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This paper focuses on addressing the issue of absolute stability for uncertain Lur'e systems with time-varying delay using a delay-segmentation approach. The approach involves decomposing the delay interval into two distinct subintervals of unequal lengths. This allows for the introduction of a delay-segmentation-based augmented Lyapunov–Krasovskii functional that ensures piecewise continuity at the partition points. By selecting two sets of Lyapunov matrices for the time-varying delay in each interval, the obtained results are less conservative, providing a more accurate assessment of absolute stability. Finally, a numerical example is given to demonstrate the superiority of the delay-segmentation approach. [ABSTRACT FROM AUTHOR]

This paper discusses optimal design of the series proportional–integral–derivative–accelerative (PIDA) controller for integral-plus-dead-time (IPDT) plants. The article starts with the design of disturbance reconstruction and compensation based on proportional-derivative-accelerative (PDA) stabilizing controllers. It shows that by introducing positive feedback by a low-pass filter from the (limited) output of the stabilizing PDA controller, one gets disturbance observer (DOB) for the reconstruction and compensation of input disturbances. Thereby, the DOB functionality is based on evaluating steady-state controller output. This DOB interpretation is in full agreement with the results of the analysis of the optimal setting of the stabilizing PDA controller and of its expanded PIDA version with positive feedback from the controller output. By using the multiple real dominant pole (MRDP) method, it confirms that the low-pass filter time constant in positive feedback must be much longer than the dominant time constant of the stabilized loop. This paper also shows that the constrained PIDA controller with the MRDP setting leads to transient responses with input and output overshoots. Experimentally, such a constrained series PIDA controller can be shown as equivalent to a constrained MRDP tuned parallel PIDA controller in anti-windup connection using conditional integration. Next, the article explores the possibility of removing overshoots of the output and input of the process achieved for MRDP tuning by interchanging the parameters of the controller transfer function, which was proven as very effective in the case of the series PID controller. It shows that such a modification of the controller can only be implemented approximately, when the factorization of the controller numerator, which gives complex conjugate zeros, will be replaced by a double real zero. Neglecting the imaginary part and specifying the feedback time constant with a smaller approximative time constant results in the removal of overshoots, but the resulting dynamics will not be faster than for the previously mentioned solutions. A significant improvement in the closed-loop performance can finally be achieved by the optimal setting of the constrained series PIDA controller calculated using the performance portrait method. This article also points out the terminologically incorrect designation of the proposed structure as series PIDA controller, because it does not contain any explicit integral action. Instead, it proposes a more thorough revision of the interpretation of controllers based on automatic reset from the controller output, which do not contain any integrator, but at the same time represent the core of the most used industrial automation. In the end, constrained structures using automatic reset of the stabilizing controller output can ensure a higher performance of transient responses than the usually preferred solutions based on parallel controllers with integral action that, in order to respect the control signal limitation, must be supplemented with anti-windup circuitry. The excellent properties of the constrained series PIDA controller are demonstrated by an example of controlling a thermal process and proven by the circle criterion of absolute stability. [ABSTRACT FROM AUTHOR]

This paper is concerned with the robust absolute stability analysis for Lurie control of the neutral singular systems. By using Lyapunov-Krasovskii function method, delay dependent criteria for absolute stability of system are derived in terms of Linear Matrix Inequality. The difference between this paper and other existing results is that the lower bounds and upper bounds of discrete-delay are considered, which will obtain some less conservative stability analysis results. Finally, some numerical examples are presented to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Simulation involving interaction with deformable objects often causes stability problems because a slowly updated force generates additional energy to the human user. This paper proposes a stable and transparent haptic rendering for simulation involving interaction between a rigid tool and deformable objects. This method computes visual and haptic feedback in the simulation and haptic feedback loops, respectively. A local stiffness matrix consisting of points around contact points is constructed based on collision detection between a virtual tool and a deformable object in the simulation loop. The haptic feedback is then computed at a higher update rate in the haptic feedback loop using the local stiffness matrix. Equivalent springs computed by using the equivalent stiffness energy are added to the boundary of the local stiffness matrix to minimize errors in the rendered force. The proposed method is compared with the virtual coupling widely used in simulation involving interaction with deformable objects. The proposed method reduces the x-, y-, and z-axis maximum force errors by up to 52%, 80%, and 70%, respectively, compared to the virtual coupling in the simulation involving interaction with the Stanford bunny object consisting of 2087 points and 9997 tetrahedrons. [ABSTRACT FROM AUTHOR]

