Adaptive stabilization of time-delay feedforward nonlinear systems

In this paper, we consider the adaptive stabilization problem for feedforward nonlinear systems with time delays. An adaptive stabilizer is proposed. Our stabilizer takes a nested saturation feedback, and a set of switching logics is designed to tune online the saturation levels in a piecewise constant or switching manner. It has been shown that under our proposed control, all closed-loop states are bounded and asymptotic regulation is achieved.     

Design of stabilizing controllers of upper triangular nonlinear time-delay systems

In this paper, it is considered the state feedback controller design for a class of upper triangular nonlinear systems with simultaneous input and state delays. By using the state transformation of nonlinear systems, the problem of designing controller can be converted into that of designing a dynamic parameter, which is dynamically regulated by a dynamic equation. Then, by appraising the nonlinear terms of the given systems, a dynamic equation can be delicately constructed. At last, with the help of Lyapunov stability theorem, it is provided the stability analysis for the closed-loop system consisting of the designed controller and the given systems. Both discrete delays and continuous delays with integral form are considered here. Different from many existing control designs for upper triangular nonlinear systems, neither forwarding recursive nor saturation computation is utilized here, and thus our design procedure is simpler. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

Output feedback control of large-scale nonlinear time-delay systems in lower triangular form

This paper is concerned with the stabilization problem for a class of large-scale nonlinear time-delay systems in lower triangular form. The uncertain nonlinearities are assumed to be bounded by continuous functions of the outputs or delayed outputs multiplied by unmeasured states or delayed states. An observer based output feedback control scheme is proposed using the dynamic gain control design approach. Based on Lyapunov stability theory, global asymptotic stability of the closed-loop control system is proved. Contrary to many existing control designs for lower triangular nonlinear systems, the celebrated backstepping method is not utilized here. An example is finally given to demonstrate the effectiveness of the proposed design procedure.     

Robust global stabilization of a class of uncertain feedforward nonlinear systems

The problem of global asymptotically stabilizing a certain class of uncertain feedforward nonlinear systems is considered. The control law is obtained by nesting saturation functions whose amplitude can be rendered arbitrarily small. With respect to previous works on the subject the design procedure is able to deal with uncertain (possibly time-varying) parameters ranging within the prescribed compact sets which can affect also the linear approximation of the system. The small gain theorem for nonlinear systems which are input to state stable “with restrictions” is shown to be a key tool for designing a state feedback saturated control law.     

Universal stabilization of feedforward nonlinear systems

In recent years, many important results on global stabilization of feedforward nonlinear systems have been proposed. However, in all previous results, to develop a global stabilizer, one must know some a priori information on system nonlinearities. For example, it is required that the system nonlinearities be known, or be bounded by known functions, or be linearly parametrized, or satisfy some high order growth rate condition, etc. The contribution of this paper is proposing a universal global stabilizer design for a class of feedforward systems, which does not require a priori knowledge of system nonlinearities.     

Global stabilization of high-order nonlinear time-delay systems by state feedback

We consider the problem of global stabilization by state feedback for a class of high-order nonlinear systems with time-delay. By developing a novel dynamic gain-based backstepping approach, a state feedback controller independent of the time-delay is explicitly constructed with the help of appropriate Lyapunov–Krasovskii functionals. The precise knowledge (even the upper bound) of the time-delay is not required. It is proved that the states of the nonlinear time-delay systems can be regulated to the origin while all the closed loop signals are globally bounded. Finally, both physical and academic examples are given to illustrate the applications of the proposed scheme.     

Global asymptotic stabilization of feedforward nonlinear systems with a delay in the input

In this paper, by constructing appropriate Lyapunov–Krasovskii functionals (LKF) and applying the model transformation of time-delay systems, a design scheme of state feedback controller for a class of feedforward nonlinear systems with a delay in the input is proposed. The designed controllers have a very simple structure and do not involve any saturation or recursive computation, which is widely applied in designing a controller of feedforward nonlinear systems. Using the transformation of coordinates and the property of Hurwitz polynomial, the problem of designing controller can be converted into the problem of finding a parameter, which can be solved by solving the optimization problem with linear matrix inequalities (LMIs) constraints. A simulation example is given to show the effectiveness of the proposed design procedure.     

Reliable stabilization of stochastic time-delay systems with nonlinear disturbances

This paper is concerned with the stabilization problem for a class of continuous stochastic time-delay systems with nonlinear disturbances, parameter uncertainties and possible actuator failures. Both the stability analysis and synthesis problems are considered. The purpose of the stability analysis problem is to derive easy-to-test conditions for the uncertain nonlinear time-delay systems to be stochastically, exponentially stable. The synthesis problem, on the other hand, aims to design state feedback controllers such that the closed-loop system is exponentially stable in the mean square for all admissible uncertainties, nonlinearities, time-delays and possible actuator failures. It is shown that the addressed problem can be solved in terms of the positive definite solutions to certain algebraic matrix inequalities. Numerical examples are provided to demonstrate the effectiveness of the proposed design method.     

