Control of Systems with Repeated Scalar Nonlinearities

A class of discrete-time nonlinear systems is studied which is described by a standard linear state-equation except that each state-component is subject to an identical odd 1-Lipschitz nonlinearity. This may represent a class of recurrent neural networks. The emphasis is on the development of an approach that makes the best use of the fact that the nonlinearity on each state-component is the same. It is shown that the origin of such a system is globally stable, and an upper bound on the `2-`2 induced gain can be deduced, if a quadratic Lyapunov/storage function exists whose Hessian matrix is positive de nite diagonally dominant. This reduces to nding such a matrix to satisfy appropriate Linear Matrix Inequalities, which is shown computationally tractable. In synthesis, it is assumed that the controller is also a nonlinear system of this form and has the same nonlinearity as the plant does. With appropriate assumptions on the stabilizability and detectability of the plant, all such controllers that would stabilize the closed-loop are parametrized. A result on performance synthesis is also given. For model reduction, the approach adopted is close to the conventional balanced truncation for stable linear systems, where the global `2-`2 error is the main concern. A model reduction algorithm is proposed based on the theoretical result obtained. The results are shown valid even when the nonlinearity is dynamic rather than static. Furthermore, using block-diagonal positive de nite diagonally dominant matrices they can be extended to the more general case that the nonlinearities on di erent state-components may be partly but not all identical. It is easy to see that such a system is in fact a constant Linear Fractional Transformation of a diagonal odd 1Lipschitz nonlinear operator, and the treatment given is an attempt to parallel the conventional treatment of an uncertain linear system described by a Linear Fractional Transformation. This is not trivial, however, due to the nonlinear nature. Still, the Linear Fractional Transformation framework of uncertain linear systems expands into the nonlinear systems to a large extent.