Absolute Stability Criterion Research Articles

Absolute stability analysis for negative-imaginary systems

This paper deals with absolute stability of a Lur’e system with positive feedback where the linear subsystem exhibits negative-imaginary frequency response and the nonlinearity connected in feedback is time-invariant, memoryless and slope-restricted. The proposed absolute stability criterion requires the linear subsystem to belong to the strongly strict negative-imaginary class. Along with that, positive definiteness of a symmetric matrix needs to be ensured, where the symmetric matrix is obtained by subtracting the dc-gain matrix of the linear subsystem from a strictly positive diagonal matrix with elements indicating the reciprocal of the maximum slope bounds of the nonlinearities. The stability criterion is proved using a Lur’e–Postnikov-type Lyapunov function. Numerical examples are presented to demonstrate the proposed results.

Less conservative robust absolute stability criteria for uncertain neutral-type Lur׳e systems with time-varying delays

This paper focuses on robust absolute stability analysis for uncertain neutral-type Lur׳e systems with time-varying delays. New delay-dependent and delay-rate-dependent stability criteria in terms of linear matrix inequalities (LMIs) are established by: (i) utilizing fully the information of delays to construct Lyapunov–Krasovskii functional (LKF) and augmented vectors; (ii) employing improved integral inequalities to estimate the derivative of LKF; and (iii) using the so-called reciprocally convex combination approach and convex combination approach simultaneously. The main contribution of the present result is to reduce conservativeness of previous results in the LMI framework. Several numerical examples are carried out to demonstrate the effectiveness and merits of the present result.

Robust absolute stability criteria for uncertain Lurie interval time-varying delay systems of neutral type

This study investigates the delay-dependent robust absolute stability analysis for uncertain Lurie systems with interval time-varying delays of neutral type. First, we divide the whole delay interval into two segmentations with an unequal width and checking the variation of the Lyapunov–Krasovskii functional (LKF) for each subinterval of delay, much less conservative delay-dependent absolute and robust stability criteria are derived. Second, a new delay-dependent robust stability condition for uncertain Lurie neutral systems with interval time-varying delays, which expressed in terms of quadratic forms of linear matrix inequalities (LMIs), and has been derived by constructing the LKF from the delayed decomposition approach (DDA) and integral inequality approach (IIA). Finally, three numerical examples are given to show the effectiveness of the proposed stability criteria.

Stability of discrete time recurrent neural networks and nonlinear optimization problems

We consider the method of Reduction of Dissipativity Domain to prove global Lyapunov stability of Discrete Time Recurrent Neural Networks. The standard and advanced criteria for Absolute Stability of these essentially nonlinear systems produce rather weak results. The method mentioned above is proved to be more powerful. It involves a multi-step procedure with maximization of special nonconvex functions over polytopes on every step. We derive conditions which guarantee an existence of at most one point of local maximum for such functions over every hyperplane. This nontrivial result is valid for wide range of neuron transfer functions.

Delayed decomposition approach to the robust absolute stability of a Lur'e control system with time-varying delay

This study considers the robust absolute stability criteria for Lur'e systems with time-varying delay and sector-bounded nonlinearity. Using a delayed decomposition approach that splits the overall delay time into three subintervals and by introducing a new Lyapunov–Krasovskii functional, some new delay-dependent and robust absolute stability criteria are obtained in terms of linear matrix inequalities. Numerical examples from four benchmark problems are presented to illustrate the superior performance of the proposed approach compared with previously reported methods.

Further improvement on delay-range-dependent robust absolute stability for Lur׳e uncertain systems with interval time-varying delays

This paper investigates improved delay-range-dependent robust absolute stability criteria for a class of Lur׳e uncertain systems with interval time-varying delays. By using delayed decomposition approach (DDA), a tighter upper bound of the derivative of Lyapunov functional can be obtained, and thus the proposed criteria give results with less conservatism compared with some previous ones. An integral inequality approach (IIA) is proposed to reduce the conservativeness in computing the allowable maximum admissible upper bound (MAUB) of the time-delay. The developed stability condition is expressed in terms of linear matrix inequality (LMI) that manipulates fewer decision variables and requires reduced computational load. Finally, three numerical examples are given to show the effectiveness of the proposed stability criteria.