Global adaptive tracking for a class of nonlinear time-delay systems

This paper aims to extend the newly developed global adaptive regulation scheme to adaptive tracking for a general class of nonlinear systems with both parameter uncertainties and unknown time delay for all the system states. With a simple yet novel decomposition of some coupling functions derived from the introduction of a reference signal, a backstepping tracking control strategy is proposed which possesses the feature that for each design step, a dynamic gain is introduced to generate an additional term to counteract the system nonlinearities and parameter uncertainties. As a result, a memoryless adaptive state-feedback controller is obtained to achieve the global adaptive tracking.     

Feedback stabilisation of time-delay nonlinear systems with continuous time-varying output function

This paper investigates the problem of global output feedback stabilisation for a class of uncertain nonlinear systems, where output function is time-varying and continuous, and multiple time delays exist in system state at the same time. A double-domination method is used to capture time-varying measurement error and delayed states. The control strategy is presented based on the choice of Lyapunov–Krasovskii functionals and the construction of a novel state observer without using the information on the output function and nonlinearities. A simulation example is given to show the efficiency of the proposed control scheme.     

Asymptotic stabilization via output feedback for nonlinear systems with delayed output

We propose a novel and simple design scheme of output feedback controller for a class of nonlinear systems with delayed output. The nonlinear systems considered here are more general than feedforward systems (upper triangular systems). By constructing an appropriate Lyapunov–Krasovskii functional (LKF) and solving linear matrix inequalities (LMIs), the delay-dependent controller making the closed-loop system globally asymptotically stable (GAS) is explicitly constructed. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

Finite-time stability and stabilization of time-delay systems

Finite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artstein’s model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input.     

On stabilization for a class of nonlinear stochastic time-delay systems： a matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally （globally） robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also given.     

On stabilization for a class of nonlinear stochastic time-delay systems:a matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally (globally) robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also eiven.     

Finite‐time output feedback stabilization of nonlinear high‐order feedforward systems

This paper considers the global finite‐time output feedback stabilization of a class of nonlinear high‐order feedforward systems. By using the homogeneous domination method together with adding a power integrator method and overcoming several troublesome obstacles in the design and analysis, a global finite‐time output feedback controller with reduced‐order observer is recursively designed to globally finite‐time stabilize nonlinear high‐order feedforward systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive observers for time-delay nonlinear systems in triangular form

A simple nonlinear observer with a dynamic gain is proposed for a class of bounded-state nonlinear systems subject to state delay. By saturating the states of the observer nonlinearities with either symmetric or non-symmetric saturation functions, we show that the observer exists, whatever the delay is. Furthermore, it will be highlighted that the observer design is free from any preliminary analysis of the time-delay system such as estimating the Lipschitz constants of nonlinearities. The proposed design encompasses a wide class of nonlinear and time-delay systems written in triangular form and generalizes previous results on delayless nonlinear systems.     

Asymptotic stabilization of dynamically quantized nonlinear systems in feedforward form

This paper studies the stabilizability of an n-dimensional quantized feedforward nonlinear system. The state of that system is first quantized into a finite number of bits, and then sent through a digital network to the controller. We want to minimize the number of transmitted bits subject to maintaining asymptotic stability. In the prior literature, n bits are used to stabilize the n-dimensional system by assigning one bit to each state variable (dimension). Under the stronger assumption of global Lipschitz continuity, this paper extends that result by stabilizing the system with a single bit. Its key contribution is a dynamic quantization policy which dynamically assigns the single bit to the most “important” state variable. Under this policy, the quantization error exponentially converges to 0 and the stability of the system can, therefore, be guaranteed. Because 1 is the minimum number of quantization bits (per sampling step), the proposed dynamic quantization policy achieves the minimum stabilizable bit number for that n-dimensional feedforward nonlinear system.     

Global stabilization of feedforward systems with exponentially unstable Jacobian linearization

The global stabilization of a class of feedforward systems having an exponentially unstable Jacobian linearization is achieved by a high-gain feedback saturated at a low level. The control law forces the derivatives of the state variables to small values along the closed-loop trajectories. This “slow control” design is illustrated with a benchmark example and its limitations are emphasized.     

An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type

An eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented, which extends the recently developed continuous pole placement method for delay equations of retarded type. The approach consists of two steps. First the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots, which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation. No restrictions are made on the dimension of the system and the number of delays. Also the effect of small delay perturbations is explicitly taken into account. As a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots. As a second step, stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi-continuous way, by applying small changes to the controller parameters, and meanwhile monitoring other characteristic roots with a large real part. A numerical example is presented.     

Filtering on nonlinear time-delay stochastic systems

In this paper, we address the filtering problem for a general class of nonlinear time-delay stochastic systems. The purpose of this problem is to design a full-order filter such that the dynamics of the estimation error is guaranteed to be stochastically exponentially ultimately bounded in the mean square. Both filter analysis and synthesis problems are considered. Sufficient conditions are proposed for the existence of desired exponential filters, which are expressed in terms of the solutions to algebraic Riccati inequalities involving scalar parameters. The explicit characterization of the desired filters is also derived. A simulation example is given to illustrate the design procedures and performances of the proposed method.