An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

Dynamic programming for a Markov-switching jump–diffusion

We consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a Markov-switching jump–diffusion stochastic differential equation. Our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems. More specifically, we provide a detailed proof of Bellman’s optimality principle (or dynamic programming principle) and obtain the corresponding Hamilton–Jacobi–Belman equation, which turns out to be a partial integro-differential equation due to the extra terms arising from the Lévy process and the Markov process. As an application of our results, we study a finite horizon consumption– investment problem for a jump–diffusion financial market consisting of one risk-free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state Markov process. We provide a detailed study of the optimal strategies for this problem, for the economically relevant families of power utilities and logarithmic utilities.

Adaptable control for electrical generation at irregural wind speeds

The main aims of this work are the development and the validation of one generic algorithm to provide the optimal control of small power wind generators. That means up to 40 kW and blades with fixed pitch angle. This algorithm allows the development of controllers to fetch the wind generators at the desired operational point in variable operating conditions. The problems posed by the variable wind intensity are solved using the proposed algorithm. This is done with no explicit measure of the wind velocity, and so no special equipment or anemometer is required to compute or measure the wind velocity.

An optimal boundary fair scheduling

Nowadays, many real-time operating systems discretize the time relying on a system time unit. To take this behavior into account, real-time scheduling algorithms must adopt a discrete-time model in which both timing requirements of tasks and their time allocations have to be integer multiples of the system time unit. That is, tasks cannot be executed for less than one time unit, which implies that they always have to achieve a minimum amount of work before they can be preempted. Assuming such a discrete-time model, the authors of Zhu et al. (Proceedings of the 24th IEEE international real-time systems symposium (RTSS 2003), 2003, J Parallel Distrib Comput 71(10):1411–1425, 2011) proposed an efficient “boundary fair” algorithm (named BF) and proved its optimality for the scheduling of periodic tasks while achieving full system utilization. However, BF cannot handle sporadic tasks due to their inherent irregular and unpredictable job release patterns. In this paper, we propose an optimal boundary-fair scheduling algorithm for sporadic tasks (named BF TeX ), which follows the same principle as BF by making scheduling decisions only at the job arrival times and (expected) task deadlines. This new algorithm was implemented in Linux and we show through experiments conducted upon a multicore machine that BF TeX outperforms the state-of-the-art discrete-time optimal scheduler (PD TeX ), benefiting from much less scheduling overheads. Furthermore, it appears from these experimental results that BF TeX is barely dependent on the length of the system time unit while PD TeX —the only other existing solution for the scheduling of sporadic tasks in discrete-time systems—sees its number of preemptions, migrations and the time spent to take scheduling decisions increasing linearly when improving the time resolution of the system.

Application of robust fixed point transformations for technological operation of robots

A robot’s drive has to exert appropriate driving forces that can keep its arm and end effector at the proper position, velocity and acceleration, and simultaneously has to compensate for the effects of the contact forces arising between the tool and the workpiece depending on the needs of the actual technological operation. Balancing the effects of a priori unknown external disturbance forces and the inaccuracies of the available dynamic model of the robot is also important. Technological tasks requiring well prescribed end effector trajectories and contact forces simultaneously are challenging control problems that can be tackled in various manners.

