This paper proposes a multiple-model adaptive control methodology, using set-valued observers (MMAC-SVO) for the identification subsystem, that is able to provide robust stability and performance guarantees for the closed-loop, when …
This paper proposes a multiple-model adaptive control methodology, using set-valued observers (MMAC-SVO) for the identification subsystem, that is able to provide robust stability and performance guarantees for the closed-loop, when the plant, which can be open-loop stable or unstable, has significant parametric uncertainty. We illustrate, with an example, how set-valued observers (SVOs) can be used to select regions of uncertainty for the parameters of the plant. We also discuss some of the most problematic computational shortcomings and numerical issues that arise from the use of this kind of robust estimation methods. The behavior of the proposed control algorithm is demonstrated in simulation.
The author was an undergraduate and graduate student at the University of California at Berkeley (Cal for short) during the initial period of the modern control theory revolution. In this …
The author was an undergraduate and graduate student at the University of California at Berkeley (Cal for short) during the initial period of the modern control theory revolution. In this paper he recounts some of the events that took place at Cal during that time and reminisce about his teachers and colleagues in the early days of optimal control theory. He attempts to interlace his personal experiences with the educational, research, and social atmosphere at Cal, so as to present a "readable tale".
Gain scheduling has proven to be a successful design methodology in many engineering applications. In the absence of a sound theoretical analysis, these designs come with no guarantees of the …
Gain scheduling has proven to be a successful design methodology in many engineering applications. In the absence of a sound theoretical analysis, these designs come with no guarantees of the robustness, performance, or even nominal stability of the overall gain-scheduled design. An analysis is presented for two types of nonlinear gain-scheduled control systems: (1) scheduling on a reference trajectory, and (2) scheduling on the plant output. Conditions which guarantee stability, robustness, and performance properties of the global gain schedule designs are given. These conditions confirm and formalize popular notions regarding gain scheduled designs, such as that the scheduling variable should vary slowly, and capture the plant's nonlinearities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
A systematic methodology for the design of control systems with multiple saturations in control magnitude and rate is introduced. A supervisor loop is introduced. When the references and/or disturbances are …
A systematic methodology for the design of control systems with multiple saturations in control magnitude and rate is introduced. A supervisor loop is introduced. When the references and/or disturbances are small enough not to cause saturations, the system operates linearly as designed. When the signals are large enough to cause saturations, the control law is modified in such a way as to preserve, to the extent possible, the behavior of the linear control design. It is shown that the operator error governor can be used to design control systems when the controls saturate in magnitude and rate. The main benefits of the methodology are that it leads to controllers with the following properties: the signals that the modified compensator produces never cause saturation; possible integrators or slow dynamics in the compensator never windup; the closed-loop system has inherent stability properties; and the online computation required to implement the control system is feasible. An example is used to illustrate the methodology and its benefits.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
A well known result for finite-dimensional time-varying linear systems is that if each 'frozen time' system is stable, then the time-varying system is stable for sufficiently slow time-variations. These results …
A well known result for finite-dimensional time-varying linear systems is that if each 'frozen time' system is stable, then the time-varying system is stable for sufficiently slow time-variations. These results are reviewed and extended to a class of Volterra integrodifferential equations, specifically, differential equations with a convolution operator in the right-hand-side. The results are interpreted in the context of robustness of time-varying linear systems with a special emphasis on analysis of gain-scheduled control systems.
In order to provide a theoretical tool well suited for use in characterizing the stability margins (e.g., gain and phase margins) of multiloop feedback systems, multiloop input-output stability results generalizing …
In order to provide a theoretical tool well suited for use in characterizing the stability margins (e.g., gain and phase margins) of multiloop feedback systems, multiloop input-output stability results generalizing the circle stability criterion are considered. Generalized conic sectors with "centers" and "radii" determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency-dependent convex set of modeling errors-including nonlinearities, gain variations, and phase variations-which the system must be able to tolerate in each feedback loop without instability. The resulting stability criterion gives sufficient conditions for closed-loop stability in the presence of such frequency-dependent modeling errors, even when the modeling errors occur simultaneously in all loops; so, for example, stability is assured as loop gains and phases vary throughout a "set of nonzero measure" whose boundaries are frequency-dependent. The stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds, or falls short of, its stability margin specifications.
Additional quantitative results are presented for the existence of optimal decision rules and stochastic stability for linear systems with white random parameters with respect to quadratic performance criteria by examining …
Additional quantitative results are presented for the existence of optimal decision rules and stochastic stability for linear systems with white random parameters with respect to quadratic performance criteria by examining a specific version of a multivariable optimization problem.
A self-contained theory of extrema (viz., suprema, maxima, minima, and infima) of differentiable functions of several (possibly infinitely many) variables mapping into finite-dimensional integrally closed directed partially ordered linear spaces …
A self-contained theory of extrema (viz., suprema, maxima, minima, and infima) of differentiable functions of several (possibly infinitely many) variables mapping into finite-dimensional integrally closed directed partially ordered linear spaces is reported. The applicability of the theory to the analysis of linear least squares vector estimation problem is demonstrated.
