General 9-instant discrete-time Zhang neural network for time-dependent applications

Zhang neural network (ZNN) is widely applied to solving time-dependent problems. For the sake of the implementation on the digital hardware platform, ZNN models need to be discretized. In this paper, as a further study of Zhang et al. discretization (ZeaD) formulas, a novel general 9-instant ZeaD formula is presented, and clear constraints are firstly given with proof. To evaluate the presented 9-instant ZeaD formula, three continuous-time models for time-dependent matrix inversion and pseudoinversion are presented with the help of Getz-Marsden dynamic system (GMDS) and ZNN. Then the corresponding discrete-time models are obtained by using the 9-instant ZeaD formula. According to the comparison experiments, the 9-instant ZeaD formula is substantiated to be effective and consistent with the theory. Furthermore, the problem of mobile angle-of-arrival (AoA) localization is investigated as a more specific and practical problem. In order to overcome the singularity problem of the tangent function in the representation of the AoA localization system, a new representation with sine and cosine functions is presented. Similarly, the continuous-time model is derived and discretized. Through comparison experiments, the discrete-time model obtained by the 9-instant ZeaD formula achieves desirable results, which further show the efficacy of the 9-instant ZeaD formula.     

Recurrent neural dynamics for handling linear equation system with rank-deficient coefficient and disturbance existence

In this paper, for handling discrete-form time-variant linear equation system (DF-TV-LES) with rank-deficient coefficient and disturbance existence, a luminous discrete-time recurrent neural dynamics (DTRND) method is presented. Firstly, the continuous-time recurrent neural dynamics (CTRND) method can be discretized to the DTRND method by using recently-developed 5-instant discretization formula. Secondly, aiming at the situations of rank-deficient coefficient and disturbance existence, corresponding handling methods are presented, respectively. Specifically, on the one hand, under the situation of rank-deficient coefficient, we present an effective method to compute the least-squares solution of DF-TV-LES; on the other hand, under the situation of disturbance existence, integral state of error function is introduced, and then the presented DTRND method possesses a certain performance for restraining different types of disturbances. Finally, comparative numerical experiment substantiates the superiority of the presented DTRND method for handling DF-TV-LES.     

Prescribed-time robust ZNN models for solving equality and inequality systems

At present, there are few studies on solving time-variant linear equality and inequality systems (TVLEIS) under noise interference, and the numerical algorithm has limitations in solving the TVLEIS problems. Therefore, to determine the online solution of the TVLEIS in a complex environment, two prescribed-time robust zeroing neural network (PTRZNN) models are proposed, investigated, and verified in this paper. The PTRZNN models have a faster convergence rate and superior robustness compared with other zeroing neural network models activated by common activation functions. In addition, the detailed theoretical derivation of the prescribed-time convergence and robustness of the PTRZNN models is provided. The effectiveness and superiority of the PTRZNN models for determining the TVLEIS are further demonstrated by simulation results. It is worth mentioning that the design idea of the PTRZNN models is applied to the multi-agent system, which shows the practical value of the PTRZNN models.     

A novel finite-time complex-valued zeoring neural network for solving time-varying complex-valued Sylvester equation

A novel finite-time complex-valued zeroing neural network (FTCVZNN) for solving time-varying Sylvester equation is proposed and investigated. Asymptotic stability analysis of this network is examined with any general activation function satisfying a condition or with an odd monotonically increasing activation function. So far, finite-time model studies have been investigated for the upper bound time of convergence using a linear activation function with design formula for the derivative of the error or with variations of sign-bi-power activation functions to zeroing neural networks. A function adaptive coefficient for sign-bi-power activation function (FA-CSBP) is introduced and examined for faster convergence. An upper bound on convergence time is derived with the two components in the function adaptive coefficients of sign-bi-power activation function. Numerical simulation results demonstrate that the FTCVZNN with function adaptive coefficient for sign-bi-power activation function is faster than applying a sign-bi-power activation function to the zeroing neural network (ZNN) and the other finite-time complex-valued models for the selected example problems.     

