User:Jiuguang Wang/George Zames

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Key Studies
(1) Nonlinear Operators for Systems Analysis, MIT Research Laboratory of Electronics Laboratory Technical Report 370, September 1960 This publication is based on Zames’s Ph.D. dissertation. It was the first systematic study of feedback by operator theoretic methods. Systems were represented by elements of a normed algebra of input-output mappings. The term gain, conventionally used to describe transducer amplification, is introduced to denote the operator norm, which depends on the sector width or Lipschitz constant of the operator. For feedback with loop gain less than one, the effects of open-loop perturbations (distortion, uncertainty, etc.) on closed-loop behavior are shown to be bounded. Since the loop gain is typically large in practice, an effort is made to catalog the various loop transformations that could be used to reduce the gain to produce a contraction. These turned out to be combinations of fractional transformations and weightings or “multipliers.” The study emphasizes robustness and gives a precise definition of physical realizability of systems. It also addresses global linearization of nonlinear systems. (2) Realizability Conditions for Nonlinear Feedback Systems, IEEE Transactions on Circuit Theory, Vol. 11, No. 2, pp. 186-194, June 1964 The question posed by this study is “what is a good enough mathematical model of a physical system—a model that does not lead to impossible results in feedback problems?” Zames defines the concepts of “generalized delay” and “generalized attenuation,” shows how such properties determine “system realizability,” and demonstrats “that they are necessary in order to avoid paradoxical and perplexing behavior of models in feedback.” This was the first fundamental treatise on the subject of well-posedness of feedback systems. (3) On the input-output stability of time-varying nonlinear feedback systems, Part one: Conditions derived using concepts of loop gain, conicity, and positivity, IEEE Transactions on Automatic Control, Vol. 11, No. 2, pp. 465-476, April 1966 On the input-output stability of time-varying nonlinear feedback systems, Part II: Conditions involving circles in the frequency plane and sector nonlinearities, IEEE Transactions on Automatic Control, Vol. 11, No. 3, pp. 465-476, July 1966 The papers develop a theory of input–output stability. They introduce extended Normed Linear Spaces, the Small Gain Theorem, the Passivity Theorem, the Circle Criterion in input–output form, and the use of multipliers. There are three main results in Part I, which follow the introduction of concepts of gain, conicity, positivity, and strong positivity: THEOREM 1: If the open loop gain is less than one, then the closed loop is bounded. THEOREM 2: If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. Part II contains applications to a loop with one nonlinear element. The basic approach is summarized by Zames at the end of Part I of this paper thus: “One of the broader implications of the theory developed here concerns the use of functional analysis for the study of poorly defined systems. It seems possible, from only coarse information about a system, and perhaps even without knowing details of internal structure, to make assessments of qualitative behavior.” (4)  Dither in nonlinear systems, IEEE Transactions on Automatic Control, Vol. 21, No. 5, pp. 660-667, October 1976, with N. Shneydor   Structural stabilization and quenching by dither in nonlinear systems, IEEE Transactions on Automatic Control, Vol. 22, No. 3, pp. 352-361, June 1977, with N. Shneydor  A dither is a high frequency signal introduced into a nonlinear system for the purpose of augmenting stability or quenching undesirable jump-phenomena. Zames and Shneydor show that the effects of a dither depend on its amplitude distribution function. The stability of a dithered system is related to that of an equivalent smoothed system, whose nonlinear element is the convolution of the dither distribution and the original nonlinearity. Furthermore, they demonstrate the ability of dithers to quench jump-phenomena (i.e., to induce continuity). A notion of structural-stability for feedback equations is introduced, and it is shown that dithers can structurally stabilize large classes of nonlinear systems subject to a second-order Lipschitz condition. The quenching properties of dithers are explained in terms of an effective narrowing of the nonlinear incremental sector. (5)  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses, IEEE Transactions on Automatic Control, Vol. 26, No. 2, pp. 301-320, April 1981 This study introduced for the first time what was to be called H∞ control theory. It studies causal linear time-invariant bounded maps from L2to L2, which, by the Foures–Segal theorem, is in one-to-one correspondence with operators which are multiplications by H∞ functions. “ H∞ is the space of matrix-valued functions that are analytic and bounded in the left-hand half of the complex plane defined by Re(s) < 0; the H∞ norm is the maximum singular value of the function over that space. (This can be interpreted as a maximum gain in any direction and at any frequency; for single input single output systems, this is effectively the maximum magnitude of the frequency response.)” Zames formulates the problem of sensitivity reduction by feedback as an optimization problem, separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity-reducing schemes are derived, and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time invariant systems characterized by n X n matrices of H∞ frequency response functions, either with or without zeros in the right half-plane. Approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. (6) Unstable systems and feedback: the gap metric, Proceedings of the 1980 Allerton Conference, pp. 380-385, October 1980, with A. El-Sakkary Zames and El-Sakkary identify the natural topology for studying questions of robustness in feedback systems. They characterize the tolerable open-loop errors in terms of a metric defined as the “gap” or “aperture” between the graphs of operators in Hilbert space, and use it to quantify distances between systems and to show that all metrics which are “continuously robust” in the above sense are equivalent to the gap metric. The introduction of a topology for robustness questions provides a unifying framework within which other approaches to robustness can be considered, and where questions of well-posedness and continuity of design techniques can be addressed. The approach addresses a major puzzle of how to deal with possible instability of the open loop while avoiding an overly structured model for uncertainty. (7) Towards a general complexity-based theory of identification and adaptive control, in Lecture Notes in Control and Information Sciences (Berlin: Springer, ISBN: 978-3-540-76097-9), Volume 222, 1997 Fundamentally the study is concerned with representations of model uncertainty which usually will have a parametric part and will also contain a residual unmodeled nonparametric part. Zames was interested in the question of robust identification of the parametric part in the presence of unmodeled nonparametric uncertainty. In his view, a resolution of this problem was a prerequisite for a satisfactory theory of adaptive control. The study is based on the solution of two problems of control theory, which are combined to generate new approaches to H∞/l1 adaptive control, and make a contribution toward a general theory of adaptation and complexity-based or information-based learning. The two problems are: (1) optimally fast identification to reduce plant uncertainty (e.g., to a weighted ball of a given radius in H∞/l1 ; and  (2) exact computation of feedback performance under large plant uncertainty (including such a ball).  By combining these two solutions and using frozen-time approximations to compute optimal feedbacks under time-varying data, Zames obtains control laws which conform to his definition of what the term “adaptive” should mean.  These laws are in a certain sense “nearly optimally adaptive,” and the results lead to concrete algorithms. Furthermore, they suggest a more general theory of adaptive control. Zames proposes definitions of the notions of machine adaptation and machine learning which are independent of the internal structure of, say, the controller in the case of a feedback system; and are independent of properties such as the presence or absence of nonlinearity, time-variation, or even feedback. Instead, these definitions are based on external performance. They address such questions as: What should the term “adaptive” and “learning” mean in the context of control? Is it possible to tell whether or not a black box is adaptive? Is adaptation synonymous with the presence of nonlinear feedback? In control system design, is it possible to determine beforehand whether it is necessary for a controller to adapt and learn in order to meet performance specifications, or is adaptation a matter of choice? 