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Neural networks can be trained to get internal working models in dynamic systems control schemes. This has usually been done designing the neural network in the form of a discrete model with delayed inputs of the NARMA type (Non-linear Auto Regressive Moving Average). In recent works the use of the neural network inside the structure of ordinary differential equations (ODE) numerical integrators has also been considered to get dynamic systems discrete models. In this paper, an extension of this latter approach, where a feed forward neural network modeling mean derivatives is used in the structure of an Euler integrator, is presented and applied in a Nonlinear Predictive Control (NPC) scheme. The use of the neural network to approximate the mean derivative function, instead of the dynamic system ODE instantaneous derivative function, allows any specified accuracy to be attained in the modeling of dynamic systems with the use of a simple Euler integrator. This makes the predictive control implementation a simpler task, since it is only necessary to deal with the linear structure of a first order integrator in the calculations of control actions. To illustrate the effectiveness of the proposed approach, results of tests in a problem of orbit transfer between Earth and Mars and in a problem of three-axis attitude control of a rigid body satellite are presented.
Redes neurais podem ser treinadas para obter o modelo de trabalho interno para esquemas de controle de sitemas dinâmicos. A forma usual adotada é projetar a rede neural na forma de um modelo discreto com entradas atrasadas do tipo NARMA (Non-linear Auto Regressive Moving Average). Em trabalhos recentes a utilização de uma rede neural inserida em uma estrutura de integração numérica tem sido também considerada para a obtenção de modelos discretos para sistemas dinâmicos. Neste trabalho, uma extensão da última abordagem é apresentada e aplicada em um esquema de controle não-linear preditivo (NPC), com uma rede feed forward modelando as derivadas médias em uma estrutura de integrador numérico de Euler. O uso de uma rede neural para aproximar a função de derivadas médias, em vez da função de derivadas instantâneas do sistema dinâmico ODE, permite que qualquer precisão desejada na modelagem discreta de sistemas dinâmicos possa ser realizada, com a utilização de um simples integrador Euler, tornando a implementação do controle preditivo uma tarefa mais simples, uma vez que ela somente necessitará lidar com a estrutura linear de um integrador de primeira ordem na determinação das ações de controle. Para ilustrar a efetividade da abordagem proposta, são apresentados resultados dos testes em um problema de transferência de órbitas Terra/Marte e em um problema de controle de atitude em três eixos de satélite comportando-se como corpo rígido

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The usual approach to nonlinear dynamic systems neural modeling has been that of training a feed forward neural network to represent a discrete nonlinear input-output NARMA (Nonlinear Auto Regressive Moving Average) type of model. In this paper, the recently developed alternative approach of combining feed forward neural networks with the structure of ordinary differential equations (ODE) numerical integrator algorithms is done in a way not yet considered. In this new approach, instead of using the neural network to learn the instantaneous derivative function of the ordinary differential equation (ODE) that describes the dynamic system, it is used to learn the dynamic system mean derivative function. This allows the use of an Euler structure to obtain a first order ODE neural integrator, which in principle can provide the same accuracy as that of any higher order integrator. The main objective is to have an approach in which the dynamic system neural modeling is simple. First in terms of the feed forward neural network training, since it has to learn only the algebraic and static functions of the system dynamic ODE mean derivatives. Second in terms of numerical complexity, since a first order integrator structure is sufficient to attain a specified accuracy. Test results of a practical problem, representing the dynamics of orbit transfer between the Earth and Mars, are used to illustrate the effectiveness of this new methodology.