Stability Analysis of AC Microgrids With Constant Power Loads Based on Popov's Absolute Stability Criterion

This brief considers nonlinear stability analysis of ac microgrids with constant power loads (CPLs). A microgrid is a subsystem consisting of small electrical resources, local loads, and power electronic devices. It should be noted that the presence of a CPL as a sink in a microgrid results in nonlinear behavior of the microgrid. CPLs are generally connected to an ac microgrid through a controlled rectifier and/or dc–dc converters. We model an ac microgrid with local RLC load and a CPL with power converter subsystem, which results in a set of nonlinear state–space equations. Then, Popov's absolute stability theorem is utilized to analyze stability conditions for an ac microgrid in the presence of CPL. Simulation results are presented, which are in agreement with the analytical approach.

Bilateral teleoperation system stability with non-passive and strictly passive operator or environment

A bilateral teleoperation system comprises a human operator, a teleoperator, and an environment. Without exact models for the teleoperator׳s terminations (i.e., human operator and the environment), it is typically assumed that they are passive but otherwise arbitrary. Based on this assumption, the stability of the teleoperation system is investigated through Llewellyn׳s absolute stability criterion for the teleoperator. However, the assumption of passivity of the terminations is less than accurate and may be violated in practice. Using Mobius transformations, this paper develops a new powerful stability analysis tool for a two-port network coupled to a passive termination and another termination that is (a) input strictly passive (ISP), (b) output strictly passive (OSP), (c) input non-passive (INP), or (b) disc-like non-passive (DNP). While this new stability criterion is applicable to any two-port network, we apply it to bilateral teleoperation systems with position-error-based (PEB) and direct-force-reflection (DFR) controllers. Simulations and experiments are reported for a pair of Phantom haptic robots.

The Yakubovich quadratic criterion, F-stability and multi-agent consensus.

Generalizing numerous results on absolute stability of nonlinear Lurie systems, Vladimir A. Yakubovich established a fundamental abstract criterion of absolute stability under uncertain nonlinearities. His Quadratic Criterion offers elegant conditions for the uniform output L2-stability in a wide class of uncertain nonlinearities, obeying anytime ("local") or integral quadratic constraints; these conditions are based on either quadratic Lyapunov functions whose existence is proved via the KYP lemma, or the method of integral quadratic constraints dating back to the works by V.M. Popov. In the present paper, we extend the Yakubovich quadratic criterion, replacing the output L2-stability by boundedness of some quadratic integral performance index, defined by a quadratic form F and referred to as F -stability. We demonstrate that our F -stability criteria enables one to obtain easily many stability criteria for the “critical” case, where the frequency-domain inequality is non-strict and usual L2-stability of the solution cannot be proved, e.g., we derive the circle and Popov's stability criteria in this degenerate case. All of these criteria are especially effective for scalar nonlinearities, nevertheless, we demonstrate that our approach is also applicable to a class of large scale nonlinear multi-agent networks, and derive the “networked” circle criterion for such networks as a consequence of our results.

Integral Sliding Mode Control of Lur’e Singularly Perturbed Systems

This paper investigates the integral sliding mode control problem for Lur’e singularly perturbed systems with sector-constrained nonlinearities. First, we design a proper sliding manifold such that the motion of closed-loop systems with a state feedback controller along the manifold is absolutely stable. To achieve this, we give a matrix inequality-based absolute stability criterion; thus the above problem can be converted into a matrix inequality feasibility problem. In addition, the gain matrix can also be derived by solving the matrix inequality. Then, a discontinuous control law is synthesized to force the system state to reach the sliding manifold and stay there for all subsequent time. Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.

