Linear Time-invariant Subsystems Research Articles

The feedback interconnection of multivariable systems: Simplifying theorems for stability

We consider the stability of the feedback interconnection of possibly unstable n-input n-output subsystems whose interconnection is described by e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> = u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> - y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> = u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> + y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> = G <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> (e <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ), i = 1,2. We give three theorems which simplify the stability tests. Theorem 1 deals with nonlinear time-varying subsystems. It gives conditions on G <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> so that the stability of u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ↦ y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> guarantees that of the feedback system. The other two theorems consider continuous-time linear time-invariant subsystems. It is noted that in the multivariable case, the stablity of u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ↦ y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , i = 1,2 is not sufficient to guarantee the stability of the feedback system, and Theorem 2 specifies some additional requited conditions. Theorem 3 shows that if G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (I + G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> are in some special stable classes, so is the transfer function of the feedback system. In both theorems, corollaries specialize the results to lumped and single-input single-output cases. The paper ends by showing how these results can be translated for the discrete-time case.

Justification of the Describing Function Method

Explicit conditions are given under which the use of the describing function method to investigate the nature of oscillations in autonomous nonlinear feedback systems is justified. When these conditions are satisfied, bounds are given for the frequency, fundamental magnitude, and higher harmonics of the oscillation based on the describing function approximation. The feedback systems considered are those which can be decomposed into a linear time-invariant subsystem, not necessarily causal, stable, or finite-dimensional, and a nonlinear frequency independent subsystem, possibly containing hysteresis. The approach taken is to consider the describing function equation as an approximation to a determining equation for periodic solutions of the autonomous system’s operator equation, and to use local degree theory to guarantee the existence of a solution to the determining equation.

Stability of a nonlinear time-invariant feedback system under almost constant inputs

We consider a multiple-input multiple-output feedback system consisting of a linear time-invariant subsystem and a memoryless time-invariant nonlinearity. The linear subsystem is represented by its impulse response which includes a unit step, i.e. an integrator. The nonlinearity is not required to be of the noninteracting type nor to be bounded away from zero by some sector condition. It is shown that for any “almost constant” input, the error e 2 is bounded and goes to zero as t → ∞.

Two-dimensional "Circle criterion" on the parameter plane

A graphical procedure using parameter plane theory is given for studying the stability of systems consisting of a linear time-invariant subsystem with two feedback time-varying gains.

Asymptotic Stability in the Large of a Class of Single-Loop Feedback Systems

Next article Asymptotic Stability in the Large of a Class of Single-Loop Feedback SystemsR. A. Baker and C. A. DesoerR. A. Baker and C. A. Desoerhttps://doi.org/10.1137/0306001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. P. O'shea, A combined frequency time-domain stability criterion for autonomous continuous systems, Trans. IEEE Automatic Control, AC-11 (1966), 477–484 10.1109/TAC.1966.1098402 CrossrefISIGoogle Scholar[2] Richard Bellman and , Kenneth L. Cooke, Differential-difference equations, Academic Press, New York, 1963xvi+462 MR0147745 0105.06402 Google Scholar[3] A. Halanay, Differential equations: Stability, oscillations, time lags, Academic Press, New York, 1966xii+528 MR0216103 0144.08701 Google Scholar[4] S. Bochner and , K. Chandrasekharan, Fourier Transforms, Annals of Mathematics Studies, no. 19, Princeton University Press, Princeton, N. J., 1949ix+219 MR0031582 0065.34101 Google Scholar[5] R. A. Baker and , C. A. Desoer, On scalar products of signals passing through memoryless nonlinearities with delay, 1966, Memo. ERL-M198, Electronics Research Laboratory, College of Engineering, University of California, Berkeley, to be published. Google Scholar[6] P. L. Falb and , G. Zames, On cross-correlation bounds and the positivity of certain nonlinear operators, IEEE Trans. Automatic Control, AC-12 (1967), 219–221 10.1109/TAC.1967.1098552 CrossrefISIGoogle Scholar[7] J. C. Willems and , M. Gruber, Comments on “A combined frequency-time domain stability criterion for autonomous continuous systems”, IEEE Trans. Automatic Control, AC-12 (1967), 217–219 10.1109/TAC.1967.1098516 CrossrefISIGoogle Scholar[8] M. A. Aizerman and , F. R. Gantmacher, Absolute stability of regulator systems, Translated by E. Polak, Holden-Day Inc., San Francisco, Calif., 1964viii+172 MR0183556 0123.28401 Google Scholar Next article FiguresRelatedReferencesCited ByDetails Volume 6, Issue 1| 1968SIAM Journal on Control History Submitted:23 January 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0306001Article page range:pp. 1-8ISSN (print):0036-1402Publisher:Society for Industrial and Applied Mathematics