Gradient-free algorithms for distributed online convex optimization

In this paper, we consider the distributed bandit convex optimization of time-varying objective functions over a network. By introducing perturbations into the objective functions, we design a deterministic difference and a randomized difference to replace the gradient information of the objective functions and propose two classes of gradient-free distributed algorithms. We prove that both the two classes of algorithms achieve regrets of $O(T^{3/4})$ for convex objective functions and $O(T^{2/3})$ for strongly convex objective functions, with respect to the time index $T$ and consensus of the estimates established as well. Simulation examples are given justifying the theoretical results.     

Linear Diophantine equation (LDE) decoder: A training-free decoding algorithm for multifrequency SSVEP with reduced computation cost

Multifrequency steady-state visual evoked potentials (SSVEPs) have been developed to extend the capability of SSVEP-based brain-machine interfaces (BMIs) to complex applications that have large numbers of targets. Even though various multifrequency stimulation methods have been introduced, the decoding algorithms for multifrequency SSVEP are still in early development. The recently developed multifrequency canonical correlation analysis (MFCCA) was shown to be a feasible training-free option to use in decoding multifrequency SSVEPs. However, the time complexity of MFCCA is shown to be $O(n^3)$, which will lead to long computation time as $n$ grows, where $n$ represents the input size in decoding. In this paper, a novel decoding algorithm is proposed with the aim to reduce the time complexity. This algorithm is based on linear Diophantine equation solvers and has a reduced computation cost $O(nlogn)$ while remaining training-free. Our simulation results demonstrated that linear Diophantine equation (LDE) decoder run time is only one fifth of MFCCA run time under respective optimal settings on 5-s single-channel data. This reduced computation cost makes it easier to implement multifrequency SSVEP in real-time systems. The effectiveness of this new decoding algorithm is validated with nine healthy participants when using dry electrode scalp electroencephalography (EEG).     

On the regional control problem of the bilinear wave equation

The present research deals with regional optimal control problem of the bilinear wave equation evolving on a spatial domain $\mathrm{\Omega }\subset {\mathrm{ℝ}}^n,n\ge 1$. Such an equation is excited by bounded controls that act on the velocity term. It addresses the tracking of a desired state all over the time interval $[0,T]$ only on a subregion $\omega$ of $\mathrm{\Omega }$ with minimum energy. Then, we prove that an optimal control exists and is characterized as a solution to an optimality system. Algorithm for the computation of such a control is given and successfully illustrated through simulations.     

Dynamic event-triggered distributed interval functional observers for interconnected systems

In this paper, we design dynamic event-triggered interval functional observers (FOs) for interconnected systems comprising $M$ $(M\ge 2)$ subsystems where each subsystem is subject to nonlinearities and output disturbances. Our design method consists of two main steps. First, we design decentralized dynamic event-triggered mechanisms (ETMs) which use only locally measured output information. We then consider the design of distributed interval FOs by using the newly proposed ETMs. Their existence conditions are established and formulated in terms of linear programming. We also derive a bound on the estimated error vector and show that this bound is the smallest. Thus, this ensures that the unknown linear functional state vector can be estimated within an upper and lower bound of its true value by the designed interval observers. Finally, we apply the obtained results to design dynamic event-triggered interval observers for linear functions of the state vectors of an $N$-machine power system.     

Active fault detection and isolation in linear time-invariant systems: A geometric approach

In this work, a novel approach on active fault detection and isolation for linear time-invariant systems, named forced diagnosability, is proposed. This approach computes a continuous state feedback law to render a fault diagnosable, even when it cannot be diagnosed by using passive diagnosis methods. To do that, this work derives novel geometric relationships between unobservability and $(A,B)$-invariant subspaces that, under certain conditions, guarantee the existence of such state feedback law. The objective of the state feedback law is to force all the faults, except the one required to be diagnosed, named $L_d$, to reside in an unobservability subspace. This effectively decouples the effect of $L_d$ on the system output, from the effect of the other faults, allowing the design of a residual generator to detect and isolate the desired fault. The proposed state feedback law continuously forces diagnosability, and it can be computed in polynomial time. This avoids testing faults only at fixed time intervals and solving complex optimization problems required in other active diagnosis approaches. A numerical example is presented to illustrate the efficiency of the proposed approach.     

