Toshiki Oguchi

Input-output linearization of retarded non-linear systems by using an extension of Lie derivative

This paper considers the input-output linearization problem for retarded non-linear systems, which have time-delays in the state. By using an extension of the Lie derivative for functional differential equations, we derive a coordinates transformation and a static state feedback to obtain linear input-output behaviour for a class of retarded non-linear systems. The obtained coordinates transformation is allowed to contain not only the current value of the state variables but also the past values of ones. In addition, we show that the coordinates transformation is invertible in a neighbourhood of the origin and examine the stability condition of the closed loop system with the static state feedback. The effectiveness of the proposed technique is demonstrated through numerical simulations.

Synchronization in networks of chaotic systems with time-delay coupling.

In this paper, we consider synchronization of N identical nonlinear systems unidirectionally or bidirectionally coupled with time delay. First we show, using the small-gain theorem, that trajectories of coupled strictly semi-passive systems converge to a bounded region. Next, we consider the network structure under which the synchronization error dynamics has a trivial solution at zero and derive a necessary condition for synchronization with respect to the network structure. Using these facts, we then derive sufficient conditions for synchronization of the systems in terms of linear matrix inequalities via the Lyapunov-Krasovskii functional approach. The obtained results are illustrated on networks of Lorentz systems with coupling delay.

Prediction of chaotic behavior

This paper considers the prediction of chaotic behavior using a master-slave synchronization scheme. Based on the stability theory for retarded systems using a Lyapunov-Krasovskii functional, we derive a sufficient condition for perfect state prediction of the master system via a time-delayed output signal of the slave system. The obtained result is based on the delay-dependent stability of time-delay systems. In addition, we derive an upper bound of the admissible time delay by using linear matrix inequality techniques. Finally, we show the effectiveness of the proposed predictor by two numerical examples.

Partial synchronization in diffusively time-delay coupled oscillator networks.

We study networks of diffusively time-delay coupled oscillatory units and we show that networks with certain symmetries can exhibit a form of incomplete synchronization called partial synchronization. We present conditions for the existence and stability of partial synchronization modes in networks of oscillatory units that satisfy a semipassivity property and have convergent internal dynamics.

Input-output linearization of nonlinear systems with time delays in state variables

The present paper is concerned with nonlinear systems that contain delays inside coupled with a part of state variables, which are often the cases in practical problems, but have not been treated yet. First we introduce an extension of the Lie derivative for a difference-differential equation; then we derive necessary and sufficient conditions for existence of a nonlinear feedback that linearizes the input-output behaviour of a system and decouples it from the delayed variables simultaneously. Discussions are given for two cases: firstly when the linearizing feedback contains only current values of state variables, and secondly when the linearizing feedback has memories to utilize the past values as well as the current values of state variables.

Sliding-Mode Control of Retarded Nonlinear Systems Via Finite Spectrum Assignment Approach

In the present study, a sliding-mode control design method based on the finite spectrum assignment procedure is proposed. The finite spectrum assignment for retarded nonlinear systems can transform retarded nonlinear systems into delay-free linear systems via a variable transformation and a feedback, which contain the past values of the state. This method can be considered to be an extension of both the finite spectrum assignment for retarded linear systems with controllability over polynomial rings of the delay operator and the exact linearization for finite dimensional nonlinear systems. The proposed method is to design a sliding surface via the variable transformation used in the finite spectrum assignment and to derive a switching feedback law. The obtained surface contains not only the current values of the state variables but also the past values of the state variables in the original coordinates. The effectiveness of the proposed method is tested by an illustrative example

A finite spectrum assignment for retarded non-linear systems and its solvability condition

This paper considers a finite spectrum assignment problem for retarded non-linear systems. First, using extensions of the Lie derivative and the Lie bracket for differential difference equations, we propose a finite spectrum assignment procedure for a class of retarded non-linear systems. Then we derive a necessary and sufficient condition for the solvability of a class of finite spectrum assignment problems for retarded non-linear systems. The obtained condition is an extension of the condition for the exact linearization of finite dimensional non-linear systems and the finite spectrum assignment of retarded linear systems with controllability over polynomial rings.

A SYNCHRONIZATION CONDITION FOR COUPLED NONLINEAR SYSTEMS WITH TIME-DELAY: A FREQUENCY DOMAIN APPROACH

This paper considers the synchronization problem for nonlinear systems with time-delay couplings. We assume that the error dynamics can be rewritten as a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for time-delay coupled systems by applying the multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain stability condition and avoids the use of the Lyapunov–Krasovskii approach. As a result, the condition based on the circle criterion does not contain the conservativeness caused by the Lyapunov–Krasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling.

Predictor-Based Tracking Control of a Mobile Robot with Time-Delays

In this paper, we consider the tracking control problem for a two-wheel mobile robot which has a feedback loop that passes through a computer network. The use of a computer network in the feedback loop causes several problems, but we focus particularly on the effect of time-delays caused by the data transmission. To compensate the performance degradation due to the delays, we propose a tracking control scheme with a state predictor based on anticipating synchronization. A sufficient condition for the convergence of the prediction error is also derived by the Lyapunov-Razumikhin approach. Numerical simulation results show that the proposed control scheme is effective for driftless non-holonomic systems such as the mobile robot.

Input-output linearization of retarded nonlinear systems by an extended Lie derivative

This paper considers the input-output linearization problem for retarded nonlinear systems which have time-delays in the state. By using an extension of the Lie derivative for functional differential equations, we derive a coordinate transformation and a state feedback to obtain the linear input-output behavior for a class of retarded nonlinear systems. We also examine the stability condition of the total systems. The effectiveness of the proposed technique is demonstrated through numerical simulations.