15 research outputs found

    Phase limitations of Zames-Falb multipliers

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    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

    Convex searches for discrete-time Zames-Falb multipliers

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    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 â„“2\ell_{2}-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

    On the exactness of a stability test for Lur'e systems with slope-restricted nonlinearities

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    In this note it is shown that the famous multiplier absolute stability test of R. O'Shea, G. Zames and P. Falb is necessary and sufficient if the set of Lur'e interconnections is lifted to a Kronecker structure and an explicit method to construct the destabilizing static nonlinearity is presented

    Exponential input-to-state stability for Lur’e systems via Integral Quadratic Constraints and Zames–Falb multipliers

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    Absolute stability criteria that are sufficient for global exponential stability are shown, under a Lipschitz assumption, to be sufficient for the a priori stronger exponential input-to-state stability property. Important corollaries of this result are as follows: (i) absolute stability results obtained using Zames–Falb multipliers for systems containing slope-restricted nonlinearities provide exponential input-to-state-stability under a mild detectability assumption; and (ii) more generally, many absolute stability results obtained via Integral Quadratic Constraint methods provide, with the additional Lipschitz assumption, this stronger property

    From classical absolute stability tests towards a comprehensive robustness analysis

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    In this thesis, we are concerned with the stability and performance analysis of feedback interconnections comprising a linear (time-invariant) system and an uncertain component subject to external disturbances. Building on the framework of integral quadratic constraints (IQCs), we aim at verifying stability of the interconnection using only coarse information about the input-output behavior of the uncertainty

    Robust Performance Analysis for Gust Loads Computation

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    In the design process of modern aircraft, a comprehensive analysis of worst case structural gust loads is imperative. Because this analysis requires to consider millions of cases, the examination is extremely time consuming. To solve this problem, a new approach based on robust performance analysis is introduced: the worst case energy-to-peak gain is used to efficiently determine worst case loads of nominal, uncertain, and linear parameter varying gust loads models

    A Behavioral Approach to Robust Machine Learning

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    Machine learning is revolutionizing almost all fields of science and technology and has been proposed as a pathway to solving many previously intractable problems such as autonomous driving and other complex robotics tasks. While the field has demonstrated impressive results on certain problems, many of these results have not translated to applications in physical systems, partly due to the cost of system fail- ure and partly due to the difficulty of ensuring reliable and robust model behavior. Deep neural networks, for instance, have simultaneously demonstrated both incredible performance in game playing and image processing, and remarkable fragility. This combination of high average performance and a catastrophically bad worst case performance presents a serious danger as deep neural networks are currently being used in safety critical tasks such as assisted driving. In this thesis, we propose a new approach to training models that have built in robustness guarantees. Our approach to ensuring stability and robustness of the models trained is distinct from prior methods; where prior methods learn a model and then attempt to verify robustness/stability, we directly optimize over sets of models where the necessary properties are known to hold. Specifically, we apply methods from robust and nonlinear control to the analysis and synthesis of recurrent neural networks, equilibrium neural networks, and recurrent equilibrium neural networks. The techniques developed allow us to enforce properties such as incremental stability, incremental passivity, and incremental l2 gain bounds / Lipschitz bounds. A central consideration in the development of our model sets is the difficulty of fitting models. All models can be placed in the image of a convex set, or even R^N , allowing useful properties to be easily imposed during the training procedure via simple interior point methods, penalty methods, or unconstrained optimization. In the final chapter, we study the problem of learning networks of interacting models with guarantees that the resulting networked system is stable and/or monotone, i.e., the order relations between states are preserved. While our approach to learning in this chapter is similar to the previous chapters, the model set that we propose has a separable structure that allows for the scalable and distributed identification of large-scale systems via the alternating directions method of multipliers (ADMM)
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