Proof of relationship between state estimates and error covariance matrices in state-space representations in linear filtering
Proof of relationship between state estimates and error covariance matrices in state-space representations in linear filtering
Existence of inflection points in cost vs terminal time curve for linear minimum energy regulator
Existence of inflection points in cost vs terminal time curve for linear minimum energy regulator
The controlled system has the transfer function G(s)=K/(s-? 1 )(s-? 2 ). The control input is u(t), |u(t)|?1, and the out put is y 1 (t). the control which forces …
The controlled system has the transfer function G(s)=K/(s-? 1 )(s-? 2 ). The control input is u(t), |u(t)|?1, and the out put is y 1 (t). the control which forces any initial state y 1 (0), ? 1 (0) to the terminal state y 1 (T), ? 1 ,(T), such that -K/? 1 ? 2 ?y 1 (T)?K/? 1 ? 2 , ? 1 (T)=0, and which minimizes the fuel, F(T)=? 0 T |u(t)|dt is determined. The phase plane is divided into three sets G - , G + , and G 0 ; if the state is in G - then u(t)=?1 is used; if the state is in G + , then u(t)=+1 is used; if the state is in G 0 , then u(t)=0 is used.
A class of differential pursuit-evasion games is examined in which the dynamics are linear and perturbed by additive white Gaussian noise, the performance index is quadratic, and both players receive …
A class of differential pursuit-evasion games is examined in which the dynamics are linear and perturbed by additive white Gaussian noise, the performance index is quadratic, and both players receive measurements perturbed independently by additive white Gaussian noise. Linear minimax solutions are characterized in terms of a set of implicit integro-differential equations. A game of this type also possesses a "certainty-coincidence" property, meaning that its minimax behavior coincides with that of the corresponding deterministic game in the event that all noise values are zero. This property is used to decompose the minimax strategies into sums of a certainty-equivalent term and error terms.
A derivation is presented in which it is shown that the optimal gain sequence of the discrete Kalman filter minimizes not only the trace of the estimation covariance matrix, but …
A derivation is presented in which it is shown that the optimal gain sequence of the discrete Kalman filter minimizes not only the trace of the estimation covariance matrix, but any linear combination of the main diagonal elements of that matrix.
An example is given to show that even if the eigenvalues of the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> -matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(t)</tex> of a linear time-varying system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}(t) = …
An example is given to show that even if the eigenvalues of the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> -matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(t)</tex> of a linear time-varying system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}(t) = A(t)x(t)</tex> are independent of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> and some of them have positive real parts, the system is asymptotically stable.
It is proved that a condition similar to the Nyquist criterion guarantees the stability (in an important sense) of a large class of feedback systems containing a single time-varying nonlinear …
It is proved that a condition similar to the Nyquist criterion guarantees the stability (in an important sense) of a large class of feedback systems containing a single time-varying nonlinear element. In the case of principal interest, the condition is satisfied if the locus of a certain complex-valued function (a) is bounded away from a particular disk located in the complex plane, and (b) does not encircle the disk.
A limiting case of great importance in engineering is that of slowly varying parameters. For systems described by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x} = A(t)x</tex> , one would intuitively expect that if, …
A limiting case of great importance in engineering is that of slowly varying parameters. For systems described by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x} = A(t)x</tex> , one would intuitively expect that if, for each <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> , the frozen system is stable, then the time-varying system should also be stable. Provided <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(t)</tex> is small enough, Rosenbrock has shown that this is the case [1]. Rosenbrock used a continuity argument [1, p. 75]. In this correspondence explicit bounds and slightly sharper results are obtained. Finally, it is pointed out that these results are useful in the study of the exact behavior of non-linear lumped systems with slowly varying operating points.
A powerful iterative descent method for finding a local minimum of a function of several variables is described. A number of theorems are proved to show that it always converges …
A powerful iterative descent method for finding a local minimum of a function of several variables is described. A number of theorems are proved to show that it always converges and that it converges rapidly. Numerical tests on a variety of functions confirm these theorems. The method has been used to solve a system of one hundred non-linear simultaneous equations.
Abstract A comprehensive discussion is given of the background to the generalized Nyquist stability criterion for linear multivariable feedback systems. This leads to a proof based on the use of …
Abstract A comprehensive discussion is given of the background to the generalized Nyquist stability criterion for linear multivariable feedback systems. This leads to a proof based on the use of the Principle of the Argument applied to an algebraic function defined on an appropriate Riemann surface. It is shown how the matrix-valued functions of a complex variable which define the loop transmittance, return-ratio and return-difference matrices of feedback systems analysis may be associated with a set of characteristic algebraic functions, each associated with a Riemann surface. These characteristic functions enable the characteristic loci, which featured in previous heuristic treatments of the generalized Nyquist stability criterion, to be put on a sound basis. The relationship between the algebraic structure of the matrix-valued functions and the appropriate complex-variable theory is carefully discussed. These extensions of the complex-variable concepts underlying the Nyquist criterion are then related to an appropriate generalization of the root locus concept. It is shown that multivariable root loci are the 180° phase loci of the characteristic functions of a return-ratio matrix on an appropriate Riemann surface, plus some possibly degenerate loci each consisting of a single point.
