# Contribuição para o Controlo Adaptativo Não Linear: Redes Neuronais Recorrentes no Contexto da Regulação da Saída

### Authors

### Abstract

The evolution of automatic control in the last few years has been characterized by some antagonism between two schools: one based on the analytical-algebraic approach and the other based on processing tools stemming from soft computing theory. Both have been working on to develop control methodologies for complex systems. The analytical-algebraic school, using rigorous methods for linear and non-linear systems, has built a coherent body of knowledge, however still fails to solve problems where it is not possible to obtain rigorous models. The other school, using mainly neural networks and fuzzy systems, has developed a number of methods and control architectures that may solve in practice difficult problems. Nevertheless, the resulting body of knowledge lacks coherence, systematization and generality.It is now becoming clear that only the collaboration of the two schools may lead to a new stage in the history of automatic control science and technology. The present work intends to be a contribution in this direction.

In this context, some analytical and experimental results have been carried out in order to developed hybrid control structures, by integrating the attributes of soft computing methodologies, like neural and fuzzy systems, with the well-established system control techniques. Particularly, in this work it is presented a structure that fuses a recurrent neural network with the output regulation control theory to obtain a control methodology for general non-linear discrete time systems. It is intended with this approach to profit from the identification capabilities of neural networks with the stability properties of the output regulation theory.

Given the universal approximation properties, as well as its intrinsic analogy to the non-linear state space form, a recurrent neural network is derived and applied for modeling non-linear plants. This network architecture has the well-known properties of multilayer perceptrons and has the ability to incorporate temporal behavior. This is the startingpoint to neural networks in identification and control of non-linear systems. The recurrent neural model can replace the unknown system, transforming the original problem of control into a non-linear control problem, suitable to be designed by a non-linear control technique, like the output regulation theory.

The main goal of the output regulation theory is to derive a control law such that simultaneously the closed loop system is stable and the tracking error converges to zero. If the system is linear time invariant and their parameters are completely known, a linear regulator can de designed. However, if the system is non-linear the problem becomes tougher. In this case, the solution of the output regulator problem leads to a set of non-linear difference equations, to which is very difficult, or even impossible, to derive a closed solution. To solve the regulator equations an iterative procedure based on a pole placement algorithm is proposed ensuring the convergence of the regulator equations.

Tough, assuming an exact knowledge of the systems parameters in many cases is not viable. In fact, if the training is performed off-line, the non-linear neural model is not exact in the presence of disturbances, parameter variations and uncertainties. In these conditions, the output regulator cannot assure convergence of the tracking error to zero. To overcome this an adaptive strategy is applied, providing on-line estimation of the network parameters. Estimation is implemented on-line, based on input-output data, ensuring the stability and convergence of the estimation error. The estimation strategy is developed following a dual Kalman filter strategy using both Lyapunov theory and non-linear observation techniques.

Simulation and experimental results collected from several laboratorial processes are used to confirm the viability and effectiveness of the proposed methodology. The control performance of the proposed technique is compared to well-established techniques, like PI controllers and state space feedback controllers.