This paper presents a numerical integration method recently proposed by means of an interpolating function involving a transcendental function of exponential type for the solution of continuous dynamical systems, that is, the initial value problems (IVPs) in ordinary differential equations (ODEs). The analysis of the local truncation error (Tn (h)), order of convergence, consistency and the stability of the proposed method have been investigated in the present study. The principal term of Tn (h) for the method has been derived via Taylor's series expansion. The standard test problem is taken into account to investigate the linear stability region and the corresponding stability interval of the method. It is observed that the newly proposed numerical integration method is second order convergent, consistent and conditionally stable. In order to test the performance measure of the proposed method, five IVPs of varying nature have been illustrated in the context of the maximum absolute global relative errors, the absolute relative errors computed at the final mesh point of the integration interval under consideration and the ℓ² error norm. Furthermore, the results are compared with two existing second order explicit numerical methods taken from the relevant literature. The so far obtained results have demonstrated that the proposed numerical integration method agrees with the second order convergence based upon the analysis conducted. Hence the proposed method is considered to be a good approach for finding the solution of different types of IVPs in ODEs. [ABSTRACT FROM AUTHOR]

This paper deals with master–slave synchronization for Lur'e systems subject to a more general sector condition by using time delay feedback control. A new Lyapunov–Krasovskii functional and a new Lur'e–Postnikov Lyapunov functional are proposed to obtain some new delay-dependent synchronization criteria, which are formulated in the form of linear matrix inequalities (LMIs). These criteria cover some existing results as their special cases. An example shows that the result derived in this paper significantly improves some existing ones. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of non-fragile sampled-data control for synchronization of complex dynamical networks with randomly coupling and time varying delay under exponential approach. By adopting an appropriate Lyapunov Krasovskii functional (LKF) and taking into consideration full information on the sampling pattern, free-matrix based integral and Wirtinger inequalities are explored leading to the establishment of sufficient conditions to guarantee the exponential synchronization stability and disturbance attenuation of the closed loop network, with a designed non-fragile controller under all randomly admissible gain variations. The results are presented in terms of Linear matrix inequalities (LMIs), which can effectively be solved by some available softwares. Finally, two simulated results are demonstrated to show the effectiveness and less conservativeness of our proposed scheme. [ABSTRACT FROM AUTHOR]

Fuel cell (FC) is emerging as a clean and nonpollutant energy source and is being used widely in most of the transport and microgrid applications. However, certain applications demanding fast dynamic response are restricting usage of FC on account of its lifetime issues. Slow dynamic response of FC is not suitable to provide transient load demands, typically arising in electrical vehicles during acceleration and in braking. This paper presents a novel control strategy and associated power electronics for such applications. We propose a Lure–Lyapunov-based controller executed in cascade mode for the control of hybrid energy management system of FC and ultracapacitor (UC). In the proposed scheme, UC responds to transients in load demand whereas FC takes care of the nominal load bus requirements. This combination helps in improving lifetime of FC as auxiliary storage device such as UC provides a cushion for sudden transients. The proposed control strategy also helps in maintaining state of charge of UC for better performance. The inherent nonlinearities in DC–DC converter are efficiently handled by the outer-loop Lure–Lyapunov framework. Mathematical modeling and a proof for absolute stability of the system is presented. The proposed controller strategy is robust by virtue of being passivity based control methodology. The paper also presents analysis of the required power electronics involving DC–DC converters and inner-loop current control action. The synthesized controller is experimentally validated and the results are presented with their analysis. [ABSTRACT FROM PUBLISHER]

This paper investigates the absolute stability problem of time-varying delay Lurie indirect control systems with variable coefficients. A positive-definite Lyapunov-Krasovskii functional is constructed. Some novel sufficient conditions for absolute stability of Lurie systems with single nonlinearity are obtained by estimating the negative upper bound on its total time derivative. Furthermore, the results are generalised to multiple nonlinearities. The derived criteria are especially suitable for time-varying delay Lurie indirect control systems with unbounded coefficients. The effectiveness of the proposed results is illustrated using simulation examples. [ABSTRACT FROM AUTHOR]

The problem of absolute stability of Lur’e systems with time-delay is investigated in this paper. By employing a new Lyapunov-Krasovskii functional with the idea of N-segmentation of delay length, less conservative delay-dependent stability criteria are obtained and formulated in the form of linear matrix inequalities (LMIs). Numerical example shows that the results obtained in this paper are better than existing ones. [ABSTRACT FROM AUTHOR]

