Stability Analysis Of Discrete-time Research Articles

Exact discrete-time stability analysis of multi-DOF haptic rendering: Impact of multi-rate, time-delay, and mechanical parameters

Previous studies on the stability analysis of haptic devices have predominantly focused on single-DOF devices, thereby limiting attention to multi-DOF devices, particularly those employing multi-rate sampling schemes. In this paper, we introduce a formulation for the coupled dynamics between the haptic device and the virtual environment for a multi-DOF haptic device controlled using a dual-rate sampling scheme. Subsequently, we analyze its stability through the application of a dynamic decoupling strategy within an exact discrete-time state-space framework while the device is engaged in rendering a virtual wall along one of its operational space coordinates. Furthermore, we explore how the combined influence of the dual-rate sampling approach, time delay, and the mechanical design affects the stability boundaries of the multi-DOF haptic device at a fixed workspace location as well as within the entire usable workspace. Additionally, we utilize a model-order reduction (MOR) framework to simplify the determination of the device’s stability limits, irrespective of the specific combinations of time delay and sampling rates employed.

Sector Bound-Dependent Matrix-Separation-Based Inequality and its Application to Stability Analysis of Discrete-Time Delayed Genetic Regulatory Networks

Sector Bound-Dependent Matrix-Separation-Based Inequality and its Application to Stability Analysis of Discrete-Time Delayed Genetic Regulatory Networks

Stability analysis of discrete-time tempered fractional-order neural networks with time delays

Stability analysis of discrete-time tempered fractional-order neural networks with time delays

Robust Stability Analysis of Discrete-Time Switched Linear Systems with Dwell Time and Mode-Dependent Dwell Time Switching

Robust Stability Analysis of Discrete-Time Switched Linear Systems with Dwell Time and Mode-Dependent Dwell Time Switching

Dynamic stability analysis of DC microgrid and construction of stability region of control parameters based on Pioncáre map

In the DC microgrid, the difficulty to maintain the stable operation of DC-DC power electronic devices will increase due to the random fluctuations or step changes of power. However, nonlinear dynamics' instability behaviour is very challenging to obtain through an average linearized stability analysis. Therefore, based on the Pioncáre map, this paper derived a global discrete iterative model and its explicit Jacobian matrix in the polynomial form using second-order Taylor series expansion, to detect nonlinear instability behaviour for DC microgrids. A global polynomial Pioncáre map for the microgrid is established, which contains the voltage-current controlled bidirectional DC-DC converter. Then, the derived discrete-time stability analysis method is compared with the average linearized stability analysis method based on the state space average model. Further discussion about the interactive effect on the parameters is conducted through the Jacobian matrix’ eigenvalues, bifurcation diagrams, Pioncáre sections and phase trajectories, leading to the 2-D and 3-D stable regions of operating parameters. Finally, dynamic simulation is carried out on MATLAB/Simulink. Results show that the method proposed can accurately capture the Neimark–Sacker bifurcation critical point in the DC microgrid, construct the stability region of the control parameters, and could help select the controller parameters of DC microgrids.

Incremental control system design and flight tests of a micro-coaxial rotor UAV

This paper applies the incremental nonlinear dynamic inversion (INDI) method to the control system design of coaxial rotor UAVs. The aerodynamic uncertainties and anti-disturbances problems are solved in the control system design. The designed controller gives the UAV excellent flight performance and control robustness. The incremental gain method (IGM) is adopted for the delay problem of the state derivatives. This method has the advantages of less calculation load and simple parameter adjustment, which provides excellent convenience for applying INDI controllers to coaxial rotor UAVs. The principle of IGM is further investigated by the discrete-time stability analysis methods, and the parameter selection strategy of the IGM is provided. The advantages of the controller design method are verified by comparative flight tests. The experimental results show that the average trajectory tracking error of INDI is only 58.3% of that of NDI under the same wind speed. When the angular acceleration delay is less than 0.06 s, IGM can easily keep the system stable. In addition, INDI has better robustness under obvious model errors and strong input disturbances.

