On the application of optimal control techniques in lossy fieldbuses

The performance of real-time networks is under continuous improvement as a result of several trends in the digital world. However, these tendencies not only cause improvements, but also exacerbates a series of unideal aspects of real-time networks such as communication latency, jitter of the latency and packet drop rate. This Thesis focuses on the communication errors that appear on such realtime networks, from the point-of-view of automatic control. Specifically, it investigates the effects of packet drops in automatic control over fieldbuses, as well as the architectures and optimal techniques for their compensation. Firstly, a new approach to address the problems that rise in virtue of such packet drops, is proposed. This novel approach is based on the simultaneous transmission of several values in a single message. Such messages can be from sensor to controller, in which case they are comprised of several past sensor readings, or from controller to actuator in which case they are comprised of estimates of several future control values. A series of tests reveal the advantages of this approach. The above-explained approach is then expanded as to accommodate the techniques of contemporary optimal control. However, unlike the aforementioned approach, that deliberately does not send certain messages in order to make a more efficient use of network resources; in the second case, the techniques are used to reduce the effects of packet losses. After these two approaches that are based on data aggregation, it is also studied the optimal control in packet dropping fieldbuses, using generalized actuator output functions. This study ends with the development of a new optimal controller, as well as the function, among the generalized functions that dictate the actuator’s behaviour in the absence of a new control message, that leads to the optimal performance. The Thesis also presents a different line of research, related with the output oscillations that take place as a consequence of the use of classic co-design techniques of networked control. The proposed algorithm has the goal of allowing the execution of such classical co-design algorithms without causing an output oscillation that increases the value of the cost function. Such increases may, under certain circumstances, negate the advantages of the application of the classical co-design techniques. A yet another line of research, investigated algorithms, more efficient than contemporary ones, to generate task execution sequences that guarantee that at least a given number of activated jobs will be executed out of every set composed by a predetermined number of contiguous activations. This algorithm may, in the future, be applied to the generation of message transmission patterns in the above-mentioned techniques for the efficient use of network resources. The proposed task generation algorithm is better than its predecessors in the sense that it is capable of scheduling systems that cannot be scheduled by its predecessor algorithms. The Thesis also presents a mechanism that allows to perform multi-path routing in wireless sensor networks, while ensuring that no value will be counted in duplicate. Thereby, this technique improves the performance of wireless sensor networks, rendering them more suitable for control applications. As mentioned before, this Thesis is centered around techniques for the improvement of performance of distributed control systems in which several elements are connected through a fieldbus that may be subject to packet drops. The first three approaches are directly related to this topic, with the first two approaching the problem from an architectural standpoint, whereas the third one does so from more theoretical grounds. The fourth approach ensures that the approaches to this and similar problems that can be found in the literature that try to achieve goals similar to objectives of this Thesis, can do so without causing other problems that may invalidate the solutions in question. Then, the thesis presents an approach to the problem dealt with in it, which is centered in the efficient generation of the transmission patterns that are used in the aforementioned approaches.

Regularization and Bang-bang conjugate times in optimal control

Consideramos o problema de controlo óptimo de tempo mínimo para sistemas de controlo mono-entrada e controlo afim num espaço de dimensão finita com condições inicial e final fixas, onde o controlo escalar toma valores num intervalo fechado. Quando aplicamos o método de tiro a este problema, vários obstáculos podem surgir uma vez que a função de tiro não é diferenciável quando o controlo é bang-bang. No caso bang-bang os tempos conjugados são teoricamente bem definidos para este tipo de sistemas de controlo, contudo os algoritmos computacionais directos disponíveis são de difícil aplicação. Por outro lado, no caso suave o conceito teórico e prático de tempos conjugados é bem conhecido, e ferramentas computacionais eficazes estão disponíveis. Propomos um procedimento de regularização para o qual as soluções do problema de tempo mínimo correspondente dependem de um parâmetro real positivo suficientemente pequeno e são definidas por funções suaves em relação à variável tempo, facilitando a aplicação do método de tiro simples. Provamos, sob hipóteses convenientes, a convergência forte das soluções do problema regularizado para a solução do problema inicial, quando o parâmetro real tende para zero. A determinação de tempos conjugados das trajectórias localmente óptimas do problema regularizado enquadra-se na teoria suave conhecida. Provamos, sob hipóteses adequadas, a convergência do primeiro tempo conjugado do problema regularizado para o primeiro tempo conjugado do problema inicial bang-bang, quando o parâmetro real tende para zero. Consequentemente, obtemos um algoritmo eficiente para a computação de tempos conjugados no caso bang-bang.

