New relationships between input-output and Lyapunov stability (original) (raw)

1982, IEEE Transactions on Automatic Control

https://doi.org/10.1109/TAC.1982.1102937

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Abstract

Theorem 2. I] such that G : = F '0, is a gcrd of (&E, Np, Dc), implying Manuscnpt reccived June 2. 1981: revhed August 18. 1981. ?vi M. Fahmy i b with the Department of Electrical Engineering and Electronics, University of Liverpool. Liverpool L69 3BX. England. on leave from the Department of Electrical Engineering. Asiut University. Auiut. Eg>~t. of Liverpool. 1.iverpool L69 3BX. England. J OReilly i5 with the Department of Electrical Engineenng and Electronlcs. Unlvcrrit)

Four Lectures On Stability

Journal of Control Engineering and Applied Informatics, 2006

Starting with the basic notions about Liapunov and input/output stability there are presented those elements that represent the minimal knowledge of Stability Theory and its methods.A particular attention is paid to absolute stability of feedback systems and connected framework (Liapunov functions, frequency domain inequalities, hyperstability and dissipativity). Acknowledgement: In 1990 the Romanian Society for Automation and Industry Applied Information Processing emerged as an initiative of some outstanding control engineers active both in Education and Research. Systems and Control had been a long date field of interest in Romania however a dedicated academic and application oriented society was missing for, mildly said, nonacademic reasons. Neither 1990 nor the following 1991 did not show very stimulating for normal daily research. For this reason the idea of Professor Ioan Dumitrache, first (and actual) President of the Society to organize a cycle of lectures on some basic (bu...

Input-Output-to-State Stability

Siam Journal on Control and Optimization, 2001

Consider the linear space n of polynomials of degree n or less over the complex field.

On the Input-to-State Stability Property

1995

The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunov-theoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation.

Lyapunov Inequalities and Applications

Springer eBooks, 2021

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

On Characterizations of the Input-to-State Stability Property

Systems & Control Letters, 1995

The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunov-theoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation.

Ignored input dynamics and a new characterization of control Lyapunov functions

Automatica, 2001

Our objective in this paper is to extend as much as possible the dissipativity approach for the study of robustness of stability in the presence of known/unknown but ignored input dynamics. This leads us to:Stability margin are given in terms of an inequality the supply should satisfy. Nevertheless, in spite of this extension, we show that the dissipativity approach cannot cope with ignored dynamics which have nonzero relative degree or are nonminimum phase.

Input-to-state stability with respect to inputs and their derivatives

International Journal of Robust and Nonlinear Control, 2003

A new notion of input-to-state stability involving infinity norms of input derivatives up to a finite order k is introduced and characterized. An example shows that this notion of stability is indeed weaker than the usual iss. Applications to the study of global asymptotic stability of cascaded non-linear systems are discussed. A function F : S ! R is positive definite if F ðxÞ > 0; 8x 2 S; x=0 and F ð0Þ ¼ 0: A function g : R 50 ! R 50 is of class K if it is continuous, positive definite, and strictly increasing. It is of class K 1 if it is also unbounded. Finally, b : R 50  R 50 ! R 50 is of class KL if for each fixed t50; bðÁ; tÞ is of class K and for each fixed s > 0; bðs; tÞ decreases to 0 as t ! 1: An important fact concerning K 1 functions which will often be used in the following sections is the socalled 'weak triangular inequality' gða þ bÞ4gð2aÞ þ gð2bÞ for all a; b50:

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Lyapunov Characterizations of Input to Output Stability

Siam Journal on Control and Optimization, 2000

This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated. The work is related to partial stability of differential equations (the particular case which arises when there are no external inputs).

New characterizations of input-to-state stability

Automatic Control, IEEE Transactions on, 1996

Abstruct-We present new characterizations of the input-tostate stability property. As a consequence of these results, we show the equivalence between the ISS property and several (apparent) variations proposed in the literature.

Remarks on input to output stability

Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304)

This paper deals with several related notions of output stability with respect to inputs (which may be thought of as disturbances). The main such notion is called input to output stability (ios), and it reduces to input to state stability (iss) when the output equals the complete state. For systems with no inputs, ios provides a generalization of the classical concept of partial stability. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions.

Separation principles for input-output and integral-input-to-state stability

We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS (input-to-state stability) framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of separation principles for input-output-to-state stability-characterizations of the property as conjunctions of weaker stability notions. When applied to the notion of integral ISS, these characterizations yield analogous results.

Notions of input to output stability

Systems & Control Letters, 1999

This paper deals with concepts of output stability. Inspired in part by regulator theory, several variants are considered, which differ from each other in the requirements imposed upon transient behavior. The main results provide a comparison among the various notions, all of which specialize to input to state stability (ISS) when the output equals the complete state.

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On a state space approach to nonlinear control

Systems & Control Letters, 1991

We study the standard Ho~ optimal control problem using state feedback for smooth nonlinear control systems. The main theorem obtained roughly states that the L2-induced norm (from disturbances to inputs and outputs) can be made smaller than a constant 3' > 0 if the corresponding Hoo norm for the system linearized at the equilibrium can be made smaller than y by linear state feedback. Necessary and sufficient conditions for the latter problem are by now well-known, e.g. from the state space approach to linear H~o optimal control. Our approach to the nonlinear Hoo optimal control problem generalizes the state space approach to the linear Hoo problem by replacing the Hamiltonian matrix and corresponding Riccati equation as used in the linear context by a Hamiltonian vector field together with a Hamilton-Jacobi equation corresponding to its stable invariant manifold.