Slope-restricted Nonlinearities Research Articles

Stability Analysis of Nonlinear Systems with Slope Restricted Nonlinearities

The problem of absolute stability of Lur'e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results.

Synchronization of Lurie system based on contraction analysis

In this paper, a new synchronization criterion for chaotic Lurie systems with sector and slope restricted nonlinearities is proposed. Without analyzing the stability of the error system, synchronization condition for general Lurie systems is firstly given based on the contraction analysis. Furthermore, the result is extended to the Chua’s system whose state equation is not differentiable. Finally, three representative examples are presented to support the theoretical results.

IQC-synthesis with general dynamic multipliers

SUMMARY Analogously to the existing μ-synthesis tools, we propose an alternative algorithm for the systematic design of robust controllers based on an iteration of standard nominal controller synthesis and integral quadratic constraint (IQC) analysis with general dynamic multipliers. The suggested algorithm enables us to perform robust controller synthesis for a significantly larger class of uncertainties if compared with the existing methods. Indeed, while the classical approaches are restricted to the use of real/complex time invariant or arbitrarily fast time-varying parametric uncertainties, the IQC framework also offers, for example, the possibility to efficiently handle sector-bounded and slope restricted nonlinearities or time-varying parametric uncertainties and uncertain time-varying time-delays, both with bounds on the rate-of-variation. Secondly, in contrast to the classical approaches, the proposed techniques completely avoid gridding and curve-fitting. We present new insights that allow us to reformulate the robust IQC analysis LMIs into a standard quadratic performance problem. This enables us to generate suitable initial conditions for each subsequent iteration step. Depending on the size of the problem, this can significantly speed up the synthesis process. The results are illustrated by means of two numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.

Observer-based output feedback control of discrete-time Luré systems with sector-bounded slope-restricted nonlinearities

SUMMARY Many well studied classes of dynamical systems such as actuator-constrained linear systems and dynamic artificial neural networks can be written as discrete-time Lure systems with sector-bounded and/or slope-restricted nonlinearities. Two types of observer-based output feedback control design methods are presented, compared, and analyzed with regard to robustness to model uncertainties and insensitivity to output disturbances. The controller designs are formulated in terms of LMIs that are solvable with standard software. The design equations are illustrated in numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.

Equivalence between classes of multipliers for slope-restricted nonlinearities

Different classes of multipliers have been proposed in the literature for obtaining stability criteria using passivity theory, integral quadratic constraint (IQC) theory or Lyapunov theory. Some of these classes of multipliers can be applied with slope-restricted nonlinearities. In this paper the concept of phase-containment is defined and it is shown that several classes are phase-contained within the class of Zames–Falb multipliers. There are two main consequences: firstly it follows that the class of Zames–Falb multipliers remains, to date, the widest class of available multipliers for slope-restricted nonlinearities; secondly further restrictions may be avoided when exploiting the parametrization of the other classes of multipliers.

LMI-Based Stability Criteria for Discrete-Time Lur'e Systems With Monotonic, Sector- and Slope-Restricted Nonlinearities

This note presents new LMI-based stability criteria for the discrete-time multivariable Lur'e system with nonlinearities that are monotonic, sector- and slope-restricted. Corresponding Lur'e-Lyapunov functions are constructed for such a system. The new criteria are expressed in a reasonably general form that can be applied to both non-diagonal and diagonal nonlinearities. We explicitly compare the new approach to the existing LMI-based Popov-like criteria in the literature, and express the results in terms of an Integral Quadratic Constraint (IQC). The applications of the new criteria to some control problems and strategies are briefly discussed. Numerical examples are included to show their performance, and they are shown to be less conservative than the previous results.

Authors Reply to “Comments on ‘On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities’”

In this reply, we show that, in certain cases, the analysis approach given by Turner et al. does indeed give “superior” results to that given by Park. We also show, briefly, how the analysis approach given by Turner et al. can be enhanced by considering both Zames-Falb and Popov multipliers (as in the papers by Jonsson and Turner and Kerr).

