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This book deals with optimization methods as tools for decision making and control in the presence of model uncertainty. It is oriented to the use of these tools in engineering, specifically in automatic control design with all its components: analysis of dynamical systems, identification problems, and feedback control design.

Developments in Model-Based Optimization and Control takes advantage of optimization-based formulations for such classical feedback design objectives as stability, performance and feasibility, afforded by the established body of results and methodologies constituting optimal control theory. It makes particular use of the popular formulation known as predictive control or receding-horizon optimization.

The individual contributions in this volume are wide-ranging in subject matter but coordinated within a five-part structure covering material on:

· complexity and structure in model predictive control (MPC);

· collaborative MPC;

· distributed MPC;

· optimization-based analysis and design; and

· applications to bioprocesses, multivehicle systems or energy management.

The various contributions cover a subject spectrum including inverse optimality and more modern decentralized and cooperative formulations of receding-horizon optimal control. Readers will find fourteen chapters dedicated to optimization-based tools for robustness analysis, and decision-making in relation to feedback mechanisms—fault detection, for example—and three chapters putting forward applications where the model-based optimization brings a novel perspective.

Developments in Model-Based Optimization and Control is a selection of contributions expanded and updated from the Optimisation-based Control and Estimation workshops held in November 2013 and November 2014. It forms a useful resource for academic researchers and graduate students interested in the state of the art in predictive control. Control engineers working in model-based optimization and control, particularly in its bioprocess applications will also find this collection instructive.



Complexity and Structural Properties of Linear Model Predictive Control


Chapter 1. Complexity Certifications of First-Order Inexact Lagrangian Methods for General Convex Programming: Application to Real-Time MPC

In this chapter, we derive the computational complexity certifications of first-order inexact dual methods for solving general smooth constrained convex problems which can arise in real-time applications, such as model predictive control. When it is difficult to project on the primal constraint set described by a collection of general convex functions, we use the Lagrangian relaxation to handle the complicated constraints and then, we apply dual (fast) gradient algorithms based on inexact dual gradient information for solving the corresponding dual problem. The iteration complexity analysis is based on two types of approximate primal solutions: the primal last iterate and an average of primal iterates. We provide sublinear computational complexity estimates on the primal suboptimality and constraint (feasibility) violation of the generated approximate primal solutions. In the final part of the chapter, we present an open-source quadratic optimization solver, referred to as DuQuad, for convex quadratic programs and for evaluation of its behavior. The solver contains the C-language implementations of the analyzed algorithms.
Ion Necoara, Andrei Patrascu, Angelia Nedić

Chapter 2. Fully Inverse Parametric Linear/Quadratic Programming Problems via Convex Liftings

In this chapter, we present in an unified manner the latest developments on inverse optimality problem for continuous piecewise affine (PWA) functions. A particular attention is given to convex liftings as a cornerstone for the constructive solution we advocate in this framework. Subsequently, an algorithm based on convex lifting is presented for recovering a continuous PWA function defined over a polyhedral partition of a polyhedron. We also prove that any continuous PWA function can be equivalently obtained by a parametric linear programming problem with at most one auxiliary one-dimensional variable.
Ngoc Anh Nguyen, Sorin Olaru, Pedro Rodriguez-Ayerbe, Morten Hovd, Ion Necoara

Chapter 3. Implications of Inverse Parametric Optimization in Model Predictive Control

Recently, inverse parametric linear/quadratic programming problem was shown to be solvable via convex liftings approach [13]. This technique turns out to be relevant in explicit model predictive control (MPC) design in terms of reducing the prediction horizon to at most two steps. In view of practical applications, typically leading to problems that are not directly invertible, we show how to adapt the inverse optimality to specific, possibly convexly non-liftable partitions. Case study results moreover indicate that such an extension leads to controllers of lower complexity without loss of optimality. Numerical data are also presented for illustration.
Martin Gulan, Ngoc Anh Nguyen, Sorin Olaru, Pedro Rodriguez-Ayerbe, Boris Rohal’-Ilkiv

Distributed-coordinated and Multi-objective Features of Model Predictive Control


Chapter 4. Distributed Robust Model Predictive Control of Interconnected Polytopic Systems