Frequency-domain criteria for consensus in multiagent systems with nonlinear sector-shaped couplings

Consideration was given to the distributed algorithms for consensus (synchronization) in the multiagent networks with identical agents of arbitrary order and unknown nonlinear couplings satisfying the sector inequalities or their multidimensional counterparts. The network topology may be unknown and varying in time. A frequency synchronization criterion was proposed which is a generalization of the circular criterion for absolute stability of the Lur'e systems.

Popov-Type Criterion for Consensus in Nonlinearly Coupled Networks.

This paper addresses consensus problems in nonlinearly coupled networks of dynamic agents described by a common and arbitrary linear model. Interagent interaction rules are uncertain but satisfy the standard sector condition with known sector bounds; both the agent's model and interaction topology are time-invariant. A novel frequency-domain criterion for consensus is offered that is similar to and extends the classical Popov's absolute stability criterion.

On the criteria of the absolute convective stability for compressible fluids

The conditions under which natural convection is absent from compressible fluids are investigated. It is shown that in the parameter “Rayleigh number-given temperature difference” plane there is a domain in which convection occurs for neither Rayleigh numbers. It is proposed to refer to this domain as the absolute convective stability region and to name the criterion determining the boundary of this region the absolute convective stability criterion. The necessary, sufficient, and necessary and sufficient conditions of the absolute convective stability for a viscous compressible fluid are derived. It is shown that in the particular case in which the thermal properties of the fluid and the adiabatic gradient are constant, these conditions coincide with the Schwarzschild criterion.

Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large

The analysis of the stability and numerical simulation of Costas loop circuits for high-frequency signals is a challenging task. The problem lies in the fact that it is necessary to simultaneously observe very fast time scale of the input signals and slow time scale of phase difference between the input signals. To overcome this difficult situation it is possible, following the approach presented in the classical works of Gardner and Viterbi, to construct a mathematical model of Costas loop, in which only slow time change of signal׳s phases and frequencies is considered. Such a construction, in turn, requires the computation of phase detector characteristic, depending on the waveforms of the considered signals. While for the stability analysis of the loop near the locked state (local stability) it is usually sufficient to consider the linear approximation of phase detector characteristic near zero phase error, the global analysis (stability in the large) cannot be accomplished using simple linear models.The present paper is devoted to the rigorous construction of nonlinear dynamical model of classical Costas loop, which allows one to apply numerical simulation and analytical methods (various modifications of absolute stability criteria for systems with cylindrical phase space) for the effective analysis of stability in the large. Here a general approach to the analytical computation of phase detector characteristic of classical Costas loop for periodic non-sinusoidal signal waveforms is suggested. The classical ideas of the loop analysis in the signal׳s phase space are developed and rigorously justified. Effective analytical and numerical approaches for the nonlinear analysis of the mathematical model of classical Costas loop in the signal׳s phase space are discussed.

Studies on Bilateral Impedance Control Method for Teleoperation Robot

The bilateral PD control method for teleoperation robot has some defects, such as poor tracking performance and force feedback performance. This paper, based on traditional bilateral PD control method, adds an impedance controller to the master and the slave, and deduces the stability condition according to the absolute stability criterion. The simulation shows that this method can assure the system stability and improve tracking performance and force feedback performance.

Absolute stability of a class of trilateral haptic systems.

