Existence of pole-zero structures in a rational matrix equation arising in a decentralized stabilization of expanding systems

summary:A necessary and sufficient condition for the existence of pole and zero structures in a proper rational matrix equation $T_{2} X = T_{1}$ is developed. This condition provides a new interpretation of sufficient conditions which ensure decentralized stabilizability of an expanded system. A numerical example illustrate the theoretical results

Optimization-based Robustness and Stabilization in Decentralized Control

This dissertation pertains to the stabilization, robustness, and optimization of Finite Dimensional Linear Time Invariant (FDLTI) decentralized control systems. We study these concepts for FDLTI systems subject to decentralizations that emerge from imposing sparsity constraints on the controller. While these concepts are well-understood in absence of an information structure, they continue to raise fundamental interesting questions regarding an optimal controller, or on suitable notions of robustness in presence of information structures. Two notions of stabilizability with respect to decentralized controllers are considered. First, the seminal result of Wang & Davison in 1973 regarding internal stabilizability of perfectly decentralized system and its connection to the decentralized fixed-modes of the plant is revisited. This seminal result would be generalized to any arbitrary sparsity-induced information structure by providing an inductive proof that verifies and shows that those mode of the plant that are fixed with respect to the static controllers would remain fixed with respect to the dynamic ones. A constructive proof is also provided to show that one can move any non-fixed mode of the plant to any arbitrary location within desired accuracy provided that they remain symmetric in the complex plane. A synthesizing algorithm would then be derived from the inductive proof. A second stronger notion of stability referred to as "non-overshooting stability" is then addressed. A key property called "feedthrough consistency" is derived, that when satisfied, makes extension of the centralized results to the decentralized case possible. Synthesis of decentralized controllers to optimize an H_Infinity norm for model-matching problems is considered next. This model-matching problem corresponds to an infinite-dimensional convex optimization problem. We study a finite-dimensional parametrization, and show that once the poles are chosen for this parametrization, the remaining problem of coefficient optimization can be cast as a semidefinite program (SDP). We further demonstrate how to use first-order methods when the SDP is too large or when a first-order method is otherwise desired. This leaves the remaining choice of poles, for which we develop and discuss several methods to better select the most effective poles among many candidates, and to systematically improve their location using convex optimization techniques. Controllability of LTI systems with decentralized controllers is then studied. Whether an LTI system is controllable (by LTI controllers) with respect to a given information structure can be determined by testing for fixed modes, but this gives a binary answer with no information about robustness. Measures have already been developed to determine how far a system is from having a fixed mode when one considers complex or real perturbations to the state-space matrices. These measures involve intractable minimizations of a non-convex singular value over a power-set, and hence cannot be computed except for the smallest of the plants. We replace these problem by equivalent optimization problems that involve a binary vector rather than the power-set minimization and prove their equality. Approximate forms are also provided that would upper bound the original metrics, and enable us to utilize MINLP techniques to derive scalable upper bounds. We also show that we can formulate lower bounds for these measures as polynomial optimization problems,and then use sum-of-squares methods to obtain a sequence of SDPs, whose solutions would lower bound these metrics

Decentralized adaptive neural network control of interconnected nonlinear dynamical systems with application to power system

Traditional nonlinear techniques cannot be directly applicable to control large scale interconnected nonlinear dynamic systems due their sheer size and unavailability of system dynamics. Therefore, in this dissertation, the decentralized adaptive neural network (NN) control of a class of nonlinear interconnected dynamic systems is introduced and its application to power systems is presented in the form of six papers. In the first paper, a new nonlinear dynamical representation in the form of a large scale interconnected system for a power network free of algebraic equations with multiple UPFCs as nonlinear controllers is presented. Then, oscillation damping for UPFCs using adaptive NN control is discussed by assuming that the system dynamics are known. Subsequently, the dynamic surface control (DSC) framework is proposed in continuous-time not only to overcome the need for the subsystem dynamics and interconnection terms, but also to relax the explosion of complexity problem normally observed in traditional backstepping. The application of DSC-based decentralized control of power system with excitation control is shown in the third paper. On the other hand, a novel adaptive NN-based decentralized controller for a class of interconnected discrete-time systems with unknown subsystem and interconnection dynamics is introduced since discrete-time is preferred for implementation. The application of the decentralized controller is shown on a power network. Next, a near optimal decentralized discrete-time controller is introduced in the fifth paper for such systems in affine form whereas the sixth paper proposes a method for obtaining the L2-gain near optimal control while keeping a tradeoff between accuracy and computational complexity. Lyapunov theory is employed to assess the stability of the controllers --Abstract, page iv

Cooperative control theory and integrated flight and propulsion control

The major contribution of this research was the exposition of the fact that airframe and engine interactions could be present, and their effects could include loss of stability and performance of the control systems. Also, the significance of two directional, as opposed to one-directional, coupling was identified and explained. A multivariable stability and performance analysis methodology was developed, and applied to several candidate aircraft configurations. In these example evaluations, the significance of these interactions was underscored. Also exposed was the fact that with interactions present along with some integrated control approaches, the engine command/limiting logic (which represents an important nonlinear component of the engine control system) can impact closed-loop airframe/engine system stability. Finally, a brief investigation of control-law synthesis techniques appropriate for the class of systems was pursued, and it was determined that multivariable techniques, including model-following formulations of LQG and/or H infinity methods, showed promise. However, for practical reasons, decentralized control architectures are preferred, which is an architecture incompatible with these synthesis methods. The major contributions of the second phase of the grant was the development of conditions under which no decentralized controller could achieve closed loop system requirements on stability and/or performance. Sought were conditions that depended only on properties of the plant and the requirement, and independent of any particular control law or synthesis approach. Therefore, they could be applied a priori, before synthesis of a candidate control law. Under this grant, such conditions were found regarding stability, and encouraging initial results were obtained regarding performance