Multi-criteria optimization manipulator trajectory planning

In the last twenty years genetic algorithms (GAs) were applied in a plethora of fields such as: control, system identification, robotics, planning and scheduling, image processing, and pattern and speech recognition (Bäck et al., 1997). In robotics the problems of trajectory planning, collision avoidance and manipulator structure design considering a single criteria has been solved using several techniques (Alander, 2003). Most engineering applications require the optimization of several criteria simultaneously. Often the problems are complex, include discrete and continuous variables and there is no prior knowledge about the search space. These kind of problems are very more complex, since they consider multiple design criteria simultaneously within the optimization procedure. This is known as a multi-criteria (or multiobjective) optimization, that has been addressed successfully through GAs (Deb, 2001). The overall aim of multi-criteria evolutionary algorithms is to achieve a set of non-dominated optimal solutions known as Pareto front. At the end of the optimization procedure, instead of a single optimal (or near optimal) solution, the decision maker can select a solution from the Pareto front. Some of the key issues in multi-criteria GAs are: i) the number of objectives, ii) to obtain a Pareto front as wide as possible and iii) to achieve a Pareto front uniformly spread. Indeed, multi-objective techniques using GAs have been increasing in relevance as a research area. In 1989, Goldberg suggested the use of a GA to solve multi-objective problems and since then other researchers have been developing new methods, such as the multi-objective genetic algorithm (MOGA) (Fonseca & Fleming, 1995), the non-dominated sorted genetic algorithm (NSGA) (Deb, 2001), and the niched Pareto genetic algorithm (NPGA) (Horn et al., 1994), among several other variants (Coello, 1998). In this work the trajectory planning problem considers: i) robots with 2 and 3 degrees of freedom (dof ), ii) the inclusion of obstacles in the workspace and iii) up to five criteria that are used to qualify the evolving trajectory, namely the: joint traveling distance, joint velocity, end effector / Cartesian distance, end effector / Cartesian velocity and energy involved. These criteria are used to minimize the joint and end effector traveled distance, trajectory ripple and energy required by the manipulator to reach at destination point. Bearing this ideas in mind, the paper addresses the planning of robot trajectories, meaning the development of an algorithm to find a continuous motion that takes the manipulator from a given starting configuration up to a desired end position without colliding with any obstacle in the workspace. The chapter is organized as follows. Section 2 describes the trajectory planning and several approaches proposed in the literature. Section 3 formulates the problem, namely the representation adopted to solve the trajectory planning and the objectives considered in the optimization. Section 4 studies the algorithm convergence. Section 5 studies a 2R manipulator (i.e., a robot with two rotational joints/links) when the optimization trajectory considers two and five objectives. Sections 6 and 7 show the results for the 3R redundant manipulator with five goals and for other complementary experiments are described, respectively. Finally, section 8 draws the main conclusions.

Optimal fuzzy PID control tuned with genetic algorithms

Fuzzy logic controllers (FLC) are intelligent systems, based on heuristic knowledge, that have been largely applied in numerous areas of everyday life. They can be used to describe a linear or nonlinear system and are suitable when a real system is not known or too difficult to find their model. FLC provide a formal methodology for representing, manipulating and implementing a human heuristic knowledge on how to control a system. These controllers can be seen as artificial decision makers that operate in a closed-loop system, in real time. The main aim of this work was to develop a single optimal fuzzy controller, easily adaptable to a wide range of systems – simple to complex, linear to nonlinear – and able to control all these systems. Due to their efficiency in searching and finding optimal solution for high complexity problems, GAs were used to perform the FLC tuning by finding the best parameters to obtain the best responses. The work was performed using the MATLAB/SIMULINK software. This is a very useful tool that provides an easy way to test and analyse the FLC, the PID and the GAs in the same environment. Therefore, it was proposed a Fuzzy PID controller (FL-PID) type namely, the Fuzzy PD+I. For that, the controller was compared with the classical PID controller tuned with, the heuristic Ziegler-Nichols tuning method, the optimal Zhuang-Atherton tuning method and the GA method itself. The IAE, ISE, ITAE and ITSE criteria, used as the GA fitness functions, were applied to compare the controllers performance used in this work. Overall, and for most systems, the FL-PID results tuned with GAs were very satisfactory. Moreover, in some cases the results were substantially better than for the other PID controllers. The best system responses were obtained with the IAE and ITAE criteria used to tune the FL-PID and PID controllers.

