Identification of nonlinear systems is a multifaceted research area, with many diverse approaches and methods. This thesis considers two different approaches: (nonparametric) local modelling, and identification of piecewise affine systems.
Local models and methods predict the system behavior by constructing function estimates from observations in a local neighborhood of the point of interest. For many local methods, it turns out that the function estimates are in practice weighted sums of the observations, so a central question is how to choose the weights. Many of the existing methods are designed using asymptotic (in the num- ber of observations) arguments, which may lead to problems when only few data are available. To avoid this, an approach named direct weight optimization is pro- posed, where an upper bound on the worst-case mean squared error is minimized directly with respect to the weights of a linear or affine estimator. It is shown that the estimator will have a finite bandwidth, and that it keeps several of the properties of an asymptotically optimal estimator.
The case when bounds on the estimated function and its derivatives are known a priori is also studied, and it is shown that one can sometimes, but not always, benefit from this extra information. The problem of estimating the function derivatives is also considered.
Another way of approaching the nonlinear system identification problem is to use a parameterized model class. Piecewise affine systems are an interesting class for this purpose. They have universal approximation properties, and are also closely related to hybrid systems. Here, an overview of different approaches appearing in the literature is presented, and a new identification method based on mixed- integer programming is proposed. One notable property of the latter method is that the global optimum is guaranteed to be found within a finite number of steps. The complexity of the mixed-integer programming approach is discussed, and its relations to existing approaches are pointed out. The special case of identification of Wiener models is considered in detail, since this model structure makes it possible to reduce the computational complexity. Some suboptimal modifications of the mixed-integer programming approach are also investigated.
As for hybrid systems in general, there has been a growing interest for piecewise affine systems in recent years, and they occur in many application areas. In many cases, safety is an important issue, and there is a need for tools that prove that certain states are never reached, or that some states are reached in finite time. The process of proving these kinds of statements is called verification. Many verification tools for hybrid systems have emerged in the last ten years. They all depend on a model of the system, which will in practice be an approximation of the real system. Therefore, it would be desirable to learn how large the model errors can be, before the verification is not valid anymore. In this thesis, a verification method for piecewise affine systems is presented, where bounds on the allowed model errors are given along with the verification.
Linköping: Linköping University , 2003. , 254 p.