Robust eigenstructure assignment in geometric control theory

In this paper we employ the Rosenbrock system matrix pencil for the computation of output-nulling subspaces of linear time-invariant systems which appear in the solution of a large number of control and estimation problems. We also consider the problem of finding friends of these output-nulling subspaces, i.e., the feedback matrices that render such subspaces invariant with respect to the closed-loop map and output-nulling with respect to the output map, and which at the same time deliver a robust closed-loop eigenstructure. We show that the methods presented in this paper offer considerably more robust eigenstructure assignment than the other commonly used methods and algorithms

Controllability of linear differential-algebraic systems - A survey

Different concepts related to controllability of differential-algebraic equations are described. The class of systems considered consists of linear differential-algebraic equations with constant coefficients. Regularity, which is, loosely speaking, a concept related to existence and uniqueness of solutions for any inhomogeneity, is not required in this article. The concepts of impulse controllability, controllability at infinity, behavioral controllability, strong and complete controllability are described and defined in time-domain. Equivalent criteria that generalize the Hautus test are presented and proved. Special emphasis is placed on normal forms under state space transformation and, further, under state space, input and feedback transformations. Special forms generalizing the Kalman decomposition and Brunovsky form are presented. Consequences for state feedback design and geometric interpretation of the space of reachable states in terms of invariant subspaces are proved

