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For a class of continuous-time Markovian jump linear uncertain
systems with partly known transition rates and input quantization, the

Recently, much attention has been devoted to the study of the stochastic stability for the Markovian jump systems, and many important results have been published [

On the other hand, in many modern engineering practices, all kinds of information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used. By the utilization of such information processing devices, some advantages have been brought, for example, lower cost, reduced weight and power, simple installation, and maintenance. However, some new phenomena have also been induced, which might cause server deterioration of system performance or even lead to system instability. Signal quantization is one of the important aspects that should be fully considered in such cases, which always exists in computer-based control systems. Nowadays, many well-known results have been published on quantized feedback control. For example, the feedback stabilization problem is considered by utilizing dynamic quantizers [

To the best of our knowledge, no result has been presented for the control design of the continuous-time Markovian jump linear uncertain systems with partly known transition rates and input signal quantization. In this paper, the

The rest of this paper is organized as follows. The problem statement and preliminaries are presented in Section

Consider a class of the continuous-time Markovian jump linear uncertain systems in the following probability space

where

In general, the Markovian process transition rates matrix

where

For clarity, we denote that

When the lower and upper bounds of the elements in

For convenience, the notations

The following assumptions are assumed to be valid.

In addition, the quantizer

where

The objective of this paper is to design the state-feedback control law

When

Some useful lemmas are firstly presented before formulating the main result.

Given a symmetric matrix

For any given

According to

In Shen and Yang [

For

For the symmetric and positive definite matrices

For the system (

Take the Lyapunov function candidate

Two cases will be considered.

Applying the Schur complement formula, one can get (

For Case 2, pre- and postmultiplying

Thus, the LMIs in (

From the above proof, one can see that

The merit of the proposed results lies in that the transition rates of the jumping process are assumed to be more general, which means that some elements in the transition rates matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. Dealing with the unknown transition rates, a less conservative method is used. At the same time, the impact of the input signal quantization on the system is also considered. Finally, the controller design conditions are presented in the framework of LMIs.

In order to make comparison with the design method using the LMIs technique in Shen and Yang [

For the system (

The proof process is similar to that in Theorem

An example is presented to illustrate the effectiveness of the proposed method.

Consider the MJLSs with four operation modes as follows:

The considered transition rates matrix is given as follows:

Comparison of optimal

Theorem |
Proposition |
---|---|

1.4518 | 2.4567 |

For simulation, one can obtain the controller gains by solving the LMIs in Theorem

The following parameters are used in the simulation:

The switching mode, the control input, and the response curves of the system states are presented in Figures

Evolution of the system mode.

The curve of the control input

The response curve of the state

This work is supported by the National Natural Science Foundation of China (Grants nos. 61273155, 61273355, 61322312, and 61273356), the Foundation of State Key Laboratory of Robotics (Grant no. Z2013-06), the New Century Excellent Talents in the University (Grant no. NCET-11-0083), a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), the Fundamental Research Funds for the Central Universities (Grant no. N120504003).