Model reduction for nonlinear control systems : with stability preservation and error bounds

Abstract

The analysis of system models forms an important tool in the design of high-tech systems. However, the increasing complexity of the designs and higher demands on their quality generally leads to complex high-order models. Model reduction is a means to obtain reduced-order approximations of these models, thus allowing for efficient analysis, e.g. by means of fast simulation. Since the reduced-order approximation is used as a substitute for the original model, it is of importance to preserve relevant system properties during the reduction process. Herein, stability is the most crucial property, especially in the context of control systems. In addition, the availability of an error bound is highly instrumental, since this provides a direct measure of the quality of the reduced- order model. For linear systems, model reduction procedures preserving stability and guaranteeing a bound on the reduction error exist. However, nonlinearities play an important role in many engineering applications. This thesis therefore focusses on model reduction for (classes of) nonlinear systems, hereby addressing the problems of stability preservation and the availability of a computable error bound. The results can be structured in two parts. The first part of the thesis deals with the reduction of nonlinear systems that can be decomposed into a feedback interconnection of a high-order linear sub-system and a nonlinear subsystem of relatively low order. In this setting, model reduction is applied to the linear subsystem only, allowing for the application of well-developed model reduction techniques for linear systems and making the approach computationally attractive. Then, conditions for stability of the reduced-order nonlinear system as well as bounds on the reduction error are given. The derivation of these error bounds relies on incremental input-output properties of the subsystems, which characterize the perturbation on an output trajectory as a function of the perturbation on an input signal. As a result, incremental input-output properties characterize the evolution of errors introduced by reduction of the linear subsystem. In particular, the incremental properties of input-to-state convergence, a bounded incremental L2 gain or incremental passivity are considered, leading to model reduction procedures for different classes of nonlinear systems. These reduction techniques have in common that they provide conditions for the preservation of certain stability properties of the nonlinear system with inputs as well as error bounds. Moreover, the preservation of the additional system properties of contractivity (an L2 gain bounded by one) and passivity can be guaranteed when dedicated model reduction techniques are applied to the linear subsystem. These ideas are also exploited in the scope of controller reduction for a class of nonlinear controlled systems, hereby again exploiting existing techniques for controller reduction for linear controlled systems. Here, controller reduction techniques focussing on the approximation of the closed-loop behavior or performance preservation are proposed. These techniques are applied in the design of a low-order controller for the temperature control within a lab-on-a-chip demonstrator, as is of importance in many biomedical applications such as disease diagnostics or drug tests. The reduced-order controller obtained using these techniques is shown to accurately track the desired temperature profiles for the lab-on-a-chip demonstrator. Contrary to the first part of the thesis, in which existing model reduction techniques for linear systems are exploited, the second part deals with a direct approach towards the reduction of nonlinear systems. In particular, the method of incremental balanced truncation is introduced, which can be considered as an extension of balanced truncation for linear systems towards the nonlinear case. Incremental balanced truncation is based on the introduction of two energy functions; the incremental observability and incremental controllability function. A model reduction procedure based on these incremental energy functions firstly guarantees the preservation of stability, and, secondly, allows for the computation of an error bound. Furthermore, an alternative to incremental balanced truncation is presented, which is based on the computation of bounds on the incremental energy functions rather than the energy functions themselves. This increases the computational feasibility of the method at the cost of a potentially larger error bound and a potentially less accurate reduced-order model. In particular, this method gives a computationally efficient approach for model reduction of piecewise affine systems. Summarizing, several methods for model reduction of (classes of) nonlinear systems are developed in this thesis, hereby focussing on the preservation of stability properties and the derivation of error bounds. These methods have in common that they rely on incremental system properties.