Optimal control and numerical optimization applied to epidemiological models

A relação entre a epidemiologia, a modelação matemática e as ferramentas computacionais permite construir e testar teorias sobre o desenvolvimento e combate de uma doença. Esta tese tem como motivação o estudo de modelos epidemiológicos aplicados a doenças infeciosas numa perspetiva de Controlo Ótimo, dando particular relevância ao Dengue. Sendo uma doença tropical e subtropical transmitida por mosquitos, afecta cerca de 100 milhões de pessoas por ano, e é considerada pela Organização Mundial de Saúde como uma grande preocupação para a saúde pública. Os modelos matemáticos desenvolvidos e testados neste trabalho, baseiam-se em equações diferenciais ordinárias que descrevem a dinâmica subjacente à doença nomeadamente a interação entre humanos e mosquitos. É feito um estudo analítico dos mesmos relativamente aos pontos de equilíbrio, sua estabilidade e número básico de reprodução. A propagação do Dengue pode ser atenuada através de medidas de controlo do vetor transmissor, tais como o uso de inseticidas específicos e campanhas educacionais. Como o desenvolvimento de uma potencial vacina tem sido uma aposta mundial recente, são propostos modelos baseados na simulação de um hipotético processo de vacinação numa população. Tendo por base a teoria de Controlo Ótimo, são analisadas as estratégias ótimas para o uso destes controlos e respetivas repercussões na redução/erradicação da doença aquando de um surto na população, considerando uma abordagem bioeconómica. Os problemas formulados são resolvidos numericamente usando métodos diretos e indiretos. Os primeiros discretizam o problema reformulando-o num problema de optimização não linear. Os métodos indiretos usam o Princípio do Máximo de Pontryagin como condição necessária para encontrar a curva ótima para o respetivo controlo. Nestas duas estratégias utilizam-se vários pacotes de software numérico. Ao longo deste trabalho, houve sempre um compromisso entre o realismo dos modelos epidemiológicos e a sua tratabilidade em termos matemáticos.

Computational methods in the fractional calculus of variations and optimal control

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.

