Absolute Stability Criterion Research Articles

Low conservative stability criteria for discrete-time Lur’e systems with sector and slope constrained nonlinearities

Lur’e system is an important nonlinear system in engineering, which is composed of a feedback connection of a linear dynamic system and a nonlinear system that satisfies the sector constraint. In this paper, for nominal and uncertain discrete Lur’e systems with time-varying delays, the absolute and robustly absolute stability are studied in detail with the nonlinearity satisfying the sector and slope constraints. Firstly, an augmented Lyapunov–Krasovskii functional (LKF) is designed, in which some augmented vectors are selected to increase the coupling information between the delay intervals and other system state variables. At the same time, some effective integral terms of nonlinear state variables are extended in the LKF according to the characteristics of sector constraint and slope constraint. Secondly, a general lemma of summation inequality for free-weighted matrices is given, which adds some coupling information between vectors and extends the search range for solving linear matrix inequalities (LMIs). Thirdly, by synthesizing the general summation inequality and the newly constructed LKF, different absolute stability criteria are derived respectively, and the absolute stability criteria are extended to robustly absolute stability cases with uncertain parameters. The stability criteria reduce the conservatism of some previously proposed results. Finally, some common numerical examples and simulations are used to verify the results of this paper.

The growth and morphological stability of a precipitate in a multi-component solution

The growth and morphological stability of a precipitate in a multi-component solution with cross diffusion between solute species is studied by using the asymptotic method. The resulting asymptotic solution for the dynamics model of the precipitate reveals the relationship between precipitate growth and solute diffusion, and the absolute and relative stability criteria of the precipitate in the multi-component solution with cross diffusion between solute species. The absolute and relative stability radii of the precipitate are determined. It is demonstrated that the growth velocity and size of the precipitate in the multi-component solution are controlled by a growth factor that is determined by the solute diffusion coefficients. In the multi-component solution with cross diffusion between solute species, via the co-action between cross diffusion of solute species and interface kinetic undercooling, the precipitate enhances or reduces the morphological stability by changing the growth factor. In the multi-component solution, the interface kinetics decreases the growth velocity of the precipitate, and enhances the stability of the precipitate. When the interface kinetic undercooling is neglected, the absolute and relative stability criteria of the precipitate remain the same whether the diffusion of solute species is intrinsic or cross diffusive.

Absolute Exponential Stability Criteria for Some Classes of Nonlinear Time-Varying Systems with Delays and Sector Nonlinearities

Absolute Exponential Stability Criteria for Some Classes of Nonlinear Time-Varying Systems with Delays and Sector Nonlinearities

Universal Absolute Stability Criterion for Single-Phase Grid-Following Inverters Equipped With Orthogonal-Normalization-Based Grid Synchronizer

The stability problem of grid-following inverters has been widely investigated. Nevertheless, existing literature has not yet proposed a strict stability analysis method for the overall grid-following inverter system with nonlinear nature. Focusing on single-phase systems, by using frequency-domain nonlinear theory, this article proposes a universal absolute stability criterion for the single-phase grid-following inverters (SGFIs) equipped with orthogonal-normalization-based grid synchronizer. It will be shown that a SGFI will be absolutely stable under small- and large-disturbances if both the traditional impedance criterion and the synchronous nonlinear criterion are satisfied. The usefulness and applicability of the proposed new criterion is demonstrated by a case study of the conventional LCL-type SGFI system. Finally, the theoretical analysis is fully supported by a series of steady-state and transient experimental tests.

Series PIDA Controller Design for IPDT Processes

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.