Observability, controllability and stability analysis of discrete time engineering dynamic systems by means of Lagrangian, Hamiltonian and dissipative functions in discrete forms

Purpose The purpose of this paper is to analyze the dynamical state of a discrete time engineering/physical dynamic system. The analysis is performed based on observability, controllability and stability first using difference equations of generalized motion obtained through discrete time equations of dissipative generalized motion derived from discrete Lagrange-dissipative model [{L,D}-model] for short of a discrete time observed dynamic system. As a next step, the same system has also been analyzed related to observability, controllability and stability concepts but this time using discrete dissipative canonical equations derived from a discrete Hamiltonian system together with discrete generalized velocity proportional Rayleigh dissipation function. The methods have been applied to a coupled (electromechanical) example in different formulation types. Design/methodology/approach An observability, controllability and stability analysis of a discrete time observed dynamic system using discrete equations of generalized motion obtained through discrete {L,D}-model and discrete dissipative canonical equations obtained through discrete Hamiltonian together with discrete generalized velocity proportional Rayleigh dissipation function. Findings The related analysis can be carried out easily depending on the values of classical elements. Originality/value Discrete equations of generalized motion and discrete dissipative canonical equations obtained by discrete Lagrangian and discrete Hamiltonian, respectively, together with velocity proportional discrete dissipative function are used to analyze a discrete time observed engineering system by means of observability, controllability and stability using state variable theory and in the method proposed, the physical quantities do not need to be converted one to another.

Stability analysis of discrete-time coupled systems with delays

Addressed in this paper is the stability issue of discrete-time delayed coupled systems consisting of a difference equation and an algebraic equation in coupled states. First, a necessary and sufficient condition of exponential stability is proposed for coupled systems with constant delays. Second, periodicity and stability are investigated for periodic coupled systems. Third, based on the Lyapunov inequality, a necessary and sufficient condition is presented for exponential stability of general time-varying coupled systems. Finally, as an application of these results, stability of discrete-time singular systems is discussed.

Stability Analysis of Discrete-Time Switched Systems With Unstable Modes: An Improved Ratio-Based Tradeoff Approach

This brief investigates the stability of switched systems with unstable modes in discrete-time context. To enlarge the room for switching scheme design, a new multiple discontinuous Lyapunov functions (MDLFs) method is developed, and a novel mode-dependent average dwell time (MDADT) tradeoff strategy is explored. Subsequently, based on the new MDLFs method, stability criteria are achieved for the systems under arbitrary switching signals with the MDADT-based tradeoff switching scheme. Finally, the proposed results are applied to a Pulse-Width-Modulation driven boost converter to demonstrate the effectiveness.

A Limit Φ-Dependent ADT Technique for Stability Analysis of Discrete-Time Switched Systems With Unstable Modes

The stability problem of discrete-time switched systems with unstable modes is concerned under two novel switching strategies, i.e., limit inferior and superior Φ-dependent average dwell time switchings. Some improved stability criteria of discrete-time switched systems with unstable modes are obtained by combining a multiple Lyapunov function method with two strategies proposed, where limit inferior and superior switchings are applied to deal with stable and unstable modes, respectively. Especially, we also give stability conditions of discrete-time switched systems comprising stable modes. It is shown that these conditions with tighter bounds of average dwell time (ADT) and mode-dependent ADT cover some existing ones. Finally, two illustrated numerical examples are presented to show the effectiveness of the obtained theoretical results.

Stability Analysis of Discrete-Time Switched Linear Time-Varying Systems Based on Function-Dependent LMIs

This paper considers the problem of uniformly asymptotic stability (UAS) for discrete-time switched linear time-varying (DSLTV) systems. Starting with discrete-time linear time-varying (DLTV) systems, some stability conditions are given by using function-dependent linear matrix inequalities (LMIs). Comparing with the existing results, the conditions obtained allow the norm and rate of variation of system matrix are unbounded. Furthermore, the obtained results are extended to study the UAS of DSLTV systems, the stability condition is given by combining the methods of function-dependent LMIs with average dwell time. Finally, some further discussions about the relaxation of conditions obtained are also proposed. Three numerical examples are given to illustrate the theoretical results.

Asymptotic Stability Analysis of Discrete-Time Switched Cascade Nonlinear Systems With Delays

This paper addresses the stability issue of a class of delayed switched cascade nonlinear systems consisting of separate subsystems and coupling terms between them. Some global and local asymptotic stability sufficient conditions are proposed, drawing stability conclusion of the overall cascade system from those of separate systems. These results essentially rely on the following observation: For a general delayed switched nonlinear system being asymptotically stable, the trajectories of the perturbed system asymptotically approach zero if so does the perturbation. This observation is one of the main results in this paper.

Stability Analysis of Discrete-Time Switched T-S Fuzzy Systems With All Subsystems Unstable

In this paper, the stability problem of discrete-time switched nonlinear systems with all subsystems unstable is investigated. The nonlinear subsystems are represented by the Takagi-Sugeno (T-S) fuzzy models. By constructing a novel piecewise multiple Lyapunov function approach, an exponential stability condition of switched T-S fuzzy systems is first established with the bounded maximum average dwell time. A numerical example is finally provided to illustrate the effectiveness of the obtained theoretical results.