On differential-algebraic control systems

In der vorliegenden Dissertation werden differential-algebraische Gleichungen (differential-algebraic equations, DAEs) der Form \ddt E x = Ax + f betrachtet, wobei $E$ und $A$ beliebige Matrizen sind. Falls $E$ nichtverschwindende Einträge hat, dann kommen in der Gleichung Ableitungen der entsprechenden Komponenten von $x$ vor. Falls $E$ eine Nullzeile hat, dann kommen in der entsprechenden Gleichung keine Ableitungen vor und sie ist rein algebraisch. Daher werden Gleichungen vom Typ \ddt E x = Ax + f differential-algebraische Gleichungen genannt. Ein Ziel dieser Dissertation ist es, eine strukturelle Zerlegung einer DAE in vier Teile herzuleiten: einen ODE-Anteil, einen nilpotenten Anteil, einen unterbestimmten Anteil und einen überbestimmten Anteil. Jeder Anteil beschreibt ein anderes Lösungsverhalten in Hinblick auf Existenz und Eindeutigkeit von Lösungen für eine vorgegebene Inhomogenität $f$ und Konsistenzbedingungen an $f$. Die Zerlegung, namentlich die quasi-Kronecker Form (QKF), verallgemeinert die wohlbekannte Kronecker-Normalform und behebt einige ihrer Nachteile. Die QKF wird ausgenutzt, um verschiedene Konzepte der Kontrollierbarkeit und Stabilisierbarkeit für DAEs mit~$f=Bu$ zu studieren. Hier bezeichnet $u$ den Eingang des differential-algebraischen Systems. Es werden Zerlegungen unter System- und Feedback-Äquivalenz, sowie die Folgen einer Behavioral-Steuerung $K_x x + K_u u = 0$ für die Stabilisierung des Systems untersucht. Falls für das DAE-System zusätzlich eine Ausgangs-Gleichung $y=Cx$ gegeben ist, dann lässt sich das Konzept der Nulldynamik wie folgt definieren: die Nulldynamik ist, grob gesagt, die Dynamik, die am Ausgang nicht sichtbar ist, d.h. die Menge aller Lösungs-Trajektorien $(x,u,y)$ mit $y=0$. Für rechts-invertierbare Systeme mit autonomer Nulldynamik wird eine Zerlegung hergeleitet, welche die Nulldynamik entkoppelt. Diese versetzt uns in die Lage, eine Behavior-Steuerung zu entwickeln, die das System stabilisiert, vorausgesetzt die Nulldynamik selbst ist stabil. Wir betrachten auch zwei Regelungs-Strategien, die von den Eigenschaften der oben genannten System-Klasse profitieren: Hochverstärkungs- und Funnel-Regelung. Ein System \ddt E x = Ax + Bu, $y=Cx$, hat die Hochverstärkungseigenschaft, wenn es durch die Anwendung der proportionalen Ausgangsrückführung $u=-ky$, mit $k>0$ hinreichend groß, stabilisiert werden kann. Wir beweisen, dass rechts-invertierbare Systeme mit asymptotisch stabiler Nulldynamik, die eine bestimmte Relativgrad-Annahme erfüllen, die Hochverstärkungseigenschaft haben. Während der Hochverstärkungs-Regler recht einfach ist, ist es jedoch a priori nicht bekannt, wie groß die Verstärkungskonstante $k$ gewählt werden muss. Dieses Problem wird durch den Funnel-Regler gelöst: durch die adaptive Justierung der Verstärkung über eine zeitabhängige Funktion $k(\cdot)$ und die Ausnutzung der Hochverstärkungseigenschaft wird erreicht, dass große Werte $k(t)$ nur dann angenommen werden, wenn sie nötig sind. Eine weitere wesentliche Eigenschaft ist, dass der Funnel-Regler das transiente Verhalten des Fehlers $e=y-y_{\rm ref}$ der Bahnverfolgung, wobei $y_{\rm ref}$ die Referenztrajektorie ist, beachtet. Für einen vordefinierten Performanz-Trichter (funnel) $\psi$ wird erreicht, dass $\|e(t)\|<\psi(t)$. Schließlich wird der Funnel-Regler auf die Klasse von MNA-Modellen von passiven elektrischen Schaltkreisen mit asymptotisch stabilen invarianten Nullstellen angewendet. Dies erfordert die Einschränkung der Menge der zulässigen Referenztrajektorien auf solche die, in gewisser Weise, die Kirchhoffschen Gesetze punktweise erfüllen.In this dissertation we study differential-algebraic equations (DAEs) of the form Ex'=Ax+f. One aim of the thesis is to derive the quasi-Kronecker form (QKF), which decomposes the DAE into four parts: the ODE part, nilpotent part, underdetermined part and overdetermined part. Each part describes a different solution behavior. The QKF is exploited to study the different controllability and stabilizability concepts for DAEs with f=Bu, where u is the input of the system. Feedback decompositions, behavioral control and stabilization are investigated. For DAE systems with output equation y=Cx, we may define the concept of zero dynamics, which are those dynamics that are not visible at the output. For right-invertible systems with autonomous zero dynamics a decomposition is derived, which decouples the zero dynamics of the system and allows for high-gain and funnel control. It is shown, that the funnel controller achieves tracking of a reference trajectory by the output signal with prescribed transient behavior. Finally, the funnel controller is applied to the class of MNA models of passive electrical circuits with asymptotically stable invariant zeros

Optimization methods for deadbeat control design: a state space approach

This thesis addresses the synthesis problem of state deadbeat regulator using state space techniques. Deadbeat control is a linear control strategy in discrete time systems and consists of driving the system from any arbitrary initial state to a desired final state infinite number of time steps. Having described the framework for development of the thesis which is in the form of a lower linear-fractional transformation (LFT), the conditions for internal stability based on the notion of coprime factorization over the set of proper and stable transfer matrices, namely RH, is discussed. This leads to the derivation of the class of all stabilizing linear controllers, which are parameterized affinely in terms of a stable but otherwise free parameter Q, usually known as the Q-parameterization. In this work, the classical Q- parameterization is generalized to deliver a parameterization for the family of deadbeat regulators. Time response characteristics of the deadbeat system are investigated. In particular, the deadbeat regulator design problem in which the system must satisfy time domain specifications and minimize a quadratic (LQG-type) performance criterion is examined. It is shown that the attained parameterization for deadbeat controllers leads to the formulation of the synthesis problem in a quadratic programming framework with Q regarded as the design variable. The equivalent formulation of this objective as a quadratic integral in the frequency domain provides the means for shaping the frequency response characteristics of the system. Using the LMI characterization of the standard H problem, a new scheme for shaping the system frequency response characteristics by minimizing the infinity norm of an appropriate closed-loop transfer function is introduced. As shown, the derived parameterization of deadbeat compensators simplifies considerably the formulation and solution of this problem. The last part of the work described in this thesis is devoted to addressing the synthesis problem of deadbeat regulators in a robust way, when the plant is subject to structured norm-bounded parametric uncertainties. A novel approach which is expressed as an LMI feasibility condition has been proposed and analysed