Trilateral haptic systems can be modeled as three-port networks. We present a criterion for absolute stability of a general class of three-port networks. Traditionally, existing (i.e., Llewellyn's) criteria have facilitated the stability analysis of bilateral haptic systems modeled as two-port networks. If the same criteria were to be used for stability analysis of a three-port network, its third port termination would need to be assumed known for it to reduce to a two-port network. This is restrictive because, for absolute stability, all three terminations of the three-port network must be allowed to be arbitrary (while passive). Extending Llewellyn's criterion, we present closed-form necessary and sufficient conditions for absolute stability of a general class of three-port networks. We first find a symmetrization condition under which a general asymmetric impedance (or admittance) matrix Z3 × 3 has a symmetric equivalent Zeq from a network stability perspective. Then, via the equivalence of passivity and absolute stability for reciprocal networks, an absolute stability condition for the original nonreciprocal network is derived. To demonstrate the convenience and utility of using this criterion for both analysis and design, it is applied to the problem of designing stabilizing controllers for dual-user haptic teleoperation systems, with simulations and experiments validating the criterion.

Analysis of Coupled Stability in Multilateral Dual-User Teleoperation Systems

In this paper, we set out a framework for the analysis of coupled stability in dual-user linear teleoperation systems. An extension of the Zeheb–Walach (ZW) criteria for absolute stability of an $n$ -port network will be stated and proven. While the original theorem states conditions for asymptotic stability of a network terminated by passive impedances, the extended version allows for poles on the imaginary axis, which makes it applicable to a larger class of systems, such as robotic applications with position feedback. The extended theorem includes conditions on the Laurent expansion of the elements and the principal minors of the network immittance matrix. A novel dual-user shared control paradigm, realizing a three-way gearbox mechanism, is presented. A numerical analysis of absolute stability of the three-port network, representing the shared control architecture, demonstrates the effectiveness of the extended Zeheb–Walach method.

On computing maximum allowable time delay of Lur’e systems with uncertain time-invariant delays

In this paper, we present an improved delay-dependent absolute stability criterion for Lur’e systems with time delays. The guarantee of absolute stability is provided by Lyapunov-Krasovskii theorem with the Lyapunov functional containing the integral of sector-bounded nonlinearities. The Lyapunov functional terms involving delay are partitioned to be associated with each equidistant fragment on the length of time delay. Employing the Jensen inequality and S-procedure, the sufficient condition is derived from time derivative of the Lyapunov functional. Then, the absolute stability criterion expressed in terms of linear matrix inequalities (LMIs) can be efficiently solved using available LMI solvers. The bisection method is used to determine the maximum allowable time delays to ensure the stability of Lur’e systems in the presence of uncertain time-invariant delays. In addition, the stability criterion is extended to Lur’e systems subject to norm-bounded uncertainties by using the matrix eliminating lemma. Numerical results from two benchmark problems show that the proposed criteria give significant improvement on the maximum allowable time delays.

Robust Absolute Stability Criteria for Uncertain Lurie System With Interval Time-Varying Delay

This paper addresses the problem of absolute stability of Lurie system with interval time-varying delay. The delay range is divided into two equal segments and an appropriate Lyapunov–Krasovskii functional (LKF) is defined. A tighter bounding technique for the derivative of LKF is developed. This bounding technique in combination with the Wirtinger inequality is used to develop the absolute stability criterion in terms of linear matrix inequalities (LMIs). The stability analysis is also extended to the Lurie system with norm-bounded parametric uncertainties. The effectiveness of the proposed approach has been illustrated through a numerical example and Chua's oscillator.

Passivity and Absolute Stability Analysesof Trilateral Haptic Collaborative Systems

Trilateral haptic systems can be modeled as three-port networks. Analysis of coupled stability of a three-port network can be accomplished in either the passivity or the absolute stability frameworks assuming all three ports are connected to passive but otherwise unknown terminations. This paper first introduces our recent results in terms of extending Raisbecks passivity criterion and Llewellyns absolute stability criterion to general three-port networks --- both criteria are founded on the properties of a positive-real Hermitian matrix. Next, we show that the absolute stability criterion is less conservative than the passivity criterion. Then, to show how the two criteria may be utilized at the system design stage, we apply them to the problem of designing controllers for a dual-user haptic teleoperation system and a triple-user collaborative haptic virtual environment. Using the two criteria, controllers are then designed and compared in terms of conservatism in simulations and experiments.