This paper provides sufficient conditions for absolute stability of an indirect control Lur'e problem of neutral type. The conditions are derived using a Lyapunov-Krasovskii functional and are given in terms of a system of matrix algebraic inequalities. From these matrix inequalities a sufficient condition for linear state feedback stabilizability follows. [ABSTRACT FROM AUTHOR]

A class of singular nonlinear systems with set-valued mappings are studied in this paper. Criteria are given based on the Lyapunov function to check the absolute stability of the systems, then the results are extended to the time delay systems and the time delay systems with uncertainty. Three examples are simulated to show the e®ectiveness of the proposed stability conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the absolute stability of a class of singularly perturbed Lur'e systems with an interval time-varying delay and sector-bounded nonlinearity. By constructing a Lyapunov-Krasovskii functional (LKF), the absolute stability criteria are proposed in terms of linear matrix inequalities (LMIs). One numerical example is presented to show the effectiveness of the result. [ABSTRACT FROM AUTHOR]

The present paper points out that a class of positive polynomials deserves special attention due to several interesting applications in signal processing, system analysis and control. We consider positive hybrid polynomials with two variables, one real, the other complex, belonging to the unit circle. We present several theoretical results regarding the sum-of-squares representations of such polynomials, treating the cases where positivity occurs globally or on domains. We also give a specific Bounded Real Lemma. All the characterizations of positive hybrid polynomials are expressed in terms of positive semidefinite matrices and can be extended to polynomials with more than two variables. On the applicative side, we show how several problems are numerically tractable via semidefinite programming (SDP) algorithms. The first problem is the minimax design of adjustable FIR filters, using a modified Farrow structure. We discuss linear-phase and approximately linear-phase designs. The second is the absolute stability of time-delay feedback systems with unknown delay, for which we treat the cases of bounded and unbounded delay. Finally, we discuss the application of our methods to checking the stability of parameter-dependent systems. The design procedures are illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses both theoretical and experimental studies of the stability and performance of haptic interfaces containing active/passive actuators. Three different realizations of haptic interfaces are introduced to investigate their stability and performance: an active system equipped with a motor; a passive system equipped with a brake; and a hybrid system equipped with both brake and motor. The first objective is to demonstrate that a hybrid system is superior in its stability and performance to an active system via passivity theorem and Z-width. The second objective of this paper is to show that the conditions for the asymptotic stability of haptic interfaces during the static friction display are investigated via the absolute stability theory. Theoretical and experimental results are compared. An alternative haptic interface is proposed that provides its highly stable haptic interaction with high performance. [ABSTRACT FROM AUTHOR]

This paper deals with the absolute stability of single input single output time delay systems with, in general, a finite number of non commensurate constant internal point delays for any nonlinearity satisfying a time positivity inequality related to the first and third quadrants. The results are obtained based on Lyapunov's stability analysis via appropriate Lyapunov's functions and the related stability study is performed to obtain both delay independent and delay dependent results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• Robust output-feedback absolute stabilizing control approach to fluid resuscitation. • Observer-based LQR design for set point tracking and disturbance rejection. • LMI-based observer design for absolute stability against physiological uncertainties. This paper presents a robust closed-loop control approach to fluid resuscitation in patients with hemorrhagic blood loss. A unique strength of the proposed approach is its robustness against uncertain and time-varying patient physiology and therapeutic effectiveness. First, we adopted an observer-based control architecture that can fulfill set point tracking and disturbance rejection objectives. Second, we determined the control gains to achieve adequate transient response performance using the linear quadratic regulator design. Third, we determined the observer gains as a solution to a set of linear matrix inequalities so that the overall closed-loop fluid resuscitation control system is (i) robust against the variability in patient physiology and (ii) absolutely stable against unknown therapeutic effectiveness. We demonstrated the initial proof-of-concept of the proposed approach by conducting rigorous in silico testing using a large number of physiologically plausible virtual patients, while ascertaining the absolute stability via the circle criterion analysis. The results suggested that the proposed approach to closed-loop control of fluid resuscitation is a promising option to advance automation of fluid resuscitation armed with stability against a large variability in patient physiology and therapeutic effectiveness as well as adequate performance in set point tracking and disturbance rejection. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Large industrial-like slave robots pose a challenging problem for telerobotic control designers focused on achieving good transparency. The human operator typically feels both the large friction forces and heavy inertial forces inherent to these robots. Force control can be used to attempt and hide these internal forces from the user, but force control is a challenging design problem, especially in situations like telerobotics where it is not clear what the environmental impedance will be at any given moment.This paper introduces a model-based force controller designed to reject the friction in slave robots to improve the overall transparency of the telerobotic system. The primary objective of the force controller is to ensure the closed loop slave 2-port is absolutely stable such that the controller maintains the robustness of the overall telerobot. An analysis is provided that shows that, in order to achieve absolute stability, a local slave-side force controller cannot hide any of the robot’s inertial forces. It is from this result that model-based force control gets its name as it uses a model of the robot’s inertial properties to reject friction forces without attempting to reject inertial forces. [ABSTRACT FROM PUBLISHER]