Super finite-Time variable parameter ZNN models for time-Variant linear matrix inequality

Stuck in the speed and dimensionality of settling time-variant linear matrix inequality (LMI), this paper for the first time proposes two finite-time variable parameter zeroing neural network (FTVPZNN) models to settle the time-variant LMI. The first model is called the FTVPZNN-C model activated by the conventional sign-bi-power (S-B-P) function. The second model is called the FTVPZNN-T model activated by a tunable parameter S-B-P function. Different from the finite-time fixed-value zeroing neural network (FTFZNN) model, the proposed FTVPZNN models with variable parameters have better convergence performance and smaller upper bounds of finite-time convergence. Three theorems are presented to guarantee the stability and finite-time convergence of the FTVPZNN models. Especially, through detailed theoretical analysis and calculations, the finite-time convergence upper bounds of the proposed FTVPZNN models are obtained. Finally, a numerical simulative example is given to affirm the effectiveness and excellent convergent performance of the proposed models for settling the time-variant LMI.     

Exponential stabilization of switched linear systems subject to actuator saturation with stabilizable and unstabilizable subsystems

This paper is concerned with the exponential stabilization of switched linear systems subject to actuator saturation with both stabilizable subsystems and unstabilizable subsystems for continuous-time case and discrete-time case, respectively. Sufficient conditions for the exponential stabilization under dwell time switching under the cases of continuous-time and discrete-time are established by using a novel class of multiple time-varying Lyapunov function. The existence conditions for stabilizing controllers are presented in terms of linear matrix inequalities (LMIs) for the continuous-time case and the discrete-time case, respectively. Two optimization problems are proposed for obtaining the maximal attraction region. The problem of exponential stabilization for switched system subject to actuator saturation with asynchronous switching controller is also studied. Several numerical examples are presented to prove the validity of the obtained results.     

Adaptive control of printing devices using fractional-order hold circuits adjusted through neural networks

A connectionist method for autotuning the free parameter of a fractional-order hold (FROH) circuit in order to improve the performance of the digitally controlled systems is proposed. Such a technique employs multilayer perceptrons to approximate the mapping between the sampling period/continuous-time parameters of the estimated plant and the optimal value of the FROH adjustable gain. In this way, adaptive discretization systems to improve the stability properties of the resulting discrete-time zeros are implemented. Simulation results are presented in order to illustrate the properties of the complete system applied to two actual digitally controlled printing devices (HP 7090A and low-cost computer printer).     

Implicit and explicit discrete-time realizations of the robust exact filtering differentiator

This paper presents explicit and implicit discrete-time realizations for the robust exact filtering differentiator, aiming to facilitate an adequate posterior implementation structure in digital devices. This paper firstly presents an analysis of an explicit discrete-time realization of the filtering differentiator based on linear systems’ exact discretization with a zero-order holder. For this case, however, high-order terms in the filter dynamics may cause instability of the estimation error for signals with unbounded derivatives. Hence, two other new discrete-time realizations of the filtering differentiator are derived by removing some high-order terms in the filter dynamics. The first one is an explicit discrete-time realization, while the second one is implicit. After a finite time, both preserve the accuracy of the continuous-time robust exact filtering differentiator in the presence of measurement noise. For each proposed discrete-time scheme, a stability analysis based on homogeneity is provided. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete-time realizations with other existing schemes. These numerical studies highlight that the implicit scheme supersedes the explicit one, consistent with the implicit and explicit realizations of other continuous-time algorithms.     

Observer-based output feedback control for networked systems with dual-channel event-triggered sampling and quantization

This paper investigates the problem of observer-based output feedback control for linear networked systems with dual-channel event-triggered mechanisms and quantization. Both continuous-time and discrete-time event detection cases are discussed. In the continuous-time case, the stability of observer error dynamics and closed-loop system are analyzed respectively, and it is proved that Zeno behavior would not occur. In order to approach engineering practice, in the discrete-time case, two types of network attacks including denial-of-service (DoS) and fault data injection (FDI) attacks are considered, whose nature property is characterized by Bernoulli variables. By combining these factors and transmission delay, a novel augmented system model is proposed, and some sufficient conditions are derived based on Lyapunov functional approach and linear matrix inequalities (LMIs). Compared with the existing results, this framework is more comprehensive and practical, and the global uniform ultimate boundedness of closed-loop systems can be guaranteed. Finally, simulation examples are given to demonstrate the effectiveness of the proposed method.     