Control design for multicast-aware class-based networks

The expectations of citizens from the Information Technologies (ITs) are increasing as the ITs have become integral part of our society, serving all kinds of activities whether professional, leisure, safety-critical applications or business. Hence, the limitations of the traditional network designs to provide innovative and enhanced services and applications motivated a consensus to integrate all services over packet switching infrastructures, using the Internet Protocol, so as to leverage flexible control and economical benefits in the Next Generation Networks (NGNs). However, the Internet is not capable of treating services differently while each service has its own requirements (e.g., Quality of Service - QoS). Therefore, the need for more evolved forms of communications has driven to radical changes of architectural and layering designs which demand appropriate solutions for service admission and network resources control. This Thesis addresses QoS and network control issues, aiming to improve overall control performance in current and future networks which classify services into classes. The Thesis is divided into three parts. In the first part, we propose two resource over-reservation algorithms, a Class-based bandwidth Over-Reservation (COR) and an Enhanced COR (ECOR). The over-reservation means reserving more bandwidth than a Class of Service (CoS) needs, so the QoS reservation signalling rate is reduced. COR and ECOR allow for dynamically defining over-reservation parameters for CoSs based on network interfaces resource conditions; they aim to reduce QoS signalling and related overhead without incurring CoS starvation or waste of bandwidth. ECOR differs from COR by allowing for optimizing control overhead minimization. Further, we propose a centralized control mechanism called Advanced Centralization Architecture (ACA), that uses a single state-full Control Decision Point (CDP) which maintains a good view of its underlying network topology and the related links resource statistics on real-time basis to control the overall network. It is very important to mention that, in this Thesis, we use multicast trees as the basis for session transport, not only for group communication purposes, but mainly to pin packets of a session mapped to a tree to follow the desired tree. Our simulation results prove a drastic reduction of QoS control signalling and the related overhead without QoS violation or waste of resources. Besides, we provide a generic-purpose analytical model to assess the impact of various parameters (e.g., link capacity, session dynamics, etc.) that generally challenge resource overprovisioning control. In the second part of this Thesis, we propose a decentralization control mechanism called Advanced Class-based resource OverpRovisioning (ACOR), that aims to achieve better scalability than the ACA approach. ACOR enables multiple CDPs, distributed at network edge, to cooperate and exchange appropriate control data (e.g., trees and bandwidth usage information) such that each CDP is able to maintain a good knowledge of the network topology and the related links resource statistics on real-time basis. From scalability perspective, ACOR cooperation is selective, meaning that control information is exchanged dynamically among only the CDPs which are concerned (correlated). Moreover, the synchronization is carried out through our proposed concept of Virtual Over-Provisioned Resource (VOPR), which is a share of over-reservations of each interface to each tree that uses the interface. Thus, each CDP can process several session requests over a tree without requiring synchronization between the correlated CDPs as long as the VOPR of the tree is not exhausted. Analytical and simulation results demonstrate that aggregate over-reservation control in decentralized scenarios keep low signalling without QoS violations or waste of resources. We also introduced a control signalling protocol called ACOR Protocol (ACOR-P) to support the centralization and decentralization designs in this Thesis. Further, we propose an Extended ACOR (E-ACOR) which aggregates the VOPR of all trees that originate at the same CDP, and more session requests can be processed without synchronization when compared with ACOR. In addition, E-ACOR introduces a mechanism to efficiently track network congestion information to prevent unnecessary synchronization during congestion time when VOPRs would exhaust upon every session request. The performance evaluation through analytical and simulation results proves the superiority of E-ACOR in minimizing overall control signalling overhead while keeping all advantages of ACOR, that is, without incurring QoS violations or waste of resources. The last part of this Thesis includes the Survivable ACOR (SACOR) proposal to support stable operations of the QoS and network control mechanisms in case of failures and recoveries (e.g., of links and nodes). The performance results show flexible survivability characterized by fast convergence time and differentiation of traffic re-routing under efficient resource utilization i.e. without wasting bandwidth. In summary, the QoS and architectural control mechanisms proposed in this Thesis provide efficient and scalable support for network control key sub-systems (e.g., QoS and resource control, traffic engineering, multicasting, etc.), and thus allow for optimizing network overall control performance.

Optimal control of affine connection control systems from the point of view of Lie algebroids

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems (ACCSs) on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

The Herglotz variational problem on spheres and its optimal control approach

The main goal of this paper is to extend the generalized variational problem of Herglotz type to the more general context of the Euclidean sphere S^n. Motivated by classical results on Euclidean spaces, we derive the generalized Euler-Lagrange equation for the corresponding variational problem defined on the Riemannian manifold S^n. Moreover, the problem is formulated from an optimal control point of view and it is proved that the Euler-Lagrange equation can be obtained from the Hamiltonian equations. It is also highlighted the geodesic problem on spheres as a particular case of the generalized Herglotz problem.

Discrete-time positive periodic systems with state and control constraints

The aim of this paper is to provide an efficient control design technique for discrete-time positive periodic systems. In particular, stability, positivity and periodic invariance of such systems are studied. Moreover, the concept of periodic invariance with respect to a collection of boxes is introduced and investigated with connection to stability. It is shown how such concept can be used for deriving a stabilizing state-feedback control that maintains the positivity of the closed-loop system and respects states and control signals constraints. In addition, all the proposed results can be efficiently solved in terms of linear programming.

Calculus of variations on time scales and discrete fractional calculus

Estudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.

Existência de minimizantes relaxados e fenómeno de Lavrentiev

Neste trabalho prova-se a existência de minimizantes relaxados em problemas de controlo óptimo não convexos usando técnicas de compactificação. Faz-se a extensão do exemplo de Manià a dimensão dois, obtendo-se uma classe de problemas variacionais em 2D que apresentam Fenómeno de Lavrentiev. Prova-se que o fenómeno persiste a certas perturbações, obtendo- -se assim uma classe de funcionais cujos Lagrangianos são coercivos e convexos em relação ao gradiente. Adicionalmente, apresentam-se exemplos de problemas do cálculo das variações com diferentes condições de fronteira, e em diferentes tipos de domínios (incluindo domínios com fronteira fractal), que exibem Fenómeno de Lavrentiev.

Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration

We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials. This problem and the corresponding optimal control problem are described in the context of higher order tangent bundles using geometric tools. The main tool, a presymplectic variant of Pontryagin’s maximum principle, allows us to study the dynamics of the control problem.