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We present a new adaptive gain robust controller for polytopic uncertain systems. The proposed adaptive gain robust controller consists of a state feedback law with a fixed gain and a compensation input with adaptive gains which are tuned by updating laws. In this paper, we show that sufficient conditions for the existence of the proposed adaptive gain robust controller are given in terms of LMIs. Finally, illustrative examples are presented to show the effectiveness of the proposed adaptive gain robust controller.

In order to design control systems, it is necessary to derive a mathematical model of the controlled system. It is well known that if the mathematical model describes the controlled system completely, one can design a satisfactory control system by using some controller design methods. However, there are some gaps between the controlled system and its mathematical model and the gaps are referred to as uncertainties. The uncertainties included in the mathematical model may occur deterioration of control performance and, at the worst, control systems become unstable. Therefore, robust stability analysis and robust stabilization for uncertain dynamical systems have received much attention for a long time (e.g., see [

By the way in most practical situations, it is desirable to design robust control systems which achieve not only robust stability but also an adequate level of performance. Thus, robust controllers with achievable performance level have also been well studied. For instance, guaranteed cost control is introduced by Chang and Peng [

In this paper, we consider a design problem of an adaptive gain robust controller for polytopic uncertain systems. The proposed adaptive gain robust controller is composed of a state feedback law with a fixed gain and a compensation input. The compensation input is defined as a state feedback with a fixed gain and one with adaptive gains tuned by updating rules. The advantage of our new adaptive gain robust control is that, for the case that conventional quadratic stabilizing controller based on Lyapunov criterion cannot be obtained, the proposed design method may be able to design stabilizing controller. This paper is organized as follows. In Section

In this section, we introduce notations which are used in this paper as well as the existing works (e.g., see [

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

Consider the following uncertain linear system:

The nominal system, ignoring the unknown parameters in (

In this paper, first of all, we consider the standard linear quadratic control problem for the nominal system of (

Now, by using the optimal feedback gain matrix

From the above, our control objective in this paper is to design the compensation input of (

In this section, we show a design method of the proposed adaptive gain robust controller such that the uncertain closed-loop system of (

Consider the uncertain linear system of (

Firstly, in order to design the fixed gain matrices

Next, the fixed gain matrix

First of all, for the design parameters

Next, we introduce the following quadratic function as a Lyapunov function candidate:

Thus from the updating rule of (

By introducing the matrix

Thus the proof of Theorem

It is well known that if the LMIs of (

In this paper, by introducing adjustable parameters

In this paper, we consider the polytopic uncertain system of (

In order to demonstrate the efficiency of the proposed control scheme, we have run two simple numerical examples. The control problems considered here are not necessarily practical. However, the simulation results stated below illustrate the distinct feature of the proposed controller design.

Consider the uncertain linear system

Now, in this example, the initial value for the uncertain system of (

Case 1:

Case 2:

Case 1:

Case 2:

The results of the simulation of this example are depicted in Figures

From Figures

Time histories of the state

Time histories of the state

Time histories of the control input

Time histories of adjustable parameters

Time histories of adjustable parameters

Consider the linear system with coefficient matrices of

On the other hand, the uncertain system with the coefficient matrices of (

This paper has dealt with a design problem of adaptive gain robust controllers for polytopic uncertain systems. The proposed adaptive gain robust controller consists of fixed gain parameters and time-varying adjustable ones. In this paper, we show an LMI-based design algorithm of the proposed robust controller and simple numerical examples are given for illustration of the proposed controller design method. The simulation result has shown that the closed-loop system is well stabilized in spite of plant uncertainties for the case such that the robust stabilizing controller based on quadratic stabilization cannot be derived.

The proposed adaptive gain robust controller can easily be obtained by solving LMIs; that is, the proposed design method is very simple. Besides, one can see that, for the case that conventional quadratic stabilizing controller based on Lyapunov criterion cannot be obtained, the proposed controller design method has possibility of robust controller design. Therefore, the proposed design approach is very useful.

The future research subjects are extensions of the proposed adaptive gain robust controller to such a broad class of systems as uncertain time-delay systems, uncertain discrete-time systems, and so on.

In this appendix, the conventional quadratic stabilizing controller via Lyapunov criterion is shown.

One can easily see that the following theorem shows the LMI-based design method of quadratic stabilizing controllers for the polytopic uncertain system of (

Consider the polytopic uncertain system of (

There exists the state feedback gain matrix

See the studies by Boyd et al. [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Professor Bin Jiang with Nanjing University of Aeronautics and Astronautics and the anonymous reviewers for their valuable and helpful comments that greatly contributed to this paper. Additionally, the authors would like to express their gratitude to CAE Solutions Corp. for providing its support in conducting this study.