Set-theoretic methods for analysis estimation and control of nonlinear systems

This thesis is concerned with the application of set-theoretical methods to problems in analysis, estimation and control of nonlinear systems. Set-theoretical concepts are often used in the formulation of various problems in science and engineering. One of the key enablers for the successful application of set-theoretical methods is the ability to enclose the image set of nonlinear multivariate systems, which is the focus of the main body of this thesis. Chapter 2 concentrates on bounding the image of factorable, vector-valued functions. To this aim, a framework is developed which enables the analysis of existing set-valued arithmetics---such as interval and polynomial model arithmetics---and the construction of new ones e.g. an ellipsoidal arithmetic for vector-valued nonlinear factorable functions. This framework also allows to study on a unified way the convergence (in the Hausdorff sense) of the enclosures to the exact image as the domain of the function shrinks. Chapters 3 and 4 are concerned with set propagation through dynamic systems (reachability analysis) defined by parametric ordinary differential equations (ODEs). Computing enclosures for the reachable set is not straightforward, since it is the image of a function which is not factorable, but it is defined implicitly by the ODEs. Nevertheless, computational methods for reachability analysis can take advantage the factorable structure of the ODE right-hand side. The focus on Chapter 3 is on discrete-time set propagation, i.e. methods where the integration horizon is discretized into finite steps, and then propagating the enclosure through each of these steps. Classical methods rely on Taylor expansions of the ODE solution and proceed in two phases. First, an a step-size and an a priori enclosure for the reachable set over the current step are determined. Then, the enclosure is tightened at the end of the step. The algorithm presented in this chapter is also based on Taylor expansions of the solution, but the order of the phases is reversed. This construction leads to a natural step-size control mechanism and eliminates the need for tightening the enclosure at the end of the time-step. Furthermore, sufficient conditions are then derived for the algorithm to be locally asymptotically stable in the neighborhood of a locally asymptotically stable periodic orbit or equilibrium point. The key requirement for stability is that the affine-set extensions used in the propagation have quadratic Hausdorff convergence order. On the other hand, Chapter 4 deals with continuous-time set propagation methods. These class of methods rely on the construction of an auxiliary system of ODEs, whose solution is guaranteed to enclose the reachable set of the original ODEs. Here, a unified framework for the construction of continuous time methods is presented. It is based on a generalized differential inequality (GDI), whose solutions describe the support function of time-varying enclosures for the reachable set. This GDI contains as special cases known continuous-time reachability methods, such as differential inequalities and ellipsoidal set propagation techniques. Although the GDI is based on the support function characterization of convex sets, an extension for nonconvex sets is provided using polynomial models with convex remainders. The framework also provides a means for analyzing the Hausdorff convergence properties of continuous-time enclosure methods. A nontrivial extension of the GDI in the form of a min-max differential inequality is also introduced for the characterization of robust forward invariant tubes. This min-max DI provides as a by product a semi-explicit nonlinear feedback control law, which can also be exploited in robust optimal control and tube-based robust model predictive control. Chapter 5 is concerned with the characterization of sets defined implicitly by systems of constraints. These problems are addressed from a set-theoretical perspective, by adopting the use of complete-search based constraint projection methods. The chapter presents a branch-and-prune algorithm which is enhanced by the use of higher order bounding strategies based on polynomial models. The use of optimization-based domain reduction strategies inspired by developments in branch-and-bound algorithms for complete-search global optimization is also studied. We also introduce a CPU time reduction strategy for polynomial models, which allows reusing the computed bounds whenever they have converged. For constraint systems that include undetermined systems of equations a domain reduction strategy in reduced space is presented. This strategy relies on the use of polynomial models in order to characterize the boundary of the set and makes use of state-of-the-art Newton-like methods for the solution of systems of nonlinear implicit algebraic equations. The algorithm is applied to two different problems: guaranteed parameter estimation and guaranteed asymptotic analysis. The methods in this thesis have been implemented in CRONOS (https://bitbucket.org/omega-icl/cronos), a C++ library that builds upon the MC++ library (https://bitbucket.org/omega-icl/mcpp) for bounding factorable functions. These algorithms have also been tested in a variety of case studies drawn from Chemical Engineering and Systems biology.