THE FEEDBACK DELAY IN DIGITAL CONTROL

It is a fact that the feedback delay actually arises in digital control systems. It is necessary to modify the structure of digital control systems and develop new control algorithms, which is done in this paper. A great number of digital computer simulation experiments have shown the obvious advantage of the new algorithms.

Capacity analysis of transmit beamforming with variable feedback delay

The feedback delay can severely affect the quality of the channel state information at the transmitter （CSIT） which is fed back from the receiver. The outdated CSIT will cause large performance loss in the transmit beamforming systems. The effect of variable feedback delay on the capacity of transmit beamforming systems over Rayleigh fading channels is studied. First, the case of fixed feedback delay is considered and a closed-form expression of system capacity is derived. Based on the results of fixed delay, the delay following certain distributions in variable delay case is assumed and the closed-form expressions of capacities are derived. The closed-form expressions show that the capacity is significantly affected by the statistical characteristics of the feedback delay. The obtained results provide an analytical insight into the effects caused by variable delay on the system capacity.

Addressing the CQI feedback delay in 5G/6G networks via machine learning and evolutionary computing

5G networks apply adaptive modulation and coding according to the channel condition reported by the user in order to keep the mobile communication quality.However,the delay incurred by the feedback may make the channel quality indicator(CQI)obsolete.This paper addresses this issue by proposing two approaches,one based on machine learning and another on evolutionary computing,which considers the user context and signal-to-interference-plus-noise ratio(SINR)besides the delay length to estimate the updated SINR to be mapped into a CQI value.Our proposals are designed to run at the user equipment(UE)side,neither requiring any change in the signalling between the base station(gNB)and UE nor overloading the gNB.They are evaluated in terms of mean squared error by adopting 5G network simulation data and the results show their high accuracy and feasibility to be employed in 5G/6G systems.

Delayed feedback control experiments on some flexible structures

In recent decades,studies on delayed system dynamics have attracted increasing attention and advances have been achieved in stability,nonlinearity,delay identification, delay elimination and application.However,most of the existing work is on the theoretical basis and little is on the experiment.This paper presents our experimental studies on delayed feedback control conducted in recent years with the focus on the discussion of a DSP-based delayed experiment system.Some phenomena in our delay experiments are discussed and a few topics of interest for further research are brought forward.

Controlling chaos in RCL-shunted Josephson junction by delayed linear feedback

The resistively-capacitively-inductively-shunted （RCL-shunted） Josephson junction （RCLSJJ） shows chaotic behaviour under some parameter conditions. Here a scheme for controlling chaos in the RCLSJJ is presented based on the linear feedback theory. Numerical simulations show that this scheme can be effectively used to control chaotic states in this junction into stable periodic states. Moreover, the different stable period states with different period numbers can be obtained by appropriately adjusting the feedback intensity and delay time without any pre-knowledge of this system required.

A survey on delayed feedback control of chaos

This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-known odd number linfitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.

H_∞ Synchronization of Chaotic Systems via Delayed Feedback Control

The H∞ synchronization problem for a class of delayed chaotic systems with external disturbance is investigated. A novel delayed feedback controller is established under which the chaotic master and slave systems are synchronized with a guaranteed H∞ performance. Based on the Lyapunov stability theory, a delay-dependent condition is derived and formulated in the form of linear matrix inequality （LMI）. A numerical simulation is also presented to validate the effectiveness of the developed theoretical results.

Hopf bifurcation analysis of Chen circuit with direct time delay feedback

Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit （oscillation） in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit.

Exponential Synchronization of Master-Slave Lur'e Systems via Intermittent Time-Delay Feedback Control

This paper concerns with the master-slave exponential synchronization analysis for a class of general Lur'esystems with time delay.Different from the previous methods based on the differential inequality technique, a newapproach is proposed to derive some new exponential synchronization criteria.The restriction that the control widthhas to be larger than the time delay is removed.This leads to a larger application scope for our method.Moreover, notranscendental equation is involved in the obtained result, which reduces the computational burden.Two examples aregiven to validate the theoretical results.

