Optimal stabilization for differential systems with delays - Malkin’s approach

The paper considers a process controlled by a system of delayed differential equations. Under certain assumptions, a control function is determined such that the zero solution of the system is asymptotically stable and, for an arbitrary solution, the integral quality criterion with infinite upper limit exists and attains its minimum value in a given sense. To solve this problem, Malkin’s approach to ordinary differential systems is extended to delayed functional differential equations, and Lyapunov’s second method is applied. The results are illustrated by examples, and applied to some classes of delayed linear differential equations.     

基于经验模态分解的中国生态足迹与生态承载力动力学预测

利用mathcad2001和matlab6.5软件,计算了中国1961年～2005年生态足迹和生态承载力,通过经验模态分解（EMD）方法对其进行分析,在此基础上,建立动力学模型对其未来进行预测,希望能通过对中国生态足迹动态研究,建立一个带有周期波动的动力学预测模型,为研究长时间序列的动态变化提供一个全新的研究方法。通过EMD分析后我们可以得到研究要素的不同时间尺度的演化曲线,将这些曲线看成是系统在不同时间尺度下的特解,则可以根据不同时间尺度的EMD分量的曲线够建起对应的动力模型,如趋势项一般对应于指数或线形动力方程,大尺度的周期分量一般可以对应于正弦或与余弦形式的动力方程,最后我们建立一个总的动力预测模型。研究结果表明：随着生态足迹的增大和生态承载力的减小,中国未来20年的生态赤字越来越大,由2006年的-0.932到2025年的-1.632,接近2倍,发展处于不可持续状态。经验模态分解（EMD）方法能很好的分解出生态足迹和生态承载力的波动周期,符合它们的发展规律,并用动力学预测了它们的发展态势,经拟合表明,预测的结果与实际值误差较小。这说明基于EMD的动力学预测模型是科学的。     

Control of a directly excited structural dynamic model of an F-15 tail section

We used two simple control laws based on linear velocity and cubic velocity feedback to suppress the high-amplitude vibrations of a structural dynamic model of the twin-tail assembly of an F-15 fighter when subjected to primary resonance excitations. We developed the nonlinear differential equations of motion and obtained an approximate solution using the method of multiple scales. Then, we conducted bifurcation analyses for the open- and closed-loop responses of the system and investigated theoretically the performance of the control strategies. The theoretical findings indicate that the control laws lead to effective vibration suppression and bifurcation control. Furthermore, we conducted experiments to verify the theoretical analysis. We built a digital control system that consists of a SIMULINK modeling software and a dSPACE controller installed in a personal computer. Actuators made of piezoelectric ceramic material were used. The results show that both laws are effective at suppressing the vibrations. To compare the performance of both techniques, we calculated the power requirements for a simple system.     

Stabilization of networked singular control systems under double-channel quantization and DoS attacks

This paper is concerned with the asymptotic stabilization of discrete singular systems over a bandwidth limited digital network, when the state measurements are periodically sampled and encoded using a finite alphabet, and the control input signals are subject to finite-alphabet encoding and Denial-of-Service attacks. It is assumed that the attack signals are uniform for all sampling periods and have been identified. A dynamic controller is designed based on a restricted equivalent model of the controlled plant. Two types of finite-level quantizers are designed for encoding: uniform and logarithmic. For both types of quantizers, dynamic encoding-decoding strategies for the plant state and the control input are proposed, which exploit the controller’s state and the origin, respectively, as the quantization centers. Sufficient conditions for asymptotic stabilizability involving the sampling period, the numbers of the state and input quantization levels, the beginning time and corresponding duration of the attack signals are established by propagating reachable sets during sampling interval. Finally, several numerical examples are given to illustrate the design procedures and the efficacy of the theoretical results.     

Buck电路求解方法的研究

滞回比较器是一种多用途的比较器,本文中选择它来控制Buck电路。利用滞回比较器的滞回特性控制Buck电路的输出在一个很小的范围内变化,从而保证了系统具有很高的稳定性。对受滞回比较器控制的Buck电路而言,其输出的求解方法有状态空间法、微分方程法以及拉普拉斯变换法等。本文所提出的算法是基于状态空间法的,并在其基础上引入一种等价无穷小的概念,从而化简了转移矩阵,避免了由于直接利用状态空间法解方程而带来的不便。通过Matlab编程以及PSPICE仿真,都有力地验证了本解法的有效性和正确性。     

The impact of static output nonlinearities on the control strategies that humans use in command-following tasks

