A unified closed-loop stability measure for finite-precision digital controller realizations implemented in different representation schemes

The closed-loop stability issue of finite-precision realizations is investigated for digital controllers implemented in three different arithmetic formats, namely fixed-point, floating-point and block-floating-point schemes. It is shown that the controller coefficient perturbations resulting from using different finite word length (FWL) representation schemes possess quite different properties. A unified FWL closed-loop stability measure is derived which is applicable to all the three arithmetic schemes. Unlike the existing works which only take into account the precision of a representation scheme with an assumption of an unlimited dynamic range, both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure. To facilitate the design of optimal finite-precision controller realizations, a computationally tractable FWL closed-loop stability measure is then introduced and the method of computing the value of this measure for a given controller realization is given. For each arithmetic scheme, the optimal controller realization is defined as the solution that maximizes the corresponding measure, and a numerical optimization approach is adopted to solve for the resulting optimal realization problem. The proposed design procedure provides a unified framework for true optimal controller implementation that requires a minimum bit length with maximum robustness to the FWL effect. Numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes

A Unifying Framework for Finite Wordlength Realizations.

A general framework for the analysis of the finite wordlength (FWL) effects of linear time-invariant digital filter implementations is proposed. By means of a special implicit system description, all realization forms can be described. An algebraic characterization of the equivalent classes is provided, which enables a search for realizations that minimize the FWL effects to be made. Two suitable FWL coefficient sensitivity measures are proposed for use within the framework, these being a transfer function sensitivity measure and a pole sensitivity measure. An illustrative example is presented

Finite Wordlength Controller Realizations using the Specialized Implicit Form

Une forme d'état implicite spécialisée est présentée pour étudier les effets de l'implantation en précision finie des régulateurs. Cette forme permet une description macroscopique des algorithmes à implanter. Elle constitue un canevas unificateur permettant de décrire les différentes structures utilisées pour l'implantation, telles que les réalisations avec l'opérateur delta, la forme directe II en rho, la forme d'état-observateur et bien d'autres formes qui sont d'habitude traitées séparément dans la littérature. Différentes mesures quantifiant les effets de l'implantation sur le comportement en boucle fermée sont définis dans ce contexte. Elles concernent aussi bien la stabilité que la performance. L'écart entre la réalisation à précision infinie et la réalisation à précision finie est évaluée selon la mesure de sensibilité des coefficients et la mesure du bruit de quantification. Le problème consistant à trouver une réalisation dont l'implantation amène un minimum de dégradation peut alors est résolut numériquement. Cette approche est illustrée avec deux exemples

Measurements, Models, Systems and Design

531 s.

Finite worldlength effects in fixed-point implementations of linear systems

Thesis (M.Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 173-194).by Vinay Mohta.M.Eng

MIT Space Engineering Research Center

The Space Engineering Research Center (SERC) at MIT, started in Jul. 1988, has completed two years of research. The Center is approaching the operational phase of its first testbed, is midway through the construction of a second testbed, and is in the design phase of a third. We presently have seven participating faculty, four participating staff members, ten graduate students, and numerous undergraduates. This report reviews the testbed programs, individual graduate research, other SERC activities not funded by the Center, interaction with non-MIT organizations, and SERC milestones. Published papers made possible by SERC funding are included at the end of the report