A robust fast convergence zeroing neural network and its applications to dynamic Sylvester equation solving and robot trajectory tracking

In order to find the theoretical solution of a dynamic Sylvester equation (DSE) in noisy environment, a robust fast convergence zeroing neural network (RFCZNN) is proposed in this paper. Unlike the original zeroing neural network (ZNN) model with existing activation functions (AF), by introducing a new AF, the proposed RFCZNN model guarantees fixed-time convergence to theoretical solution of DSE and robustness against noise simultaneously. The effectiveness and robustness of the proposed RFCZNN model are investigated in theory and demonstrated through simulation results. In addition, its effectiveness and robustness are further verified by a successful robotic trajectory tracking application in noisy environment.     

A modified noise-tolerant ZNN model for solving time-varying Sylvester equation with its application to robot manipulator

Recently, Xiao et al. (2021) proposed an efficient noise-tolerant zeroing neural network (NTZNN) model with fixed-time convergence for solving the time-varying Sylvester equation. In this paper, we propose a modified version of their NTZNN model, named the modified noise-tolerant zeroing neural network (MNTZNN) model. It extends the NTZNN model to a more general form and then we prove that, with appropriate parameter selection, our new MNTZNN model can significantly accelerate the convergence of the NTZNN model. Numerical experiments confirm that the MNTZNN model not only maintains fixed-time convergence and noise-tolerance but also has a faster convergence rate than the NTZNN model under certain conditions. In addition, the design strategy of the MNTZNN is also successfully applied to the path tracking of a 6-link planar robot manipulator under noise disturbance, which demonstrates its applicability and practicality.     

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.     

Enhanced Finite Spectrum Assignment with disturbance compensation for LTI systems with input delay

This paper presents a Finite Spectrum Assignment (FSA) with a generalized feedforward control for Linear Time-Invariant (LTI) systems with input delay and bounded unmeasured disturbances. A novel two-layer feedforward strategy is proposed in order to deal with matched and unmatched disturbances. The proposed control law is based on a filtered disturbance estimator and a generalized feedforward compensation which can be applied to any Artstein based predictor. An optimization design procedure is presented to improve disturbance attenuation properties in the presence of band-limited disturbances. The conditions to achieve disturbance rejection are also shown to deal with deterministic disturbance models. Furthermore, the proposed solution can be used to define either continuous-time or discrete-time control algorithms. Two case studies are presented to illustrate the benefits of the new approach.     

A relaxed MSIO iteration algorithm for solving coupled discrete Markovian jump Lyapunov equations

In this paper, combining the multi-step Smith-inner-outer (MSIO) iteration framework with some tunable parameters, a relaxed MSIO iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. The convergence properties of the relaxed MSIO iteration method are investigated, and the choices of the parameters are also discussed. In order to accelerate the convergence rate of the relaxed MSIO iteration method for solving the CLMEs, a current-estimation-based and a weighted relaxed MSIO iteration algorithms are presented, respectively. Finally, several numerical examples are given to verify the superiorities of the proposed relaxed algorithms.     

Output reversibility in linear discrete-time dynamical systems

Output reversibility involves dynamical systems where for every initial condition and the corresponding output there exists another initial condition such that the output generated by this initial condition is a time-reversed image of the original output with the time running forward. Through a series of necessary and sufficient conditions, we characterize output reversibility in linear discrete-time dynamical systems in terms of the geometric symmetry of its eigenvalue set with respect to the unit circle in the complex plane. Furthermore, we establish that output reversibility of a linear continuous-time system implies output reversibility of its discretization. In addition, we present a control framework that allows to alter the system dynamics in such a way that a discrete-time system, otherwise not output reversible, can be made output reversible. Finally, we present numerical examples involving a discretization of a Hamiltonian system that exhibits output reversibility and an example of a controlled system that is rendered output reversible.     

Design of digital controllers for uncertain chaotic systems using fuzzy logic

A new and systematic method to design digital controllers for uncertain chaotic systems with structured uncertainties is presented in this paper. Takagi-Sugeno (TS) fuzzy model is used to model the chaotic dynamic system, while the uncertainties are decomposed such that the uncertain chaotic system can be rewritten as a set of local linear models with an additional disturbed input. Conventional control techniques are utilized to develop the continuous-time controllers first. Then, the digital controllers are obtained as the digital redesign of the continuous-time controllers using the state-matching approach. The performance of the proposed controller design is illustrated through numerical examples.     