The results of a human-in-the-loop experiment are used to investigate the control strategies that humans use to interact with nonlinear dynamic systems. Two groups of human subjects interact with a dynamic system and perform a command-following task. The first group interacts with a linear time-invariant (LTI) dynamic system. The second group interacts with a Wiener system, which consists of the same LTI dynamics cascaded with a static output nonlinearity. Both groups exhibit improved performance over the trials, but the average of the linear group’s performance is better on more than three-fourths of the trials. A new nonlinear subsystem identification algorithm is presented and used to identify the feedback and feedforward control strategies used by the subjects in both groups. The identification results for the linear group agree with prior studies suggesting that adaptive feedforward inversion is a primary control strategy used by humans for command-following tasks. The main results of this paper address an open question of whether a similar control strategy is used for nonlinear systems. The identification results for the nonlinear group suggest that those subjects also use adaptive feedforward inversion. However, the static output nonlinearity inhibits the human’s ability to approximate the inverse.     

Adaptive data-driven fault-tolerant control for nonlinear systems: Koopman-based virtual actuator approach

This paper proposes an adaptive data-driven fault-tolerant control scheme using the Koopman operator for unknown dynamics subjected to nonlinearities, time-varying loss of effectiveness, and additive actuator faults. The main objective of this method is to design a virtual actuator to hide actuator faults from the view of the system’s nominal controller without having any prior knowledge about the system’s underlying dynamics. The designed virtual actuator is placed between the faulty plant and the nominal controller of the system to keep the dynamical system’s performance consistent before and after the occurrence of actuator faults. Based on the Koopman operator theory, an equivalent Koopman predictor is first obtained using the process data only, without knowing the governing equations of the underlying dynamics. Koopman operator is an infinite-dimensional, linear operator which takes the nonlinear process data into an infinite-dimensional feature space where the dynamic data correlations have linear behavior. Next, based on the approximated system’s Koopman operator, a virtual actuator is designed and implemented without knowing the system’s nominal controller. Needless to use a separate fault detection, isolation, and identification module to perform fault-tolerant control, the current method leverages the adaptive framework to keep the system’s desired performance in facing time-varying additive and loss of effectiveness actuator faults. Finally, the approach’s efficacy is demonstrated using simulation on a two-link manipulator benchmark, and a comparison study is presented.     

Free final time fractional optimal control problems

A formulation and solution scheme of free final time fractional optimal control problems is presented in this paper. The dynamic constraint is described by a fractional differential equation. Performance index considered is a function of both the state and control variables. The necessary conditions of optimality and the transversality condition are obtained using Lagrange multiplier technique. A numerical technique similar to Shooting method is used for solving the optimal conditions. Numerical example is provided to show the effectiveness of the formulation and numerical solution scheme. It is interesting to note that the final time changes with the interchange of the boundary conditions, which does not occur in classical optimal control problems.     

Observability of singular Boolean control networks with state delays

This paper investigates the observability of singular Boolean control networks (SBCNs) with state delays. First, via Cheng product, the considered system is converted into its algebraic form. Then, under the uniqueness of the solution, two methods are proposed to transform the system into a general Boolean control network (BCN). Therefore, the considered system’s observability is equivalent to the BCN’s observability. Subsequently, the restricted input-state incidence matrix is introduced and some necessary and sufficient conditions are presented for the observability of BCNs with state delays. Finally, two illustrative examples are presented to show that the main results obtained are effective in analyzing the observability of SBCNs with state delays.     

Design and analysis of a novel cascade control algorithm for braking-by-wire system based on electromagnetic direct-drive valves

In order to improve the response speed and control precision of the braking system with parameters uncertainty and nonlinear friction, a braking-by-wire system based on the electromagnetic direct-drive valve and a novel cascade control algorithm was proposed in this paper. An electromagnetic linear actuator directly drives the valve spool and rapidly adjusts the pressure of braking wheel cylinders. A dynamic model of electromagnetic direct-drive valve considering improved LuGre dynamic friction is established. A novel cascade control algorithm with an outside loop pressure fuzzy controller and an inside loop electromagnetic direct-drive valve position controller was proposed. An adaptive integral robust inside loop controller is designed by combining friction compensation adaptive control law, linear feedback, and integral robust control. The uncertainty parameters and the friction state are estimated online. The stability of the cascade controller is proved by the Lyapunov method. Then a multi-objective opitimizemization design method of control parameters is proposed, which combines a multi-objective game theory and a technique for order preference by similarity to ideal solution (TOPSIS) based on entropy weight. The results show that the pressurization time of cascade control is less than 0.09 s under the 15 MPa step target signal. The control precision is improved effectively by the cascade controller under the ARTEMIS condition.     

