Lyapunov characterization of forced oscillations

Abstract

This paper develops a Lyapunov approach to the analysis of input-output characteristics for systems under the excitation of a class of oscillatory inputs. Apart from sinusoidal signals, the class of oscillatory inputs include multi-tone signals and periodic signals which can be described as the output of an autonomous system. The Lyapunov approach is developed for linear systems, homogeneous systems (differential inclusions) and nonlinear systems (differential inclusions), respectively. In particular, it is established that the steady-state gain can be arbitrarily closely characterized with Lyapunov functions if the output response converges exponentially to the steady-state. Other output measures that will be characterized include the peak of the transient response and the convergence rate. Tools based on linear matrix inequalities (LMIs) are developed for the numerical analysis of linear differential inclusions (LDIs). This paper's results can be readily applied to the evaluation of frequency responses of general nonlinear and uncertain systems by restricting the inputs to sinusoidal signals. Guided by the numerical result for a second order LDI, an interesting phenomenon is observed that the peak of the frequency response can be strictly larger than the L 2 gain. • x 1 , x 2 : the inner product of x 1 and x 2 .• X : Matrix norm induced from the Euclidean norm.• u : ess.sup t 0 |u(t)| for an essentially bounded func- increasing. It is said to belong to class K ∞ if, in addition, it is unbounded.• A function : R 0 × R 0 → R 0 is said to belong to class KL if, for each t 0, (·, t) is nondecreasing and lim s→0 + (s, t) = 0, and for each s 0, (s, ·) is nonincreasing and lim t→∞ (s, t) = 0.

Model Reduction by Moment Matching for Linear and Nonlinear Systems

“…The linear system and the reduced order model, both driven by the signal generator, have been numerically integrated from zero initial conditions. Figure 2 (left) displays the output y(t) of the linear system when driven by the signal generator, and the signals ψ [1] (t), ψ [2] (t) and ψ [3] (t), obtained by truncating the formal power series defining ψ(t) to the first, second and third order terms, respectively. Figure 2 (right) displays the approximation errors y(t) − ψ [1] (t), y(t) − ψ [2] (t), y(t) − ψ [3] (t).…”

“…For example, the methods, known as moment matching methods, which zero the transfer function of the error system for specific frequencies, belong to this class [1]. This approach does not have a direct nonlinear counterpart, despite the recent developments in this direction [2] (see also the early contributions [3], [4], [5], [6], [7], [8]). Alternatively, approximation errors expressed in terms of the H 2 or H ∞ norm of the error system have been considered both in the linear case [9], [10], [11], [12] and in the nonlinear A. Astolfi is with the Department of Electrical and Electronic Engineering, Imperial College London, and with the Dipartimento di Informatica, Sistemi e Produzione, Università di Roma "Tor Vergata".…”

Robust generalized asymptotic regulation against non‐stationary sinusoidal disturbances with uncertain frequencies

“…The notion of generalized asymptotic regulation adopted in this paper is generalized from 17. Recall that the classical asymptotic regulation problem is formulated with κ = 0.…”

“…A necessary and sufficient condition is provided for generalized asymptotic regulation in LTI systems by Hu et al . 17. This condition can be rephrased as a sufficient condition for parameter‐dependent systems considered in this paper as follows:…”

“…Inspired by Hu et al . 17, such a generalization has been considered in 18 on top of an ℋ︁ ∞ synthesis problem, where a structure is identified for the candidate controllers similar to the previous works on exact regulation with additional performance objectives 19, 20. It is also recently established in 21 how generalized asymptotic regulation constraints can be integrated into ℋ︁ 2 as well as other synthesis problems that can be solved based on linear matrix inequalities (LMI).…”