High-performance tracking systems require both accurate and robust control. An accurate model of the plant is needed in order to design a model-based controller or adaptive controller that satisfies these requirements. However, due to factors like process uncertainties, process non-linearity or time varying parameters, the identification and modelling that is needed might be difficult, expensive or sometimes even impossible. To overcome this kind of difficulties, several learning control techniques have been proposed. The origin of learning feedforward control can be traced back to a scheme known as Feedback Error Learning (FEL). This scheme was proposed by Kawato et al. (1987). The main objective of FEL is to implement the feedforward controller as a function approximator, instead of designing linear control on the basis of a plant model. The type of function approximator used by Kawato (1993) and Brown (1994), as an example, are Multi Layer Perception (MLP) and B-splines neural network, respectively. Several problems exist for theses kinds of function approximators such as slow convergence, local minima, computational load and high memory requirement. However, applications have shown that the FEL controller gives a considerable improvement on the performance of the system. Moreover, FEL can obtain a high tracking performance without extensive modeling. The novelty of FEL lies in its use of feedback error as a teaching signal, which is essentially new in control literature. Recently, Miyamura and Kimura (2002) have established a control theoretical validity of the FEL method in the frame of adaptive control for SISO, proving its stability based on strictly positive realness, whereas Muramatsu and Watanabe (2004) have relaxed the positive realness condition of FEL by using the error signal between the reference and the output signal as well as the feedback input. They have used an adaptive linear filter for function approximator, and did not use MLP nor B-spline neural networks any more, but still the learning of the inverse model has been effectively used. In view of the fact that most of the process control applications are multi-input multi-output (MIMO), we generalize FEL for a MIMO system in this paper using simple linear system parameterization of the feedforward control as a function approximator. The parameter has a matrix form instead of a vector form, but we can derive a learning law in a similar way as in the SISO case. Also, we relax the biproper condition even in the MIMO case. The concept of the interactor (Mutoh and Nikiforuk, 1992) is used for our purpose. This concept was originally introduced to generalize the relative degree of scalar transfer functions to MIMO systems. Furthermore, MIMO-FEL online learning algorithm provides a powerful set of tools for automatically fine-tuning of a feedforward controller to improve the performance while in operation, or for automatically adapting to the changing dynamics. For MIMO-FEL to be widely adopted for applications such as airplane flight, it is critical that we have to have safety guarantees, specifically stability guarantees. Therefore, stability analysis of the derived MIMO-FEL learning law will be proved using Lyapunov satiability theory.
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