Conditions For Stability Of Systems Research Articles

Sufficient conditions for uniform global asymptotic stabilization of affine discrete-time systems with periodic coefficients

Sufficient conditions for uniform global asymptotic stabilization of affine discrete-time systems with periodic coefficients

Stability and bifurcation analysis for a fractional-order cancer model with two delays

Considering that healthy tissue cells and tumor cells restrain each other’s the average growth rate with a corresponding delay, a fractional-order cancer model with time delays is established. Firstly, by selecting different delays as bifurcation parameters, the delay-independent stability and Hopf bifurcation conditions of such system are derived. Then, some interesting dynamical phenomena of the system under different initial values can be observed, such as a pair of coexisted stable periodic solution, equilibrium point transition, and even chaos. Finally, some numerical simulations are exhibited to substantiate the obtained results. The work reveals that the fractional order and delays play an important role in the stability behavior and bifurcation characteristics of the system, and that a suitable control strategy of the system can be designed by selecting reasonable system parameters (fractional orders and time delays).

Simple criterion for evaluating stability of hydraulic oscillation based on water-hammer reflection coefficients

Hydraulic oscillation is a fluctuating phenomenon of pressure and discharge in pipes, which can threaten the safety of hydropower, pumping and water conveyance systems. To analyse the associated problems, new methods with simple form and clear physics are needed. This paper presents a water-hammer reflection coefficient-based criterion for stability evaluation of free-vibration of hydraulic systems. The stability (or attenuation) condition for a single pipe system is that the modulus of the product of the reflection coefficients at the inlet and outlet should be smaller than 1. For a complex pipe system, the condition necessary for stability is that every single pipe system is stable. To apply this new criterion to stabilizing the reservoir-single pipe-constant power turbine system by adding a head loss valve in the pipe, a formula for the critical head loss is proposed and verified. This new method is theoretically consistent with conventional methods, but more convenient in application.

Periodic-Disturbance Observer Using Spectrum-Selection Filtering Scheme for Cross-Coupling Suppression in Atomic Force Microscopy

Repetitive disturbances exist widely within the automation systems, which is one of the major issues that hinder the achievement of precision operations. Dedicated to mitigating these disturbances, a generalized periodic-disturbance observer (PDOB) using spectrum-selection filtering scheme is proposed in this paper with an application to a non-minimum phase system. The design process of the spectrum-selection filter that derives from the comb-like notch filter for the proposed PDOB is presented in detail. The variants of the proposed PDOB are also presented for the disturbances distributed only in odd-and even-harmonics. To achieve tracking of the desired trajectories, the proposed PDOB is combined in parallel with a baseline Proportional-Integral (PI) controller. The stability condition of the closed-loop system is derived to provide criteria for parameters selection. The experimental validation of the generalized PI+PDOB is conducted via real-time cross-coupling suppression in raster scanning of atomic force microscope (AFM), where the coupling induced root-mean-square tracking errors are reduced from 141.8 <inline-formula> <tex-math notation="LaTeX">$nm$</tex-math> </inline-formula> to 2.4 <inline-formula> <tex-math notation="LaTeX">$nm$</tex-math> </inline-formula> by employing the proposed PDOB for the scanning rate of 100 Hz. The results of convergence testing, raster scanning and AFM imaging are compared to illustrate the significant improvements achieved with the proposed PDOB. In addition, experimental results show that the tracking performance using PI+PDOB in the case of existing periodic disturbances resembles with that using PI control without periodic disturbances, which implies that the employment of the proposed PDOB does not interfere with the tracking-based control schemes for staircase trajectories, showing the advantages of the proposed PDOB. <i>Note to Practitioners</i>&#x2014;Repetitive motions facilitate diverse advanced functions within the automation systems, including the scanning electron microscope, atomic force microscope, manipulators, and robotic systems. These operations would unavoidably introduce repetitive disturbances, hindering its achievable precision. One particular issue is the widely existent cross-coupling effect that results in these repetitive disturbances. In this paper, a generalized periodic-disturbance observer (PDOB) based on the spectrum-selection filtering scheme is developed to address the repetitive disturbances for real-time coupling suppression in raster scanning of Atomic Force Microscopy. Consequently, the offline learning procedures required for other learning based schemes can be avoided. Without loss of generalization, the design of PDOB is demonstrated with an application to non-minimum phase systems. A scheme of the varying parameter is also developed to achieve both fast convergence and better performance at rejecting the periodic disturbances. The detailed variations to handle the disturbances distributed only in odd-and even-harmonics are also demonstrated. Different from other disturbance observers, the inclusion of the spectrum-selection filtering scheme provides more flexibility in rejecting specific repetitive disturbances. In terms of tracking accuracy and the simple structure, this development can also be easily implemented to other systems that suffer from repetitive disturbances.

