Design of multi-parametric NCO-tracking controllers for linear continuous-time systems

Process optimization for industrial applications aims to achieve performance enhancements while satisfying system constraints. A major challenge for any such method lies in the problem of uncertainty stemming from model mismatch and process disturbances. Classical approaches such as model predictive control usually handle the uncertainty by repeatedly solving the optimization problem on-line, which may prove a rather computationally demanding task nonetheless and cause serious delays for fast dynamic systems. Existing approaches for mitigating the on-line computational burden via off-line optimization include multi-parametric programming and NCO-tracking. Multi-parametric programming aims to generate a mapping of control strategies as a function of given parameters; whereas NCO-tracking involves tracking the necessary conditions of optimality (NCOs) based on a precomputed control switching structure, which enables a dynamic real-time optimization problem to be transferred into an on-line tracking problem using a feedback controller. A methodology, called multi-parametric (mp-)NCO-tracking is developed in this thesis, whereby multi-parametric dynamic optimization and NCO-tracking methods are combined into a unified framework. An algorithm for the design of mp-NCO-tracking controllers for continuous-time, linear-quadratic optimal control problems is presented in Chapter 2. The off-line step defines the multi-parametric control structure mapped to given uncertain (measurable) parameters in terms of so-called critical regions and feedback laws. Specifically, each critical region corresponds to a unique control switching structure in terms of the sequence of active constraints. The on-line step involves determining the current critical region once the parameter value has been revealed, and then applying the corresponding feedback control laws in a receding horizon manner. The mp-NCO-tracking approach provides a means for relaxing the invariant switching structure assumption in NCO-tracking by constructing critical regions for various switching structures. Moreover, addressing the problem directly in continuous-time can potentially reduce the number of critical regions compared with standard multi-parametric programming based on a time discretization and a control vector parameterization. The methodology and its benefits are illustrated for a number of simple case studies. To obtain the mathematical representation of the generally nonlinear critical regions, Chapter 3 investigates a machine learning model as a classifier, based on deep neural network. This feed-forward network is selected for its representational power as a universal approximator for arbitrary continuous functions. Here, the classifier takes the unknown parameter as input and maps the corresponding critical regions in terms of their switching structures. An algorithm for training the classifier is presented, which involves generating the training data set, setting up a neural network architecture, and applying optimization based training. By using a Softmax classifier in the output layer of the network, a normalized probability distribution is obtained, which consist of a vector with as many elements as the total number of critical regions, and each element representing the likelihood for a region to be the correct one. The classifier is conveniently embedded into the multi-parametric NCO-tracking controller for choosing the real-time switching structure in on-line control. Lastly, a robustification of the mp-NCO-tracking methodology is developed in Chapter 4, where constraints are guaranteed to be satisfied under all possible uncertainty scenarios, which leads to a min-max formulation. A robust counterpart formulation of the multi-parametric dynamic optimization problem is presented, which considers both additive or multiplicative time-varying disturbances. The approach involves backing-off the path and terminal constraints of the linear-quadratic optimal control problem based on a worst-case uncertainty propagation computed using either interval or ellipsoidal reachability tubes. The uncertain system state is decomposed into a nominal reference and a perturbed component, and a convex enclosure of the reachable set for the perturbed component is precomputed via some auxiliary differential equations. Conservative constraint back-offs are obtained from the precomputed reachability tubes, which enables the controller design procedure in the nominal case to be directly applied for the robust control problem, and to retain the same computational effort as in the nominal case. These developments are demonstrated by numerical case studies, and ways of extending this approach to more general, nonlinear optimal control problems are discussed in Chapter 5.Open Acces

ROAM: a Rich Object Appearance Model with Application to Rotoscoping

Rotoscoping, the detailed delineation of scene elements through a video shot, is a painstaking task of tremendous importance in professional post-production pipelines. While pixel-wise segmentation techniques can help for this task, professional rotoscoping tools rely on parametric curves that offer the artists a much better interactive control on the definition, editing and manipulation of the segments of interest. Sticking to this prevalent rotoscoping paradigm, we propose a novel framework to capture and track the visual aspect of an arbitrary object in a scene, given a first closed outline of this object. This model combines a collection of local foreground/background appearance models spread along the outline, a global appearance model of the enclosed object and a set of distinctive foreground landmarks. The structure of this rich appearance model allows simple initialization, efficient iterative optimization with exact minimization at each step, and on-line adaptation in videos. We demonstrate qualitatively and quantitatively the merit of this framework through comparisons with tools based on either dynamic segmentation with a closed curve or pixel-wise binary labelling

Adjoint-based predictor-corrector sequential convex programming for parametric nonlinear optimization

This paper proposes an algorithmic framework for solving parametric optimization problems which we call adjoint-based predictor-corrector sequential convex programming. After presenting the algorithm, we prove a contraction estimate that guarantees the tracking performance of the algorithm. Two variants of this algorithm are investigated. The first one can be used to solve nonlinear programming problems while the second variant is aimed to treat online parametric nonlinear programming problems. The local convergence of these variants is proved. An application to a large-scale benchmark problem that originates from nonlinear model predictive control of a hydro power plant is implemented to examine the performance of the algorithms.Comment: This manuscript consists of 25 pages and 7 figure