Slope-restricted Nonlinearities Research Articles

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

Abstract 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.

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

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

Generalized Lyapunov Functions for Discrete-Time Lurie Systems With Slope-Restricted Nonlinearities

A class of Lyapunov functions for discrete-time Lurie systems with monotonic non-linearities is proposed. The Lyapunov functions are composed of quadratic terms on the states and of the system's non-linearities as well as Lurie-Postnikov type integral terms. Crucially, positive definiteness of the matrix in the generalised quadratic form and positivity of the scaling terms of the Lurie-Postnikov integrals are relaxed in the stability conditions. Furthermore, they are used for regional stability analysis and performance assessment. Numerical examples show that the proposed Lyapunov function structure matches or outperforms existing ones for these systems.

Further results on the design of an observer for an electro-hydraulic actuation system with deadzone nonlinearity

In this paper we present further results on the design a non-linear observer for estimating the null current in an electro-hydraulic actuation system. In previous work it was shown that the observer achieves ultimate boundedness of the observation error based on results on the stability of control loops with slope restricted non-linearities. In this work we give a complete characterisation of the asymptotic convergence of the observation error considerably tightening the results obtained earlier.

Stability Analysis Using Quadratic Constraints for Systems With Neural Network Controllers

A method is presented to analyze the stability of feedback systems with neural network controllers. Two stability theorems are given to prove asymptotic stability and to compute an ellipsoidal innerapproximation to the region of attraction (ROA). The first theorem addresses linear time-invariant systems, and merges Lyapunov theory with local (sector) quadratic constraints to bound the nonlinear activation functions in the neural network. The second theorem allows the system to include perturbations such as unmodeled dynamics, slope-restricted nonlinearities, and time delay, using integral quadratic constraint (IQCs) to capture their input/output behavior. This in turn allows for off-by-one IQCs to refine the description of activation functions by capturing their slope restrictions. Both results rely on semidefinite programming to approximate the ROA. The method is illustrated on systems with neural networks trained to stabilize a nonlinear inverted pendulum as well as vehicle lateral dynamics with actuator uncertainty.

Construction of Periodic Counterexamples to the Discrete-Time Kalman Conjecture

This paper considers the Lurye system of a discrete-time, linear time-invariant plant in negative feedback with a nonlinearity. Both monotone and slope-restricted nonlinearities are considered. The main result is a procedure to construct destabilizing nonlinearities for the Lurye system. If the plant satisfies a certain phase condition then a monotone nonlinearity can be constructed so that the Lurye system has a non-trivial periodic cycle. Several examples are provided to demonstrate the construction. This represents a contribution for absolute stability analysis since the constructed nonlinearity provides a less conservative upper bound than existing bounds in the literature.

Applying negative imaginary systems theory to non-square systems with polytopic uncertainty

This paper aims to apply Negative Imaginary (NI) systems theory to non-square LTI systems, for example, flexible structure systems with redundant sensors and actuators. The paper proposes static pre- and post-compensation schemes to transform stable fat (i.e. no. of inputs more than the no. of outputs) and tall (i.e. no. of outputs more than the no. of inputs) LTI plants into the class of Strongly Strict Negative Imaginary (SSNI) systems, a subset of the Strictly Negative Imaginary (SNI) systems. The proposed compensators can also stabilise a non-square or a non-NI square plant in a positive feedback loop. Moreover, the compensators can be utilised to design a simple constant reference tracking scheme exploiting the integral controllability propoerty of SSNI systems. Subsequently, observer-based pre- and post-compensators are developed to handle the cases when some of the system states are not available for direct measurement. Then, the compensation schemes are extended to stable, non-square or non-NI square systems with polytopic uncertainty. Finally, the post-compensation technique is applied to stabilise a class of tall/square uncertain LTI plants preceded by a slope-restricted nonlinearity.

Discrete‐time systems with slope restricted nonlinearities: Zames–Falb multiplier analysis using external positivity

AbstractThis article exploits positive systems theory in the search for Zames–Falb multipliers for the analysis of discrete‐time Lurie systems, where the nonlinearity is assumed to be slope‐restricted. Although a similar problem has been tackled in a continuous time context, the results in discrete‐time take a different form and require a somewhat different approach to overcome certain technical problems. The work has two compelling features: (i) the arising algorithms are completely convex; and (ii) numerical results compare well with the state‐of‐the‐art in some cases, providing the least conservative result in one instance.

On Lyapunov-Lur’e functional based stability criterion for discrete-time Lur’e systems

In this manuscript, we consider the stability problem of Lur’e systems with slope-restricted nonlinearities. We focus on a specific parametrisation of the Lyapunov-Lur’e functional in the literature, and extend it to a higher order. Meanwhile, we show this Lyapunov-Lur’e functional based stability criterion is equivalent to the search for noncausal FIR multipliers with a restricted form for the SISO case. Finally, we discuss the restrictions of this Lyapunov-Lur’e functional approach with some numerical examples.

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 $\ell_{2}$-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.