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node （SN） or Hopf bifurcation （HB） as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

Effective suppression of beta oscillation in Parkinsonian state via a noisy direct delayed feedback control scheme

This work explores the function of the noisy direct delayed feedback(NDDF)control strategy in suppressing the pathological oscillations in the basal ganglia(BG)with Parkinson’s disease(PD).Deep brain stimulation(DBS)alleviates the PD state fantastically.However,due to its unclear mechanism and open-loop characteristic,it is challenging to further improve its effects with lower energy expenditure.The noise stimulus performs competitively in alleviating the PD state theoretically,but it cannot adapt to the neural condition timely and automatically due to its open-loop control scheme.The direct delayed feedback(DDF)control strategy is able to disturb excessive synchronous effectively.Therefore,the NDDF control strategy is proposed and researched based on a BG computational model,which can reflect the intrinsic properties of the BG neurons and their connections with thalamic neurons.Simulation results show that the NDDF control strategy with optimal parameters is effective in removing the pathological beta oscillations.By comparison,we find the NDDF control strategy performs more excellent than DDF in alleviating PD state.Additionally,we define the multiple-NDDF control strategy and find that the multiple-NDDF with appropriate parameters performs better than NDDF.The obtained results contribute to the cure for PD symptoms by optimizing the noise-induced improvement of the BG dysfunction.

Dynamical behaviour of Liu system with time delayed feedback

This paper investigates the dynamical behaviour of the Liu system with time delayed feedback. Two typical situations are considered and the effect of time-delay parameter on the dynamics of the system is discussed. It is shown that the Liu system with time delayed feedback may exhibit interesting and extremely rich dynamical behaviour. The evolution of the dynamics is shown to be complex with varying time-delay parameter. Moreover, the strange attractor like ‘wormhole＇ is detected via numerical simulations.

Directed transport of coupled Brownian ratchets with time-delayed feedback

A time-delayed feedback ratchet consisting of two Brownian particles interacting through the elastic spring is consid ered. The model describes the directed transport of coupled Brownian particles in an asymmetric two-well ratchet potential which can be calculated theoretically and implemented experimentally. We explore how the centre-of-mass velocity is af fected by the time delay, natural length of the spring, amplitude strength, angular frequency, external force, and the structure of the potential. It is found that the enhancement of the current can be obtained by varying the coupling strength of the delayed feedback system. When the thermal fluctuation and the harmonic potential match appropriately, directed current evolves periodically with the natural length of the spring and can achieve a higher transport coherence. Moreover, the external force and the amplitude strength can enhance the directed transport of coupled Brownian particles under certain conditions. It is expected that the polymer of large biological molecules may demonstrate a variety of novel cooperative effects in real propelling devices.

Towards frequency adaptation for delayed feedback deep brain stimulations

In neurodegenerative disorders such as Parkinson＇s disease （PD）, deep brain stimulation （DBS） is a desirable approach when the medication is less effective for treating the symptoms. DBS incorporates transferring electrical pulses to a specific tissue of the central nervous system, obtaining therapeutic results by modulating the neuronal activity of that region. DBS has certain advantages such as reversibility and adjustability features over medication, since the neuronal firing patterns can be recorded and used to alter the parameters of the DBS signal （Benabid et al., 2009）. One of the DBS indications is its ability to suppress the abnormal neuronal activity to treat symptoms like tremor, akinesia and dystonia.

Experimental study of delayed positive feedback control for a flexible beam

Recently, some researches indicate that positive feedback can benefit the control if appropriate time delay is intentionally introduced into control system. However, most work is theoretical one but few are experimental. This paper presents theoretical and experimental studies of delayed positive feedback control technique using a flexible beam as research object. The positive feedback weighting coefficient is designed by using the optimal control method. The available time delay is determined by analyzing the maximal real part of characteristic roots of the system. A DSP-based experiment system is introduced. Simulation and experimental results indicate that the delayed positive feedback control may effectively reduce the beam vibration if time delay is appropriately selected.