The evolution and morphological stability of a particle in a binary alloy melt

The evolution and morphological stability of a particle in a binary alloy melt is re-examined using the asymptotic method. The resulting asymptotic solution of the dynamics model of the particle reveals that the temporal evolution and morphological stability of the particle vary continuously as functions of solute concentration. In particular, (1) the critical nucleation radius of the particle in the binary alloy melt is determined. When solute is rejected from solid phase to liquid phase, the initial concentration increases the critical nucleation radius and depresses the growth velocity of the particle. When solute is incorporated into the solid, the initial concentration decreases the critical nucleation radius. (2) The absolute and relative stability criteria of the particle in the binary alloy melt are shown on the basis of Mullins and Sekerka’s work in the pure melt. The critical stability radii for the absolute and relative stability criteria of the particle in the binary alloy melt are determined. The growth ratio of the critical stability radius to the critical nucleation radius in the binary melt does not exceed that in the pure melt. (3) When the segregation coefficient is less than one, with the growth of the particle, the initial concentration in the binary alloy melt first suppresses and then enhances the morphological stability of the particle. There exists a minimum solute concentration in the binary alloy melt such that when the initial concentration is lower than the minimum solute concentration, the stability of the particle decreases with the increase of solute concentration, whereas when the initial concentration is higher than the minimum solute concentration, the stability of the particle increases with the solute concentration. As the initial concentration increases, the critical stability radius first decreases to the minimum solute concentration at a certain solute concentration, and then increases with the solute concentration. This may control the morphological stability of particles by changing the addition of solute in the alloy melt.

Synthesis of a Fuzzy Controller According to the Degree of Stability of the Control System

The article is devoted to solving the problem of analysis and synthesis of a control system with a fuzzy controller (fuzzy control system) in terms of the degree of stability. As interest in fuzzy control systems grows, various approaches to the study of such systems are being developed more and more actively. One of the most actively developing areas is based on the modification of methods for assessing the stability regions of nonlinear ACS, however, when solving most practical problems, this knowledge is not enough, since the developer needs to provide the required qualitative characteristics of the transient process (and, in particular, the control time). Therefore, the synthesis of fuzzy ACS in terms of the degree of stability seems to be quite constructive, since it allows quite accurately assessing the performance of a fuzzy ACS. The proposed solution is based on the criterion of absolute stability for a system with an approximated nonlinear characteristic of a fuzzy controller, which can be obtained directly on the basis of the Sugeno method. The article develops Yakubovich’s circular criterion of absolute stability, modified for a fuzzy control system, using a shifted AFC of the linear part. With this approach, it is possible to obtain a completely constructive solution to the problem of synthesizing the parameters of a fuzzy controller in the frequency domain. On the example of fuzzy control systems with static and astatic linear parts, the features of the application of the developed approach are shown and methods for synthesizing the parameters of a fuzzy controller are proposed. The analysis of the influence of individual components of the nonlinear transformation on the quality of the transient process is carried out, and on the basis of this a number of practical recommendations are given for correcting the fuzzy controller settings that provide the required performance.

Absolute stability and absolute hyperbolicity in systems with discrete time-delays

An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.

An improved stability criterion for discrete-time time-delayed Lur'e system with sector-bounded nonlinearities

The absolute stability problem of discrete-time time-delayed Lur'e systems with sector bounded nonlinearities is investigated in this paper. Firstly, a modified Lyapunov-Krasovskii functional (LKF) is designed with augmenting additional double summation terms, which complements more coupling information between the delay intervals and other system state variables than some previous LKFs. Secondly, some improved delay-dependent absolute stability criteria based on linear matrix inequality form (LMI) are proposed via the modified LKF and the relaxed free-matrix-based summation inequality technique application. The stability criteria are less conservative than some results previously proposed. The reduction of the conservatism mainly relies on the full use of the relaxed summation inequality technique based on the modified LKF. Finally, two common numerical examples are presented to show the effectiveness of the proposed approach.

