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This paper is concerned with disturbance observer-based input-output finite-time control of a class of nonlinear systems with one-sided Lipschitz condition, as well as multiple disturbances. Firstly, a disturbance observer is constructed to estimate the disturbance generated by an exogenous system. Secondly, by integrating the estimation of disturbance with a classical state feedback control law, a composite control law is designed and sufficient conditions for input-output finite-time stability (IO-FTS) of the closed-loop system are attained. Such conditions can be converted into linear matrix inequalities (LMIs). Finally, two examples are given to show the effectiveness of the proposed method.

The robust Lyapunov stability reflects the asymptotic behavior; that is, the result only is achieved in an infinite-time interval. However, in many practical applications (for example, in biochemical reaction systems, communication network systems, or robot control systems), one is more interested in what happens over a finite-time interval rather than the asymptotical property. To discuss this transient performance, Dorato [

On the other hand, the complex systems include multiple disturbances, such as unknown frictions or loads, harmonic disturbances, modeling uncertainties, and stochastic noises. The presence of different types of disturbances will seriously affect control accuracy. Therefore, how to design a controller to suppress disturbances is a hot topic. So disturbance observer-based control technique is proposed as an effective approach, and many related meaningful results are presented [

This paper considers disturbance observer-based input-output finite-time control of a class of nonlinear systems with one-sided Lipschitz condition, as well as disturbances. The system model includes two parts of disturbances. One part is a norm-bounded disturbance. The other part is supposed by an exogenous system, which is supposed to have a modeling perturbation. Firstly, a reduced-order disturbance observer is designed to estimate the disturbance generated by this exogenous system. Secondly, a composite control law is designed, which includes the estimation of disturbance and the state feedback control law. Moreover, sufficient conditions are derived to guarantee that the closed-loop system is IO-FTS. Such conditions can be converted into linear matrix inequalities (LMIs). Finally, two examples are given to show the effectiveness of the proposed method.

Consider the following nonlinear system:

The following concepts about Lipschitz property, the one-sided Lipschitz property, and quadratic inner-boundedness property for the nonlinear function

The nonlinear function

The nonlinear function

From Definitions

The nonlinear function

From the definition, any Lipschitz function is quadratically inner-bounded with

The disturbance

In (

The disturbance observer is constructed as

Let the estimation error be

In this work, a class of norm-bounded square integrable signals

Given a time interval

In [

In (

In this section, we will give some sufficient conditions for IO-FTS of the closed-loop system (

Given a scalar

Consider the following Lyapunov functional candidate:

The time derivative of

From (

If (

The proof is completed.

Given a scalar

For (

For (

In this section, two examples are given to illustrate the effectiveness of the proposed scheme.

Consider system (

It is shown that

The matrix

Choose

Therefore, we have

Figures

The weighted system output

Disturbance estimation error

Consider system (

Let

The matrix

Choose

Therefore, we have

Figures

The weighted system output

Disturbance estimation error (

In Example

This paper investigates the problem of disturbance observer-based input-output finite-time control of a class of nonlinear systems with one-sided Lipschitz condition, as well as multiple disturbances. Firstly, a reduced-order disturbance observer is designed to estimate the disturbance generated by this exogenous system. Secondly, by integrating the estimation of disturbance with the classical state feedback control law, a composite control law is designed to guarantee that the closed-loop system is IO-FTS. The obtained sufficient conditions can be converted into linear matrix inequalities (LMIs). Finally, two examples are given to show the effectiveness of the proposed method.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are grateful for the support of the National Natural Science Foundation of China under Grants U1404610, 61473115, and 61374077.