Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations

Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples

Alternatives for jet engine control

Research centered on basic topics in the modeling and feedback control of nonlinear dynamical systems is reported. Of special interest were the following topics: (1) the role of series descriptions, especially insofar as they relate to questions of scheduling, in the control of gas turbine engines; (2) the use of algebraic tensor theory as a technique for parameterizing such descriptions; (3) the relationship between tensor methodology and other parts of the nonlinear literature; (4) the improvement of interactive methods for parameter selection within a tensor viewpoint; and (5) study of feedback gain representation as a counterpart to these modeling and parameterization ideas

Application of system theory to power processing problems

The work in power processing is reported. Input-output models, and Lie groups in control theory are discussed along with the methods of analysis for time invariant electrical networks

Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction

In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to reformulate the original nonlinear dynamics into a different, more simplified format. Additionally, we study the problem of complexity/model reduction of large-scale lifted models of nonlinear systems from data. The method under consideration is the Loewner framework, an established data-driven approach that requires samples of input–output mappings. The latter are known as generalized transfer functions, which are appropriately defined for both bilinear and quadratic-bilinear systems. We show connections between these mappings as well as between the matrices of reduced-order models. Finally, we illustrate the theoretical discussion with two numerical examples

Krylov Subspace Model Order Reduction for Nonlinear and Bilinear Control Systems

The use of Krylov subspace model order reduction for nonlinear/bilinear systems, over the past few years, has become an increasingly researched area of study. The need for model order reduction has never been higher, as faster computations for control, diagnosis and prognosis have never been higher to achieve better system performance. Krylov subspace model order reduction techniques enable this to be done more quickly and efficiently than what can be achieved at present. The most recent advances in the use of Krylov subspaces for reducing bilinear models match moments and multimoments at some expansion points which have to be obtained through an optimisation scheme. This therefore removes the computational advantage of the Krylov subspace techniques implemented at an expansion point zero. This thesis demonstrates two improved approaches for the use of one-sided Krylov subspace projection for reducing bilinear models at the expansion point zero. This work proposes that an alternate linear approximation can be used for model order reduction. The advantages of using this approach are improved input-output preservation at a simulation cost similar to some earlier works and reduction of bilinear systems models which have singular state transition matrices. The comparison of the proposed methods and other original works done in this area of research is illustrated using various examples of single input single output (SISO) and multi input multi output (MIMO) models

On algebraic estimation and systems with graded polynomial structure

In the first half of this thesis the algebraic properties of a class of minimal, polynomial systems on IRn are considered. Of particular interest in the sequel are the results that (i) a tensor algebra generated by the observation space and strong accessibility algebra is equal to the Lie algebra of polynomial vector fields on IRn and (ii) the observation algebra of such a system is equal to the ring of polynomial functions on IRn. The former result is proved directly, but to establish the second we construct a canonical form for which the claim is trivial, the general case then following from the properties of the diffeomorphism relating the two realisations. It is also shown that, as a consequence of the structure of the observation space, any system in the class considered has a finite Volterra series solution, thereby showing that the canonical form developed is dual to that of Crouch. The second part of the work is devoted to the algebraic aspects of nonlinear filtering. The fundamental question that this 'algebraic estimation theory' seeks to answer is the existence of a homomorphism between a Lie algebra A of differential operators and a Lie algebra of vector fields. By restricting A to be finite dimensional we obtain a restrictive condition on the system generating A. Results of Ocone and Hijab are extended and connections with the work of Omori and de la Harpe established thus showing A seldom has a Banach structure. Finally, using an observability condition, we develop a further canonical form and thus define a class of systems for which A is isomorphic to the Weyl algebra on n-generators and hence cannot satisfy the above homomorphism principle