A New Approach to the Lur'e Problem in the Theory of Absolute Stability
01 May 2003-Siam Journal on Control and Optimization (Society for Industrial and Applied Mathematics)-Vol. 42, Iss: 5, pp 1895-1904
TL;DR: A new approach is developed providing a sufficient stability criterion for systems with time-variable coefficients, expressed in the transfer function of the linear block and the sector margins of the nonlinear block, which is shown to be guaranteed by stability of the system with a limit linear feedback.
Abstract: The classical Lur'e problem consists of finding conditions for absolute stability of a linear system with a nonlinear feedback contained within a prescribed sector. Most of the results obtained on this problem are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach providing a sufficient stability criterion for systems with time-variable coefficients, which is expressed in the transfer function of the linear block and the sector margins of the nonlinear block. The systems for which this criterion is precise are found. It is shown that stability of a system with a sign-constant transfer function is guaranteed by stability of the system with a limit linear feedback (so that, for such systems, the famous Aizerman conjecture is true). This, in particular, is the case for systems with a linear block consisting of an arbitrary number of first order time-dependent links. As an example, the stability criterion is applied to a second order system for which the obtained results are contrasted with ones delivered by the Popov criterion.
Citations
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TL;DR: A brief historical background of the development of the absolute stability theory was presented, and some of its methods and approaches, as well as the results obtained using them, were discussed as discussed by the authors.
Abstract: A brief historical background of the development of the absolute stability theory was presented, and some of its methods and approaches, as well as the results obtained using them, were discussed. The kinds of systems under consideration are named, and applications of the methods and results of the absolute stability theory to other scientific and practical fields for solution of the engineering, mechanical, physical, and other problems were presented.
99 citations
10 Oct 2005
TL;DR: The absolute stability of time-delay systems with non-commensurate internal point delays for any non-linearity satisfying a standard time positivity inequality is discussed in this article.
Abstract: The absolute stability, independent of the delays, of time-delay systems with non-commensurate internal point delays for any non-linearity satisfying a standard time positivity inequality is discussed. That property holds if an associate delay-free system is absolutely stable and the size of the delayed dynamics is sufficiently small. The results are obtained for non-linearities belonging to sectors [0, k] and [h, k+h], k≥0 and are based on a parabola test type.
12 citations
TL;DR: It is shown that if the transfer function is sign-constant, asymptotic stability of the system with the margin-linear feedback guarantees absolute Stability of the considered system; thus, such systems satisfy the known Aizerman conjecture.
Abstract: A nonautonomous linear system controlled by a nonlinear sector-restricted feedback with a time-varying delay is considered. Delay-independent sufficient conditions for absolute stability and instability (expressed in the transfer function of the linear part and the sector bounds) are established. For a system with an exponentially stable linear part, an upper bound for the Lyapunov exponent is found. It is shown that if the transfer function is sign-constant, asymptotic stability of the system with the margin-linear feedback guarantees absolute stability of the considered system; thus, such systems satisfy the known Aizerman conjecture. They include, in particular, a closed-loop system consisting of any number of time-varying first order links and feedback with arbitrary delay. Under some additional condition (which is certainly true for a time-invariant linear block), the obtained stability criterion is precise. The approach employed in the proofs is based on a direct analysis of the corresponding Volterra equation which contains only the transfer function of the linear block and, therefore, embraces a wide range of control systems. As an example, a second order system is considered; it is shown that here the obtained stability bound is reached for a linear feedback with a periodic delay function.
9 citations
01 Jan 2007
TL;DR: The first results on the Kalman-Yakubovich-Kalman-Popov lemma were due to Yakubovich and Popov as mentioned in this paper, who used general factorization of matrix polynomials.
Abstract: The Kalman–Yakubovich–Popov Lemma (also called the Yakubovich–Kalman–Popov Lemma) is considered to be one of the cornerstones of Control and Systems Theory due to its applications in absolute stability, hyperstability, dissipativity, passivity, optimal control, adaptive control, stochastic control, and filtering. Despite its broad field of applications, the lemma has been motivated by a very specific problem which is called the absolute stability Lur’e problem [1, 2], and Lur’e’s work in [3] is often quoted as +being the first time the so-called KYP Lemma equations have been introduced. The first results on the Kalman–Yakubovich–Popov Lemma are due to Yakubovich [4, 5] . The proof of Kalman [6] was based on factorization of polynomials, which were very popular among electrical engineers. They later became the starting point for new developments. Using general factorization of matrix polynomials, Popov [7, 8] obtained the lemma in the multivariable case. In the following years, the lemma was further extended to the infinite-dimensional case (Yakubovich [9], Brusin [10], Likhtarnikov and Yakubovich [11]) and discrete-time case (Szego and Kalman [12]).
4 citations
12 Dec 2005
TL;DR: In this article, the authors developed a new approach to stability analysis of a system containing a linear part and a scalar nonlinear sector restricted function based on a direct analysis of the corresponding integral Volterra equation about the input of the nonlinear block.
Abstract: Finding conditions for absolute stability of a system containing a linear part and a scalar nonlinear sector restricted function is a classical Lur'e problem. Most of the corresponding results are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach to stability analysis of the problem based on a direct analysis of the corresponding integral Volterra equation about the input of the nonlinear block. The obtained sufficient stability criterion is applicable to non-autonomous systems with arbitrary time-varying delay in the feedback. The approach is extended to general time-varying systems including a linear block and norm bounded vector nonlinear terms with uncertain time-varying delays. The obtained delay-independent stability conditions are formulated in the terms of the transition matrix of the linear part and the norms of the nonlinear terms. The systems are indicated for which the obtained criteria are not only sufficient but also necessary for any delay function. The obtained results are applied to stability analysis of some systems previously studied in the literature; in all cases less conservative stability bounds are found.
3 citations
References
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Book•
01 Jan 1978TL;DR: In this article, the authors consider non-linear differential equations with unique solutions, and prove the Kalman-Yacubovitch Lemma and the Frobenius Theorem.
Abstract: Introduction. Non-linear Differential Equations. Second-Order Systems. Approximate Analysis Methods. Lyapunov Stability. Input-Output Stability. Differential Geometric Methods. Appendices: Prevalence of Differential Equations with Unique Solutions, Proof of the Kalman-Yacubovitch Lemma and Proof of the Frobenius Theorem.
3,388 citations
TL;DR: A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality.
Abstract: This paper introduces a unified approach to robustness analysis with respect to nonlinearities, time variations, and uncertain parameters. From an original idea by Yakubovich (1967), the approach has been developed under a combination of influences from the Western and Russian traditions of control theory. It is shown how a complex system can be described, using integral quadratic constraints (IQC) for its elementary components. A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality. A systematic computational approach is described, and relations to other methods of stability analysis are discussed. Last, but not least, the paper contains a summarizing list of IQCs for important types of system components.
1,547 citations
584 citations
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Book•
15 Jan 1971
TL;DR: "Parts of this monograph appeared in the author's doctoral dissertation entitled 'Nonlinear harmonic analysis' ... 1968."
Abstract: "Parts of this monograph appeared in the author's doctoral dissertation entitled 'Nonlinear harmonic analysis' 1968"
477 citations
Book•
01 Jan 1973
454 citations
Additional excerpts
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