Structure evolving systems and control in integrated design

Existing methods in Systems and Control deal predominantly with Fixed Systems, that have been designed in the past, and for which the control design has to be performed. The new paradigm of Structure Evolving Systems (SES), expresses a new form of system complexity where the components, interconnection topology, measurement-actuation schemes may not be fixed, the control scheme also may vary within the system-lifecycle and different views of the system of varying complexity may be required by the designer. Such systems emerge in many application domains and in the engineering context in problems such as integrated system design, integrated operations, re-engineering, lifecycle design issues, networks, etc. The paper focuses on the Integrated Engineering Design (IED), which is revealed as a typical structure evolution process that is strongly linked to Control Theory and Design type problems. It is shown, that the formation of the system, which is finally used for control design evolves during the earlier design stages and that process synthesis and overall instrumentation are critical stages of this evolutionary process that shapes the final system structure and thus the potential for control design. The paper aims at revealing the control theory context of the evolutionary mechanism in overall system design by defining a number of generic clusters of system structure evolution problems and by establishing links with existing areas of control theory. Different aspects of model evolution during the overall design are identified which include cases such as: (i) Time-dependent evolution of system models from “early” to “late” stages of design. (ii) Design stage-dependent evolution from conceptualisation to process synthesis and to overall instrumentation. (iii) Redesign of given systems and constrained system evolution. Within each cluster a number of well defined new Control Theory problems are introduced, which may be studied within the structural methodologies framework of Linear Systems. The problems posed have a general systems character, but the emphasis here is on Linear Systems; an overview of relevant results is given and links with existing research topics are established. The paper defines the Structural Control Theoretic context of an important family of complex systems emerging in engineering design and defines a new research agenda for structural methods of Control Theory

Systems and control problems in early systems design

This thesis is concerned with the evaluation of properties of early design models, the control structure selection and the representation and properties of composite system models. The work is motivated by the need to introduce a Systems and Control Theory based framework for early design stages of the overall system design of engineering processes, and in particular chemical processes, such as process synthesis and control structure selection. The overall spirit of the work is that engineering design is an evolutionary process, the different stages of which shape the structure of the resulting system models and precondition the potential for design at the next stage. The work identifies a number of key problems in the overall design, which are of a generic, systemic character, and then deals with working out solutions for such problems. The results contribute in the development of a framework for systems integration using as criteria and tools, those provided by Systems and Control Theory. The work aims to provide a control theoretic dimension to the rules and practices currently used in the specific application areas. The thesis contributes in the development of a Systems and Control conceptual and tools framework for integrated design of engineering processes by providing results in the following areas: - Specification of a number of generic problems in the field of integrated design and identification of relevant control theoretic concepts and tools. - Study of Model Orientation for linear implicit state-space models and definition of classes of oriented realisations. - Development of solutions to two problems of Structural Identification for uncertain early process models related to infinite zero structure and McMillan degree. - Development of a generic representation of composite systems that allows the study of transition from the aggregate to composite system properties, as a generalised Control Design and characterisation of some key system properties. -Specification of a framework for integrated Control Structure Selection and development of software for many approaches of the "interaction analysis" indicators. The results contribute in the area of development of the systems and control ideas for the problems of systems integration and early design. The work emphasises the strong links between Modelling, Selection of Control Structures and Control Structural methodologies

Structural decomposition of general singular linear systems and its applications

Ph.DDOCTOR OF PHILOSOPH

Exact Feedback Linearization of Systems with State-Space Modulation and Demodulation

The control theory of nonlinear systems has been receiving increasing attention in recent years, both for its technical importance as well as for its impact in various fields of application. In several key areas, such as aerospace, chemical and petrochemical industries, bioengineering, and robotics, a new practical application for this tool appears every day. System nonlinearity is characterized when at least one component or subsystem is nonlinear. Classical methods used in the study of linear systems, particularly superposition, are not usually applied to the nonlinear systems. It is necessary to use other methods to study the control of these systems. For a wide class of nonlinear systems, a rather important structural feature comes from the strong nonlinearity appearing as coupling between spectrally decoupled parts of the system. Even in the case of low frequencies, where lumped models can still be employed the nonlinear coupling between parts of the system requires specific treatment, using advanced mathematical tools. In this context, an alternative, frequency domain approach is pursued here. In the rest of this work, a specific system form of linearly decoupled but nonlinearly coupled subsystems is examined. The mathematical toolbox of the Hilbert transform is appropriately introduced for obtaining two low-pass subsystems that form an equivalent description of the essential overall system dynamics. The nonlinear coupled dynamics is investigated systematically by partitioning the coupled system state vector in such a way as to fully exploit the low-pass and the band-pass intrinsic features of free dynamics. In particular, by employing the Hilbert Transform, a low-pass equivalent system is derived. Then, a typical case is investigated thoroughly by means of numerical simulation of the original coupled low and band-pass, real-state-variable system and the low-pass-equivalent, complex-state-variable derived one. The nonlinear model equations considered here pave the way for a systematic investigation of nonlinear feedback control options designed to operate mechatronic transducers in energy harvesting, sensing or actuation modes

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

Algebraic and Geometric Methods and Problems for Implicit Linear Systems

This thesis investigates a number of problems of the implicit linear systems framework. First, the problem of realisations of nonproper transfer functions is considered. The main result obtained here is the generalisation of the realisation method from MFDs to the case of the nonproper transfer functions. The obtained realisations are singular systems. The method treats both finite and infinite frequency behaviour in a unified way and generalises the results related to the minimality of the realisation and coprimeness and column reducedness of the MFD. Furthermore, it displays transparently the relation between the extended MacMillan degree of the transfer function and the minimal realisation. The next problem considered is the problem of canonical forms of minimal singular systems under restricted system equivalence transformations. For systems with outputs a canonical form is obtained and it is shown that it is directly related to the echelon form of the composite matrix of an MFD of the transfer function of the system. This result is a direct generalisation of the results of Popov and Forney for strictly proper systems. The canonical form obtained is of Popov type and may be considered as a direct generalisation of the well known form for strictly proper systems. The second canonical form is for systems without outputs. A Popov type canonical form for a class of these systems is obtained. This class is that of systems with equal reachability indices. For both canonical forms, the sequence of the transformations yielding the canonical description is described in detail. In the general case of systems without outputs a semi canonical Popov type form is obtained. Another problem considered in the thesis is the problem of first order realisations of autoregressive equations within the external equivalence framework. An alternative to the existing methods is provided; in fact, the proposed method is simpler than the existing ones and allows the derivation of the realisation by inspection of the autoregressive equations. A generalisation of the observability indices is proposed for nonsquare descriptor systems and their connection to the autoregressive equations is established. The problem of model matching for implicit systems is considered next. This is a generalisation of the model matching problem for systems described by transfer functions. Here a controller is interconnected to the given plant such that the overall system has a desired external behaviour. The problem is studied within the framework of external and A-external (input-output) equivalence. Necessary as well as sufficient conditions for the solvability of the problem are derived and the equations of the controllers are found in a constructive way. The last problem considered here is the generalised dynamic cover problem of geometric theory i.e. the problem of finding the family of (A, B)-invariant subspaces covering a given subspace. This problem is formulated here by using the matrix pencil approach of the geometric theory. This approach allows the unification of the problem for state-space and nonsquare descriptor systems. An extension of the problem to the case of infinite spectrum spaces is also obtained. The solution of the above problems is reduced to the solution of appropriately defined systems of linear equations. Finally, an alternative method for the solution involving systems of multilinear equations is proposed using the mathematical tool of Groebner bases

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

Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance

Issued as Progress report, and Final report, Project no. E-21-67

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