A suboptimal approach to distributed robust MPC for uncertain systems consisting of polytopic subsystems with coupled dynamics subject to both state and input constraints is proposed. The robustness is defined in terms of the optimization of a cost function accumulated over the uncertainty and satisfying state constraints for a finite subset of uncertainties. The approach reformulates the original centralized robust MPC problem into a quadratic programming problem, which is solved by distributed iterations of the dual accelerated gradient method. A stopping condition is used that allows the iterations to stop when the desired performance, stability, and feasibility can be guaranteed. This allows for the approach to be used in an embedded robust MPC implementation. The developed method is illustrated on a simulation example of an uncertain system consisting of two interconnected polytopic subsystems.
Alexandra Grancharova, Sorin Olaru

Chapter 5. Optimal Distributed-Coordinated Approach for Energy Management in Multisource Electric Power Generation Systems

In the context of distributed power generation systems, the energy management and coordination of generators are imperative tasks to be done. Such systems, typically considered as large-scale systems, can include different dynamical and functional characteristics in both, generators and loads. In this sense, the use of distributed-coordinated control strategies, including operational constraints, becomes an interesting alternative for these applications. This chapter proposes a novel price-driven coordination technique. The approach considers that a centralized optimal control problem can be splitted into several unconstrained controlled subsystems, all coordinated by an agent which is intended for accomplishing the global performance, while assuring the system constraints. The approach is applied to a microgrid that combines different generation technologies and load profiles.
John Sandoval-Moreno, John Jairo Martínez, Gildas Besançon

Chapter 6. Evolutionary Game-Based Dynamical Tuning for Multi-objective Model Predictive Control

Model predictive control (MPC) is one of the most used optimization-based control strategies for large-scale systems, since this strategy allows to consider a large number of states and multi-objective cost functions in a straightforward way. One of the main issues in the design of multi-objective MPC controllers, which is the tuning of the weights associated to each objective in the cost function, is treated in this work. All the possible combinations of weights within the cost function affect the optimal result in a given Pareto front. Furthermore, when the system has time-varying parameters, e.g., periodic disturbances, the appropriate weight tuning might also vary over time. Moreover, taking into account the computational burden and the selected sampling time in the MPC controller design, the computation time to find a suitable tuning is limited. In this regard, the development of strategies to perform a dynamical tuning in function of the system conditions potentially improves the closed-loop performance. In order to adapt in a dynamical way the weights in the MPC multi-objective cost function, an evolutionary game approach is proposed. This approach allows to vary the prioritization weights in the proper direction taking as a reference a desired region within the Pareto front. The proper direction for the prioritization is computed by only using the current system values, i.e., the current optimal control action and the measurement of the current states, which establish the system cost function over a certain point in the Pareto front. Finally, some simulations of a multi-objective MPC for a real multivariable case study show a comparison between the system performance obtained with static and dynamical tuning.
Julián Barreiro-Gomez, Carlos Ocampo-Martinez, Nicanor Quijano

Collaborative Model Predictive Control


Chapter 7. A Model Predictive Control-Based Architecture for Cooperative Path-Following of Multiple Unmanned Aerial Vehicles

This chapter proposes a sampled-data model predictive control (MPC) architecture to solve the decentralized cooperative path-following (CPF) problem of multiple unmanned aerial vehicles (UAVs). In the cooperative path-following proposed scenario, which builds on previous work on CPF, multiple vehicles are required to follow pre-specified paths at nominal speed profiles (that may be path dependent) while keeping a desired, possibly time-varying, geometric formation pattern. In the proposed framework, we exploit the potential of optimization-based control strategies with significant advantages on explicitly addressing input and state constraints and on the ability to allow the minimization of meaningful cost functions. An example consisting of three fixed wing UAVs that are required to follow a given desired maneuver illustrates the proposed framework. We highlight and discuss some features of the UAVs trajectories.
Alessandro Rucco, António Pedro Aguiar, Fernando A. C. C. Fontes, Fernando Lobo Pereira, João Borges de Sousa

Chapter 8. Predictive Control for Path-Following. From Trajectory Generation to the Parametrization of the Discrete Tracking Sequences