Global stability analysis for discrete-time coupled systems with both time delay and multiple dispersal and its application

This paper is concerned with the global stability analysis of discrete-time coupled systems with both time delay and multiple dispersal (DCSTMs). We provide a systematic method to constructing a global Lyapunov function for DCSTMs. By employing the graph-theoretic method on multiple digraphs and Lyapunov method, a Lyapunov-type theorem and a coefficients-type criterion for a DCSTM are derived to guarantee global stability for the DCSTM. The main results are applied to a discrete-time predator-prey model with time delay where both preys and predators disperse among l patches, which is discretized by a nonstandard finite difference scheme. A numerical example is provided to show the effectiveness of the proposed results.

Stability Analysis for Discrete-Time Switched Nonlinear System Under MDADT Switching

The problem of stability analysis for discrete-time switched nonlinear system is investigated with mode-dependent average dwell time (MDADT) method in this paper. A slow switching strategy is adopted in the discrete-time nonlinear stable subsystems and unstable subsystems are handled by a fast switching strategy. Takagi-Sugeno (T-S) fuzzy model is utilized to approximate the switched nonlinear system. By constructing a multiple discontinuous Lyapunov function approach, the stability condition of switched T-S fuzzy system is built to get tighter bound on MDADT, which shows that our proposed method outperforms the classical one. Finally, through a numerical example, the effectiveness of the presented control approach is illustrated by comparison with result from classical one.

A Time Delay Compensation Method Based on Area Equivalence For Active Damping of an <italic>LCL</italic> -Type Converter

Control of the LCL -type three-phase grid-connected converter is difficult due to high resonance peak of the LCL filter. Active damping is the state-of-the-art solution to this problem, but the damping performance will be affected by the inherent time delay of digital control, especially for high-power low switching frequency applications. Based on a discrete-time stability analysis of an LCL -type converter with capacitor-current-feedback active damping, a simple and effective time delay compensation method, which is based on area equalization concept, is proposed. The method can reduce the negative impact of the computation delay significantly. It has the potential to serve as a general solution to time delay compensation of a digitally controlled PWM converter. The validity of the proposed method is proved by experimental results.

Stability Analysis for Continuous T-S Fuzzy Models Having Uncertainties: A Systematic Approach

Stability analysis of continuous-time unforced T-S fuzzy systems is considered. Based on the pairwise commutative feature that usually occurs among the state matrices in switching systems, here we reformularize an existing discrete-time stability analysis approach to its continuous-time domain version. Using a common quadratic Lyapunov function, we first present a systematic approach for the asymptotic stability of such systems in case where the state matrices of sub-systems follow pairwise commutative feature. We then show that the method is not only limited to systems following such feature but rather can be applied to wider category. Finally, we investigate the maximum permissible uncertainty bound for holding the stability when the uncertainties in the system belong to convex sets.

Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application

In this paper, we investigate the global stability of discrete-time coupled systems with multi-diffusion (DCSMDs). By utilizing a multi-digraph theory, we construct a global Lyapunov function for DCSMDs. Consequently, some sufficient conditions are presented to ensure the stability of a general DCSMDs. Then the proposed theory is successfully applied to analyze the global stability for a discrete-time predator-prey model which is discretized by a nonstandard finite difference scheme. Finally, an example with numerical simulation is given to demonstrate the effectiveness of the obtained results.

Local stability analysis of discrete-time, continuous-state, complex-valued recurrent neural networks with inner state feedback.

Recurrent neural networks (RNNs) are well known for their capability to minimize suitable cost functions without the need for a training phase. This is possible because they can be Lyapunov stable. Although the global stability analysis has attracted a lot of interest, local stability is desirable for specific applications. In this brief, we investigate the local asymptotical stability of two classes of discrete-time, continuous-state, complex-valued RNNs with parallel update and inner state feedback. We show that many already known results are special cases of the results obtained here. We also generalize some known results from the real-valued case to the complex-valued one. Finally, we investigate the stability in the presence of time-variant activation functions. Complex-valued activation functions in this brief are separable with respect to the real and imaginary parts.

Neural Network Inverse Model Control Strategy: Discrete-Time Stability Analysis for Relative Order Two Systems

This paper discusses the discrete-time stability analysis of a neural network inverse model control strategy for a relative order two nonlinear system. The analysis is done by representing the closed loop system in state space format and then analyzing the time derivative of the state trajectory using Lyapunov’s direct method. The analysis shows that the tracking output error of the states is confined to a ball in the neighborhood of the equilibrium point where the size of the ball is partly dependent on the accuracy of the neural network model acting as the controller. Simulation studies on the two-tank-in-series system were done to complement the stability analysis and to demonstrate some salient results of the study.