Optimal intelligent supervisory control system in cyber-physical intelligence

This paper studies Optimal Intelligent Supervisory Control System (OISCS) model for the design of control systems which can work in the presence of cyber-physical elements with privacy protection. The development of such architecture has the possibility of providing new ways of integrated control into systems where large amounts of fast computation are not easily available, either due to limitations on power, physical size or choice of computing elements.

Stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

Optimal controllers with complex order derivatives

This paper studies the optimization of complex-order algorithms for the discrete-time control of linear and nonlinear systems. The fundamentals of fractional systems and genetic algorithms are introduced. Based on these concepts, complexorder control schemes and their implementation are evaluated in the perspective of evolutionary optimization. The results demonstrate not only that complex-order derivatives constitute a valuable alternative for deriving control algorithms, but also the feasibility of the adopted optimization strategy.

Optimal tuning of fractional controllers using genetic algorithms

This study addresses the optimization of fractional algorithms for the discrete-time control of linear and non-linear systems. The paper starts by analyzing the fundamentals of fractional control systems and genetic algorithms. In a second phase the paper evaluates the problem in an optimization perspective. The results demonstrate the feasibility of the evolutionary strategy and the adaptability to distinct types of systems.

Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms

This study addresses the optimization of rational fraction approximations for the discrete-time calculation of fractional derivatives. The article starts by analyzing the standard techniques based on Taylor series and Padé expansions. In a second phase the paper re-evaluates the problem in an optimization perspective by tacking advantage of the flexibility of the genetic algorithms.

Probability and statistics: selected problems

Probability and Statistics—Selected Problems is a unique book for senior undergraduate and graduate students to fast review basic materials in Probability and Statistics. Descriptive statistics are presented first, and probability is reviewed secondly. Discrete and continuous distributions are presented. Sample and estimation with hypothesis testing are presented in the last two chapters. The solutions for proposed excises are listed for readers to references.

Otimização dos sistemas biológicos e melhorias nos tratamentos da ETARI da Refinaria de Matosinhos