Finite settling time stabilization for linear multivariable time-invariant discrete-time systems: An algebraic approach

The problem of Total Finite Settling Time Stabilization of linear time-invariant discrete-time systems is investigated in this thesis. This problem falls within the same area of the well-known deadbeat (time-optimal) control and in particular, constitutes a generalization of this problem. That is, instead of seeking time-optimum performance, it is required that all internal and external variables (signals) of the closed-loop system settle to a new steady state after a finite time from the application of a step change to any of its inputs and for every initial condition. The state/output deadbeat control is a special case of the Total FSTS problem. Using a mathematical and system theory framework based on sequences and the polynomial equation (algebraic) approach, we are able to tackle the FSTS problem in a unifying manner. The one-parameter (unity) feedback configuration is mainly used for the solution of the FSTS problem and FSTS related control strategies. The whole problem is reduced to the solution of a polynomial matrix Diophantine equation which guarantees not only internal stability but also internal FSTS and is further reduced to the solution of a linear algebra problem over R. This approach enables the parametrizat ion of the family of all FSTS controllers, as well as those which are causal, in a Youla-Bongiorno-Kucera type parametrization. The minimal McMillan degree FSTS problem is completely solved for vector plants and a parametrization of the FSTS controllers according to their McMillan degree is obtained. In the MIMO case bounds of the minimum McMillan degree controllers are derived and families of FSTS controllers with given lower/upper McMillan degree bounds are provided in parametric form. Having parametrized the family of all FSTS controllers, the state deadbeat regulation is treated as a special case of FSTS and complete parametrization of all the deadbeat regulators is presented. In addition, further performance criteria, or design constraints are imposed such as, FSTS tracking and/or disturbance rejection, partial assignment of controller dynamics, l1-, l∞-optimization and robustness to plant parameter variations. Finally, the Simultaneous-FSTS problem is formulated, and necessary as well as sufficient conditions for its solution are derived. Also, a two-parameter control scheme is introduced to alleviate some of the drawbacks of the one-parameter control. A parametrization of the family of FSTS controllers as well as the FSTS controllers for tracking and/or disturbance rejection is given as an illustration of the particular advantages of the two-parameter FSTS controllers

Approximate controllability and observability measures in control systems design

The selection of systems of inputs and outputs (input and output structure) forms part of early system design, which is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance from uncontrollability (unobservability). The thesis introduces novel measures for evaluating and estimating the distance to uncontrollability and relatively unobservability. At first, the modal measuring approach is studied in detail, providing a framework for the ”best” structure selection. Although controllability (observability) is invariant under state feedback (output injection), the corresponding degrees expressing distance from uncontrollability (unobservability) are not. Hence, the thesis introduces new criteria for the distance problem from uncontrollability (unobservability) which is invariant under feedback transformations. The approach uses the restricted input-state (state-output) matrix pencil and then deploys exterior algebra that reduces the overall problem to the standard problem of distance of a set of polynomials from non-coprimeness. Results on the distance of the Sylvester Resultants from singularity provide the new measures. Since distance to singularity of the corresponding Sylvester matrix is the key in evaluating the distance to uncontrolability it is of the particular interest in the present work. In order to find the solution two novel methods are introduced in the thesis, namely the alternating projection algorithm and a structured singular value approach. A least-squares alternating projection algorithm, motivated by a factorisation result involving the Sylvester resultant matrix, is proposed for calculating the ”best” approximate GCD of a coprime polynomial set. The properties of the proposed algorithm are investigated and the method is compared with alternative optimisation techniques which can be employed to solve the problem. It is also shown that the problem of an approximate GCD calculation is equivalent tothe solution of a structured singular value (µ) problem arising in robust control for which numerous techniques are available. Motivated by the powerful concept of the structured singular values, the proposed method is extended to the special case of an implicit system that has a wide application in the behavioural analysis of complex systems. Moreover, µ-value approach has a potential application for the general distance problem to uncontrollability that is numerically hard to obtain. Overall, the proposed framework significantly simplifies and generalises the input-output structure selection procedure and evaluates alternative solutions for a variety of distance problems that appear in Control Theory