Lumped uncertainty alleviation in output channels of MIMO nonlinear systems based on robust disturbance observer-based control strategy

Disturbances and uncertainties can produce unsatisfactory responses in many industrial and engineering systems. Besides, the practical systems and processes are multiple-input multiple-output (MIMO). Hence, achieving a good control performance with adequate output responses is not simple. Many different methods were provided for control of industrial processes in some references. However, in this paper, the primary goal is to design an appropriate tracking controller for alleviating the destructive effects of uncertainties in output channels of MIMO nonlinear systems. For this purpose, a robust mechanism has been introduced according to the optimal design of centralized extended proportional-derivative (CEPD) and disturbance observer (DOB). By designing the derivative part $K_d$ based on famous Vandermonde matrix and DOB gain $\mathrm{\Gamma }$, the robust criterion $R={\left(I+CG{K}_d\right)}^{-1}$ is obtained to tackle the undesirable factors such as nonlinear functions and uncertainties in error dynamics. The closed-loop stability is guaranteed by tuning the proportional part $K_p$ under linear matrix inequality. The proposed scheme in this paper can be used for a wide range of MIMO nonlinear systems in practical situations.     

Interval estimation and fault detection for switched nonlinear systems based on zonotope method

This paper investigates the state interval estimation and fault detection (FD) problems for a Lipschitz nonlinear switched system based on the combination of the $H_{\infty }$ observer and the zonotope method. To begin with, an $H_{\infty }$ observer that is robust to the disturbance is designed, and the stability of the observer error dynamic system is analyzed under the average dwell time (ADT) concept. After this, the zonotope theory is applied on the $H_{\infty }$ observer error dynamic system such that the interval state estimation can be calculated iteratively when the system suffers from no fault. Furthermore, for FD purpose, a residual is constructed based on the $H_{\infty }$ observer. Because the constructed residual contains disturbance, it cannot be used for FD directly. To overcome this drawback, the interval estimation method of the residual is proposed by using the zonotope method, and a residual-based FD scheme is developed. Finally, a simulation example is given to verify the effectiveness of the proposed method.     

An improved output feedback controller design for linear discrete-time systems using a matrix decomposition method

This paper presents the design of output feedback controllers for discrete-time (DT) linear systems. New sufficient LMI conditions are derived for designing static $H_2$ and $H_{\infty }$ controllers using decomposition of an auxiliary matrix. The decomposition facilitates linearization of nonlinear term of reduced size to obtain linear matrix inequality criteria. This leads to less conservative results as shown in the numerical examples. In addition, the proposed static output feedback criteria is also used for designing dynamic output feedback controllers for DT systems. Furthermore, a comparative study is also made for the proposed design method with the results existing in the literature. Finally, a DT static output feedback $H_{\infty }$ controller is designed for a quarter-car suspension system. Simulation results are provided to show the efficacy of the proposed design method.     

H∞$$ {H}_{\infty } $$ constraint Pareto optimal control for discrete-time Markov jump linear stochastic systems in finite horizon

This paper mainly focuses on discrete-time Markov jump linear stochastic systems (DMJLSSs) with external disturbances and multiplicative noise. Due to the multi-player in the systems and multi-objective cost function, the strategy design for these systems is more difficult than the traditional Markov jump stochastic systems. We combine $H_{\infty }$ theory with Pareto control and use the weighted sum method of the objective functional of each player to solve the problem of Pareto cooperative game. Four coupled difference matrix-valued recursion equations (CDMREs) are derived by stochastic bounded real lemma (SBRL). Then, a sufficient and necessary condition for the existence of the Pareto optimal strategy is obtained, and the Pareto solution for every controller is derived under this strategy. A simulation example is given to verify the correctness of the theory.     

