Nonlinear Delay Systems Research Articles

State‐Feedback Stabilization of Stochastic Nonlinear Systems with Time‐Varying Delay and Different Orders

AbstractThis paper studies the problem of state feedback stabilization for a class of stochastic time‐varying delay nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input. Based on the backstepping design method and the adding of a power integrator technique, a state feedback controller is constructed to ensure the origin of closed‐loop system is globally asymptotically stable in probability. The main design difficulty is how to deal with the different power orders, time‐varying delay and the nonsmooth system perturbations. The efficiency of the state feedback controller is demonstrated by a simulation example.

New incremental Takagi–Sugeno state model for optimal control of multivariable nonlinear time delay systems

In this work, a novel approach based on incremental state models has been proposed for the modeling of multivariable nonlinear delayed systems expressed by a generalized version of Takagi–Sugeno (T–S) fuzzy model. One of the key features of the new approach is that the proposed incremental state model compared with the no incremental one, naturally solves the problem of computing the target state, since for a desired output vector, a zero incremental state can be taken as an objective. Moreover, the control action in an incremental form is equivalent to introduce an integral action, thereby cancelling the steady state errors. Among other advantages using incremental models are the disappearance of the affine terms. Then, a fuzzy based linear quadratic regulator (FLC-LQR) is designed. Furthermore, a new optimal observer for multivariable fuzzy systems is developed, because not all states of the nonlinear system are fully available or measured. A multivariable thermal mixing tank system is chosen to evaluate the robustness of the proposed controller. The results obtained show a robust, well damped response with zero steady state error in the presence of disturbances and modeling errors.

Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback

This paper studies the robust finite-time Ho., control for a class of nonlinear systems with time-varying delay and disturbances via output feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent conditions for the existence of output feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow us to design the output feedback controllers which robustly stabilize the closed-loop system in the finite-time sense. An application to $H_\infty$ control of uncertain linear systems with interval time-varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.

A separation principle for the stabilization of a class of time delay nonlinear systems

In this paper, we establish a separation principle for a class of time-delay nonlinear systems satisfying some relaxed triangular-type condition. Under delay independent conditions, we propose a nonlinear time-delay observer to estimate the system states, a state feedback controller and we prove that the observer-based controller stabilizes the system.

Delay-Dependent Criteria for Absolute Stability of Time-Varying Delay Lurie Control Systems

A new method for constructing a Lyapunov-Razumikhn function to deal with the stability problem of time-varying delay nonlinear uncertain system is presented in this paper. A quadratic trinomial with two variables（ξ,Τ）is obtained, and then the upper bound of the allowable delay Τ can be obtained by solving the optimization problem with varying positive matrix Q. That is to say, we can obtain the optimal combination of Τ and Q matrix.

A new stability result for nonlinear cascade time-delay system and its application in chaos control

Based on the converse Lyapunov stability theorem and invariant set theory, this paper presents a new theorem for the nonlinear cascade delay system. With this new proposed method, a lot of coupled items can be taken as zero items. So the whole system can be converted to a very simple form. Also, a simple chaos control technique is proposed for the uncertain time-delay Lorenz chaotic system via this new method. The controller designed is linear and easy to be implemented. Simulation results for uncertain chaotic systems are provided to illustrate the effectiveness of the proposed scheme.

The generating equations method: Constructing exact solutions to delay reaction–diffusion systems and other non-linear coupled delay PDEs

We present a new method for constructing exact solutions to non-linear systems of coupled delay reaction–diffusion equations of the formut=k1uxx+F(u,u¯,w,w¯),wt=k2wxx+G(u,u¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), and τ is the delay time. The method relies on using generalized and functional separable solutions, obtained for single delay reaction–diffusion equations (called generating equations), to construct exact solutions of more complex systems of coupled delay equations. All of the systems considered contain two or more arbitrary functions of two or three arguments. We show that the generating equations method can also be used for (i) non-linear reaction–diffusion systems with time-varying delay τ=τ(t), (ii) delay reaction–diffusion systems with varying transfer coefficients, (iii) multicomponent delay reaction–diffusion systems, (iv) multidimensional delay reaction–diffusion systems, and (v) non-linear delay systems of higher-order PDEs. The results may be suitable for solving certain model problems and testing approximate analytical and numerical methods for some classes of similar and more complex non-linear coupled delay systems.

