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A neural net (NN)-based actuator saturation compensation scheme for the nonlinear systems in Brunovsky canonical form is presented. The scheme that leads to stability, command following, and disturbance rejection is rigorously proved and verified using a general "pendulum type" and a robot manipulator dynamical systems. Online weights tuning law, the overall closed-loop system performance, and the boundedness of the NN weights are derived and guaranteed based on Lyapunov approach. The actuator saturation is assumed to be unknown and the saturation compensator is inserted into a feedforward path. Simulation results indicate that the proposed scheme can effectively compensate for the saturation nonlinearity in the presence of system uncertainty.

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One of the most important properties of neural nets (NNs) for control purposes is the universal approximation property. Unfortunately,, this property is generally proven for continuous functions. In most real industrial control systems there are nonsmooth functions (e.g., piecewise continuous) for which approximation results in the literature are sparse. Examples include friction, deadzone, backlash, and so on. It is found that attempts to approximate piecewise continuous functions using smooth activation functions require many NN nodes and many training iterations, and still do not yield very good results. Therefore, a novel neural-network structure is given for approximation of piecewise continuous functions of the sort that appear in friction, deadzone, backlash, and other motion control actuator nonlinearities. The novel NN consists of neurons having standard sigmoid activation functions, plus some additional neurons having a special class of nonsmooth activation functions termed "jump approximation basis function." Two types of nonsmooth jump approximation basis functions are determined- a polynomial-like basis and a sigmoid-like basis. This modified NN with additional neurons having "jump approximation" activation functions can approximate any piecewise continuous function with discontinuities at a finite number of known points. Applications of the new NN structure are made to rigid-link robotic systems with friction nonlinearities. Friction is a nonlinear effect that can limit the performance of industrial control systems; it occurs in all mechanical systems and therefore is unavoidable in control systems. It can cause tracking errors, limit cycles, and other undesirable effects. Often, inexact friction compensation is used with standard adaptive techniques that require models that are linear in the unknown parameters. It is shown here how a certain class of augmented NN, capable of approximating piecewise continuous functions, can be used for friction compensation.

We present a distance metric based upon the notion of minimum-cost injective mappings between sets. Our function satisfies metric properties as long as the cost of the minimum mappings is derived from a semimetric, for which the triangle inequality is not necessarily satisfied. We show that the Jaccard distance (alternatively biotope, Tanimoto, or Marczewski-Steinhaus distance) may be considered the special case for finite sets where costs are derived from the discrete metric. Extensions that allow premetrics (not necessarily symmetric), multisets (generalized to include probability distributions), and asymmetric mappings are given that expand the versatility of the metric without sacrificing metric properties. The function has potential applications in pattern recognition, machine learning, and information retrieval.

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