Two-time Scale Systems Research Articles

Moderate Deviations for Two-Time Scale Systems with Mixed Fractional Brownian Motion

Moderate Deviations for Two-Time Scale Systems with Mixed Fractional Brownian Motion

Mixed-mode oscillations in a three-timescale coupled Morris-Lecar system.

Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.

Bias and Refinement of Multiscale Mean Field Models

Mean field approximation is a powerful technique which has been used in many settings to study large-scale stochastic systems. In the case of two-timescale systems, the approximation is obtained by a combination of scaling arguments and the use of the averaging principle. This paper analyzes the approximation error of this 'average' mean field model for a two-timescale model (X, Y), where the slow component X describes a population of interacting particles which is fully coupled with a rapidly changing environment Y. The model is parametrized by a scaling factor N, e.g. the population size, which as N gets large decreases the jump size of the slow component in contrast to the unchanged dynamics of the fast component. We show that under relatively mild conditions, the 'average' mean field approximation has a bias of order O(1/N) compared to E[X]. This holds true under any continuous performance metric in the transient regime, as well as for the steady-state if the model is exponentially stable. To go one step further, we derive a bias correction term for the steady-state, from which we define a new approximation called the refined 'average' mean field approximation whose bias is of order O(1/N2). This refined 'average' mean field approximation allows computing an accurate approximation even for small scaling factors, i.e., N ~10 -50. We illustrate the developed framework and accuracy results through an application to a random access CSMA model.

Tube-Based Robust MPC for Two-Timescale Systems Using Reduced-Order Models

A tube-based robust Model Predictive Control (MPC) formulation is presented for systems with slow and fast timescale dynamics. The controller uses a reduced-order model that approximates the slow timescale dynamics and is implemented with a relatively large time step size. By analyzing the error between the full- and reduced-order models, the MPC optimization problem is proven to be recursively feasible and to produce closed-loop trajectories that satisfy state and input constraints. By relating bounds on all model errors to bounds on the change of the input in time, input change bounds are included as decision variables in the MPC problem to achieve a time-varying balance between fast transients with large model error and steady-state operation with zero model error. Zonotopes are used to make the approach practical and a numerical example demonstrates the benefits of optimizing time-varying input change bounds as part of the MPC formulation.

Distributed Event-Triggered Synchronization of Interconnected Linear Two-Time-Scale Systems With Switching Topology.

This article investigates the synchronization problem of interconnected linear two-time-scale systems (TTSSs) with switching topology. By utilizing the Chang transformation, a distributed synchronization protocol is proposed with event-triggered communication. Static and dynamic event-triggered mechanisms are proposed successively, which both contain two separated event-triggering conditions corresponding to the slow and the fast subsystems. The existence of a strictly positive time period between any two successive transmissions is ensured regardless of the initial states. The main difficulty of this study lies in that the state jump and parametric uncertainty appear because of the system transformation. To overcome the difficulty, the system is first modeled as an uncertain hybrid system. Then, the control gain is properly designed by solving Riccati-like equations dependent on the rough bounds of the eigenvalues of communication graph Laplacians, and a piecewise quadratic Lyapunov function is proposed with which the jump caused by the switching topology is subtly evaluated. Sufficient conditions are thus established to achieve the event-triggered synchronization. Furthermore, the results are also extended to solve the synchronization problem of the interconnected impulsive linear TTSSs. Finally, three numerical examples are provided to demonstrate the effectiveness of the proposed theoretical results.

Asynchronous event-triggered H... control problem for two-time-scale systems based on double-rate sampled-data

This work focuses on the asynchronous event-triggered H∞ control problem for two-time-scale systems (TTSSs) with double-rate sampling. Firstly, by fully considering the positive exponential items and the quadratic items including the state information, an improved triggering condition is constructed. Based on the improved condition, an asynchronous event-triggered mechanism (ETM) is derived, which realizes the independent update on the triggering signals of the fast and slow subsystems. Secondly, an asynchronous event-triggered control scheme is established by co-designing the H∞ controller and the asynchronous ETM. The designed asynchronous control scheme mainly has two features: 1) reducing the frequency of signal transmission at the whole stage of system operation; 2) being applied to the TTSSs with double-rate sampled-data. Finally, examples are presented to demonstrate the validity and merits of the obtained methods.

Distributed Nash Equilibrium Seeking in a Multi-Coalition Noncooperative Game Under Incomplete Decision Information

This brief studies a distributed Nash equilibrium (NE) seeking strategy for a multi-coalition noncooperative game with local decision sets. In the game model, each coalition consists of some agents and its cost funciton is determined by the sum of the local cost functions subject to the agents in the coalition, where each coalition serves as a virtual player and the actual decision-makers are the agents in the coalition. The cost of each coalition is to minimize its own cost function under the case that the agents in each coalition only know their local information and can not directly access the opponent’s decision information in other coalitions. To this end, a distributed seeking strategy that can be viewed a two-time scale system is developed to search the NE of the formulated multi-coalition game. The fast time-scale system is used to estimate each coalition’s pseudo-gradient by a connected interference graph that illustrates the interactions among the agents in each coalition. The slow time-scale system based on a projected psedudo-gradient dynamics is implemented to seek the NE, where the coalitions estimate other coalitions’ decisions by interacting only with their neighbors via a weight-balanced digraph. For this distributed NE seeking strategy, asymptotic convergence result is given by Lyapunov stability analysis. Finally, the theoretical results is demonstrated via a numerical example.

