Slow Subsystem Research Articles

Study on Urban Proximity Prediction Based on Doppler Radar Gust Front Characteristics and Urban Microclimate Characteristics

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Controllability and Observability Criteria for Systems Described by Fractional Differential Equations

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Robust iterative learning control for random switched singular systems in time domain

A robust iterative learning control algorithm is proposed for continuous-time switched singular systems disturbed by random measurement noise. Firstly, the switched singular systems is transformed into a differential-algebraic equation composed of slow subsystem and fast subsystem by nonsingular transformation. Then, the control input is continuously modified by the state measurement errors and the learning gain with attenuation characteristics. The convergence of each subsystem is strictly proved in theory, and the sufficient conditions for the convergence of this algorithm are given. Theoretical analysis shows that the algorithm can effectively suppress the measurement noise and make the system state completely track the desired trajectory in a finite time interval. Finally, compared with the fixed gain iterative learning control method, the simulation results show that the tracking accuracy of the proposed algorithm is significantly improved.

Adaptive neural network vibration suppression control of flexible joints space manipulator based on H∞ theory

Considering the control problems caused by uncertainties such as inaccurate modeling, external disturbance and joint flexibility, a neural network control method based on H∞ is proposed. By establishing the dynamic model of the free-floating space robot with flexible joints, according to its dynamic characteristics, it is split into a slow subsystem model representing the rigid characteristics and a fast subsystem model representing the flexible characteristics. Based on the H∞ robust control theory, a robust controller based on neural network is designed to realize the decoupling control of the rigid dynamic model, The designed weight adaptive learning rate can ensure the online and real-time adjustment of parameters. Based on Lyapunov theory, it is proved that the designed controller can ensure that the L2 gain of the system is less than the given index. A feedback controller based on velocity differential is designed to compensate the angle error caused by joint flexibility. The experimental simulation results verify that the proposed control method is effective and has good engineering application value.

Research on Constraint Following Control of Flexible Joint Manipulators Based on Singular Perturbation

For the control of 2-link flexible joint manipulators, a control method based on the singular perturbation theory and the Udwadia-Kalaba (U-K) method was proposed. The control design was implemented by 2 steps. First, the system order was reduced based on the singular perturbation method and the system was divided into fast and slow sub-systems, to simplify the solution process and overcome the system flexibility. Second, the state feedback constraint following control law for the fast and slow sub-systems was designed with the U-K method, to make the constraint following errors of the fast and slow sub-systems converge to zero, even if the system can't initially satisfy the constraints. The proposed method can deal with holonomic and nonholonomic servo constraints at the same time without the auxiliary variables of the Lagrange multiplier and the pseudo generalized velocity. The method was applied to 2-link flexible joint manipulator systems to solve the flexible oscillation and constraint following problems. Through simulations on MATLAB, and was compared with the traditional PID control, to verify the effectiveness and superiority.

SPT-Based Composite Hierarchical Antidisturbance Control Applied to a Quadrotor UAV

Although the flight control of quadrotor unmanned aerial vehicles have been widely studied, it is still very challenging to suppress multiple disturbances and efficiently utilize the inherent multi time-scale characteristics. As a well-known antidisturbance strategy, composite hierarchical antidisturbance control (CHADC) can compensate and attenuate disturbances simultaneously, thus enhancing the control performance of complex systems significantly. In the meantime, the singular perturbation theory (SPT) is employed to analyze the multi time-scale properties, and the original system is decomposed into the slow and fast subsystems. In this article, we present an antidisturbance control scheme based on the SPT and CHADC to enhance the capability of antidisturbance. A novel composite hierarchical antidisturbance (CHAD) predictor–corrector is proposed for the slow subsystem to integratively handle the disturbances from guidance and control perspective, and a CHAD <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{H}_{\infty }}$</tex-math></inline-formula> controller is employed for the fast subsystem subject to external disturbances and inertial uncertainties. Meanwhile, the compensability of disturbance is explicitly analyzed. Finally, experiments are carried out to validate the effectiveness of the proposed scheme.