Computing the Density Function for a Nonlinear Stochastic Delay System

We develop a numerical method to compute the density of a specific nonlinear stochastic delay system, with no sampling. This system arises as a switch-type control model for human balance. Numerical tests against the Euler-Maruyama method show that our method is capable of computing accurate solutions. In particular, the method captures the covariance of the solution at the present and delayed times. This is accomplished through the time-evolution of a Gaussian approximation of the joint density at the present and delayed times. Issues of circularity prevent the numerical solution of the Fokker-Planck equation for stochastic delay systems. Our method bypasses these issues and offers one of the first deterministic algorithms to compute the density of a nonlinear stochastic delay system.

Photonic delay systems as machine learning implementations

Nonlinear photonic delay systems present interesting implementation platforms for machine learning models. They can be extremely fast, offer great degrees of parallelism and potentially consume far...

Adaptive neural control for a class of nonlinear time-varying delay systems with unknown hysteresis.

This paper investigates the fusion of unknown direction hysteresis model with adaptive neural control techniques in face of time-delayed continuous time nonlinear systems without strict-feedback form. Compared with previous works on the hysteresis phenomenon, the direction of the modified Bouc-Wen hysteresis model investigated in the literature is unknown. To reduce the computation burden in adaptation mechanism, an optimized adaptation method is successfully applied to the control design. Based on the Lyapunov-Krasovskii method, two neural-network-based adaptive control algorithms are constructed to guarantee that all the system states and adaptive parameters remain bounded, and the tracking error converges to an adjustable neighborhood of the origin. In final, some numerical examples are provided to validate the effectiveness of the proposed control methods.

State Feedback Guaranteed Cost Controller for Nonlinear Time-Varying Delay Systems

This paper considers the problem of designing state feedback guaranteed cost controllers for a class of nonlinear systems with time-varying delay. The time delay is a continuous function belonging to a given interval and not required to be differentiable. By applying Lyapunov function method and linear matrix inequality (LMI) technique, new delay-dependent sufficient conditions for designing the state feedback guaranteed cost controller are derived. The method is also extended to handle system uncertainties in a straightforward way. A numerical example is given to illustrate the design procedure.

Stability Properties of Switched Nonlinear Delay Systems with Synchronous or Asynchronous Switching

AbstractThis paper is concerned with the problem of input‐to‐state stability (ISS) for a class of switched nonlinear delay systems. The cases where the switching signal of the system and the switching signal of the corresponding controller are synchronous and asynchronous are both considered. To study two asynchronous switching signals in a unified framework, we adopt the technique of the merging switching signal. Based on a piecewise Lyapunov–Krasovskii functional method, some sufficient conditions are explicitly given to guarantee the ISS of the switched nonlinear delay system under the average dwell time scheme. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed theory.

Simultaneous compensation of input and state delays for nonlinear systems

The problem of compensation of arbitrary large input delay for nonlinear systems was solved recently with the introduction of the nonlinear predictor feedback. In this paper we solve the problem of compensation of input delay for nonlinear systems with simultaneous input and state delays of arbitrary length. The key challenge, in contrast to the case of only input delay, is that the input delay-free system (on which the design and stability proof of the closed-loop system under predictor feedback are based) is infinite-dimensional. We resolve this challenge and we design the predictor feedback law that compensates the input delay. We prove global asymptotic stability of the closed-loop system using two different techniques—one based on the construction of a Lyapunov functional, and one using estimates on solutions. We present two examples, one of a nonlinear delay system in the feedforward form with input delay, and one of a scalar, linear system with simultaneous input and state delays.

