Spatiotemporal sampled-data fuzzy control for switched singularly perturbed PDE systems with average dwell-time switching mechanism

This paper investigates a spatiotemporal sampled-data fuzzy control strategy for switched singularly perturbed partial differential equation (PDE) systems, where the systems’ operation modes obey average dwell-time switching mechanism. To efficiently deal with nonlinear terms and guarantee the system stability for the considered systems, a spatiotemporal sampled-data fuzzy control scheme is developed. Furthermore, based on the fact that mode mismatch phenomena during switching and sampling, through formulating novel Lyapunov functionals (LFs) with the discontinuous terms and mode-dependent two-sided looped-functionals, which can fully utilize the state information of the sampling period, a new exponential stability criterion is provided for the target systems. Finally, an example is provided to prove the validity of the proposed control approach.     

Factor gradient iterative algorithm for solving a class of discrete periodic Sylvester matrix equations

At present, gradient iteration methods have been used to solve various Sylvester matrix equations and proved effective. Based on this method, we generalize the factor gradient iterative method (FGI) for solving forward periodic Sylvester matrix equations (FPSME) and backward periodic Sylvester matrix equations (BPSME). To accelerate the convergence of the iterative method, we refer to Gauss-Seidel and Jacobi iterative construction ideas and use the latest matrix information in the FGI iterative method to obtain the modified factor gradient iterative (MFGI) method. Then, the convergence of the proposed methods and the selection of optimal factors are proved. The last numerical examples illustrate the effectiveness and applicability of the iterative methods.     

A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations

The purpose of this paper is to present an iterative algorithm for solving the general discrete-time periodic Sylvester matrix equations. It is proved by theoretical analysis that this algorithm can get the exact solutions of the periodic Sylvester matrix equations in a finite number of steps in the absence of round-off errors. Furthermore, when the discrete-time periodic Sylvester matrix equations are consistent, we can obtain its unique minimal Frobenius norm solution by choosing appropriate initial periodic matrices. Finally, we use some numerical examples to illustrate the effectiveness of the proposed algorithm.     

Impact angle,speed and acceleration control guidance via polynomial trajectory shaping

An Impact Angle, Speed and Acceleration Control Guidance (IASAG) law against the stationary target is proposed, which is critical for the effectiveness of the air-to-surface guided weapons. It is hard to address multiple terminal constraints problem for unpowered missile, especially including terminal speed constraint, which is uncontrollable state. Based on Line-of-Sight (LOS) angle, a fourth-order polynomial function is designed to make the number of coefficients of the function equal to number of boundary conditions. Through analytic calculation and transformation, the relation between the specified boundary conditions and the coefficients are established. The coefficient equations are reduced to a univariate nonlinear equation whose solution is determined by terminal speed constraint. Based on the characteristic of the nonlinear equation, we propose a Particle Swarm Optimization(PSO) method to find the coefficient that satisfies terminal speed constraint. According to Lyapunov stability theory, an asymptotically stable trajectory tracking controller is designed to track the reference leading angle with respect to range-to-go to guarantee the impact angle, speed and acceleration constraints. The effectiveness of the proposed guidance law is verified through numerical simulations.     

Model-Free distributed optimal consensus control of nonlinear multi-agent systems: A graphical game approach

In this paper, the optimal consensus control problem of nonlinear multi-agent systems(MASs) with completely unknown dynamics is considered. The problem is formulated in a differential graphical game approach which can be solved by Hamilton-Jacobi (HJ) equations. The main difficulty in solving the HJ equations lies in the nonlinear coupling between equations. Based on the Adaptive Dynamic Programming (ADP) technique, an VI-PI mixed HDP algorithm is proposed to solve the HJ equations distributedly. With the PI step, a suitable iterative initial value can be obtained according to the initial policies. Then, VI steps are run to get the optimal solution with exponential convergence rate. Neural networks (NNs) are applied to approximate the value functions, which makes the data-driven end-to-end learning possible. A numerical simulation is conducted to show the effectiveness of the proposed algorithm.     

Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equations

This paper focuses on constructing a conjugate gradient-based (CGB) method to solve the generalized periodic coupled Sylvester matrix equations in complex space. The presented method is developed from a point of conjugate gradient methods. It is proved that the presented method can find the solution of the considered matrix equations within finite iteration steps in the absence of round-off errors by theoretical derivation. Some numerical examples are provided to verify the convergence performance of the presented method, which is superior to some existing numerical algorithms both in iteration steps and computation time.     

A robust fractional-order controller design with gain and phase margin specifications based on delayed Bode’s ideal transfer function

In this study, a robust fractional-order controller design methodology for a type of fractional-order or integer-order model with dead time is proposed using phase and gain margin specifications. The delayed Bode’s ideal transfer function is used as a reference model to design the controller analytically. The delay term in delayed Bode’s ideal transfer function provides the exact determination of these frequency domain specifications when the system owns a dead time. The analytical robust controller design problem is transformed to solving four nonlinear equations with four unknown variables, two of which are the desired specifications; namely, phase and gain margins. The remaining two are the phase and gain cross-over frequencies. Next, some conditions are set based on the desired specifications so that nonlinear equations provide a unique solution. The proposed method is compared with the other existing robust controller methods based on the same frequency domain specifications. The simulation results reveal that the proposed method outperforms the other methods and also gives closer outcomes to the desired specifications.     