Comments on “On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities”

The above technical note presents a novel convex search within the Zames-Falb multiplier class. The aim of this correspondence is to correct the misuse of a relaxation on one condition in the search. This relaxation leads to nontrivial numerical errors in all six examples discussed in the technical note. Correct application of the conditions still gives an improvement over absolute stability criteria in the literature for at least one example, but some of the claims for lack of conservativeness in the above technical note should be moderated.

Adaptive observers-based synchronization of a class of Lur'e systems with delayed outputs for chaotic communications

In this paper, we propose an adaptive observer-based synchronization approach for a class of chaotic Lur'e systems with slope restricted nonlinearities and delayed outputs. The delay is assumed bounded and time varying and the information to be transmitted is assumed piecewise constant. Based on the Lyapunov-Krasovskii approach, we show that for sufficiently small values of the time-delay upper bound, both synchronization and information reconstruction objectives are ensured under a condition of persistent excitation and after solving a convex optimization problem. The result is illustrated via a numerical example of a chaotic communication system subject to a transmission delay.

Systems with monotone and slope restricted nonlinearities

Abstract The paper starts from the suggestion of R. E. Kalman that additional information on nonlinearity slope may improve the sufficient conditions for absolute stability. This leads to the so called systems with augmented dynamics. Motivated also by the problem of the PIO II aircraft oscillations-self sustained oscillations induced by the saturation nonlinearities, which are both sector and slope restricted-the paper considers a generalization of the Yakubovich criterion to the case of the systems with critical and unstable linear part. The same generalization concerns a quite well known stability criterion where only slope restrictions are taken into account: the published version is improved by using all advantages of the Liapunov method and of the frequency domain stability inequalities. The results are illustrated by several applications.

Synchronization for Time-Delay Lur’e Systems with Sector and Slope Restricted Nonlinearities Under Communication Constraints

This paper presents a new approach to the output feedback control problem of master-slave synchronization of time-delay chaotic Lur’e systems with sector and slope restricted nonlinearities under communication constraints. The communication constraints involve measurement quantization and signal transmission delay. By constructing an appropriate Lyapunov functional with the idea of a discretized Lyapunov–Krasovskii functional method and utilizing the sector bound of the logarithmic quantizer, a delay-dependent synchronization criterion is derived. The desired synchronization controller can be obtained by solving a set of linear matrix inequalities. Finally, numerical simulations for Chua’s circuit are proposed to show the effectiveness of the proposed approach.

Observers for descriptor systems with slope-restricted nonlinearities

This paper addresses the observer design problem for a class of nonlinear descriptor systems whose nonlinear terms are slope-restricted. The full-order observer is formulated as a nonlinear descriptor system. A linear matrix inequality (LMI) condition is derived to construct the full-order observer. The existence and uniqueness of the solution to the obtained observer system are guaranteed. Furthermore, under the same LMI condition and a common assumption, a reduced-order observer is designed. Finally, the design methods are reduced to a strict LMI problem and illustrated by a numerical example.

Robust [formula omitted] control for uncertain Lur’e systems with sector and slope restricted nonlinearities by PD state feedback

In this paper, the design of a PD controller for robust H ∞ stabilization of uncertain Lur’e systems with sector and slope restricted nonlinearities is considered. A PD controller is utilized to not only guarantee stability of systems but also reduce the effect of external disturbance to a H ∞ norm constraint. Sector bounds and slope bounds are used on a Lyapunov–Krasovskii functional through convex representation of the nonlinear function so that a less conservative H ∞ stability criterion is obtained. Using Finsler’s Lemma, the criterion is derived in terms of linear matrix inequalities (LMIs) that can be easily solved by various convex optimization techniques. Several examples show the effectiveness of the proposed methods.

Design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities

In this paper, design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities is presented. A new synchronization criterion is proposed based on Lyapunov functions with quadratic form of states and nonlinear functions of the systems. Sector and slope bounds are employed to the Lyanunov–Krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained. The criteria is given in terms of linear matrix inequalities (LMIs) by using Finsler’s lemma. A numerical example is provided to illustrate the effectiveness of the method.