Sliding-mode control for teleoperation system with uncertain kinematics and dynamics: An observer-based approach

This study concentrates on the tracking control of teleoperation system subjected to robot uncertainties. The coupling of kinematic and dynamic uncertainties poses a challenge to construct the teleoperation controller. To overcome this difficulty, an observer-based approach is designed to ensure position tracking while compensating for the unfavorable effects arising from the uncertainties. First, two sliding-mode observers together with a novel power reaching law are constructed, upon which, the uncertainties will be estimated in finite time. Next, a controller is proposed to solve the finite-time convergence of the tracking errors. The settling time and the stability of the closed-loop system are derived by Lyapunov’s direct method. Simulation results are presented to testify the tracking performance of the suggested control.     

A maximum principle approach to optimal control for one-dimensional hyperbolic systems with several state variables

The maximum principle developed by Sloss et al. [Optimal control of structural dynamic systems in one space dimension using a maximum principle, J. Vibr. Control 11 (2005) 245–261] is used to determine the optimal control functions for a class of one-dimensional distributed parameter structures. The distributed parameter structures are governed by systems of fourth order hyperbolic equations with constant coefficients. A quadratic performance index is formulated as the cost functional of the problem and can be used to represent the energy of the structure and the force spent in the control process. The developed maximum principle establishes a theoretical foundation for the solution of the optimal control problem and relates the optimal control vector to an adjoint variable vector. The method of solution is outlined which involves reducing the original problem to a system of ordinary differential equations. The solution of the general problem is given and a structural control problem is solved to illustrate the solution procedure. The effectiveness of the proposed control solution is shown by comparing the behavior of controlled and uncontrolled systems.     

Stabilizing a class of mixed states for stochastic quantum systems via switching control

The global stabilization of a class of mixed states for finite dimensional stochastic quantum systems with degenerate measurement operator is investigated in this paper. We construct a measurement operator and control Hamiltonian that make the target state one of the system equilibria. Based on the proposed Lyapunov function, a control law is designed following Lyapunov’s method to steer system state to the target mixed state from an initial state in the convergence domain, which is obtained through the analyses of invariant set based on LaSella’s invariance principle. When the initial state isn’t in the convergence domain, a constant control is used to steer the system state so that it enters the convergence domain in finite time. The constant control and the control designed by Lyapunov method compose a switching control strategy, which can steer system state to the target mixed state from any arbitrary state in the state space, i.e., the target mixed state is globally stable under the switching control. The convergence of the switching control is proved based on state sample trajectories. Moreover, the numerical experiments on a three dimensional stochastic quantum system are performed to demonstrate the effectiveness of switching control.     

Exponential synchronization of multilayer networks with white-noise-based coupling via intermittent periodic event-triggered control

In this paper, the mean-square exponential synchronization of stochastic multilayer networks with white-noise-based time-varying coupling is investigated via intermittent dynamic periodic event-triggered control. The existence of a dynamic term can reduce the number of event triggers. Furthermore, by introducing periodic sampling mechanism, a minimum inter-execution time is guaranteed to avoid Zeno phenomenon. Additionally, by employing Lyapunov method, graph theory, and stochastic analysis techniques, synchronization criteria for multilayer networks under intermittent dynamic periodic event-triggered control are established. To clarify the process of synchronization of multilayer networks, a brief framework is developed on the basis of Tajan’s algorithm. Ultimately, theoretical results are applied into Chua’s circuits and corresponding numerical simulations are given to illustrate the effectiveness and feasibility of the results.     

Fault diagnosis and fault tolerant control for the non-Gaussian time-delayed stochastic distribution control system

The purpose of fault diagnosis of stochastic distribution control (SDC) systems is to use the measured input and the system output probability density functions (PDFs) to obtain the fault information of the SDC system. When the target PDF is known, the purpose of fault tolerant control of stochastic distribution control system is to make the output PDF still track the given distribution using the fault tolerant controller. However, in practice, time delay may exist in the data (or image) processing, the modeling and transmission phases. When time delay is not considered, the effectiveness of the fault detection, diagnosis and fault tolerant control of stochastic distribution systems will be reduced. In this paper, the rational square-root B-spline is used to approach the output probability density function. In order to diagnose the fault in the dynamic part of such systems, it is then followed by the novel design of a nonlinear neural network observer-based fault diagnosis algorithm. The time delay term will be deleted in the stability proof of the observation error dynamic system. Based on the fault diagnosis information, a new fault tolerant controller based on PI tracking control is designed to make the post-fault probability density function still track the given distribution, which is dependent of the time delay term. Finally, simulations for the particle distribution control problem are given to show the effectiveness of the proposed approach.     