This paper is a survey of the recent results of the author on the stability of linear and nonlinear neutral type functional differential equations. Mainly, vector equations are considered. In particular, equations whose nonlinearities are causal mappings are investigated. These equations include neutral type, ordinary differential, differential-delay, integro-differential and other traditional equations. Explicit conditions for the Lyapunov, exponential, input-to-state and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide us the bounds for the regions of attraction of steady states. A part of the paper is devoted to the Aizerman type problem from the the absolute stability theory. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the generalized Bohl - Perron principle, positivity conditions for fundamental solutions of scalar equations and properties of the so called generalized norm. [ABSTRACT FROM AUTHOR]

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-infinite optimization (LSIO) problem, in the sense of maintaining the strong uniqueness property under sufficiently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (affinely) stable problems, i.e., those LSIO problems whose feasible set (its affine hull, respectively) remains constant under sufficiently small perturbations. [ABSTRACT FROM AUTHOR]

This paper considers the synchronization problem for nonlinear systems with time-delay couplings. We assume that the error dynamics can be rewritten as a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for time-delay coupled systems by applying the multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain stability condition and avoids the use of the Lyapunov-Krasovskii approach. As a result, the condition based on the circle criterion does not contain the conservativeness caused by the Lyapunov-Krasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling. [ABSTRACT FROM AUTHOR]

In this paper, a more general third-order chaotic system with attraction/repulsion function is introduced on the basis of [Duan et al., 2005]. A gallery of chaotic attractors, bifurcation diagrams and Lyapunov exponent spectra are presented to show the interesting phenomena of the given system. Based on the absolute stability theory and linear matrix inequality (LMI), a simple method of chaos control for the system is proposed and a stabilizing controller is derived such that chaos oscillations of the system disappear and all chaotic trajectories of it are led to certain equilibrium. Numerical simulations are provided to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: The absolute stability of Lurie system with multiple time delays and nonlinearities is considered in this paper. Based on the Lyapunov stability theory, using the descriptor system approach and the method of decomposing matrices, a novel delay-dependent sufficient condition for the absolute stability of Lurie system is derived and expressed in the form of the linear matrix inequality (LMI). The maximum upper bound of the allowable delay is obtained by solving a convex optimization problem. A numerical example shows that the result obtained in this paper improved the estimate of the stability limit over some existing results. [Copyright &y& Elsevier]

In this paper, in order to show some interesting phenomena of fourth-order Chua's circuit with a piecewise-linear nonlinearity and with a smooth cubic nonlinearity and compare dynamics between them, different kinds of attractors and corresponding Lyapunov exponent spectra of systems are presented, respectively. The frequency-domain condition for absolute stability of a class of nonlinear systems is transformed into linear matrix inequality (LMI) by using the celebrated Kalman–Yakubovich–Popov (KYP) lemma. A stabilizing controller based on LMI is designed so that chaos oscillations of fourth-order Chua's circuit with the piecewise-linear nonlinearity disappear and chaotic or hyperchaotic trajectories of the system are led to the origin. Simulation results are provided to demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, the absolute stability theory and methodology for nonlinear control systems are employed to study the well-known Chua's circuit. New results are obtained for the globally exponent synchronization of two Chua's circuits. The explicit formulas can be easily applied in practice. With the aid of constructing Lyapunov functions, sufficient conditions are derived, under which two (drive-response) Chua's circuits are globally and exponentially synchronized, even if the motions of the systems are divergent to infinity. Numerical simulation results are given to illustrate the theoretical predictions. [ABSTRACT FROM AUTHOR]