Novel Stability and Stabilization Conditions of Linear Fractional-Order Time-Delay Systems Using Free Matrix Approach

The stability and stabilization problem of linear fractional-order time-delay systems is investigated. By introducing a free matrix, the linear fractional-order time-delay systems is transformed to a delay-free system interconnected with uncertainties. New stability and stabilization criteria are derived based on the small gain theorem. The criteria are formulated as linear matrix inequalities and are computationally efficient. Since the free matrix is utilized, the new conditions are less conservative than the existing ones. Two numerical examples indicate that the proposed criteria are less conservative than existing ones.

Analysis of a Heterogeneous Queuing Model with Intermittently Obtainable Servers under a Hybrid Vacation Schedule

This paper investigates the concept of a Markovian queueing model with heterogeneous, intermittently available servers with feedback under a hybrid vacation policy. Both the asymmetric transition representation and the hybrid vacation policy are addressed in this article. The necessary and sufficient conditions for system stability are presented. In addition, the steady-state probability distribution of the queueing model was derived by employing the matrix geometric method. Furthermore, a few formulae were constructed to determine the model’s performance indicators. Finally, the influence of system parameters was also investigated using some numerical examples.

Dissipativity-based conditions for the feedback stabilization of systems with time-varying input delays

This note is concerned with presenting new dissipativity-based conditions for the stabilizability of discrete-time systems with time-varying delays by linear static output feedback (SOF). We demonstrate that the feasibility of nonlinear matrix inequalities for the design of feedback stabilizing gains derived from the literature is equivalent to the feasibility of a linear matrix inequality establishing dissipativity of the system plus one matrix inequality constraint. An iterative strategy allowing the computation of stabilizing SOF gains is then developed based on a new relaxed sufficient condition. Compared with other popular approaches in the literature, such as the celebrated cone complementarity linearization (CCL) method, the strategy avoids providing initial “guesses” for the matrix variables, which is especially complicated when dealing with a large number of matrices. Due to being a particular case of SOF with an identity output matrix, static state feedback (SSF) gains can also trivially be computed by exploiting the developed conditions.

Two networked predictive control methods for output tracking of networked systems with plant-model mismatch

This paper addresses the output tracking control problem for networked systems with plant-model mismatch, where two-channel random network delays are considered. According to the different ways of obtaining future control commands, two networked predictive control (NPC) methods are designed. One is the NPC method using one-way time delays, called ONPC, and the other is the NPC method using round-trip time delays, called RNPC. Furthermore, they are compared in terms of the way of compensation for two-channel delays, prediction process, and communication load, respectively. Then, the closed-loop stability conditions of the RNPC system and the ONPC system for the plant-model mismatch and match cases are derived, and their output tracking performance is also theoretically analysed. Finally, the results of the theoretical analysis are verified through numerical simulation that illustrates the effectiveness of the proposed two NPC methods as well as their separate advantages.

On the Thermodynamic Stability of Multicomponent Reactive Systems

Derivation and analysis of conditions of the thermodynamic stability for multicomponent reactive systems are proposed based on transformations and different representations of the stability matrix. Stability boundaries and spinodal conditions in systems with chemical interaction of compounds are discussed.

Event-Triggered Networked Predictive Output Tracking Control of Cyber–Physical Systems With Model Uncertainty and Communication Constraints

This brief investigates the output tracking control problem of cyber-physical systems with model uncertainty and random communication constraints. An event-triggered networked predictive output tracking control method is presented to actively compensate random communication constraints in the two-channel, namely the feedback channel and forward channel. To handle the communication overload issue caused by communication constraints in networked predictive control approach, two kinds of event-triggered mechanisms are designed in both sensor and controller to save network resources. Then, the stability conditions of closed-loop systems with model certainty and uncertainty cases are derived. Finally, numerical simulation is performed to verify the effectiveness of presented method.

Robot Policy Improvement With Natural Evolution Strategies for Stable Nonlinear Dynamical System.

Robot learning through kinesthetic teaching is a promising way of cloning human behaviors, but it has its limits in the performance of complex tasks with small amounts of data, due to compounding errors. In order to improve the robustness and adaptability of imitation learning, a hierarchical learning strategy is proposed: low-level learning comprises only behavioral cloning with supervised learning, and high-level learning constitutes policy improvement. First, the Gaussian mixture model (GMM)-based dynamical system is formulated to encode a motion from the demonstration. We then derive the sufficient conditions of the GMM parameters that guarantee the global stability of the dynamical system from any initial state, using the Lyapunov stability theorem. Generally, imitation learning should reason about the motion well into the future for a wide range of tasks; it is significant to improve the adaptability of the learning method by policy improvement. Finally, a method based on exponential natural evolution strategies is proposed to optimize the parameters of the dynamical system associated with the stiffness of variable impedance control, in which the exploration noise is subject to stability conditions of the dynamical system in the exploration space, thus guaranteeing the global stability. Empirical evaluations are conducted on manipulators for different scenarios, including motion planning with obstacle avoidance and stiffness learning.