Finite-time convergent zeroing neural network for solving time-varying algebraic Riccati equations

Various forms of the algebraic Riccati equation (ARE) have been widely used to investigate the stability of nonlinear systems in the control field. In this paper, the time-varying ARE (TV-ARE) and linear time-varying (LTV) systems stabilization problems are investigated by employing the zeroing neural networks (ZNNs). In order to solve the TV-ARE problem, two models are developed, the ZNNTV-ARE model which follows the principles of the original ZNN method, and the FTZNNTV-ARE model which follows the finite-time ZNN (FTZNN) dynamical evolution. In addition, two hybrid ZNN models are proposed for the LTV systems stabilization, which combines the ZNNTV-ARE and FTZNNTV-ARE design rules. Note that instead of the infinite exponential convergence specific to the ZNNTV-ARE design, the structure of the proposed FTZNNTV-ARE dynamic is based on a new evolution formula which is able to converge to a theoretical solution in finite time. Furthermore, we are only interested in real symmetric solutions of TV-ARE, so the ZNNTV-ARE and FTZNNTV-ARE models are designed to produce such solutions. Numerical findings, one of which includes an application to LTV systems stabilization, confirm the effectiveness of the introduced dynamical evolutions.     

A comparative study of numerical methods for solving quadratic Riccati differential equations

In this paper, we use Legendre wavelet method for solving quadratic Riccati differential equations and perform a comparative study between the proposed method and other existing methods. Our results show that in comparison with other existing methods, the Legendre wavelet method provides a fast convergent series of easily computable components. The present study is illustrated by exploring two kinds of nonlinear Riccati differential equations that shows the pertinent features of the Legendre wavelet method.     

Stabilization of highly nonlinear hybrid systems driven by Lévy noise and delay feedback control based on discrete-time state observations

There are many hybrid stochastic differential equations (SDEs) in the real-world that don’t satisfy the linear growth condition (namely, SDEs are highly nonlinear), but they have highly nonlinear characteristics. Based on some existing results, the main difficulties here are to deal with those equations if they are driven by Lévy noise and delay terms, then to investigate their stability in this case. The present paper aims to show how to stabilize a given unstable nonlinear hybrid SDEs with Lévy noise by designing delay feedback controls in the both drift and diffusion parts of the given SDEs. The controllers are based on discrete-time state observations which are more realistic and make the cost less in practice. By using the Lyapunov functional method under a set of appropriate assumptions, stability results of the controlled hybrid SDEs are discussed in the sense of pth moment asymptotic stability and exponential stability. As an application, an illustrative example is provided to show the feasibility of our theorem. The results obtained in this paper can be considered as an extension of some conclusions in the stabilization theory.     

Stability analysis of a class of variable fractional-order uncertain neutral-type systems with time-varying delay

Variable fractional-order (VFO) differential equations are a beneficial tool for describing the nonlinear behavior of complex dynamical phenomena. In comparison with the constant FO derivatives, it describes the memory properties of such systems that can vary in the time domain and spatial location. This article investigates the stability and stabilization of VFO neutral systems in the presence of time-varying structured uncertainties and time-varying delay. FO Lyapunov theorem is adopted to achieve order-dependent and delay-dependent criteria for both nominal and uncertain VFO neutral delay systems. The obtained conditions are given in respect of linear matrix inequality by designing a delayed state feedback controller. Simulations verify the main results.     

Output regulation control for stochastic systems with additive noise

This paper investigates the robust output regulation problem for stochastic systems with additive noises. As is known, for the output regulation control problem, a general method is to regard that the system is disturbed by an autonomous exosystem (which is consisted by external disturbances and reference signals), and for the system disturbed by the white noise, the stochastic differential equations (SDEs) should be utilized in modeling, accordingly, a controller with a feedforward regulator is constructed for the stochastic system with an exosystem, which can not only cancel the external disturbance, but also transform the trajectory tracking problem into the stabilization problem; In consideration of the state variables in stochastic systems cannot be measured completely, we embed an observer to the controller, such that the random interference can be suppressed, and the trajectory tracking can be achieved. Based on the stochastic control theory, the criteria of the exponential practical stability in the mean square is presented for the closed-loop system, finally, through tuning the controller parameters, the mean square of the tracking error can converge to an arbitrarily small neighborhood of the origin.     

Exponential stabilization of semilinear parabolic systems

This work studies the local exponential stabilization of semilinear parabolic systems using multiplicative controls. We build an exponential feedback control that drives the solution to equilibrium. Using the stabilization result, we design a bilinear controller to drive the system solution to the desired state. Applications and simulations are given for the Fitzhugh–Nagumo equation and the heat equation.     