Diverse Bursting Oscillations in an Asymmetric Memristive Sallen-Key Filter

The asymmetric memristive diode bridge-based the Sallen-Key Filter (AMSKF) achieved by introducing an asymmetry diode bridge connected with a capacitor and an inductor to the Sallen-Key Filter, is studied in this paper. Through the analysis of equilibrium point, the stability of the circuit is discussed. Combined with the bifurcation diagram, it can be seen that the negative feedback gain of the Sallen-Key Filter greatly affects the dynamics of the circuit. Diverse dynamic behaviors are discovered by means of phase portraits and Lyapunov exponents, which include six types of typical oscillating behaviors: Period, Quasi-period, Quasi-periodic bursting, Period-2 bursting, Chaotic bursting and Period-1 bursting. Especially, bifurcation mechanisms of various bursting oscillating behaviors are researched by separating the formal equations into two parts, consisting of the fast spiking subsystem and the slow spiking subsystem, based on the fast-slow analysis method. Finally, Multisim experiments are carried out to validate the results of numerical simulations. Compared with the Sallen-Key Filter cascaded with the symmetric memristive diode bridge, bursting oscillating behaviors with larger amplitude and wider periodic range are obtained.

Almost global stability results for a class of singularly perturbed systems

We consider a classical singularly perturbed system consisting of a slow subsystem with state x and a fast subsystem with state z, where the small parameter ε multiplies the derivative of z. We assume that the fast system is almost globally asymptotically stable (aGAS) for each fixed x, and the slow system, with z replaced by its equilibrium value, is globally asymptotically stable. We show that there exists a sufficiently small ε* > 0 such that the equilibrium point of the overall system is aGAS for all ε ∈ (0, ε*). We use the theory of density functions as developed by A. Rantzer, D. Angeli and R. Rajaram.

Autonomous three-dimensional oscillator with five terms: spiking oscillations generation mechanism, microcontroller implementation and controls

This paper presents a report on the microcontroller implementation of an autonomous three-dimensional oscillator with five terms (ATDOFT) and performance analysis based on partial and total amplitude controls. ATDOFT displays periodic spiking behaviors, period-tripling bifurcation to chaos, chaotic spiking attractors, coexisting attractors and bistable attractors. ATDOFT is divided into two subsystems; namely the fast and slow subsystems to investigate the mechanism of the spiking dynamics. Relying on the stability analysis based on the fast subsystem with respect to the slow variable, it is shown that the spiking oscillations present in the ATDOFT arise from the system switching between the unstable state and the stable state of the lone equilibrium point of the fast subsystem. By inserting two controller parameters into the rate equations of the ATDOFT, total and partial amplitude controls are achieved. Finally, the dynamical behaviors found in ATDOFT are validated by the microcontroller implementation.

Event-triggered and self-triggered impulsive control for two-time-scale systems

This paper studies the stability problem of two-time-scale system via event-triggered impulsive control and self-triggered impulsive control. The overall system is modeled with the hybrid formalism. Two Chang transformations are introduced to completely decouple the hybrid system states into flow set and jump set. A composite impulsive controller based on slow and fast system states is proposed, under which the slow and fast subsystems are simultaneously triggered by event-triggered and self-triggered mechanism, respectively. As a result, the stability conditions are derived for the system under event-triggered and self-triggered impulsive control, respectively. Furthermore, the theoretical result of self-triggered impulsive control is applied to the consensus of the interconnected two-time-scale systems. Finally, simulation examples and comparison study show the effectiveness of the proposed control strategies.

Finite time extended state observer based nonsingular fast terminal sliding mode control of flexible-joint manipulators with unknown disturbance

To solve the problems concerning with finite time trajectory tracking of the flexible-joint manipulator subject to unknown disturbance, a singular perturbation theory (SPT) based decoupling controller is proposed in this paper. SPT is employed to decouple the basic dynamic model of the flexible-joint manipulator into a fast subsystem and a slow subsystem. For the slow subsystem, a finite time extended state observer (FTESO) is proposed to estimate the unknown disturbance. In order to avoid the singularity and accelerate the convergence speed of terminal sliding mode controller (TSMC), a novel nonsingular fast terminal sliding mode controller (NFTSMC) is designed to achieve finite time trajectory tracking. Besides, to attenuate the chattering effect, a nonsingular terminal sliding mode controller (NTSMC) is introduced for the fast subsystem. Finally, the finite time stability of the closed-loop system is proven by means of the Lyapunov theory and some comparative simulations are performed to verify the effectiveness of the proposed controller.