Robust tracking-surveillance and landing over a mobile target by quasi-integral-sliding mode and Hopf bifurcation

This paper addresses the problem of a robust UAV tracking, surveillance and landing of a mobile ground target. The translational and angular dynamics of the vehicle are affected by bounded uncertainties; a Quasi-Integral Sliding Mode control is designed to obtain robustness from nearly the initial time. The flying mission considers three different dynamics of movement: the take-off to the desired altitude, the relative circular surveillance motion around the mobile ground target and eventually precise landing over the ground vehicle. This paper introduces a novel dynamic motion planning generator to perform such tracking maneuvers. It is based on the solution of a second order nonlinear differential equation, whose solution is force to move in a set of new parameterized ‘Bifurcation Sliding Mode Surfaces’ that exploit the Hopf Bifurcation properties to change the dynamic around the equilibrium point. A temporal switching technique is introduced for changing between three different bifurcation sliding surfaces at different time intervals. To illustrate that the quadcopter effectively performs the desired maneuvers, a computer animation is provided at the end of the paper.     

Distributed set-membership estimation for state-saturated systems with mixed time-delays via a dynamic event-triggered scheme

This paper is focused on the distributed estimation issue in the form of set-membership (SM) for a class of discrete time-varying systems suffering mix-time-delays and state-saturations. The phenomena of time-delays and state-saturations are introduced to better describe insightful engineering. During local measurements transmission between sensors over a resource-limited sensor network, to prevent data collisions and resource-consumption, a newly dynamic event-triggering strategy (DETS) is designed to dispatch the local measurements transmission for each sensor to its neighbors. Compared with the most existing static ETSs, this DETs can mitigate the total number of triggering times and enlarge interval time between consecutive triggering instants. Then, some novel adequate criteria for designing the desired event-based SM estimators are derived such that the plant’s true state always resides in each sensor’s ellipsoidal region regardless of the simultaneous presence of bounded noises, mixed time delays and state-saturations. Subsequently, a recursive optimization algorithm is formulated such that the minimal ellipsoids, the estimators gains and event-triggering weighted matrices are acquired simultaneously. A verification simulation is presented to illustrate the advantages of the design approach of the developed state estimator.     

Chaos control of the permanent magnet synchronous motor with time-varying delay by using adaptive sliding mode control based on DSC

This paper focuses on the problem of chaos control for the permanent magnet synchronous motor with chaotic oscillation, unknown dynamics and time-varying delay by using adaptive sliding mode control based on dynamic surface control. To reveal the mechanism of motor system and facilitate controller design, the dynamic behavior of the system is investigated. Nonlinear items of system model, upper bounds of time delays and their derivatives are taken as unknown in the overall process. A RBF neural network with an adaptive law, which eliminates restrictions on accurate model and parameters, is employed to cope with unknown dynamics. In order to solve issues such as chaotic oscillation, ‘explosion of complexity’ of backstepping, and chattering associated with sliding mode control, a sliding mode controller is developed within the framework of dynamic surface control by the hybrid of adaptive technology and RBF neural network. In addition, an appropriate Lyapunov function is employed to demonstrate the system stability. Finally, the feasibility of the proposed scheme is testified by simulation.     

Switched observer-based dynamic event-triggered consensus of multi-agent systems under DoS attacks

This paper investigates the observer-based consensus control for high-order nonlinear multi-agent systems (MASs) under denial-of-service (DoS) attacks. When the DoS attacks appear, the communication channels are destroyed, and the blocked information may ruin the consensus of MASs. A switched state observer is designed for the followers to observe the leader’s state whether the DoS attacks occur or not. Then, a dynamic event-triggered condition is proposed to reduce the consumption of communication resources. Moreover, an observer-based and dynamic event-triggered controller is formulated to achieve leader-following consensus through the back-stepping method. Additionally, the boundedness of all closed-loop signals is obtained based on the Lyapunov stability theory. Finally, the simulation results demonstrate the effectiveness of the presented control strategy under DoS attacks.     

Fixed-time stabilization of IT-2 T-S fuzzy control systems with discontinuous interconnections: Indefinite derivative Lyapunov method

Using the interval type-2 Takagi–Sugeno (IT-2 T-S) fuzzy control method, this paper formulates a class of non-autonomous interconnected dynamical system (IDS) with discontinuities. Under the differential inclusion (DI) framework, the fixed-time stabilization (FXTS) problem is studied via indefinite derivative Lyapunov approach, where the time-derivative of constructed Lyapunov function doesn’t have to be negative/semi-negative. By designing novel IT-2 T-S fuzzy switching control protocol possessing time-varying control gain coefficients, several sufficient stabilization conditions are obtained to determine the system’s stability in fixed time. Furthermore, the settling time (ST) of FXTS is estimated. Due to the time-varying property of control gain coefficients and indefiniteness of system’s parameters, the advantage of the IT-2 T-S fuzzy switching control protocol designed in this paper is that its control gain coefficients are not only more flexible, but also can affect the estimation of ST. Finally, the designed control protocols and FXTS results are confirmed by numerical example.