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This paper deals with a design problem of an adaptive robust controller for a class of nonlinear systems with specified input saturations. For the nonlinear system under consideration, the nonlinearity means unknown perturbations and satisfies the matching condition. In this paper, we show that sufficient conditions for the existence of the proposed adaptive robust controller giving consideration to input saturations are given in terms of linear matrix inequalities (LMIs). Finally, simple illustrative examples are shown.

It is well known that robust control for uncertain dynamical systems is very important topic for the control engineering community, and therefore various robust control problems have been well studied (see [

On the other hand in practical systems, there are some constraints such as some limit of actuators and electric saturations for electronic circuits. If the constraint conditions are violated, they cannot generate the desired response, and at worst the system becomes unstable. From these viewpoints, analysis and/or controller design of dynamical systems with constraint conditions are very important issue, and there are a large number of the existing results such as reachable and controllable sets [

In this paper, on the basis of the work of Oya and Hagino [

This paper is organized as follows. In Section

In this section, notations and useful and well-known lemmas (see [

In the paper, the following notations are used: For a matrix

Next, we show some useful lemmas which are used in this paper.

For arbitrary vectors

Let

For a given constant real symmetric matrix

Let us consider the nonlinear system represented by the following state equation:

Now for the nonlinear system of (

Next we consider a state reachable set

Therefore, our objective in this paper is to develop an LMI-based design procedure of the proposed adaptive robust controller which guarantees the internal stability of the resultant closed-loop system of (

In this section, we show an LMI-based design method of the fixed feedback gain

The time derivative of the quadratic function

Firstly, we consider the case of

On the other hand, if

From the above discussion, in order for the state reachable set

Consequently, if there exist the symmetric positive definite matrix

In Section

From (

Firstly, we consider the 1st term of the right hand side of (

From (

Now, by introducing the complementary variables

From the above discussion, one can see that, for given positive constants

Consequently, our main result is summarized as the following theorem.

Consider the nonlinear system of (

For given positive constants

In this paper, we introduce the assumption

By the way, for the control input

On the other hand, for the case of

In this paper, we deal with a stabilization problem of an adaptive robust controller for a class of nonlinear systems with input constraints. On the other hand, in the work of Oya and Hagino [

In order to demonstrate the efficiency of the proposed robust controller, we have run a simple numerical example.

Let us consider the nonlinear system

Firstly, in order to derive the proposed adaptive robust controller, we select the design parameter

By using the symmetric positive definite matrix

In this example, the initial value of the uncertain nonlinear system of (

The simulation result of this numerical example is shown in Figures

Time histories of

Time histories of

Time histories of

Time histories of

Time histories of

Time histories of

Ellipsoidal set and trajectory of

Ellipsoidal set and trajectory of

From these figures, the proposed robust controller and the conventional adaptive robust controller based on the existing result [

On the other hand, one can see from Figure

Therefore the effectiveness of the proposed adaptive robust controller giving consideration to input constraints for a class of nonlinear systems has been shown.

In this paper, we have proposed an adaptive robust controller for a class of nonlinear systems with input constraints. The proposed design method is based on LMIs, and thus the proposed adaptive robust controller can easily be obtained by using software such as MATLAB and Scilab. In addition, the effectiveness of the proposed robust controller has been shown by simple numerical examples. Note that the proposed adaptive robust controller is an extension of the existing results [

In the proposed adaptive robust control strategy, the control input consists of a state feedback with fixed gains and an adaptive compensation input, and the adaptive compensation input is defined as a state feedback with time-varying adjustable parameters.. The advantages of the proposed adaptive robust control scheme are as follows: the proposed robust controller is more flexible and adaptive than conventional fixed gain robust controller, which is designed for the worst case of unknown parameter variations, and satisfies the given input saturations.

In our future work, we will extend the proposed adaptive robust controller synthesis to such a broad class of systems as linear systems with mismatched uncertainties, uncertain nonlinear systems with time delays, and so on. Furthermore, analysis of the conservativeness of the proposed adaptive robust controller giving consideration to input saturations is also an important issue for our future works. Additionally, the extension of the proposed approach to more general types for constraints such as (

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

The authors would like to thank Professor Mohammad D. Aliyu, Associate Editor, with Ecole Polytechnique de Montreal for his valuable and helpful comments that greatly contributed to this paper.