Guaranteed cost control for uncertain large-scale systems with time-delays via delayed feedback

In this paper, we propose a design method for guaranteed cost controllers for uncertain large-scale systems with time-delays in subsystem interconnections using delayed feedback. Based on the Lyapunov method, an LMI (Linear Matrix Inequality) optimization problem is formulated to design the delayed feedback controller which minimizes the upper bound of a given quadratic cost function. A numerical example is included to illustrate the design procedures.     

Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems

In this paper, we propose a memory state feedback model predictive control (MPC) law for a discrete-time uncertain state delayed system with input constraints. The model uncertainty is assumed to be polytopic, and the delay is assumed to be unknown, but with a known upper bound. We derive a sufficient condition for cost monotonicity in terms of LMI, which can be easily solved by an efficient convex optimization algorithm. A delayed state dependent quadratic function with an estimated delay index is considered for incorporating MPC problem formulation. The MPC problem is formulated to minimize the upper bound of infinite horizon cost that satisfies the sufficient conditions. Therefore, a less conservative sufficient conditions in terms of linear matrix inequality (LMI) can be derived to design a more robust MPC algorithm. A numerical example is included to illustrate the effectiveness of the proposed method.     

Optimal feedback control of the forced van der Pol system

A simple feedback control strategy for chaotic systems is investigated using the forced van der Pol system as an example. The strategy regards chaos control as an optimization problem, where the maximum magnitude Floquet multiplier of a target unstable periodic orbit (UPO) is used as a cost function that needs to be minimized. Thus, the method obtains the optimal control gain in terms of the stability of the target UPO. This strategy was recently proposed for the proportional feedback control (PFC) method. Here, it is extended to the highly popular delayed feedback control (DFC) method. Since the DFC method treats the system as a delay-differential equation whose phase space is infinite-dimensional, the characteristic multipliers are found through a truncation in the number of delayed states. Control of a target UPO is achieved for several values of the forcing amplitude. We compare the DFC and PFC methods in terms of stability of the controlled orbit, steady state error and control effort.     

LMI Optimization Approach to Observer-Based Controller Design of Uncertain Time-Delay Systems via Delayed Feedback

In this paper, we propose a design method of an observer-based controller for uncertain time-delay systems by delayed feedback. Based on the Lyapunov method, an LMI (linear matrix inequality) criterion is derived to design an observer-based controller which makes the system stable. A numerical example is included to illustrate the design procedure.     

Control studies of time-delayed dynamical systems with the method of continuous time approximation

This paper presents control studies of delayed dynamical systems with the help of the method of continuous time approximation (CTA). The CTA method proposes a continuous time approximation of the delayed portion of the response leading to a high and finite dimensional state space formulation of the time-delayed system. Various controls of the system such as LQR and output feedback controls are readily designed with the existing design tools. The properties of the method in frequency domain are also discussed. We have found that time-domain methods such as semi-discretization and CTA, and other numerical integration algorithms can produce highly accurate temporal responses and dominant poles of the system, while missing all the fast and high frequency poles, which explains why many numerical methods can be applied to study the stability of time-delayed systems, and may not be a good tool for control design. Optimal feedback controls for a linear oscillator, collocated and non-collocated feedback controls of an Euler beam, and an experimental demonstration are presented in the paper.     

Minimum entropy control of chaos via online particle swarm optimization method

One of the recently developed approaches for control of chaos is the minimum entropy (ME) control technique. In this method an entropy function based on the Shannon definition, is defined for a chaotic system. The control action is designed such that the entropy as a cost function is minimized which results in more regular pattern of motion for the system trajectories. In this paper an online optimization technique using particle swarm optimization (PSO) method is developed to calculate the control action based on ME strategy. The method is examined on some standard chaotic maps with error feedback and delayed feedback forms. Considering the fact that the optimization is online, simulation results show very good effectiveness of the presented technique in controlling chaos.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

Numerical time-delayed feedback control in a neural network with delay

In this article, by a nonstandard finite-difference method we obtain the general time delayed feedback control numerical discrete scheme for a delayed neural network model. Firstly, the local stability of the equilibria point is discussed according to the Neimark–Sacker bifurcation theory. Then, from the point of view of control, for any step-size, a general time delayed feedback control numerical algorithm is introduced to delay the onset of the Neimark–Sacker bifurcation at a desired point by choosing appropriate control parameters. This controller can deal with the general system that the natural equilibrium cannot be given by analytic expression. Finally, numerical examples are provided to illustrate the theoretical results. The results show that the time delayed feedback numerical scheme is better than a polynomial function time delayed feedback method.     

A class of optimal state-delay control problems

We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.     