An augmented approach to absolute stability for uncertain Lur'e system with time‐varying delay

This paper investigates the absolute stability criteria of Lur'e system with time‐varying delays, uncertainties, and sector bounded nonlinearities. By constructing suitable Lyapunov–Krasovskii functionals (LKFs) and utilizing some useful mathematical techniques, an improved delay‐dependent stability criterion is introduced in Theorem 1. Based on the result of Theorem 1, a further enhanced criterion is proposed in Theorem 2 by employing the augmented zero equality approach. Finally, two numerical examples show the improved performance of the criteria by comparing maximum delay bounds.

Stability of single-stage single-bit ∑Δ modulator

A feedback loop of sigma-delta modulator (∑ΔM) is analysed, including the following parameters: operational amplifier gain factor, leakages of integrators, internal scale coefficients. The aspects of stability were presented for both chaotic and non-chaotic ∑ΔM mode. A criterion close to the Tsypkin absolute stability criterion was obtained for the chaotic mode. This criterion provides a system of inequalities of internal parameters for the ∑ΔM configuration. The coefficients down to the 5th order, satisfying the criterion, are given. A stability bound, according to which different configurations of the same order are compared to one another, appears as a useful innovation. The bound is also compared to a common non-invertible area. Based on the non-invertible area for non-chaotic case and ADC (analog-to-digital converter), the ranges were found for internal integrators and input, sufficient for smooth operation. In discrete-time systems, the role of necessary criterion for absolute stability in respect to both chaotic and non-chaotic cases is under discussion.

Morphological stability of spherical particles - Extension of the Mullins-Sekerka criteria to multi-component alloys under a non-stationary diffusive regime

An extension of the pioneer work by Mullins and Sekerka is proposed to analyze the stability conditions of a spherical particle growing in a matrix phase. The present model describes the onset of instabilities in an unsteady growth regime for multi-component alloys with cross-diffusion of chemical species. The developments are based on the decomposition of initial perturbations in spherical harmonics to determine their time evolution. Expressions for the threshold particle radii associated to the limits for absolute and relative stability criteria are derived. They depend on the degree of the spherical harmonic functions, the growth parameter for a sphere (proportional to the interface velocity times the square root of time) and the eigenvalues of the diffusion matrix. A large effect of the interface velocity on the stability domains is demonstrated considering the time-dependent solutions. Comparisons are provided regarding more recent solutions proposed in the literature. A similar evolution is only observed for slow regimes. An application for solidification of a ternary alloy is finally given.

Infinite-dimensional Lur\u2019e systems with almost periodic forcing

We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.

Analysis of time-stepping methods for the monodomain model

To a large extent, the stiffness of the bidomain and monodomain models depends on the choice of the ionic model, which varies in terms of complexity and realism. In this paper, we compare and analyze a variety of time-stepping methods: explicit or semi-implicit, operator splitting, exponential, and deferred correction methods. We compare these methods for solving the monodomain model coupled with three ionic models of varying complexity and stiffness: the phenomenological Mitchell–Schaeffer model, the more realistic Beeler–Reuter model, and the stiff and very complex ten Tuscher–Noble–Noble-Panfilov (TNNP) model. For each method, we derive absolute stability criteria of the spatially discretized monodomain model and verify that the theoretical critical time steps obtained closely match the ones in numerical experiments. We also verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. The efficiency of the different methods is also considered by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured, respectively, by the eigenvalues of the model’s Jacobian and the number of variables, and based on strict stability and accuracy criteria.

Semi-global incremental input-to-state stability of discrete-time Lur’e systems

We present sufficient conditions for semi-global incremental input-to-state stability of a class of forced discrete-time Lur’e systems. The results derived are reminiscent of well-known absolute stability criteria such as the small gain theorem and the circle criterion. We derive a natural sufficient condition which guarantees that asymptotically (almost) periodic inputs generate asymptotically (almost) periodic state trajectories. As a corollary, we obtain sufficient conditions for the converging-input converging-state property to hold.

Is knowledge reliance in source guessing a cognitive trait? Examining stability across time and domain.