Process and systems based methodologies related to control structure selection

This thesis is concerned with an important aspect of process control design, that is, the synthesis of the control structures. A review of the rapidly growing process methodologies' literature is presented and this leads to the identification of wider issues and new problems which are referred to as global instrumentation and forms the main subject of this thesis. The main objective has been the integration of existing process based tools and methodologies with a much more general approach of a systems and control theory character. The problem of Global Process Instrumentation concerns the selection of systems of measurement and actuation variables, found during the synthesis/design and operation of large-scale industrial processes/systems. The role of traditional instrumentation was considered but the emphasis has been on the systems aspects. In fact, instrumentation leads to the shaping of the final system and thus, is crucial in defining the control quality properties and operability characteristics of the final design. The development of these system aspects led to the emergence of an integrated framework for Global Instrumentation. An attempt was also made to abstract some results and formulate generic issues and problems, that would provide a wider scenario for activities in the future. Development of CAD to support the selection of control structures has been a major task undertaken here. The system aspects of Global Instrumentation are demonstrated by studying two specific problems that involve the study of the structural properties of interconnected systems as a function of local selection of sensors and actuators and the problem of well-conditioning badly structured transfer functions. The role of selection of inputs and outputs, on the overall shaping of composite structure properties, at the subsystem level, was examined, and the significance of an assumption related to interconnections, referred to as the completeness assumption, was investigated. Specifically, the significance of the deviations from the completeness, was the subject of the investigation. Matrix Pencil Theory was used to examine the controllability, observability and zero structure related properties of composite systems under partial or total loss of inputs/outputs at the subsystem level. Selecting subsets of the original sets of inputs, outputs to guarantee full rank transfer function, was also an issue that was examined. The above problems were presented as part of an integrated design philosophy that aims to explore the system structure. An integrated approach to the overall problem of control structure selection was formulated and open issues and problems were identified. It was based on the assumption that there exists a progenitor model of the linear type for the process, which, however, may not be well defined. Structural analysis of the system theoretic framework, the interaction measures and the results for evaluation of alternative decentralisation schemes were then used, to specify a step by step approach to the control structure selection. The problem of handling alternative criteria was also considered and basic elements of a system procedure were given. There are many open issues, which were identified and are still open and thus the proposed structural approach should be considered as the first step to the development of an integrated methodology that involves the following major steps: (a) Classification of system model variables and definition of well structured progenitor model. (b) Definition of effective input, output structure based on operability, controllability criteria. (c) Determining the structure of the control scheme by evaluation of alternative decentralised structures. An important part of the integrated methodology for control structure selection is the - so called - interaction analysis. It consists of a number of diagnostics and structural tests that help to restrict the choice of the best scheme. Several of these tests/methodologies were reviewed and some of them were further expanded. The outcomes obtained by these methodologies provided promising results. These results gave the motivation for the construction of a complete CAD package, the "Interaction Analysis Toolbox", written in MATLAB®t. This Toolbox provides many tools and diagnostics that can be applied during the design stages, for the evaluation of the various alternative control structures