Slope-restricted Nonlinearities Research Articles

A unified [formula omitted] adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities

This paper investigates the problem of the H ∞ adaptive observer design for a class of nonlinear dynamical systems. The main contribution consists in providing a unified synthesis method for systems with both Lipschitz and monotone nonlinearities (not necessarily Lipschitz). Thanks to the innovation terms into the nonlinear functions [M. Arcak, P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities, IEEE Transactions on Automatic Control 46 (7) (2001) 1146–1150] and to the differential mean value theorem [A. Zemouche, M. Boutayeb, G.I. Bara, Observers for a class of Lipschitz systems with extension to H ∞ performance analysis, Systems and Control Letters 57 (1) (2008) 18–27], the stability analysis leads to the solvability of a Linear Matrix Inequality (LMI) with several degrees of freedom. For simplicity, we start by presenting the result in an H ∞ adaptive-free context. Furthermore, we propose an H ∞ adaptive estimator that extends easily the obtained results to systems with unknown parameters in the presence of disturbances. We show, in particular, that the matching condition in terms of an equality constraint required in several works is not necessary and therefore allows reducing the conservatism of the existing conditions. Performances of the proposed approach are shown through a numerical example with a polynomial nonlinearity.

Improved delay-dependent stability criteria for uncertain Lur’e systems with sector and slope restricted nonlinearities and time-varying delays

This paper deals with the absolute stability analysis for uncertain time delay Lur’e systems that have time-varying delays and 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 optimization techniques. Sector bounds and slope bounds are employed to the Lyapunov–Krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained. Numerical examples show effectiveness of the proposed stability condition.

Delay-independent absolute stability for time-delay Lur'e systems with sector and slope restricted nonlinearities

This Letter presents an analysis method for a delay-independent absolute stability of time-delay Lur'e systems with sector and slope restrictions. The proposed method is based on the Lyapunov functions with quadratic form of states and nonlinear functions of the systems. Several criteria are derived in terms of linear matrix inequalities (LMIs). A numerical example is illustrated to show the effectiveness of the proposed method.

Robust stabilization of discrete-time nonlinear Lur’e systems with sector and slope restricted nonlinearities

In this paper, we present a novel method for the robust control problem of uncertain nonlinear discrete-time linear systems with sector and slope restrictions. The nonlinear function considered in this paper is expressed as convex combinations of sector and slope bounds. Then the equality constraint is derived by using convex properties of the nonlinear function. A stabilization criterion for the existence of the state feedback controller is derived in terms of linear matrix inequalities (LMIs) by using Finsler’s lemma. The proposed method is demonstrated by a system with saturation nonlinearity.

Systems with sector bound nonlinearities: A behavioral approach

The class of nonlinear dynamical systems obtained by interconnection of a linear, time-invariant dynamical system and a memoryless, time-invariant, sector-bound nonlinearity has been widely studied in literature. In this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems. Systems with slope-restricted nonlinearities are also considered.

Improved asymptotic stability analysis for Lur'e systems with sector and slope restricted nonlinearities

In this Letter, we propose new criteria for asymptotic stability of Lur'e systems with sector and slope restrictions. The criteria are expressed in terms of linear matrix inequality which has full block scaling matrix. By interpreting the criteria in the frequency domain using KYP lemma, the less conservativeness of the proposed analysis method is investigated. Furthermore, we analyze the dynamic system with input saturation using sector and slop properties of the saturation nonlinearity. A numerical example shows the effectiveness of the proposed method.

Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

ABSTRACTThis paper considers robust stability and robust performance analysis for discrete‐time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time‐invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur'e‐Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H2 and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.

Observer-based control of systems with slope-restricted nonlinearities

An observer design is presented which makes use of bounds on the slope of system nonlinearities. Necessary and sufficient conditions are derived for the feasibility of the design. A class of state feedback control laws are characterized which, when implemented with the observer states, ensure global asymptotic stability. One such certainty-equivalence design is illustrated on an active magnetic bearing example.

New results for analysis of systems with repeated nonlinearities

This work presents new results for analysis of systems with repeated monotonic or slope restricted nonlinearities. We first give new integral quadratic constraints (IQCs) that are satisfied by the inputs and outputs of diagonal operators with equal, monotonic or slope restricted, diagonal entries. Then, we present new analysis results for systems with repeated nonlinearities, obtained using the new IQCs. For the type of nonlinearities considered in this work, the new results are less conservative than previously known results. The computation of the new conditions is discussed. The results in this paper apply also to problems with nonrepeated nonlinearities whenever these are representable as the interconnection of multiple repeated nonlinearities and a constant matrix.