The Institution of Engineering and Technology

Abstract: In a debate paper, Keel and Bhattacharyya have suggested, by means of simple examples taken from the open literature, that optimal and robust controllers can be fragile in the sense that a minute perturbation in the controller parameters can make the closed-loop system unstable. However, is it true that the optimal and robust controllers presented by Keel and Bhattacharyya are actually fragile? It is demonstrated that the particular parametric stability margin used by Keel and Bhattacharyya can be very conservative and to overcome this problem, two non-conservative measures of controller fragility are proposed. In addition, it will be shown that the examples in Keel and Bhattacharyya's paper are very special and the resulting fragility cannot be linked to the H 1 optimisation but to non-appropriate H 1 optimisation criterions and to bad choice of weights. Introduction In . Different explanations for the fragility problem can be found in the literature. Mäkilä [4] examine Examples 3, 4 and 5 of [1] and present a procedure for assessing the fragility on the basis of the inherent robustness of the closed-loop system to perturbation in the physical parameters that make up implementation, using first-and second-order active RC filters in the implementation of continuous-time controllers and considering the effects of floating point erros in the implementation of digital controllers. More recently, Examples 1 and 2 of [1] have been revisited In spite of all the works listed in the previous paragraph, some questions still remain to be answered. Is it true that the optimal and robust controllers presented in [1] are actually so fragile? More importantly, is it true that the controllers obtained as solutions of the simple optimisation criteria presented in [1] are necessarily fragile? In this paper, these questions are answered and it is demonstrated that the particular stability margin used by Keel and Bhattacharyya can be very conservative and to overcome this problem, two non-conservative measures, based on necessary and sufficient conditions, are proposed here. In addition, it will be shown that the examples presented in [1] are very special and the resulting fragility cannot be associated with H 1 optimisation but to non-appropriate H 1 optimisation criterions and to bad choice of weights. This paper is organised as follows: in section 2, the relative parametric stability margin is reviewed, and an example that suggests the conservativeness of this measure is presented. In section 3, two nonconservative measures of controller fragility are proposed and a comparison between the relative parametric stability margin and the two nonconservative measures introduced in this paper is drawn. In section 4, the examples used in [1] to label H 1 controllers as fragile are re-examined. Finally, conclusions are drawn in section 5. 2 Relative parametric stability margin Definition Consider a closed-loop system with unit negative feedback, wher

Flexible structure control laboratory development and technology demonstration

An experimental structure is described which was constructed to demonstrate and validate recent emerging technologies in the active control and identification of large flexible space structures. The configuration consists of a large, 20 foot diameter antenna-like flexible structure in the horizontal plane with a gimballed central hub, a flexible feed-boom assembly hanging from the hub, and 12 flexible ribs radiating outward. Fourteen electrodynamic force actuators mounted to the hub and to the individual ribs provide the means to excite the structure and exert control forces. Thirty permanently mounted sensors, including optical encoders and analog induction devices provide measurements of structural response at widely distributed points. An experimental remote optical sensor provides sixteen additional sensing channels. A computer samples the sensors, computes the control updates and sends commands to the actuators in real time, while simultaneously displaying selected outputs on a graphics terminal and saving them in memory. Several control experiments were conducted thus far and are documented. These include implementation of distributed parameter system control, model reference adaptive control, and static shape control. These experiments have demonstrated the successful implementation of state-of-the-art control approaches using actual hardware

Fundamental limits in Gaussian channels with feedback: confluence of communication, estimation, and control

The emerging study of integrating information theory and control systems theory has attracted tremendous attention, mainly motivated by the problems of control under communication constraints, feedback information theory, and networked systems. An often overlooked element is the estimation aspect; however, estimation cannot be studied isolatedly in those problems. Therefore, it is natural to investigate systems from the perspective of unifying communication, estimation, and control;This thesis is the first work to advocate such a perspective. To make Matters concrete, we focus on communication systems over Gaussian channels with feedback. For some of these channels, their fundamental limits for communication have been studied using information theoretic methods and control-oriented methods but remain open. In this thesis, we address the problems of characterizing and achieving the fundamental limits for these Gaussian channels with feedback by applying the unifying perspective;We establish a general equivalence among feedback communication, estimation, and feedback stabilization over the same Gaussian channels. As a consequence, we see that the information transmission (communication), information processing (estimation), and information utilization (control), seemingly different and usually separately treated, are in fact three sides of the same entity. We then reveal that the fundamental limitations in feedback communication, estimation, and control coincide: The achievable communication rates in the feedback communication problems can be alternatively given by the decay rates of the Cramer-Rao bounds (CRB) in the associated estimation problems or by the Bode sensitivity integrals in the associated control problems. Utilizing the general equivalence, we design optimal feedback communication schemes based on the celebrated Kalman filtering algorithm; these are the first deterministic, optimal communication schemes for these channels with feedback (except for the degenerated AWGN case). These schemes also extend the Schalkwijk-Kailath (SK) coding scheme and inherit its useful features, such as reduced coding complexity and improved performance. Hence, this thesis demonstrates that the new perspective plays a significant role in gaining new insights and new results in studying Gaussian feedback communication systems. We anticipate that the perspective could be extended to more general problems and helpful in building a theoretically and practically sound paradigm that unifies information, estimation, and control