Matrix approach to verification of finite multi-potential games

This paper studies the verification of multi-potential games with a given partition, and proposes a kind of multi-potential equation based on semi-tensor product of matrices. Firstly, the basis of each player’s potential function is constructed, based on which, a new formula is established for the calculation of each player’s potential function. Secondly, a new potential equation is proposed for the verification of potential games via each player’s potential function. Using the new potential equation, a kind of multi-potential equation is constructed for the verification of multi-potential games with a given partition. It is proved that a finite game is a multi-potential game with a given partition if and only if the multi-potential equation has solution. In addition, all possible potential functions are obtained for each group of the partition. An illustrative example is worked out to show the effectiveness of the obtained new results.     

Boundary control for synchronization of fractional-order complex spatiotemporal networks based on PDEs with multiple delays and its application in image encryption

This paper addresses the synchronization problem of fractional-order complex spatiotemporal networks (CSNs) based on partial differential equations with delays via boundary control. First, fractional-order CSNs with time-invariant and time-varying delays are studied separately due to the widespread existence of time delays in complex networks. Moreover, two boundary controllers are proposed to solve the synchronization issue of fractional-order CSNs, in which nodes communicate with each other only on the spatial boundary. Furthermore, according to the fractional-order inequality, the synchronization criteria of fractional-order CNSs with multiple delays are obtained. Finally, the numerical simulations are given to verify the feasibility of the presented results. A case provides the application of CSNs in image encryption.     

Discrete control of nonlinear stochastic systems driven by Lévy process

In this paper, the stabilization is studied for a complex dynamic model which involves nonlinearities, uncertainty, and Lévy noises. This paper also discusses the controller discretization and presents a new algorithm to obtain the upper bound for the sample interval through which the exponential stability of the discrete system can still be guaranteed. Firstly, an integral sliding surface is designed to obtain the sliding mode dynamics for the considered stochastic Lévy process. By using Lyapunov theory, generalized Itô formula and some inequality techniques, the exponential stability is proved in the sense of mean square for sliding mode dynamics. The reachability of the sliding mode surface is also ensured by designing a sliding mode control law. Secondly, the continuous-time controller is discretized from the point of control cost, and the squared difference is analyzed for the states before and after the discretization. Different from those classical stochastic differential equations driven by Brownian motions, the noise is supposed to be Lévy type and the squared difference is analyzed in different cases. Furthermore, we obtain the largest sampling interval through which the discretized controller can still stabilize the Lévy process driven stochastic system. Finally, a simulation for a drill bit system is given to demonstrate the results under the algorithms.     

Optimal control of stochastic functional neutral differential equations with time lag in control

In this work, we consider an optimal control problem of a class of stochastic differential equations driven by additive noise with aftereffect appearing in control. We develop a semigroup theory of the driving deterministic neutral system and identify explicitly the adjoint operator of the corresponding infinitesimal generator. We formulate the time delay equation under consideration into an infinite dimensional stochastic control system without time lag by means of the adjoint theory established. Consequently, we can deal with the associated optimal control problem through the study of a Hamilton–Jacob–Bellman (HJB) equation. Last, we present an example whose optimal control can be explicitly determined to illustrate our theory.     

Stochastic chaining and strengthened information-theoretic generalization bounds

We propose a new approach to apply the chaining technique in conjunction with information-theoretic measures to bound the generalization error of machine learning algorithms. Different from the deterministic chaining approach based on hierarchical partitions of a metric space, previously proposed by Asadi et al., we propose a stochastic chaining approach, which replaces the hierarchical partitions with an abstracted Markovian model borrowed from successive refinement source coding. This approach has three benefits over deterministic chaining: (1) the metric space is not necessarily bounded, (2) facilitation of subsequent analysis to yield more explicit bound, and (3) further opportunity to optimize the bound by removing the geometric rigidity of the partitions. The proposed approach includes the traditional chaining as a special case, and can therefore also utilize any deterministic chaining construction. We illustrate these benefits using the problem of estimating Gaussian mean and that of phase retrieval. For the former, we derive a bound that provides an order-wise improvement over previous results, and for the latter we provide a stochastic chain that allows optimization over the chaining parameter.     

Quantum dynamics of two photons in the coherent feedback system composing of two separated resonators coupling to a waveguide

Quantum coherent feedback based on waveguide quantum electrodynamics plays important roles in quantum devices design and quantum information processing. In this paper, we study the coherent dynamics of the quantum network composed of two separated resonators coupling to a waveguide. Using the dynamical feedback equation in Markovian regime, we obtain the analytical solutions to the output states for the two photons incident from the same or different directions of the waveguide. The two- photon scattering properties are analyzed based on the analytical expressions. It is found that the output states of the two photons can be controlled well by the phase induced by the single photon propagating between the two separated resonators. The results given in this paper are helpful in designing quantum devices at few-photon level.