Parameters optimization of grounded dynamic vibration absorber with pendulum connected via the lever mechanism

This paper investigates the control performance of grounded dynamic vibration absorber (DVA) with pendulum connected via the lever mechanism. These optimal parameters are found based on the extended fixed point theory. However, in the optimal equivalent stiffness derivation process, it was found that the amplified mass ratio could only take limited values to guarantee the stability condition of the coupled system and make positive equivalent stiffness. As a result, the best working range of the amplified mass ratio was established. The results analysis of the influence of system parameters on the primary system response shown that the amplified mass ratio has greatly vibration reduction effect, which push the peak resonant vibration of the primary system smaller than its static response without control. To better understand the control performance of the proposed DVA, three other dynamic vibration absorbers with or without negative stiffness are considered. The control performance comparison show that under harmonic excitation or even random excitation of the primary system, the proposed DVA model has a significant vibration reduction effect compared to the considered its DVA counterparts. The interest of these relevant results can be used in the field of vibration control engineering.

Band-stop-filter-based repetitive control of fast tool servos for diamond turning of micro-structured functional surfaces

Fast tool servo (FTS) is a promising technique for the high-speed diamond turning of micro-structured functional surfaces due to its fast response capacity. However, the inherent actuator non-linearity, the lightly damped vibration mode and the external disturbance of the FTS would degrade its tracking performance severely and thus restrict its machining ability. To overcome this limitation, a band-stop-filter-based repetitive control (BSF-RC) scheme is proposed in this article to improve the operating performance of the FTS. Firstly, the spectra of the FTS tool paths (FTS-TPs) for machining typical micro-structured functional surfaces with the cylindrical coordinate machining method are analyzed in detail. The results reveal that the frequency components concentrate on the tiny neighborhoods of the rotational frequency of the spindle and its harmonic frequencies. Afterwards, the comb-like band-stop filter used by the BSF-RC is elaborately designed via the cascade of three comb-like notch filters to generate a comb-like band-stop loop shape for covering main spectral components of the FTS-TPs. The analytical expression of the comb-like band-stop filter amplitude as well as the stability condition of the closed-loop control system are derived to provide effective criteria for the parameter selection of the BSF-RC. The experimental validation of the developed BSF-RC scheme is conducted on a five-axis ultra-precision lathe equipped with a self-developed FTS via the diamond turning of three typical micro-structured functional surfaces. The experimental results show that good surface qualities with the maximal form error of a few hundred nanometers and the surface roughness of a several nanometers are achieved for all the obtained micro-structured surfaces by using the BSF-RC scheme under a high spindle speed.

Reachable set estimation and stochastic sampled-data exponential synchronization of Markovian jump neural networks with time-varying delays

In this paper, the stochastic sampled-data exponential synchronization problem for Markovian jump neural networks (MJNNs) with time-varying delays and the reachable set estimation (RSE) problem for MJNNs subjected to external disturbances are investigated. Firstly, assuming that two sampled-data periods satisfy Bernoulli distribution, and introducing two stochastic variables to represent the unknown input delay and the sampled-data period respectively, the mode-dependent two-sided loop-based Lyapunov functional (TSLBLF) is constructed, and the conditions for the mean square exponential stability of the error system are derived. Furthermore, a mode-dependent stochastic sampled-data controller is designed. Secondly, by analyzing the unit-energy bounded disturbance of MJNNs, a sufficient condition is proved that all states of MJNNs are confined to an ellipsoid under zero initial condition. In order to make the target ellipsoid contain the reachable set of the system, a stochastic sampled-data controller with RSE is designed. Eventually, two numerical examples and an analog resistor–capacitor network circuit are provided to show that the textual approach can obtain a larger sampled-data period than the existing approach.