Two-time scaled identification for multi-energy systems

Multi-energy systems have been leaping forward for its various benefits, e.g., energy conservation and emission reduction. Multi-energy systems exhibit multi-time scaled characteristic and broad bandwidth, thereby causing difficulties in dynamic modeling. In this work, two-time scaled system identification is studied. A method is developed to solve the problem, which uses proper test signal design, signal pre-filtering and subtraction. The high and low frequency parts of the two-time scale system are identified separately and then combined to form the total system in a parallel structure. The consistency of the method is proved and case studies are used to verify the effectiveness of the method.

An efficient numerical method for a nonlinear system of singularly perturbed differential equations arising in a two-time scale system

An efficient numerical method for a nonlinear system of singularly perturbed differential equations arising in a two-time scale system

Loss of Determinacy at Small Scales, with Application to Multiple Timescale and Nonsmooth Dynamics

A singularity is described that creates a forward time loss of determinacy in a two-timescale system, in the limit where the timescale separation is large. We describe how the situation can arise in a dynamical system of two fast variables and three slow variables or parameters, with weakly coupling between the fast variables. A wide set of initial conditions enters the [Formula: see text]-neighborhood of the singularity, and explodes back out of it to fill a large region of phase space, all in finite time. The scenario has particular significance in the application to piecewise-smooth systems, where it arises in the blow up of dynamics at a discontinuity and is followed by abrupt recollapse of solutions to “hide” the loss of determinacy, and yet leave behind a remnant of it in the global dynamics. This constitutes a generalization of a “micro-slip” phenomenon found recently in spring-coupled blocks, whereby coupled oscillators undergo unpredictable stick-slip-stick sequences instigated by a higher codimension form of the singularity. The indeterminacy is localized to brief slips events, but remains evident in the indeterminate sequencing of near-simultaneous slips of multiple blocks.

Reduced‐order event‐triggered controller for a singularly perturbed system: An active suspension case

For two-time scale systems, singular perturbation theory is often used for designing a controller based only on an approximate model of its slow dynamics, assuming the fast model to be stable. In this context, the authors investigate and implement a stabilising event-triggered feedback law for a networked singularly perturbed system, based only on an approximate model of its slow dynamics. Triggering rule guarantees the stability and the existence of a positive lower bound between two consecutive transmissions. The proposed approach has been validated for a laboratory-scale hardware setup of an active suspension system of a quarter-car model. The presence of fast and slow modes in a vehicle suspension system is utilised to model it as a singularly perturbed system. Experimental results indicate that in spite of the simplified structure of the controller and event-triggered feedback, its performance is comparable to that of the full-state feedback design with continuous feedback with the significant reduction in control execution events.

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

Predictive Cascaded Speed and Current Control for PMSM Drives With Multi-Timescale Optimization

This paper proposes a predictive speed and current control with multi-timescale optimization in a cascade architecture for a permanent-magnet synchronous motor. Considering the difference of timescale characteristics for speed loop and current loop, different sampling times are assigned to the respective subsystem. In the prediction step of the conventional two-timescale system, the coupling between slow and fast sampling models is ignored and the output of the slow-sampling model at asynchronous sampling period is missing, which both weaken the prediction performance of the system. In this paper, the predictions of both slow and fast models for all the prediction instants are analyzed in detail. Besides, a linear estimation method based on virtual instants is proposed to improve the performance of the slow-sampling model for fast prediction instants. The data stream of the proposed method is designed based on the cascaded structure. The strategies are implemented on a field-programmable gate arrays taking advantages of parallel and pipeline processing techniques. Experimental results show that the proposed strategies have a better dynamic performance compared to the conventional method.

Robust Course Keeping Control of a Fully Submerged Hydrofoil Vessel with Actuator Dynamics: A Singular Perturbation Approach

This paper presents a two-time scale control structure for the course keeping of an advanced marine surface vehicle, namely, the fully submerged hydrofoil vessel. The mathematical model of course keeping control for the fully submerged hydrofoil vessel is firstly analyzed. The dynamics of the hydrofoil servo system is considered during control design. A two-time scale model is established so that the controllers of the fast and slow subsystems can be designed separately. A robust integral of the sign of the error (RISE) feedback control is proposed for the slow varying system and a disturbance observer based state feedback control is established for the fast varying system, which guarantees the disturbance rejection performance for the two-time scale systems. Asymptotic stability is achieved for the overall closed-loop system based on Lyapunov stability theory. Simulation results show the effectiveness and robustness of the proposed methodology.