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

NASA Workshop on Distributed Parameter Modeling and Control of Flexible Aerospace Systems

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach

Modeling and analysis of non-isothermal chemical reaction networks:A port-Hamiltonian and contact geometry approach

In dit proefschrift worden verschillende benaderingen gebruikt voor de meetkundige modellering en analyse van chemische reactienetwerken met varierende temperatuur. deze benaderingen kunnen in twee klassen worden verdeeld: de ene gebaseerd op poort-Hamiltonse systeemtheorie, en de ander gebaseerd op de theorie van contactsystemen. De eerste aanpak is de irreversibele poort-Hamiltonse formulering op basis van de interne energie. Beginnend met een overzicht van de wiskundige structuur van chemische reactienetwerken in het niet-isothermische geval wordt een irreversibele poort-Hamiltonse formulering van niet-isothermische reactienetwerken gegeven. Daarna volgt een thermodynamische analyse, inclusief de voorwaarden voor het bestaan van een thermodynamisch evenwicht en de asymptotische stabiliteit van de verzameling van thermodynamische evenwichtspunten. De tweede benadering betreft de quasi poort-Hamiltonse modellering met behulp van de totale entropie. In dit poort-Hamiltonse systeem wordt niet alleen de energiebalans maar ook de entropiebal- ansvergelijking gebruikt. Ook de thermodynamische analyse wordt in dit kader uitgevoerd, in het bijzonder de karakterisatie van evenwichtspunten en hun asymptotische stabiliteit.Gebaseerd op deze nieuwe quasi poort-Hamiltonse formulering wordt verder de interconnectie van chemische reactienetwerken bestudeerd. Tenslotte wordt de regeling van contactsystemen door middel van structuurbe-houdende terugkoppeling bestudeerd. Een aantal regelontwerpen die hierop gebaseerd zijn worden bestudeerd. Een lokale stabiliteitsanalyse wordt uitgevoerd om de structuurbehoudende terugkoppeling te bepalen, op basis van evenwichtsvoorwaarden en de Jacobimatrix van het teruggekoppelde systeem. Verder worden voorwaarden voor lokale en gedeeltelijke stabiliteit ten opzichte van de gesloten-lus invariant Legendre deelvarieteit gegeven, alsmede de gesloten-lus contact Hamiltonfunctie

The approximate determinantal assignment problem

The Determinantal Assignment Problem (DAP) is one of the central problems of Algebraic Control Theory and refers to solving a system of non-linear algebraic equations to place the critical frequencies of the system to specied locations. This problem is decomposed into a linear and a multi-linear subproblem and the solvability of the problem is reduced to an intersection of a linear variety with the Grassmann variety. The linear subproblem can be solved with standard methods of linear algebra, whereas the intersection problem is a problem within the area of algebraic geometry. One of the methods to deal with this problem is to solve the linear problem and then and which element of this linear space is closer - in terms of a metric - to the Grassmann variety. If the distance is zero then a solution for the intersection problem is found, otherwise we get an approximate solution for the problem, which is referred to as the approximate DAP. In this thesis we examine the second case by introducing a number of new tools for the calculation of the minimum distance of a given parametrized multi-vector that describes the linear variety implied by the linear subproblem, from the Grassmann variety as well as the decomposable vector that realizes this least distance, using constrained optimization techniques and other alternative methods, such as the SVD properties of the so called Grassmann matrix, polar decompositions and mother tools. Furthermore, we give a number of new conditions for the appropriate nature of the approximate polynomials which are implied by the approximate solutions based on stability radius results. The approximate DAP problem is completely solved in the 2-dimensional case by examining uniqueness and non-uniqueness (degeneracy) issues of the decompositions, expansions to constrained minimization over more general varieties than the original ones (Generalized Grassmann varieties), derivation of new inequalities that provide closed-form non-algorithmic results and new stability radii criteria that test if the polynomial implied by the approximate solution lies within the stability domain of the initial polynomial. All results are compared with the ones that already exist in the respective literature, as well as with the results obtained by Algebraic Geometry Toolboxes, e.g., Macaulay 2. For numerical implementations, we examine under which conditions certain manifold constrained algorithms, such as Newton's method for optimization on manifolds, could be adopted to DAP and we present a new algorithm which is ideal for DAP approximations. For higher dimensions, the approximate solution is obtained via a new algorithm that decomposes the parametric tensor which is derived by the system of linear equations we mentioned before

Proceedings of the Fifth NASA/NSF/DOD Workshop on Aerospace Computational Control

The Fifth Annual Workshop on Aerospace Computational Control was one in a series of workshops sponsored by NASA, NSF, and the DOD. The purpose of these workshops is to address computational issues in the analysis, design, and testing of flexible multibody control systems for aerospace applications. The intention in holding these workshops is to bring together users, researchers, and developers of computational tools in aerospace systems (spacecraft, space robotics, aerospace transportation vehicles, etc.) for the purpose of exchanging ideas on the state of the art in computational tools and techniques