Guidelines for data-driven approaches to study transitions in multiscale systems: The case of Lyapunov vectors.

This study investigates the use of covariant Lyapunov vectors and their respective angles for detecting transitions between metastable states in dynamical systems, as recently discussed in several atmospheric sciences applications. In a first step, the needed underlying dynamical models are derived from data using a non-parametric model-based clustering framework. The covariant Lyapunov vectors are then approximated based on these data-driven models. The data-based numerical approach is tested using three well-understood example systems with increasing dynamical complexity, identifying properties that allow for a successful application of the method: in particular, the method is identified to require a clear multiple time scale structure with fast transitions between slow subsystems. The latter slow dynamics should be dynamically characterized by invariant neutral directions of the linear approximation model.

Visual Servoing of Flexible-Link Manipulators by Considering Vibration Suppression Without Deformation Measurements.

Visual servoing and vibration suppression of spatial flexible-link manipulators with a fixed camera setup are addressed in this article. The singular perturbation method is adopted to decouple the dynamic equations of the flexible manipulator; hence, two subsystems that represent the rigid robot motion and flexible-link vibration are obtained, respectively. Then, for the slow subsystem related to the rigid motion, an image-based controller is designed to converge the image errors with the consideration of compensating for the errors of approximating the Jacobian matrix. For the fast subsystem corresponding to the elastic vibration, to eliminate the requirements of measuring the vibration states, an observer is designed to estimate the fast states and then a feedback controller of the fast subsystem is presented to suppress the vibration of the flexible manipulator by using the estimation values. The closed-loop stabilities of the slow and fast subsystem are both proved by employing the Lyapunov theory. Numerical simulations demonstrate the effectiveness of the proposed controller, which shows that the image errors approach zero with the vibration of the flexible manipulator damped out simultaneously.

An Improved Design of Event-Triggered Feedback Controllers for Linear Systems Based on Fast and Slow Dynamics

In this article, an event-triggered technique is proposed to implement state feedback controllers for continuous-time linear systems. Toward this end, the closed-loop dynamic is decomposed into the fast and slow subsystems. Then, two individualized event-triggered mechanisms (ETMs) are designed to compute two sequences of updates. The first ETM determines when the fast subsystem should be sampled and its measured data should be sent to the controller. The second ETM decides the execution times for sampling all the system states, including both the fast and slow states. The proposed event-triggered technique leads to sampling policies in which the data of the slow states are sent to the controller with long interevent intervals whereas the fast dynamics are simultaneously allowed to be sampled with relatively shorter interevent intervals. In comparison with conventional methods with one ETM, the proposed event-triggered technique, equipped with an extra degree of freedom, leads to better results both in achieving smoother closed-loop responses and reducing the number of the transmitted packages from the system to the controller. Theoretical aspects of the proposed method including the boundedness of the system and its Zeno free property are investigated. Two simulation and experimental case studies are provided to show the effectiveness of the proposed event-triggered controller.

Second-Order Sliding Mode Control of Permanent Magnet Synchronous Motor Based on Singular Perturbation

The second-order sliding mode control has strong robustness. Its application has greatly improved the anti-jamming ability of permanent magnet synchronous motor(PMSM) speed control systems. However, the influence of noise is unavoidable due to the introduction of a differentiator in the second-order sliding mode control, and the steady-state performance of the system is poor due to the absence of a q-axis current loop. This paper proposes a second-order sliding mode control method based on singular perturbation, which decouples the PMSM speed control system into two subsystems, the slow subsystem, including the speed variable, adopts second-order sliding mode control, and the fast subsystem, including the current variable of q-axis, adopts linear control. The design of a sliding surface for the slow subsystem avoids the application of the differentiator, which reduces the chattering better. Besides this, the steady-state performance of the system is improved due to the introduction of the feedback current of the q-axis. Experimental results show that the proposed method has strong robustness and can achieve high-precision control.

