This paper describes the synthesis of a robust and nonfragile

The Kalman filtering approach is a popular state estimation method for a class of systems presented in linear models and thus has been widely applied in previous research on control engineering and target tracking, among others [

The aforementioned research has motivated our research to construct the filter for practical application with better performance against uncertainties, various disturbances, and time-delay. This paper examines additional methods to design a robust and nonfragile Kalman-type filter for a class of uncertain systems with disturbances and time-delay by taking the PLMIs (parameterized linear matrix inequalities) approach. Here, we represent the uncertain linear system by polytopic uncertainties, filter-gain variation, time-delay, and disturbances. Next, we rewrite the sufficient condition for filter existence as PLMIs and use a relaxation technique to find finite feasible solutions. Finally, a numerical example is provided to verify the performance of the proposed filter design method. The main contributions of this article are summarized as follows.

Since uncertainties in the system and filter are designed as structured uncertainties, it is possible to find filter-gain region which guarantees filter performance using PLMI and relaxation technique with less computational efforts.

The design method included all of factor increasing estimation error such as system uncertainties, filter-gain variations, disturbances, and time-delay. Therefore, it is more suitable for practical filter applications.

Consider the following linear system with parameter uncertainties and time-varying delay:

Here, the value of

We can express a full-order filter for system (

Here, we consider a robust and nonfragile filter for the system (

In this section, we consider system performance and obtain the filter-gain

The estimation error (

We define a Lyapunov candidate functional as

When assuming the zero input, we have

In the next place, assume the zero initial condition and introduce

Noting

Let

Therefore, when

We present the proposed condition for a robust and nonfragile filter by PLMIs, which involve an infinite number of LMIs, and thus transform PLMIs into a finite number of LMIs by using, relaxation technique.

The estimation error from linear parameter uncertain system (

The inequality (

Since the filter implementation accompanied by imprecision inherent in analog-digital and digital-analog conversion, finite word length, and finite resolution measuring instruments and round-off errors in numerical computation, we have to consider a design procedure which has sufficient space for coefficients readjustment. Inequality (

Consider nominal system matrices for parameter uncertain linear system with

To solve this problem, we use the LMI toolbox with a parameter relaxation technique and obtain all feasible solutions as follows:

In this way, we obtain robust and nonfragile filter-gain with time-delay and vertices of the perturbation satisfying the nonfragility condition as follows:

For the simulation, we define the time-varying parameter as follows:

If the initial values of all states are zero and the value of

We simulate that the proposed nominal filter-gain

Simulation results of disturbances

Trajectories of the

Trajectories of the

Simulation results of states for the nominal system and filter.

System state

System state

Simulation results of output for the nominal system and filter.

Simulation result of estimation error for the nominal system and filter.

Next, it is demonstrated that the practical filter

Simulation results of states for the practical system and filter.

System state

System state

Simulation results of output for the practical system and filter.

Simulation result of estimation error for the practical system and filter.

Comparisons of estimation error between nominal filter and practical filter.

This paper proposes a method for designing a robust and nonfragile

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

This research was financially supported by the Ministry of Education (MOE) and National Research Foundation of Korea (NRF) through the Human Resource Training Project for Regional Innovation (No. 2013H1B8A2032081).