The new modular approach to constrained control of higher-order processes with dominant first-order dynamics using generalized controllers with automatic resets (ARCs) is addressed. The controller design is based on the multiple real dominant pole (MRDP) method for the integrator plus dead time (IPDT) process models. The controller output constraints are taken into account by inserting the smallest numerator time constant of the controller transfer function into the positive feedback loop representing the automatic reset (integral) term. In the series realization of the proportional–integral–derivative–acceleration (PIDA) controller (and other controllers with even derivative degree), the time constant mentioned is complex, so only the real part of the time constant has been used so far. Other possible conversions of a complex number to a real number, such as the absolute value (modulus), can be covered by introducing a tuning parameter that modifies the calculated real time constant and generalizes the mentioned conversion when designing controllers with constraints. In this article, the impact of the tuning parameter on the overall dynamics of the control loop is studied by simulation. In addition, an evaluation of the stability of the closed-loop control system is performed using the circle criterion in the frequency domain. The analysis has shown that the approximation of the complex zero by its real part and modulus leads to a near optimal response to the set point tracking. The disturbance rejection can be significantly improved by increasing the tuning parameter by nearly 50%. In general, the tuning parameter can be used to find a compromise between servo and regulatory control. The robustness and applicability of the proposed controller is evaluated using a time-delayed process with first-order dominant dynamics when the actual transfer function is much more complicated than the IPDT model. A comparison of the proposed MRDP-PIDA controller with series PI, PID and PIDA controllers based on a modified SIMC method has shown that the MRDP-PIDA controller performs better than the SIMC method, although the SIMC uses a more complex process model. [ABSTRACT FROM AUTHOR]

This paper addresses fundamental stability and performance issues associated with haptic interaction. It generalizes and extends the concept of a virtual coupling network, an artificial link between the haptic display and a virtual world, to include both the impedance and admittance models of haptic interaction. A benchmark example exposes an important duality between these two cases. Linear circuit theory is used to develop necessary and sufficient conditions for the stability of a haptic simulation, assuming the human operator and virtual environment are passive. These equations lead to an explicit design procedure for virtual coupling networks which give maximum performance while guaranteeing stability. By decoupling the haptic display control problem from the design of virtual environments, the use of a virtual coupling network frees the developer of haptic-enabled virtual reality models from issues of mechanical stability. [ABSTRACT FROM PUBLISHER]

This paper is a survey of the recent results of the author on the stability of linear and nonlinear neutral type functional differential equations. Mainly, vector equations are considered. In particular, equations whose nonlinearities are causal mappings are investigated. These equations include neutral type, ordinary differential, differential-delay, integro-differential and other traditional equations. Explicit conditions for the Lyapunov, exponential, input-to-state and absolute stabilities are derived. Moreover, solution estimates for the considered equations are established. They provide us the bounds for the regions of attraction of steady states. A part of the paper is devoted to the Aizerman type problem from the the absolute stability theory. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the generalized Bohl - Perron principle, positivity conditions for fundamental solutions of scalar equations and properties of the so called generalized norm.

Active disturbance rejection control (ADRC) is a relatively new, quite different but very practical control technology, which shows much promise in replacement of proportion-integration-differentiation (PID) with unmistakable advantage in performance and practicality. This paper mainly concerns with the robust absolute stability of the ADRC based control system with parameter perturbations of the plant, i.e., ADRC based interval control system. Firstly, the system is transformed into a perturbed indirect Lurie system. Then, the Popov criterion and its robust version are presented and some new methods are developed to analyze the (robust) absolute stability for the interval control system. Furthermore, an example is presented to illustrate (robust) absolute stability analysis via the above methods, which verifies the convenience and practicability of these methods and shows the strong stability robustness of ADRC in the presence of parametric uncertainties. [ABSTRACT FROM AUTHOR]

The article presents absolute stability analysis for time-delay Lur'e systems with both sector- and slope-restricted nonlinearities. Topics include a new delay-dependent stability criterion given by using a Lur'e-Postnikov functional, the additional slope restrictions on the nonlinearities stated to play important roles in improving the absolute stability conditions and development of control theory impelling many advanced control techniques, such as neural network and sliding mode control.

This paper focuses on addressing the issue of absolute stability for uncertain Lur’e systems with time-varying delay using a delay-segmentation approach. The approach involves decomposing the delay interval into two distinct subintervals of unequal lengths. This allows for the introduction of a delay-segmentation-based augmented Lyapunov–Krasovskii functional that ensures piecewise continuity at the partition points. By selecting two sets of Lyapunov matrices for the time-varying delay in each interval, the obtained results are less conservative, providing a more accurate assessment of absolute stability. Finally, a numerical example is given to demonstrate the superiority of the delay-segmentation approach.