et al., 2012).
A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties.
Abstract-Well-known model predictive control (MPC) theory for constrained linear time-invariant (LTI) systems is extended to accommodate hard constraints and cost penalizations on the spectra of the system's output trajectories. Thus the proposed method facilitates enforcing constraints, and placing weights, on the harmonic content of input-, state-and output-trajectories, in addition to the usual constrained control objectives. The proposed methods are demonstrated by the example problem of reducing torsional vibrations in a drive-shaft.
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