Analysis of Systems With Slope Restricted Nonlinearities Using Externally Positive Zames–Falb Multipliers

This paper proposes an approach for assessing the stability of feedback interconnections where one element is a static slope-restricted nonlinearity and the other element is a linear system. The approach is based on the use of Zames–Falb multipliers where the dynamic portion of the multiplier is chosen as an externally positive noncausal transfer function. By restricting attention to a subset of these multipliers, a set of pure linear matrix inequality conditions is obtained which requires no initial parameterization by the user. A useful byproduct of using externally positive systems is that the results are applicable to nonodd slope restricted nonlinearities, which is not the case for all classes of Zames–Falb multipliers.

Absolute stability analysis of discrete time feedback interconnections

The problem of absolute stability of a discrete time feedback interconnection is revisited. For the case of a slope-restricted nonlinearity in feedback with a linear time invariant system it is shown that several of the recently proposed stability tests are special cases of the Zames-Falb criterion. Moreover a new computationally tractable stability test is proposed that simultaneously employs full-block circle, Yakubovich and Zames-Falb multipliers.

An Improved Absolute Stability Criterion for Time-Delay Lur’e Systems and its Frequency Domain Interpretation

This paper is concerned with absolute stability analysis for time-delay Lur'e systems with both sector- and slope-restricted nonlinearities. A new delay-dependent stability criterion is given by using a Lur'e-Postnikov functional. The additional slope restrictions on the nonlinearities play important roles in improving the absolute stability conditions. Numerical examples are presented to demonstrate the effectiveness of the proposed criterion. Moreover, a frequency domain interpretation for the criterion is presented as well.

Modified static anti-windup for saturated systems with sector-bounded and slope-restricted nonlinearities

Summary This paper proposes modified static anti-windup techniques for saturated systems with sector-bounded and slope-restricted nonlinearities by augmenting the pre-designed controller with the so-called differential compensator to process the slope restriction. By using a purely quadratic Lyapunov function and with a modified sector condition dealing with actuator saturation, LMI-based synthesis conditions are presented to address the problems of the estimates of the region of attraction and performance analysis of the closed-loop system. Numerical examples illustrate the effectiveness of the proposed approaches. Copyright © 2016 John Wiley & Sons, Ltd.

Zames–Falb multipliers for absolute stability: From O׳Shea׳s contribution to convex searches

Absolute stability attracted much attention in the 1960s. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov׳s stability theories, led to one of the most important results in control engineering: the KYP Lemma.For Lurye11Also written as Lur׳e or Lurie. systems this work culminated in the class of multipliers proposed by O׳Shea in 1966 and formalized by Zames and Falb in 1968. The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came 20 years after the initial proposal of the class. A further 20 years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O׳Shea׳s contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class.This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user׳s viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O׳Shea himself.

Networked Reliable Synchronization of Chaotic Lur’e Systems with Sector- and Slope-Restricted Nonlinearities

This paper investigates the networked reliable synchronization problem of chaotic Lur'e systems containing the sector- and slope-restricted nonlinearities. The reliable synchronization of chaotic Lur'e systems is achieved based on networked control techniques. Moreover, some sufficient conditions are obtained and these conditions depend on various network-related parameters, fault indicating matrices, sector and slope bounds. Finally, a numerical example is provided to illustrate the validity of the proposed method.

A Less Conservative LMI Condition for Stability of Discrete-Time Systems With Slope-Restricted Nonlinearities

Many conditions have been found for the absolute stability of discrete-time Lur'e systems in the literature. It is advantageous to find convex searches via LMIs where possible. In this technical note, we construct two less conservative LMI conditions for discrete-time systems with slope-restricted nonlinearities. The first condition is derived via Lyapunov theory while the second is derived via the theory of integral quadratic constraints (IQCs) and noncausal Zames-Falb multipliers. Both conditions are related to the Jury-Lee criterion most appropriate for systems with such nonlinearities, and the second generalizes it. Numerical examples demonstrate a significant reduction in conservatism over competing approaches.

Frequency Domain Performance Analysis of Lur'e Systems

In this brief, a performance analysis of harmonically forced Lur'e systems with a slope-restricted nonlinearity is presented. The approach is based on the method of harmonic linearization (describing functions method). Conditions of applicability are derived and an estimate of the accuracy of this method is found. Theoretical results are illustrated by a case study and experimental results.

Lyapunov functions and [formula omitted] gain bounds for systems with slope restricted nonlinearities

The stability and L2performance analysis of systems consisting of an interconnection of a linear-time-invariant (LTI) system and a static nonlinear element which is Lipschitz, slope restricted and sector bounded is revisited. The main thrust of the paper is to improve and extend an existing result in the literature to enable (i) concise and correct conditions for asymptotic stability of the interconnection and (ii) reasonably tight bounds on the L2gain between an exogenous input and a given output to be obtained. Numerical examples indicate that the proposed algorithm performs well compared to competing results in the literature.

On multipliers for bounded and monotone nonlinearities

Recent results in equivalence between classes of multipliers for slope-restricted nonlinearities are extended to multipliers for bounded and monotone nonlinearities. This extension requires a slightly modified version of the Zames–Falb theorem and a more general definition of phase-substitution. The results in this paper resolve apparent contradictions in the literature on classes of multipliers for bounded and monotone nonlinearities.