Synchronization Criterion for Lur’e Systems via Delayed PD Controller

In this paper, the effects of a time varying delay on a chaotic drive-response synchronization are considered. Using a delayed feedback proportional-derivative (PD) controller scheme, a delay-dependent synchronization criterion is derived for chaotic systems represented by the Lur’e system with sector and slope restricted nonlinearities. The derived criterion is a sufficient condition for the absolute stability of the error dynamics between the drive and the response systems. By the use of a convex representation of the nonlinearity and the discretized Lyapunov-Krasovskii functional, stability condition is obtained via the LMI formulation. The condition represented in the terms of linear matrix inequalities (LMIs) can be solved by the application of convex optimization algorithms. The effectiveness of the work is verified through numerical examples.

On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities

The stability of a feedback interconnection of a linear time invariant (LTI) system and a slope-restricted nonlinearity is revisited. Unlike the normal treatment of this problem, in which multipliers are explicitly chosen and then stability conditions checked, this technical note derives existence conditions for a sub-class of these multipliers, namely those which are L 1 bounded, stable, causal and of order equal to the LTI part of the system. It is proved that for the single-input-single-output case, these existence conditions can be expressed as a set of linear matrix inequalities and thus can be solved efficiently with modern optimization software. Examples illustrate the effectiveness of the results.

Delay-dependent criteria for absolute stability of uncertain time-delayed Lur’e dynamical systems

Abstract This paper deals with the absolute stability analysis for uncertain time-delayed Lur systems with sector and slope restricted nonlinearities. New delay-dependent stability criteria are derived via linear matrix inequality (LMI) formulation that can be easily solved by various convex optimization techniques. Sector bounds and slope bounds are employed to a Lyapunov–Krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained. A numerical example shows effectiveness of the proposed stability condition over some existing ones.

Formula omitted] control of Lur'e systems with sector and slope restricted nonlinearities

This Letter considers H ∞ controller design scheme for Lur'e systems with sector/slope restrictions and external disturbance. Based on Lyapunov theory and linear matrix inequality (LMI) formulation, a state feedback controller is designed to not only guarantee stability of systems but also reduce the effect of external disturbance to an H ∞ norm constraint. The nonlinearities are expressed as convex combinations of sector and slope bounds so that equality constraints are converted into inequality constraints using convex properties of the nonlinear function. Then, the stabilizing feedback gain matrix is derived through LMI formulation. Finally, a numerical example shows the effectiveness of the proposed method.

Multiobjective Algebraic Synthesis of Neural Control Systems by Implicit Model Following

The advantages brought about by using classical linear control theory in conjunction with neural approximators have long been recognized in the literature. In particular, using linear controllers to obtain the starting neural control design has been shown to be a key step for the successful development and implementation of adaptive-critic neural controllers. Despite their adaptive capabilities, neural controllers are often criticized for not providing the same performance and stability guarantees as classical linear designs. Therefore, this paper develops an algebraic synthesis procedure for designing dynamic output-feedback neural controllers that are closed-loop stable and meet the same performance objectives as any classical linear design. The performance synthesis problem is addressed by deriving implicit model-following algebraic relationships between model matrices, obtained from the classical design, and the neural control parameters. Additional linear matrix inequalities (LMIs) conditions for closed-loop exponential stability of the neural controller are derived using existing integral quadratic constraints (IQCs) for operators with repeated slope-restricted nonlinearities. The approach is demonstrated by designing a recurrent neural network controller for a highly maneuverable tailfin-controlled missile that meets multiple design objectives, including pole placement for transient tuning, H(infinity) and H(2) performance in the presence of parameter uncertainty, and command-input tracking.

Master-slave synchronization of Lur'e systems with sector and slope restricted nonlinearities

This Letter presents a synchronization method for Lur'e systems with sector and slope restricted nonlinearities. A static error feedback controller based on the Lyapunov stability theory is proposed for asymptotic synchronization. The Lyapunov function candidate is chosen as a quadratic form of the error states and nonlinear functions of the systems. The nonlinearities are expressed as convex combinations of sector and slope bounds by using convex properties of the nonlinear function so that equality constraints are converted into inequality constraints. Then, the feedback gain matrix is derived through a linear matrix inequality (LMI) formulation. Finally, a numerical example shows the effectiveness of the proposed method.