On the poles of two‐time‐scale systems

In this note, the authors will study how the poles of a linear time-invariant singularly perturbed system are varied versus singular parameter ε, where the poles of the slow and fast subsystems are assumed to be placed in the prescribed regions. An explicit ε $\ \varepsilon $ bound is derived such that the poles of the original full order system remain in these regions. Finally, two examples are illustrated to show the applications of the proposed results.

Integrated sliding mode control with input restriction, output feedback and repetitive learning for space robot with flexible-base, flexible-link and flexible-joint

AbstractIn the control of space robots, flexible vibrations exist in the base, links and joints. When building a motion control scheme, the following three aspects should be considered: (1) the complexity in dynamic modeling; (2) the low accuracy of motion control and (3) the simultaneous suppression of multiple flexible vibrations. In this paper, we propose a motion vibration integrated saturation control scheme. First, the dynamic model of space robot with flexible-base, flexible-link and flexible-joint is established according to the assumed modes method and Lagrange equation. Second, singular perturbation theory is used to decompose the model into two subsystems: a slow subsystem containing the rigid motions of base and joints as well as the vibration of links, and a fast subsystem containing vibrations of base and joints. Third, an integrated sliding mode control with input restriction, output feedback and repetitive learning (ISMC-IOR) is designed, which can track the desired trajectories of base and joints with −3 orders of magnitude accuracy, while suppressing the multiple flexible vibrations of base, links and joints 50%–80% and 37% performance improvement over ISMC-IOR-NV were achieved. Finally, the algorithm is verified by simulations.

A mathematical model for cancer treatment based on combination of anti-angiogenic and immune cell therapies

In this study, we investigated a mathematical model that integrated mechanisms of tumor angiogenesis and tumor-targeted cytotoxicity in immune cells. The model considered the interaction of cancer cells with the immune system and the treatment that combines unlicensed dendritic cells and anti-vascular endothelial growth factor antibodies. We proposed an unconventional treatment protocol; for now, treatment is purely theoretical and has not been practically attempted. The treatment has been described by analytical functions that depend on time and dose. This was a function of t and q, where q is the amount (dose) of the drug. A customized treatment is a significant advantage of explicit analytical functions. We compared the results obtained from the mathematical model with the proposed treatment to the clinical results and observed an almost complete match between the treatments.In addition, we applied the global quasi-linearization GQL method to the mathematical model, which enabled us to expose the hierarchy of the system and, hence, split the system into fast and slow subsystems. This decomposition enabled us to analytically compute the equilibrium points of the model and their stability, which is crucial for the application.

Dual-Time-Scale Sliding Mode Control for Surface-Mounted Permanent Magnet Synchronous Motors

The permanent magnet synchronous motors (PMSMs) as the completely symmetrical three-phase machines, which are usually driven by symmetrical voltage signals. Unfortunately, a PMSM system usually suffers from the different lumped disturbances, such as internal parametric perturbations and external load torques, the speed regulation problem should be addressed within the different operation situations. Characterizing by the current variation speed of the motor winding is much faster than that of the mechanical dynamic velocity, a dual-time-scale sliding mode control (SMC) method for the surface-mounted PMSMs is proposed in this paper. Firstly, the mathematical model of PMSMs is established in the two-phase synchronous rotating orthogonal reference coordinate system, and the slow and fast variation subsystems are obtained based on the quasi-steady-state theory. Secondly, a tracking differentiator (TD)-based and exponential reaching law-based sliding mode controllers are individually designed within dual-time-scale, respectively. As a result, the eventual SMC strategy is presented, and the stability of control system is analyzed by applying the Lyapunov stability theory. The main contribution of this study is to present an alternative control framework for the PMSMs servo system, where the dual-time-scale characteristic is involved, and thus a non-cascade control structure that different from the traditional drive strategy is proposed in the motor community. Finally, the model of whole system is built and carried out on the simulation platform. Research results demonstrate that the presented servo control system can accurately track the reference angle velocity signal, while the strong robustness and fast response performance are guaranteed in the presence of external disturbances. In addition, the three-phase current transient response values are completely symmetrical with the rapid adjustment characteristic.

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.