Positive real control for 2-D discrete delayed systems via output feedback controllers

This paper considers the problem of positive real control for two-dimensional (2-D) discrete delayed systems in the Fornasini–Marchesini second local state-space model. Attention is focused on the design of dynamic output feedback controllers, which guarantee that the closed-loop system is asymptotically stable and the closed-loop transfer function is extended strictly positive real. We first present a sufficient condition for extended strictly positive realness of 2-D discrete delayed systems. Based on this, a sufficient condition for the solvability of the positive real control problem is obtained in terms of a linear matrix inequality (LMI). When the LMI is feasible, an explicit parametrization of a desired output feedback controller is presented. Finally, we provide a numerical example to demonstrate the application of the proposed method.     

Guaranteed cost control of time-delay chaotic systems via memoryless state feedback

This paper studies the problem of guaranteed cost control for a class of time-delay chaotic systems via memoryless state feedback. A design procedure is proposed to construct a memoryless state feedback controller, which guarantees that the resulting closed-loop system is asymptotically stable and achieves an adequate level of performance. A numerical example is provided to demonstrate the effectiveness of the proposed method.     

Delayed feedback control of chaotic spinning disk via minimum entropy approach

Chaos control of a spinning disk model via delayed feedback method is presented. The feedback gain is obtained and adapted according to a minimum entropy (ME) algorithm. In this method, stabilizing an unstable fixed point of the system Poincare map is achieved by minimizing the entropy of point distribution on the Poincare section. Simulation results show the feasibility of the proposed method in applying the delayed feedback technique for chaos control of spinning disks.     

线性离散不确定时滞系统滞后相关H∞鲁棒控制

基于Lyapunov泛函方法,对存在状态时滞的线性离散不确定系统,给出了滞后相关型无记忆H∞状态反控制器设计方案,通过求解相应的线性矩阵不等式即可求得满足设计要求的控制器．     

Control of chaos in atomic force microscopes using delayed feedback based on entropy minimization

Active chaos control of a tapping mode atomic force microscope (AFM) model via delayed feedback method is presented. The feedback gain is obtained and adapted according to a minimum entropy (ME) algorithm. In this method, stabilizing an unstable fixed point of the system Poincare map is achieved by minimizing the entropy of points distribution on the Poincare section. Simulation results show the feasibility of the proposed method in applying the delayed feedback technique for chaos control of an AFM system.     

Optimal syntheses for infinite-dimensional linear delayed state-output systems: A semicausality approach

Quadratic optimal control synthesis for infinite-dimensional delayed dynamical systems with output time delay involved in the cost functional is described. By introducing a truncation operator and associated semicausal trajectory, a new dynamical optimality principle was established. The closed-loop optimal control is given in three parts as a linear feedback: real-time state feedback, retarded state integral feedback, and initial data feedback which effects only a small time interval. The main feedback operator can be determined by solving a linear Fredholm integral equation.This research was supported by the National Science Foundation under Grant No. 8607687 and the Airforce Office of Scientific Research under Grant No. 860088.     

Event-triggered control for sampled-data set stabilization of switched delayed boolean control networks

This paper considers the event-triggered control design for the uniform sampled-data set stabilization of switched delayed Boolean control networks (SDBCNs). First, using the algebraic state space representation method, SDBCNs are converted into the equivalent algebraic form. Second, using the algebraic form, the uniform sampled-data reachable sets are constructed, based on which, a necessary and sufficient condition is obtained for the uniform sampled-data set stabilization of SDBCNs. Finally, the event-triggered mechanism is presented, and a sufficient condition is proposed to design the time-variant state feedback event-triggered controller for the uniform sampled-data set stabilization of SDBCNs.     

Delayed feedback stabilization and the Huijberts–Michiels–Nijmeijer problem

A short survey on delayed feedback stabilization is given. The Huijberts–Michiels–Nijmeijer problem on the delayed feedback stabilization of unstable equilibria of two- and three-dimensional dynamical systems is considered. It is shown that the methods of delayed feedback stabilization of unstable periodic orbits can be used with advantage for the stabilization of unstable equilibria. An analytical study based on the D-decomposition method is given. Efficient necessary and/or sufficient conditions for the stabilizability of the systems in question are obtained in the form of explicit analytic expressions. These conditions define the boundaries of stabilizability domains in terms of system parameters. It follows from these conditions that the introduction of a delayed feedback control generally extends the possibilities of stationary stabilization of linear systems with delay-free feedback.     

An iterative method for the finite-time bilinear-quadratic control problem

For bilinear control systems with quadratic cost, the so-called bilinear-quadratic problems, a feedback controller for the finite-time case is designed. An iteration procedure in close proximity to the Riccati approach is presented, and the proof of convergence is outlined. The potential of the new method is discussed, and the design procedure is illustrated for two examples.