New criteria on delayed state feedback stabilization for stochastic systems with time-varying delay

The problems of robust exponential stability in mean square and delayed state feedback stabilization for uncertain stochastic systems with time-varying delay are studied. By using Jensen＇s integral inequality and combining with the free weighting matrix approach, new delay-dependent stability conditions and delayed state feedback stabilization criteria are obtained in terms of linear matrix inequalities. Meanwhile, the proposed delayed state feedback stabilization criteria are more convenient in application than the existing ones since fewer tuning parameters are involved. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Position and Velocity Time Delay for Suppression Vibrations of a Hybrid Rayleigh-Van der Pol-Duffing Oscillator

In this paper,we used time delay feedback to minimize the vibrations of a hybrid Rayleigh–van der Pol–Duffing oscillator.This system is a one-degree-offreedom containing the cubic and fifth nonlinear terms and an external force.We applied the multiple scales method to get the solution from first approximation.Graphically and numerically,we studied the system before and after adding time delay feedback at the primary resonance case(
ffi!).We used MATLAB program to simulate the efficacy of different parameters and the time delay on the main system.

Feedback control in a coupled Brownian ratchet

On the basis of the double-well ratchet potential which can be calculated theoretically and implemented experimentally, the influences of the time delay, the coupling constant, and the asymmetric parameter of the potential on the performance of a delayed feedback ratchet consisting of two Brownian particles coupled mutually with a linear elastic force are investigated. The centre-of-mass velocity of two coupled Brownian particles, the average effective diffusion coefficient, and the Pe number are calculated. It is found that the parameters are affected by not only the time delay and coupling constant but also the asymmetric parameter of the double-well ratchet potential. It is also found that the enhancement of the current may be obtained by varying the coupling constant of the system for the weak coupling case. It is expected that the results obtained here may be observed in some physical and biological systems.

Linear Precoder Design for Correlated Rician Fading Channel Under Imperfect CSI

In this paper,we have modeled a linear precoder for indoor multiuser multiple input multiple output(MU-MIMO)system with imperfect channel state information(CSI)at transmitter.The Rician channel is presumed to be mutually coupled and spatially,temporarily correlated.The imperfection with CSI is primarily due to the channel estimation error at receiver and feedback delay amidst the receiver and transmitter in CSI transmission.Along with,the insufficient spacing between the antenna at transmitter and receiver persuades mutual coupling(MC)among the array elements.In addition,the MIMO channel is presumed to be jointly correlated(Weichselberger correlation model).When we look back on the existing precoder design,it considered spatial correlation alone disregarding joint correlation of antenna array elements.With all above assumption,we have designed a linear precoder which minimizes mean squared error(MSE)subjected to total transmit power constraint for MUMIMO system.The simulation results proven that proposed precoder shows substantial enhancement in bit error rate(BER)performance in comparison with the existing technique.The mathematical analysis corroborates the simulation results.

Stabilization of a class of nonlinear discrete time systems with time varying delay

The stability and stabilization of a class of nonlinear discrete time delayed systems(NDTDS) with time-varying delay and norm-bounded nonlinearity are investigated. Based on discrete time Lyapunov–Krasovskii functional method, a sufficient delaydependent condition for asymptotic stability of nonlinear systems is offered. Then, this condition is used to design a new efficient delayed state feedback controller(DSFC) for stabilization of such systems. These conditions are in the linear matrix inequality(LMI) framework. Illustrative examples confirm the improvement of the proposed approach over the similar cases. Furthermore, the obtained stability and stabilization conditions will be extended to uncertain discrete time delayed systems(UDTDS) with polytopic parameter uncertainties and also with norm-bounded parameter uncertainties.