H∞ containment of networked MASs with disturbances,position, and velocity constraints

The objective is to achieve the constrained containment of discrete-time multiagent systems with external disturbances in directed networks. In contrast to the existing constrained containment results that ignore disturbances, this work incorporates the disturbances, the position, and velocity constraints of followers simultaneously. A projection-based control protocol is developed to pull all followers into a target area formed by some leaders, and meanwhile, a certain level of disturbance attenuation performance is satisfied. By utilizing the multiple model transformations, convexity analysis, and the Lyapunov approach, a sufficient condition is derived to realize the constrained containment with $H_{\infty }$ performance. Some simulation results are provided to demonstrate the disturbance-rejection performance of the proposed control method.     

Convergence on iterative learning control of Hilfer fractional impulsive evolution equations

In this paper, impulsive fractional differential equations with Hilfer fractional derivatives of order and type $0\le \nu \le 1$ is considered. Convergence analysis of $P$-type and $P{I}^{\mu }$-type open-loop iterative learning scheme is studied in the sense of $\lambda$-norm. Examples are provided to explain the theory developed.     

Event-triggered H∞ control of discrete-time switched linear systems with delays

This paper studies the $H_{\infty }$ control problem of discrete-time delayed switched systems with average dwell time switching signals using event-triggered mechanism. The event-triggered mechanism can reduce redundant information transmission. First, employing state-feedback controller, two sufficient conditions are proposed to guaranteeing (i) exponential stability of the disturbance-free closed-loop system and (ii) $H_{\infty }$ performance of the closed-loop system with disturbance. Moreover, the controller gain is explicitly computed by solving a set of linear matrix inequalities. Second, results obtained for state-feedback controller are extended to those for observer-based state-feedback controller, and both the controller and observer gains are explicitly computed. All conditions are derived by utilizing a properly constructed decay-rate-dependent Lyapunov functional. Finally, a numerical example illustrates the validity of the obtained theoretical results.     

An improved nonlinear extended state observer with adaptive variable gain

This paper proposes an improved nonlinear extended observer with adaptive gain (ANESO), which can be applied to systems with large disturbance amplitude changes without parameter re-tuning. An ANESO is added an adaptive factor $A_i$ to the gain of the original NESO. The expression of $A_i$ is obtained by assuming that nonlinear extended states observer (NESO) and linear extended states observer (LESO) has the same characteristic of steady-state error when the amplitude of disturbance turns. On this basis, an adaptive adjustment algorithm based on tracking error is designed. In addition, the stability of the proposed ANESO is analyzed by using the Routh-Hurwitz criterion. Finally, through simulation experiments, the observation performance of ANESO was compared with NESO and switching extended state observer (SESO). The experimental results demonstrate that the observation performance of ANESO is hardly affected by the disturbance amplitude, and moreover, its observation precision is higher than that of NESO and SESO.     

Bifurcation and control of disease spreading networks model with two delays

In this paper, the dynamical behaviors are investigated for a complex network with two independent delays. Instead of taking time delays as bifurcation parameters, we choose probability $p$ and parameter $\mu$ as the control parameters to study their effects on local stability and Hopf bifurcation, respectively. Moreover, the conditions for generating Hopf bifurcation are given. Furthermore, we further discuss the effects of two time delays on the critical values of parameters $p$ and $\mu$. Finally, numerical simulations are used to illustrate the validity of the obtained results.     

Observer-based event-triggered H-infinity sliding control of Markovian jump system suffer from actuator attacks

This article investigates the issue of observer-based $H_{\infty }$ sliding mode control for Markovian jump systems suffer from actuator attacks through an adaptive technique. During the communication channel from the plant output to estimator, a dynamic event-triggered generator is employed in enhancing communication efficiency. Taking consideration of malicious attacks on the plant actuator, an adaptive compensator is put forward for security purposes. By designing a state observer, the desired sliding mode dynamics can be derived based on an integral-type sliding surface. Further, maintaining the sliding motion with uncertain mode information is ensured in finite time by proposing a feasible sliding mode control law. In addition, both stochastic stability and $H_{\infty }$ performance conditions are established for closed-loop systems in terms of linear matrix inequalities. Finally, a numerical example is offered to illustrate the validity of the constructed strategy.     