This chapter discusses a series of developments on predictive control for path following via a priori generated trajectory for autonomous aerial vehicles. The strategy partitions itself into offline and runtime procedures with the assumed goal of moving the computationally expensive part into the offline phase and of leaving only tracking decisions to the runtime. First, it will be recalled that differential flatness represents a well-suited tool for generating feasible reference trajectory. Next, an optimization-based control problem which minimizes the tracking error for the nonholonomic system is formulated and further enhanced via path following mechanisms. Finally, possible changes of the selection of sampling times along the path and their impact on the predictive control formulation will be discussed in detail.
Ionela Prodan, Sorin Olaru, Fernando A.C.C. Fontes, Fernando Lobo Pereira, João Borges de Sousa, Cristina Stoica Maniu, Silviu-Iulian Niculescu

Chapter 9. Formation Reconfiguration Using Model Predictive Control Techniques for Multi-agent Dynamical Systems

The classical objective for multiple agents evolving in the same environment is the preservation of a predefined formation because it reinforces the safety of the global system and further lightens the supervision task. One of the major issues for this objective is the task assignment problem, which can be formulated in terms of an optimization problem by employing set-theoretic methods. In real time the agents will be steered into the defined formation via task (re)allocation and classical feedback mechanisms. The task assignment calculation is often performed in an offline design stage, without considering the possible variation of the number of agents in the global system. These changes (i.e., including/excluding an agent from a formation) can be regarded as a typical fault, due to some serious damages on the components or due to the operator decision. In this context, the present chapter proposes a new algorithm for the dynamical task assignment formulation of multi-agent systems in view of real-time optimization by including fault detection and isolation capabilities. This algorithm allows to detect whether there is a fault in the global multi-agent system, to isolate the faulty agent and to integrate a recovered/healthy agent. The proposed methods will be illustrated by means of a numerical example with connections to multi-vehicle systems.
Minh Tri Nguyen, Cristina Stoica Maniu, Sorin Olaru, Alexandra Grancharova

Applications of Optimization-Based Control and Identification


Chapter 10. Optimal Operation of a Lumostatic Microalgae Cultivation Process

This chapter proposes the optimization of batch microalgae cultures in artificially lighted photobioreactors. The strategy consists in controlling the incident light intensity so that the microalgae growth rate is maximized. Two approaches were developed and compared. In the first one, the ratio between the incident light intensity and the cell concentration (light-to-microalgae ratio) is optimized, either offline or online, and then maintained at its optimal value. In the second approach, the cells growth rate is maintained at its optimal value by means of nonlinear model predictive controller (NMPC). The proposed control strategies are illustrated and their efficiency is assessed, in simulation, for Chlamydomonas reinhardtii batch cultures. The proposed lumostatic operation strategies are shown to lead to a higher cell productivity and to a more efficient light utilization in comparison to conventional constant light operation approach.
Sihem Tebbani, Mariana Titica, George Ifrim, Marian Barbu, Sergiu Caraman

Chapter 11. Bioprocesses Parameter Estimation by Heuristic Optimization Techniques

This work presents a bioprocesses parameter estimation method based on heuristic optimization approaches. The identification problem is formulated as a multimodal numerical optimization problem in a high-dimensional space. Then, the optimization problem is split in simpler sub-problems that require fewer computational resources. The main results are obtained using genetic algorithms (GA) and particle swarm optimization (PSO) methods. One applies three global-search metaheuristic algorithms for numerical optimization: two variants of PSO and one type of genetic algorithm. The estimation procedures are applied for identification of a bacterial growth model associated with the enzymatic catalysis where reaction kinetics is described by Monod and Haldane models. The performances of the proposed methods are analysed by numerical simulations. The simulation results indicate that the proposed metaheuristic algorithms are effective and efficient, and demonstrate that the applied techniques exhibit a significant performance improvement over classical optimization methods.
Dorin Şendrescu, Sihem Tebbani, Dan Selişteanu

Chapter 12. Real-Time Experimental Implementation of Predictive Control Schemes in a Small-Scale Pasteurization Plant

This chapter proposes three closed-loop control topologies based on model predictive control (MPC) for a small-scale pasteurization plant. The topologies are designed taking into account the role of the predictive controller within the loop: (i) as supervisor control for the computation of the references for regulatory controllers, (ii) as unique controller within the closed loop and (iii) acting simultaneously as supervisor and regulatory controllers together with other regulatory controllers. All control designs have been applied in real time to a test bench station, then experimental results are both presented and discussed. The main advantages and drawbacks for each topology are presented for the regulation of the temperature of the output product, while the energy consumption of the overall system is minimized.
Albert Rosich, Carlos Ocampo-Martinez

