Fernando A. C. C. Fontes

A general framework to design stabilizing nonlinear model predictive controllers

Abstract We propose a new model predictive control (MPC) framework to generate feedback controls for time-varying nonlinear systems with input constraints. We provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered. The supplied sufficient conditions for stability can also be used to analyse the stability of most previous MPC schemes. The class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system. Some important classes of nonlinear systems, including some nonholonomic systems, can now be stabilized by MPC. In addition, we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem, and thereby reduce the computational effort in the optimization algorithms used to implement MPC.

Special section technical notes and correspondence. Min-max model predictive control of nonlinear systems using discontinuous feedbacks

This note proposes a model predictive control (MPC) algorithm for the solution of a robust control problem for continuous-time systems. Discontinuous feedback strategies are allowed in the solution of the min-max problems to be solved. The use of such strategies allows MPC to address a large class of nonlinear systems, including among others nonholonomic systems. Robust stability conditions to ensure steering to a certain set under bounded disturbances are established. The use of bang-bang feedbacks described by a small number of parameters is proposed, reducing considerably the computational burden associated with solving a differential game. The applicability of the proposed algorithm is tested to control a unicycle mobile robot.

Model predictive control of vehicle formations

We propose a two-layer scheme to control a set of vehicles moving in a formation. The first layer, the trajectory controller, is a nonlinear controller since most vehicles are nonholonomic systems and require a nonlinear, even discontinuous, feedback to stabilize them. The trajectory controller, a model predictive controller, computes centrally a bang-bang control law and only a small set of parameters need to be transmitted to each vehicle at each iteration. The second layer, the formation controller, aims to compensate for small changes around a nominal trajectory maintaining the relative po- sitions between vehicles. We argue that the formation control can be, in most cases, adequately carried out by a linear model predictive controller accommodating input and state constraints. This has the advantage that the control laws for each vehicle are simple piecewise affine feedback laws that can be pre-computed off-line and implemented in a distributed way in each vehicle. Although several optimization problems have to be solved, the control strategy proposed results in a simple and efficient implementation where no optimization problem needs to be solved in real-time at each vehicle.

Sampled-Data Model Predictive Control for Nonlinear Time-Varying Systems: Stability and Robustness

We describe here a sampled-data Model Predictive Control framework that uses continuous-time models but the sampling of the actual state of the plant as well as the computation of the control laws, are carried out at discrete instants of time. This framework can address a very large class of systems, nonlinear, time-varying, and nonholonomic.

An Euler–Lagrange Inclusion for Optimal Control Problems with State Constraints

New first-order necessary conditions for optimality for control problems with pathwise state constraints are given. These conditions are a variant of a nonsmooth maximum principle which includes a joint subdifferential of the Hamiltonian – a condition called Euler–Lagrange inclusion (ELI). The main novelty of the result provided here is the ability to address state constraints while using an ELI.The ELI conditions have a number of desirable properties. Namely, they are, in some cases, able to convey more information about minimizers, and for the normal convex problems they are sufficient conditions of optimality. It is shown that these strengths are retained in the presence of state constraints.

Concave minimum cost network flow problems solved with a colony of ants

In this work we address the Single-Source Uncapacitated Minimum Cost Network Flow Problem with concave cost functions. This problem is NP-Hard, therefore we propose a hybrid heuristic to solve it. Our goal is not only to apply an ant colony optimization (ACO) algorithm to such a problem, but also to provide an insight on the behaviour of the parameters in the performance of the algorithm. The performance of the ACO algorithm is improved with the hybridization of a local search (LS) procedure. The core ACO procedure is used to mainly deal with the exploration of the search space, while the LS is incorporated to further cope with the exploitation of the best solutions found. The method we have developed has proven to be very efficient while solving both small and large size problem instances. The problems we have used to test the algorithm were previously solved by other authors using other population based heuristics. Our algorithm was able to improve upon some of their results in terms of solution quality, proving that the HACO algorithm is a very good alternative approach to solve these problems. In addition, our algorithm is substantially faster at achieving these improved solutions. Furthermore, the magnitude of the reduction of the computational requirements grows with problem size.

A hybrid biased random key genetic algorithm approach for the unit commitment problem

This work proposes a hybrid genetic algorithm (GA) to address the unit commitment (UC) problem. In the UC problem, the goal is to schedule a subset of a given group of electrical power generating units and also to determine their production output in order to meet energy demands at minimum cost. In addition, the solution must satisfy a set of technological and operational constraints. The algorithm developed is a hybrid biased random key genetic algorithm (HBRKGA). It uses random keys to encode the solutions and introduces bias both in the parent selection procedure and in the crossover strategy. To intensify the search close to good solutions, the GA is hybridized with local search. Tests have been performed on benchmark large-scale power systems. The computational results demonstrate that the HBRKGA is effective and efficient. In addition, it is also shown that it improves the solutions obtained by current state-of-the-art methodologies.

A biased random key genetic algorithm approach for unit commitment problem

A Biased Random Key Genetic Algorithm (BRKGA) is proposed to find solutions for the unit commitment problem. In this problem, one wishes to schedule energy production on a given set of thermal generation units in order to meet energy demands at minimum cost, while satisfying a set of technological and spinning reserve constraints. In the BRKGA, solutions are encoded by using random keys, which are represented as vectors of real numbers in the interval [0,1]. The GA proposed is a variant of the random key genetic algorithm, since bias is introduced in the parent selection procedure, as well as in the crossover strategy. Tests have been performed on benchmark large-scale power systems of up to 100 units for a 24 hours period. The results obtained have shown the proposed methodology to be an effective and efficient tool for finding solutions to large-scale unit commitment problems. Furthermore, from the comparisons made it can be concluded that the results produced improve upon some of the best known solutions.

An ant colony optimization algorithm to solve the minimum cost network flow problem with concave cost functions

In this work we address the Singe-Source Uncapacitated Minimum Cost Network Flow Problem with concave cost functions. Given that this problem is of a combinatorial nature and also that the total costs are nonlinear, we propose a hybrid heuristic to solve it. In this type of algorithms one usually tries to manage two conflicting aspects of searching behaviour: exploration, the algorithms ability to search broadly through the search space; and exploitation, the algorithm ability to search locally around good solutions that have been found previously. In our case, we use an Ant Colony Optimization algorithm to mainly deal with the exploration, and a Local Search algorithm to cope with the exploitation of the search space. Our method proves to be very efficient while solving both small and large size problem instances. The problems we have used to test the algorithm were previously solved by other authors using other population based heuristics and our algorithm was able to improve upon their results, both in terms of computing time and solution quality.

Normality and Nondegeneracy for Optimal Control Problems with State Constraints

In this paper, we investigate normal and nondegenerate forms of the maximum principle for optimal control problems with state constraints. We propose new constraint qualifications guaranteeing nondegeneracy and normality that have to be checked on smaller sets of points of an optimal trajectory than those in known sufficient conditions. In fact, the constraint qualifications proposed impose the existence of an inward pointing velocity just on the instants of time for which the optimal trajectory has an outward pointing velocity.