A Refinaria de Matosinhos é um dos complexos industriais da Galp Energia. A sua estação de tratamento de águas residuais industriais (ETARI) – designada internamente por Unidade 7000 – é composta por quatro tratamentos: o pré-tratamento, o tratamento físico-químico, o tratamento biológico e o pós-tratamento. Dada a interligação existente, é fundamental a otimização de cada um dos tratamentos. Este trabalho teve como objetivos a identificação dos problemas e/ou possibilidades de melhoria do pré-tratamento, tratamento físico-químico e pós-tratamento e principalmente a otimização do tratamento biológico da ETARI. No pré-tratamento verificou-se que a separação de óleos e lamas não era eficaz uma vez que se formam emulsões destas duas fases. Como solução, sugeriu-se a adição de agentes desemulsionantes, que se revelou economicamente inviável. Assim, sugeriu-se como alternativa o recurso a técnicas de tratamento da emulsão gerada, tais como a extração com solvente, centrifugação, ultrassons e micro-ondas. No tratamento físico-químico constatou-se que o controlo da unidade de saturação de ar na água era feito com base na análise visual dos operadores, o que pode conduzir a condições de operação afastadas das ótimas para este tratamento. Assim, sugeriu-se a realização de um estudo de otimização desta unidade com vista à determinação da razão ar/sólidos ótima para este efluente. Para além disto, constatou-se, ainda, que os consumos de coagulante aumentaram cerca de -- % no último ano, pelo que foi sugerido o estudo da viabilidade do processo de eletrocoagulação como substituto do sistema de coagulação existente. No pós-tratamento identificou-se o processo de lavagem dos filtros como sendo a etapa com possibilidade de ser otimizada. Através de um estudo preliminar concluiu-se que a lavagem contínua de um filtro por cada turno melhorava o desempenho dos mesmos. Constatou-se, ainda, que a introdução de ar comprimido na água de lavagem promove uma maior remoção de detritos do leito de areia, no entanto esta prática parece influenciar negativamente o desempenho dos filtros. No caso do tratamento biológico, identificaram-se problemas ao nível do tempo de retenção hidráulico do tratamento biológico II, que apresentou elevada variabilidade. Apesar de identificado concluiu-se que este problema era de difícil solução. Verificou-se, também, que o oxigénio dissolvido não era monitorizado, pelo que se sugeriu a instalação de uma sonda de oxigénio dissolvido numa zona de baixa turbulência do tanque de arejamento. Concluiu-se que o oxigénio era distribuído de forma homogénea por todo o tanque de arejamento e tentou-se identificar quais os fatores que influenciariam este parâmetro, no entanto, dada a elevada variabilidade do efluente e das condições de tratamento, tal não foi possível. Constatou-se, também, que o doseamento de fosfato para o tratamento biológico II era pouco eficiente já Otimização dos sistemas biológicos e melhorias nos tratamentos da ETARI da Refinaria de Matosinhos que em -- % dos dias se verificaram níveis baixos de fosfato no licor misto (< - mg/L). Foi, por isso, proposta a alteração do atual sistema de doseamento por gravidade para um sistema de bomba doseadora. Para além disso verificou-se que os consumos deste nutriente aumentaram significativamente no último ano (cerca de --%), situação que se constatou estar relacionada com um aumento da população microbiana para este período. Foi possível relacionar-se o aparecimento frequente de lamas à superfície dos decantadores secundários com incrementos repentinos de condutividade, pelo que se sugeriu o armazenamento do efluente nas bacias de tempestade, nestas situações. Verificou-se que a remoção de azoto era praticamente ineficaz uma vez que a conversão de azoto amoniacal em nitratos foi muito baixa. Assim, sugeriu-se o recurso à técnica de bio-augmentação ou a transformação do sistema de lamas ativadas num sistema bietápico. Por fim, constatou-se que a temperatura do efluente à entrada da ETARI apresenta valores bastante elevados para o tratamento biológico (aproximadamente de --º C) pelo que se sugeriu a instalação de uma sonda de temperatura no tanque de arejamento de modo a controlar de forma mais eficaz a temperatura do licor misto. Ainda no que diz respeito ao tratamento biológico, foi possível desenvolver-se um conjunto de ferramentas que visaram o funcionamento otimizado deste tratamento. Nesse sentido, foram apresentadas várias sugestões de melhoria: a utilização do índice volumétrico de lamas como indicador da qualidade das lamas em alternativa à percentagem de lamas; foi desenvolvido um conjunto de fluxogramas para a orientação dos operadores de exterior na resolução de problemas; foi criada uma “janela de operação” que pretende ser um guia de apoio à operação; foi ainda proposta a monitorização frequente da idade das lamas e da razão alimento/microrganismo.

An efficient method for optimal location and sizing of fixed and switched shunt capacitors in large distribution systems

This paper proposes a computationally efficient methodology for the optimal location and sizing of static and switched shunt capacitors in large distribution systems. The problem is formulated as the maximization of the savings produced by the reduction in energy losses and the avoided costs due to investment deferral in the expansion of the network. The proposed method selects the nodes to be compensated, as well as the optimal capacitor ratings and their operational characteristics, i.e. fixed or switched. After an appropriate linearization, the optimization problem was formulated as a large-scale mixed-integer linear problem, suitable for being solved by means of a widespread commercial package. Results of the proposed optimizing method are compared with another recent methodology reported in the literature using two test cases: a 15-bus and a 33-bus distribution network. For the both cases tested, the proposed methodology delivers better solutions indicated by higher loss savings, which are achieved with lower amounts of capacitive compensation. The proposed method has also been applied for compensating to an actual large distribution network served by AES-Venezuela in the metropolitan area of Caracas. A convergence time of about 4 seconds after 22298 iterations demonstrates the ability of the proposed methodology for efficiently handling large-scale compensation problems.