Cooperative output regulation problem of multi-agent systems with stochastic packet dropout and time-varying communication delay

In this paper, we deal with the cooperative output regulation problem of linear multi-agent systems on a directed network topology subject to both stochastic packet dropout and time-varying communication delay. On the basis of introducing a queuing mechanism, a distributed state feedback control algorithm is proposed. Then the continuous-time multi-agent systems with piece-wise constant control are converted into discrete-time systems. Under some standard assumptions, the necessary and sufficient conditions under which the tracking errors of followers approach to the origin asymptotically are proposed for different exosystems. Finally, the proposed results are verified via two examples.     

Q-learning solution for optimal consensus control of discrete-time multiagent systems using reinforcement learning

This paper investigates a Q-learning scheme for the optimal consensus control of discrete-time multiagent systems. The Q-learning algorithm is conducted by reinforcement learning (RL) using system data instead of system dynamics information. In the multiagent systems, the agents are interacted with each other and at least one agent can communicate with the leader directly, which is described by an algebraic graph structure. The objective is to make all the agents achieve synchronization with leader and make the performance indices reach Nash equilibrium. On one hand, the solutions of the optimal consensus control for multiagent systems are acquired by solving the coupled Hamilton–Jacobi–Bellman (HJB) equation. However, it is difficult to get analytical solutions directly of the discrete-time HJB equation. On the other hand, accurate mathematical models of most systems in real world are hard to be obtained. To overcome these difficulties, Q-learning algorithm is developed using system data rather than the accurate system model. We formulate performance index and corresponding Bellman equation of each agent i. Then, the Q-function Bellman equation is acquired on the basis of Q-function. Policy iteration is adopted to calculate the optimal control iteratively, and least square (LS) method is employed to motivate the implementation process. Stability analysis of proposed Q-learning algorithm for multiagent systems by policy iteration is given. Two simulation examples are experimented to verify the effectiveness of the proposed scheme.     

The solutions of vibration control problems using artificial neural networks

This paper introduces an alternative method artificial neural networks (ANN) used to obtain numerical solutions of mathematical models of dynamic systems, represented by ordinary differential equations (ODEs) and partial differential equations (PDEs). The proposed trial solution of differential equations (DEs) consists of two parts: The initial and boundary conditions (BCs) should be satisfied by the first part. However, the second part is not affected from initial and BCs, but it only tries to satisfy DE. This part involves a feedforward ANN containing adjustable parameters (weight and bias). The proposed solution satisfying boundary and initial condition uses a feedforward ANN with one hidden layer varying the neuron number in the hidden layer according to complexity of the considered problem. The ANN having appropriate architecture has been trained with backpropagation algorithm using an adaptive learning rate to satisfy DE. Moreover, we have, first, developed the general formula for the numerical solutions of nth-order initial-value problems by using ANN.For numerical applications, the ODEs that are the mathematical models of linear and non-linear mass-damper-spring systems and the second- and fourth-order PDEs that are the mathematical models of the control of longitudinal vibrations of rods and lateral vibrations of beams have been considered. Finally, the responses of the controlled and non-controlled systems have been obtained. The obtained results have been graphically presented and some conclusion remarks are given.     

Non-fragile control of positive Markovian jump systems

This paper investigates the non-fragile control for positive Markovian jump systems both in continuous-time and discrete-time cases with actuator uncertainty. It is assumed that the coefficient matrices of the non-fragile controller is unknown and bounded. The state-feedback controller gain consists of nominal controller gain and gain perturbation. First, a set of state-feedback controllers for the considered system are designed by using a stochastic co-positive Lyapunov function integrated with linear programming approach. Under the designed controllers, the resulting closed-loop systems are positive and stochastically stable. Then, the proposed controller design approach is extended to discrete-time systems. Through comparisons, it is shown that existing results are special cases of the presented ones in the paper. Finally, two examples are given to illustrate the effectiveness of the proposed design.