Design of excitations for structure discrimination of nonlinear systems, using the best linear approximation

A mathematically rigorous concept of relative stability based on the v-functions of the direct method of Lyapunov is introduced. Two systems of the type representable by x˙ = f(x) are considered, where under the proper restrictions on f(x), a Lyapunov function, v(x) is uniquely determined by a positive definite error criterion r(x) and the equation v˙(x) = −r(x). The definition of the relative stability proposed, eventually leads to conditions on the linear approximation systems which are sufficient to assure the relative stability of the nonlinear systems. This leads to conditions on the eigenvalues of the linear approximation system which are necessary but not sufficient for relative stability. Additional conditions on the choice of the error criteria are needed. The present definition permits the gap between concepts of stability in classical control theory and that due to the direct method of Lyapunov to be at least partially bridged.

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Stabilization of Nonlinear Systems Using Linear Observers

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3426775 ◽

1974 ◽

Vol 96 (1) ◽

pp. 55-60 ◽

Cited By ~ 3

Author(s):

R. E. Strane ◽

W. G. Vogt

Keyword(s):

Nonlinear System ◽

Nonlinear Systems ◽

Asymptotic Stability ◽

Nonlinear Equations ◽

Linear Approximation ◽

Asymptotically Stable ◽

State Variables ◽

Finite Region ◽

Linear Portion ◽

Linear Observer

In this paper, it is shown that a linear observer can always be designed to stabilize a nonlinear system which contains a Lur’e type nonlinearity in the sector [0, k], where k is finite, if both the output of the nonlinearity and a completely observable output of the linear portion are available as inputs to the observer. In case a completely observable output is not available from the linear portion, stabilization is shown to be possible if the original linear approximation of the system is asymptotically stable or those state variables corresponding to the unstable eigenvalues are available. It is also established that a linear observer can be used to guarantee that a finite region of asymptotic stability exists for a plant described by a more general set of nonlinear equations, and in some cases the domain of asymptotic stability can be made as large as desired.

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THE USE OF BINARY SEQUENCES IN DETERMINING THE BEST LINEAR APPROXIMATION OF NONLINEAR SYSTEMS

IFAC Proceedings Volumes ◽

10.3182/20120711-3-be-2027.00051 ◽

2012 ◽

Vol 45 (16) ◽

pp. 1323-1328

Author(s):

Hin Kwan Wong ◽

Johan Schoukens ◽

Keith Godfrey

Keyword(s):

Nonlinear Systems ◽

Linear Approximation ◽

Binary Sequences

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Estimating Parameterized Scalable Models From the Best Linear Approximation of Nonlinear Systems for Accurate High-Level Simulations

IEEE Transactions on Instrumentation and Measurement ◽

10.1109/tim.2006.876580 ◽

2006 ◽

Vol 55 (4) ◽

pp. 1186-1191 ◽

Cited By ~ 3

Author(s):

L. De Locht ◽

G. Vandersteen ◽

Y. Rolain ◽

R. Pintelon

Keyword(s):

Nonlinear Systems ◽

Linear Approximation ◽

High Level

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The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.20.201803.304-317 ◽

2018 ◽

Vol 20 (3) ◽

pp. 304-317

Author(s):

Pavel A. Shamanaev ◽

Olga S. Yazovtseva

Keyword(s):

Banach Space ◽

Nonlinear System ◽

Nonlinear Systems ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Linear Approximation ◽

Critical Case ◽

Sufficient Conditions ◽

Asymptotic Equivalence ◽

Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

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