469 research outputs found
Phase limitations of Zames-Falb multipliers
Phase limitations of both continuous-time and discrete-time Zames-Falb
multipliers and their relation with the Kalman conjecture are analysed. A phase
limitation for continuous-time multipliers given by Megretski is generalised
and its applicability is clarified; its relation to the Kalman conjecture is
illustrated with a classical example from the literature. It is demonstrated
that there exist fourth-order plants where the existence of a suitable
Zames-Falb multiplier can be discarded and for which simulations show unstable
behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is
developed. Its application is demonstrated with a second-order counterexample
to the Kalman conjecture. Finally, the discrete-time limitation is used to show
that there can be no direct counterpart of the off-axis circle criterion in the
discrete-time domain
Exponential Convergence Bounds using Integral Quadratic Constraints
The theory of integral quadratic constraints (IQCs) allows verification of
stability and gain-bound properties of systems containing nonlinear or
uncertain elements. Gain bounds often imply exponential stability, but it can
be challenging to compute useful numerical bounds on the exponential decay
rate. In this work, we present a modification of the classical IQC results of
Megretski and Rantzer that leads to a tractable computational procedure for
finding exponential rate certificates
Convex searches for discrete-time Zames-Falb multipliers
In this paper we develop and analyse convex searches for Zames--Falb
multipliers. We present two different approaches: Infinite Impulse Response
(IIR) and Finite Impulse Response (FIR) multipliers. The set of FIR multipliers
is complete in that any IIR multipliers can be phase-substituted by an
arbitrarily large order FIR multiplier. We show that searches in discrete-time
for FIR multipliers are effective even for large orders. As expected, the
numerical results provide the best -stability results in the
literature for slope-restricted nonlinearities. Finally, we demonstrate that
the discrete-time search can provide an effective method to find suitable
continuous-time multipliers.Comment: 12 page
A stability criterion for systems with neutrally stable modes and deadzone nonlinearities
Stability analysis is considered for feedback interconnections of deadzone nonlinearities with linear systems that has a neutrally stable mode. Such systems do not have a unique equilibrium point and the standard techniques from passivity and Lyapunov theory cannot be applied. A stability criterion that generalizes the Popov criterion for this class of systems is derived in this report and several examples will prove its applicability
Conditions for the equivalence between IQC and graph separation stability results
This paper provides a link between time-domain and frequency-domain stability
results in the literature. Specifically, we focus on the comparison between
stability results for a feedback interconnection of two nonlinear systems
stated in terms of frequency-domain conditions. While the Integral Quadratic
Constrain (IQC) theorem can cope with them via a homotopy argument for the
Lurye problem, graph separation results require the transformation of the
frequency-domain conditions into truncated time-domain conditions. To date,
much of the literature focuses on "hard" factorizations of the multiplier,
considering only one of the two frequency-domain conditions. Here it is shown
that a symmetric, "doubly-hard" factorization is required to convert both
frequency-domain conditions into truncated time-domain conditions. By using the
appropriate factorization, a novel comparison between the results obtained by
IQC and separation theories is then provided. As a result, we identify under
what conditions the IQC theorem may provide some advantage
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