Reconfigurable Control of Two-Time Scale Systems in Presence of Additive Faults

This work presents an adaptive approach for fault tolerant control of singularly perturbed systems, where both actuator and sensor faults are examined in presence of external disturbances. For sensor faults, an adaptive controller is designed based on an output-feedback control scheme. The feedback controller gain is determined in order to stabilize the closed-loop system in the fault free case and vanishing disturbance, while the additive gain is updated using an adaptive law to compensate for the sensor faults and the external disturbances. To correct the actuator faults, a state-feedback control method based on adaptive mechanism is considered. The both proposed controllers depend on the singular perturbation parameter ε leading to ill-conditioned problems. A well-posed problem is obtained by simplifying the Lyapunov equations and subsequently the controllers using the singular perturbation method and the reduced subsystems yielding to an ε-independent controller. The control scheme, designed based on the Lyapunov stability theory, guarantees asymptotic stability in presence of additive faults and external disturbances provided the singular perturbation parameter is sufficiently small. Finally, a numerical example is presented to demonstrate the effectiveness of the obtained results.DOI: http://dx.doi.org/10.5755/j01.itc.44.4.8532

A density-dependent model describing age-structured population dynamics using hawk–dove tactics

In this paper we deal with a nonlinear two-timescale discrete population model that couples age-structured demography with individual competition for resources. Individuals are divided into juvenile and adult classes, and demography is described by means of a density-dependent Leslie matrix. Adults compete to access resources; every time two adults meet, they choose either being aggressive (hawk) or non-aggressive (dove) to get the best pay-off. Individual encounters occur much more frequently than demographic events, what yields that the model takes the form of a two-timescale system. Approximate aggregation methods allow us to reduce the system while preserving at the same time crucial asymptotic information for the whole population. In this way, we are able to describe the total population size as function of individual aggressiveness level and environmental richness. Model analysis shows a general trend with species that look for richer environment having smaller proportions of hawk individuals with larger costs.

Single-Hopf Bursting in Periodic Perturbed Belousov—Zhabotinsky Reaction with Two Time Scales

By introducing weak periodic perturbation to the Oregonator, we investigate a mathematical model with two time scales. The novel dynamical phenomena called the single-Hopf bursting with unusually quiescent state are observed. The related bifurcation mechanism is presented by means of the transition portrait and slow-fast analysis, which has rarely been reported in the previous works associated with the Oregonator model. Furthermore, the influence of forcing amplitude on bursting behaviors is studied. Further investigation finds that excitation amplitude may play an important role in a two-timescale system with the slow procedure dominated by periodic perturbation.

Kinetic State Tracking for a Class of Singularly Perturbed Systems

The trajectory-following control problem for a general class of nonlinear multi-input, multi-output two-time scale system is revisited. While most earlier works used singular perturbation theory and assumed that an isolated real root exists for the nonlinear set of algebraic equations that constitute the slow subsystem, here two-time scale systems are analyzed in the context of integral manifolds. This accounts for the existence of multiple roots. It is shown that the slow subsystem has a center manifold, and for small values of the slow state, an approximate solution of the nonlinear set of transcendental equations can be computed. Geometric singular perturbation theory is used as the model reduction technique, and dynamic inversion is used to formulate stabilizing control laws for slow state tracking. The control laws are independent of the scalar perturbation parameter, and an upper bound for it is determined such that boundedness of all the closed-loop signals is guaranteed. The methodology is validated through numerical simulation of a generic twodegree of freedom kinetic model and a nonlinear, coupled, six degree-of-freedom model of the F/A-18A Hornet. Results presented in the paper show that this methodology permits close tracking of the reference trajectory while maintaining all the control signals well within their specied bounds.

Stabilisation of two-time scale systems with a finite feedback data rate

This study investigates a stabilisation problem of two-time scale systems with a finite feedback data rate. The uniform estimate of the state growth rate over each sampling interval is first provided, which is shown to be independent of the small parameter. Then, based on the given uniform estimate and the decoupled form of the two-time scale systems, the requirement, which is independent of the small parameter, on the necessary data rate for the channel is derived through the design of a quantiser. Based on this requirement, the authors obtain the detecting systems, which are two-time scale hybrid systems with impulsive. Then, under the assumption of strong controllability of the original two-time scale systems, it is shown that there exists a bound of the small parameter such that the original two-time scale systems are stabilised by a quantised state via a finite feedback data rate channel for a small parameter within the bound. Finally, a simulation example illustrates the result.

Robust H8 Fuzzy Control Design for Nonlinear Two-Time Scale System with Markovian Jumps based on LMI Approach

Robust H8 Fuzzy Control Design for Nonlinear Two-Time Scale System with Markovian Jumps based on LMI Approach