Optimization-Based Analysis and Design for Particular Classes of Dynamical Systems


Chapter 13. An Optimization-Based Framework for Impulsive Control Systems

This chapter concerns a discrete-time sampling state feedback control optimizing framework for dynamic impulsive systems. This class of control systems differs from the conventional ones in that the control space is enlarged to contain measures and, thus, the associated trajectories are merely of bounded variation. In other words, it may well exhibit jumps. We adopt the most recent impulsive control solution concept that pertains to important classes of engineering systems and, in this context, present impulsive control theory results on invariance, stability, and sampled data trajectories having in mind the optimization-based framework that relies on an MPC-like scheme. The stability of the proposed MPC scheme is addressed.
Fernando Lobo Pereira, Fernando A. C. C. Fontes, António Pedro Aguiar, João Borges de Sousa

Chapter 14. Robustness Issues in Control of Bilinear Discrete-Time Systems—Applied to the Control of Power Converters

Recently, a controller design method based on Sum of Squares programming has been developed for the control of discrete-time bilinear systems, and applications to power converters have been studied. In the present work, robustness issues arising in these designs are studied. First, the issue of change of operating point is addressed, and relevant stability analysis is developed. For linear systems, one can simply “shift the origin” of the deviation variables to obtain the same behavior for a new operating point. For nonlinear systems, in contrast, one will experience changed dynamics when applying the same controller at a new operating point (even after “shifting the origin”). New criteria are introduced to verify the stability of designed controller for other desired operating points. A related topic that is covered is the introduction of integral action in the bilinear controller design, giving offset-free control for persistent disturbances. The effectiveness of the proposed methods are evaluated based on time-domain simulations of a boost converter.
Mohsen Vatani, Morten Hovd, Sorin Olaru

Chapter 15. On the LPV Control Design and Its Applications to Some Classes of Dynamical Systems

In this chapter, a control design approach based on linear parameter-varying (LPV) systems, which can be exploited to solve several problems typically encountered in control engineering, is presented. By means of recent techniques based on Youla–Kucera parametrization, it is shown how it is possible not only to design and optimize stabilizing controllers, but also to exploit the structure of the Youla–Kucera parametrized controller to face and solve side problems, including: (a) dealing with nonlinearities; (b) taking into account control input constraints; (c) performing controller commutation or online adaptation, e.g., in the presence of faults; and (d) dealing with delays in the system. The control scheme is observer-based, namely a prestabilizing observer-based precompensator is applied. Consequently, a Youla–Kucera parameter is applied to produce a supplementary input ignition, which is a function of the residual value (the difference between the output and the estimated output). Based on the fact that any stable operator which maps the residual to the supplementary input preserves stability, several additional features can be added to the compensator, without compromising the loop stability.
Franco Blanchini, Daniele Casagrande, Giulia Giordano, Stefano Miani

Chapter 16. Ultimate Bounds and Robust Invariant Sets for Linear Systems with State-Dependent Disturbances

The objective of this chapter is to present a methodology for computing robust positively invariant sets for linear, discrete time-invariant systems that are affected by additive disturbances, with the particularity that these disturbances are subject to state-dependent bounds. The proposed methodology requires less restrictive assumptions compared to similar established techniques, while it provides the framework for determining the state-dependent (parameterized) ultimate bounds for several classes of disturbances. The added value of the proposed approach is illustrated by an optimization-based problem for detecting the mode of functioning of a switching system.
Sorin Olaru, Vasso Reppa

Chapter 17. RPI Approximations of the mRPI Set Characterizing Linear Dynamics with Zonotopic Disturbances

In this chapter we provide a robust positive invariance (RPI) outer-approximation of the minimal RPI (mRPI) set associated to linear dynamics with zonotopic disturbances. We prove that the candidate sets considered are either RPI or become so with a scaling factor. The results base on the concomitant computation of extremal points and their extremal hyperplanes. Further, we consider the equivalence with ultimate bounds constructions and show that successive RPI representations become monotonically “tighter” as their complexity increases. The results are tested in illustrative examples.
Florin Stoican, Cristian Oară, Morten Hovd


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