Design of adaptive fuzzy-based tracking control of input time delay nonlinear systems

This paper proposes an adaptive fuzzy logic tracking control for unknown nonlinear systems having input time delay. In the controller design, a filtered tracking error is introduced to facilitate handling of the time delay. Approximation capability of fuzzy logic systems are exploited to provide approximations of unknown nonlinear functions that appear in the tracking controller. Lyapunov stability analysis shows that the designed tracking control yields bounded signals of the closed-loop system. Simulation examples are provided to validate the effectiveness of the proposed controller in achieving desired tracking with bounded control signals.

Impulsive control for stabilisation of discrete delay systems and synchronisation of discrete delay dynamical networks

This study investigates impulsive stabilisation of non-linear discrete-time systems with time-delay. By employing Razumikhin technique and Lyapunov functions, several global exponential stability criteria are established. It is shown that, even if the non-linear discrete delay systems, in the absence of impulses, are unstable, impulses can exponentially stabilise the system with an estimated convergence rate. Moveover, these impulsive stabilisation results can be applied to design impulsive control scheme to achieve exponential synchronisation of discrete dynamical networks with coupling time-delays. Synchronisation criteria are obtained by using some simple forms of Lyapunov functions. Some numerical examples are given to illustrate the authors' results.

Optimal Control of a Delayed HIV Infection Model via Fourier Series

We present a delayed optimal control which describes the interaction of the immune system with the human immunodeficiency virus (HIV) and CD4+ T-cells. In order to improve the therapies, treatment and the intracellular delays are incorporated into the model. The optimal control in this model represents the efficiency of drug treatment in preventing viral production and new infections. The optimal pair of control and trajectories of this nonlinear delay system with quadratic cost functional is obtained by Fourier series approximation. The method is based on expanding time varying functions in the nonlinear delay system into their Fourier series with unknown coefficients. Using operational matrices for integration, product, and delay, the problem is reduced to a set of nonlinear algebraic equations.

Output-feedback adaptive neural control for stochastic nonlinear time-varying delay systems with unknown control directions.

This paper presents an adaptive output-feedback neural network (NN) control scheme for a class of stochastic nonlinear time-varying delay systems with unknown control directions. To make the controller design feasible, the unknown control coefficients are grouped together and the original system is transformed into a new system using a linear state transformation technique. Then, the Nussbaum function technique is incorporated into the backstepping recursive design technique to solve the problem of unknown control directions. Furthermore, under the assumption that the time-varying delays exist in the system output, only one NN is employed to compensate for all unknown nonlinear terms depending on the delayed output. Moreover, by estimating the maximum of NN parameters instead of the parameters themselves, the NN parameters to be estimated are greatly decreased and the online learning time is also dramatically decreased. It is shown that all the signals of the closed-loop system are bounded in probability. The effectiveness of the proposed scheme is demonstrated by the simulation results.

Input-to-State Stability for Nonlinear Systems With Large Delay Periods Based on Switching Techniques

The paper is concerned with the problem of input-to-state stability (ISS) for nonlinear delay systems with large delay periods (LDP). The concepts of the length rate of LDP and the frequency of LDP are introduced. First the considered system is converted into a switched delay system which may include an unstable subsystem. Then based on a piecewise Lyapunov functional, the ISS properties of the systems are developed under the constraints of the length rate and the frequency of LDP. ISS for a special kind of nonlinear delay systems is also considered and the conditions of nonlinear matrix inequalities (NLMIs) are proposed to guarantee the ISS properties of such a system. A numerical example and an example of a series DC motor are given to show the effectiveness of the proposed methods.

Networked H∞ filtering for T–S fuzzy systems with quantization and data dropouts

This paper studies the problem of H∞ filtering problem for a class of nonlinear time-varying delay systems with unideal communication links. Two defectives are concerned including measurement quantization and packet dropouts. The quantized measurements are transmitted to the filter via network, where the phenomena of packet loss are taken into account. By using Lyapunov–Krasovskii functional, a less conservative delay-dependent stability condition for the closed-loop NCSs is derived. The H∞ fuzzy controller, which is designed in terms of linear matrix inequalities (LMIs), is developed for the asymptotic stabilization of the closed-loop NCSs. Numerical examples are given to illustrate the effectiveness of the proposed method.

A Discretization Method for the Nonlinear State Delay System

A Discretization Method for the Nonlinear State Delay System