In order to provide a theoretical tool well suited for use in characterizing the stability margins (e.g., gain and phase margins) of multiloop feedback systems, multiloop input-output stability results generalizing …
In order to provide a theoretical tool well suited for use in characterizing the stability margins (e.g., gain and phase margins) of multiloop feedback systems, multiloop input-output stability results generalizing the circle stability criterion are considered. Generalized conic sectors with "centers" and "radii" determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency-dependent convex set of modeling errors-including nonlinearities, gain variations, and phase variations-which the system must be able to tolerate in each feedback loop without instability. The resulting stability criterion gives sufficient conditions for closed-loop stability in the presence of such frequency-dependent modeling errors, even when the modeling errors occur simultaneously in all loops; so, for example, stability is assured as loop gains and phases vary throughout a "set of nonzero measure" whose boundaries are frequency-dependent. The stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds, or falls short of, its stability margin specifications.
Considered is the asymptotic property of the discrete-time matrix Riccati equation arising in the optimal control of linear systems with a random gain. The instability and stability conditions are derived …
Considered is the asymptotic property of the discrete-time matrix Riccati equation arising in the optimal control of linear systems with a random gain. The instability and stability conditions are derived in terms of the degree of stability of the state transition matrix.
An iterative method for finding stationary values of a function of several variables is described. In many ways it is similar to the method of steepest descents; however this new …
An iterative method for finding stationary values of a function of several variables is described. In many ways it is similar to the method of steepest descents; however this new method has second order convergence.
This paper introduces a nonconservative measure of performance for linear feedback systems in the face of structured uncertainty. This measure is based on a new matrix function, which we call …
This paper introduces a nonconservative measure of performance for linear feedback systems in the face of structured uncertainty. This measure is based on a new matrix function, which we call the Structured Singular Value.
A quotient algebra of transfer functions of distributed linear time-invariant subsystems is proposed; this algebra is a generalization of the algebra of proper rational functions. Its main virtue is that …
A quotient algebra of transfer functions of distributed linear time-invariant subsystems is proposed; this algebra is a generalization of the algebra of proper rational functions. Its main virtue is that it allows the algebraic manipulation of distributed systems within the algebra. Series, parallel, and, under some regularity conditions, feedback interconnection of transfer functions in the algebra remain in the algebra. The relation of our algebra to the algebras proposed by Morse, Dewilde, and Kamen is discussed and the algebras are compared. Finally, applications of the algebra are indicated.
A well known result for finite-dimensional time-varying linear systems is that if each 'frozen time' system is stable, then the time-varying system is stable for sufficiently slow time-variations. These results …
A well known result for finite-dimensional time-varying linear systems is that if each 'frozen time' system is stable, then the time-varying system is stable for sufficiently slow time-variations. These results are reviewed and extended to a class of Volterra integrodifferential equations, specifically, differential equations with a convolution operator in the right-hand-side. The results are interpreted in the context of robustness of time-varying linear systems with a special emphasis on analysis of gain-scheduled control systems.
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 28 June 1966Published online: 14 July 2006Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society …
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 28 June 1966Published online: 14 July 2006Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam
Let ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> denote the set of N-vector-valued functions of t defined on [0, ∞) such that for any real positive number y, the square of the modulus …
Let ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> denote the set of N-vector-valued functions of t defined on [0, ∞) such that for any real positive number y, the square of the modulus of each component of any element is integrable on [0, y], and let L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) denote the subset of ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> with the property that the square of the modulus of each component of any element is integrable on [0, ∞). In the study of nonlinear physical systems, attention is frequently focused on the properties of one of the following two types of functional equations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eqalignno{g &= f + KQf \cr g &= Kf + Qf}$</tex> in which K and Q are causal operators, with K linear and Q nonlinear, g ε ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> , and f is a solution belonging to ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> . Typically, f represents the system response and g takes into account both the independent energy sources and the initial conditions at t = 0. It is often important to determine conditions under which a physical system governed by one of the above equations is stable in the sense that the response to an arbitrary set of initial conditions approaches zero (i.e., the zero vector) as t → ∞. In a great many cases of this type, g belongs to L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) and approaches zero as t → ∞ for all initial conditions, and, in addition, it is possible to show that if f ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞), then f(t) → 0 as t → ∞. In this paper we attack the stability problem by deriving conditions under which g ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) and f ε ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> imply that f ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞). From an engineering viewpoint, the assumption that f ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> is almost invariably a trivial restriction. As a specific application of the results, we consider a nonlinear integral equation that governs the behavior of a general control system containing linear time-invariant elements and an arbitrary finite number of time-varying nonlinear elements. Conditions are presented under which every solution of this equation belonging to ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> in fact belongs to L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) and approaches zero as t → ∞.