New results for analysis of systems with repeated nonlinearities

This work presents new results for analysis of systems with repeated monotonic or slope restricted nonlinearities. We first give new integral quadratic constraints (IQCs) that are satisfied by the inputs and outputs of diagonal operators with equal, monotonic or slope restricted, diagonal entries. Then, we present new analysis results for systems with repeated nonlinearities, obtained using the new IQCs. For the type of nonlinearities considered in this work, the new results are less conservative than previously known results. The computation of the new conditions is discussed. The results in this paper apply also to problems with nonrepeated nonlinearities whenever these are representable as the interconnection of multiple repeated nonlinearities and a constant matrix.

Concepts of Strict Positive Realness and the Absolute Stability Problem of Continuous-Time Systems

After the conditions for absolute stability and positive realness are reviewed, it is shown that the current definition of positive realness is inadequate to establish less conservative criteria for absolute stability. In this paper, the concepts of wide positive realness (WPR) and wide strict positive realness (WSPR) are proposed. Both WPR and WSPR functions may have poles in the open left and right half-planes, and WPR functions may have single poles on the imaginary axis including ±j∞. Using the concepts of WPR and WSPR, less conservative criteria, including algebraic and frequency domain criteria, are presented for absolute stability of slope-restricted multiple nonlinearity feedback systems. Refined results for non-slope-restricted nonlinearity feedback systems are also proposed. It is pointed out that there is no spectral factorization for WPR and WSPR functions if they have poles at both j∞ and −j∞, this can be used to explain the deficiencies of algebraic criteria presented in previous publications.

An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities

In this brief we present a new sufficient condition for global asymptotic stability of Lur'e systems with sector and slope restricted nonlinearities. It is expressed as a matrix inequality and is based on an extended Lur'e-Postnikov Lyapunov function. It is compared based on numerical examples to conditions of Josselson-Raju and Haddad-Kapila.

Stability analysis with Popov multipliers and integral quadratic constraints

It is shown that a general form of Popov multipliers can be used in stability analysis based on integral quadratic constraints (IQC). The Popov multiplier is nonproper and a condition that the nominal plant is strictly proper will be imposed in order to ensure boundedness of the IQC corresponding to the Popov multiplier. A consequence of our main result is that the classical Popov criterion can be combined with a stability criterion for slope restricted nonlinearities developed by Zames and Falb. An example shows that the combination of these two criteria is useful in applications.

A new approach to the absolute stability of nonlinear feedback systems with slope-restricted nonlinearity

A novel frequency-domain criterion is presented for the absolute stability of nonlinear feedback systems. The criterion requires sector and slope informations of the nonlinearity. The result is derived by artificially increasing the system order by one and applying a Lure type Lyapunov function to the resulting equivalent system.

Lyapunov-based proof of Jury - Lee's criterion: Some appraisals

This note presents some observations concerning the question of proving (using a Lyapunov function) the well-known Jury-Lee's absolute stability criterion for Lure type discrete-time systems with sloperestricted nonlinearity. Specifically, the inaccuracies of a previously reported Lyapunov approach to account for the sector as well as slope informations of the nonlinearity are pointed out. An alternative Lyapunov approach that correctly leads to Jury-Lee's criterion is discussed.

A stability inequality for nonlinear discrete-time systems with slope-restricted nonlinearity

A novel frequency domain criterion is presented for the absolute stability of Lure type single-input single-output discrete-time systems with slope-restricted nonlinearity.

A stability inequality for nonlinear feedback systems with slope-restricted nonlinearity

Using Lyapunov's direct method, a stability inequality in frequency domain is derived for nonlinear feedback systems with slope-restricted nonlinearity. In the present approach, a transformed system that involves the slope of the nonlinearity is considered, thus leading to a stability inequality that incorporates the slope information (but not the sector information) of the nonlinearity. The approach is essentially an extension of a recent idea of the author.

Absolute stability criterion for a class of nonlinear systems with slope-restricted nonlinearity

Using Lyapunov's direct method, a stability inequality in frequency domain which incorporates the slope information of the nonlinearity is derived for a class of nonlinear feedback systems. The present Lyapunov function contains an integral involving the slope of the nonlinearity, i.e., unlike the conventional Lure-type Lyapunov function.

Absolute stability of control systems with many sector and slope-restricted non-linearities

The absolute stability of free control systems with many sector and slope-restricted non-linearities is considered. The sufficient conditions derived include a new stability criterion in the frequency domain. By the inclusion of the information on the slopes of the non-linearities it is shown that improved results can be obtained over the Popov criterion for several examples.

A criterion for the bounded-input bounded-output stability of a discrete-data system with a slope-restricted non-linearity†

This paper presents and proves a criterion for the bounded-input bounded-output (BIBO) stability of a class of discrete-data systems with a slope-restricted non-linearity. The proof is based on the principle of contraction mapping. A numerical example is given to illustrate the application of the criterion, and to discuss its implication in relation to some existing results.