When people need to infer the source of information in the absence of memory, they may rely on general knowledge (e.g., stereotypes) to guess the source. Prior research documented task-related determinants and individual differences of stereotype reliance in source guessing, but little is known about the underling nature of this process. In two experiments, we tested whether a cognitive trait could account for the knowledge reliance in source guessing. Participants performed two distinct study-test cycles of a classical source-monitoring paradigm in which two person sources present stereotypical information that in a later test phase had to be attributed to its origin. In Experiment 1, both tasks used item material from the same knowledge domain (age stereotypes) and were either separated by 10 minutes or 7 days. In Experiment 2, we used item material from two different knowledge domains (Task 1: age stereotypes; Task 2: gender stereotypes). Although cross-task correlations of source-guessing parameters from Bayesian-hierarchical multinomial processing tree model analyses showed only weak positive correlations, absolute source guessing remained fairly stable within individuals across time (Experiment 1) and knowledge domains (Experiment 2). Considering statistical challenges of the assessment of relative stability via correlations, we suggest based on the stricter absolute stability criterion that source guessing rather encompasses trait-like features. We discuss implications regarding the generalizability and nature of source guessing in comparison to other cognitive processes involved in source attribution, which were highly stable in both experiments.

On stability of linear dynamic systems with hysteresis feedback

The stability of linear dynamic systems with hysteresis in feedback is considered. While the absolute stability for memoryless nonlinearities (known as Lure’s problem) can be proved by the well-known circle criterion, the multivalued rate-independent hysteresis poses significant challenges for feedback systems, especially for proof of convergence to an equilibrium state correspondingly set. The dissipative behavior of clockwise input-output hysteresis is considered with two boundary cases of energy losses at reversal cycles. For upper boundary cases of maximal (parallelogram shape) hysteresis loop, an equivalent transformation of the closed-loop system is provided. This allows for the application of the circle criterion of absolute stability. Invariant sets as a consequence of hysteresis are discussed. Several numerical examples are demonstrated, including a feedback-controlled double-mass harmonic oscillator with hysteresis and one stable and one unstable poles configuration.

Experimental Assessment of Absolute Stability in Bilateral Teleoperation.

Absolute stability analysis of bilateral teleoperation systems are typically model-based. Under borderline conditions of absolute stability, depending on the degree of uncertainty in the dynamic model of the teleoperator and existing noise, the system may behave as potentially unstable when the model-based analysis predicts otherwise. In this article, we propose a methodology to experimentally verify the absolute stability of master-slave teleoperation systems. Since absolute stability demands bounds of all possible environments, we achieve this by conducting only three experiments that are often experienced in teleoperation: free slave, mass-carrying slave and locked slave (rigid environment). We will validate and compare our proposed method with the benchmark Llewellyns absolute stability criterion. Furthermore, we will examine the robustness of the proposed method and will provide guidelines for choosing the mass for the mass-carrying load condition.

Stability analysis of Lur’e systems with a pulse-modulated feedback

A nonlinear system with a sector bound nonlinearity is considered. The system is subject to a stabilizing sampled feedback with finite width impulses. An impulsive counterpart of the circle criterion for absolute stability is obtained with the help of the Gelig’s averaging method.

PID-Like Adaptive Fuzzy Controller Design Based on Absolute Stability Criterion

This paper describes a method of PID-like adaptive fuzzy controller design for a linear time-invariant single-input and single-output dynamic plant. The plant transfer function is assumed to be partially known. The controller operates in the direct adaptation mode with the output feedback and a nonlinear reference model which provides better closed-loop response parameters than the system with a PID linear controller. The closed-loop stability is guaranteed during the adaptation process. The fuzzy controller is assumed to be a nonlinearity in a bounded sector and is defined by highly interpretable fuzzy rules. The absolute stability criterion for the system, containing PID-like fuzzy controller, is delivered. A design procedure that generalizes and significantly extends results reported so far in the literature and considerably automates the searching process of a nonlinear adaptive fuzzy controller is provided.