A spatiotemporal fuzzy control approach for complex distributed parameter system with application to thermal processes

Many industrial processes belong to distributed parameter systems (DPSs) with unknown spatiotemporal dynamics and complex boundary conditions. However, attentions to both temporal and spatial dynamics in the control law are less paid in the existing control methods, which often makes them challenging to achieve accurate control. Here, a spatiotemporal fuzzy control approach with consideration of both temporal and spatial dynamics is proposed for unknown and complex DPSs. First, a spatiotemporal fuzzy model is developed to reconstruct dynamics of nonlinear DPS. Since it constructs a two-layer fuzzy rules (i.e., temporal fuzzy rule and spatial fuzzy rule) to consider both the nonlinear dynamics on time and the nonlinear energy transfer on space, it can effectively achieve the model of DPSs. On this basis, a spatiotemporal fuzzy controller is developed according to the developed two-layer fuzzy rules, which means that this derived control law considers both temporal and spatial dynamics. Then, a two-step solution method is designed for the spatiotemporal control law. Stability conditions of the control system are rigorously deduced by transforming the solving process to a series of linear matrix inequalities (LMIs). The stability and effect of the proposed spatiotemporal control strategy are demonstrated by mathematical proof and case studies. Comparison results demonstrate that the proposed method has better control performance (relative errors and mean absolute percentage errors are bounded in 4%).

Robust control strategy for networked vehicle system with fake data injection attack

This article investigates the safety control problem of intelligent networked vehicle and utilizes the sliding-mode control to solve the system with time delay, system disturbance, and parameter uncertainty. First, the modeling of automobile system is introduced. What’s more, time delay, parameter uncertainty, disturbance, and attack are added to the original model. Then, designed an integral sliding-mode surface and given the sufficient conditions for system stability using linear matrix inequalities. Furthermore, the controller is designed according to the slide approach rate, and it is proved that the designed controller can make the system state reach the sliding surface in finite time. Finally, the simulation is used to verify the effectiveness of the arithmetic.

Asymptotic Properties of a Class of Systems with Linear Delay

Sufficient conditions for the asymptotic stability of linear systems of differential equations with linear delay are obtained. On the basis of these conditions, some systems of linear differential equations are studied, and one of them is stabilized on an infinite time interval.

Interconnected Systems Modelling in Food Industry: General Solution Scheme and Stability Conditions for Linear Time-Invariant Systems

The problem of simulating complex systems, such as production lines, industrial plants, food processing, etc., today represents an opportunity that brings with it the great advantage of limiting design costs. However, nowadays the designer, after defining and implementing the mathematical models of the studied process, may need to rebuild the whole simulation framework because he needs to modify the model of even just one subsystem. It is for this reason that in this paper, a new framework for the use of Individual Subsystem Models (ISM) for the modelling and simulation of interconnected systems has been studied and implemented. Furthermore, the study of the state of the art has revealed the lack of efficient and sufficiently general numerical algorithms, but, at the same time, it is simple to use to solve the algebraic-differential equations deriving from the ISM simulation. The proposed new approach follows the paradigm of co-simulation methods, including graph theory methods, to solve the general ISM simply and efficiently. In this approach, each subsystem is required to have its own representation independently of the other subsystems. In this way, it is always possible to replace any subsystem whenever an updated representation becomes available, making maintenance and evolution of the entire ISM flexible. Our framework calls each subsystem separately in an optimal (suboptimal) order based on the structure of the graph. Each calculated output is transferred to the input of the next subsystem in the chosen. The general procedure has been validated in the context of Linear and Time-Invariant ISMs: in these hypotheses, the stability conditions have been calculated and numerical tests have been performed which show the effectiveness of the proposed approach.

A chaos control strategy for the fractional 3D Lotka–Volterra like attractor

In this paper, we have considered a three-dimensional Lotka–Volterra attractor in the frame of the Caputo fractional derivative to examine its dynamics. The theoretical concepts like existence and uniqueness and boundedness of the solution are analyzed. To regulate the chaos in this fractional-order system, we have developed a sliding mode controller and conditions for global stability of the controlled system with and without uncertainties and outside disruptions are derived. The ability of the designed controller is examined in terms of both commensurate and non-commensurate fractional order derivatives for all the aspects. The Lyapunov exponent is the novelty of this paper which is used to illustrate the behavior of the chaos and demonstrate the dissipativeness of the considered chaotic system. We have examined the effect of fractional order derivatives in this system. With the help of numerical simulations, the theoretical claims regarding the impact of the controller on the system are established.

Fixed-Time and Predefined-Time Stability of Impulsive Systems

Dear Editor, This letter establishes several criteria for fixed-time stability and predefined-time stability of impulsive systems. First, sufficient conditions for fixed-time stability of impulsive systems are presented to treat the destabilizing impulses and hybrid impulses involving multiple jump maps by fixed-time control without linear feedback regulation. It determines the robustness of nonlinear systems against impulsive disturbance which has destabilizing and hybrid effect to dynamics. Then, the predefined-time stability of impulsive systems is ensured under the requirement that the hybrid impulses have stabilizing accumulative effect. Finally, the validity of theoretical results is verified by the allocation of mobile agents on a segment via fixed-time control with impulsive regulation.