An optimal robust design method for fractional-order reset controller

In this paper, a novel design method for the reset controller structure (i.e., fractional-order proportional and integral plus Clegg integrator (PI ${}^{\alpha }$ + CI ${}^{\alpha }$)), is proposed for a second-order plus time delay plant. To this end, the designer can get an optimal fractional reset controller that gives the control system more phase margin over the base linear PI controller and robust to loop gain variation. The describing function method is used to investigate the capability of phase lead and the frequency domain properties of PI ${}^{\alpha }$ + CI ${}^{\alpha }$. The gain crossover frequency and phase margin specifications ensure the stability of the control system, and the flat phase constraint makes the control system robust to loop gain variations. Meanwhile, the integral of time and absolute error (ITAE) value is applied to achieve the optimal dynamic performance as the cost function. PI ${}^{\alpha }$ + CI ${}^{\alpha }$ is compared with its integer-order counterpart (i.e., proportional and integral plus Clegg integrator (PI + CI) controller) and their base controllers (i.e., integer-order PI and fractional-order PI controllers) in terms of the step response and robustness to loop gain variations. The simulation results illustrate that the PI ${}^{\alpha }$ + CI ${}^{\alpha }$ control system obtains lower overshoot and oscillation and better robustness to loop gain variations than others. The experiments are performed on the speed control of an air bearing stage. Experimental results show that the designed PI ${}^{\alpha }$ + CI ${}^{\alpha }$ control system behaves better than others. The proposed PI ${}^{\alpha }$ + CI ${}^{\alpha }$ design method can be applied to other general control plants easily.     

Observer-based finite-time H∞ control of Itô-type stochastic nonlinear systems

This paper investigates the finite-time $H_{\infty }$ control of Itô-type stochastic nonlinear systems. Considering the transient performance and anti-interference ability within a given limited time interval, the control strategy of stochastic nonlinear systems is researched. A new sufficient condition for mean-square finite-time boundedness for stochastic nonlinear system is developed. Due to the state components are not available, a state observer is designed. Based on the estimated state, an $H_{\infty }$ controller is proposed and Linear Matrix Inequality (LMI) conditions are given, which not only guarantee the mean-square finite-time boundedness but also satisfy the corresponding $H_{\infty }$ performance. Finally, two examples are given to demonstrate the effectiveness of the results.     

Risk-sensitive large-population linear-quadratic-Gaussian Games with major and minor agents

This paper studies large-population dynamic games involving a linear-quadratic-Gaussian (LQG) system with an exponential cost functional. The parameter in the cost functional can describe an investor's risk attitude. In the game, there are a major agent and a population of $N$ minor agents where $N$ is very large. The agents in the games are coupled via both their individual stochastic dynamics and their individual cost functions. The mean field methodology yields a set of decentralized controls, which are shown to be an $ϵ$-Nash equilibrium for a finite $N$ population system where $ϵ=O\left(\frac{1}{\sqrt{N}}\right)$. Numerical results are established to illustrate the impact of the population's collective behaviors and the consistency of the mean field estimation.     

Distributed dynamic matrix control with constrained optimization for collision and obstacle avoidance of simulated multiple quadcopters

A system of fast moving quadcopters has a high risk of collisions with neighboring quadcopters or obstacles. The objective of this work is to develop a control strategy for collision and obstacle avoidance of multiple quadcopters. In this paper, the problem of distributed dynamic matrix control (DMC) for collision avoidance among a team of multiple quadcopters attempting to reach consensus in the horizontal plane and yaw direction ( $x,y$, and $\psi$) is investigated. Violations of a predetermined safety radius generates output constraints on the DMC optimization function, which has not been dealt with in the literature. Different from past works, the proposed strategy can perform collision avoidance in the $x$, $y$, and $z$-directions. In addition, logarithmic barrier functions are implemented as input rate constraints on the control actions. Extensive simulation studies for a team of quadcopters illustrate promising results of the proposed control strategy and case variations. In addition, DMC parameter effects on the system performance are studied, and a successful study for obstacle avoidance is presented.