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This paper proposes a new method to design the observer for state delay systems such that (i) state estimation errors converge to zero quickly and (ii), at the same time, a quadratic performance measurement of the deviation of estimates from the actual states is minimized for reducing large error during the transient period of observation. The proposed new approach fuses the merits of both the orthogonal functions approach and evolutionary optimization. One illustrative example is given to verify the effectiveness and efficiency of the proposed new optimization method on performance improvement of state estimations. From the illustrative example, in addition to the asymptotical convergence of the estimated state errors, the performance index for the proposed optimal design approach is clearly much lower than that of the nonoptimal design method.

Control systems are commonly used in a variety of machines, products, and processes to regulate system output. Most control system design approaches that exploit state feedback control theory are based on the assumption of having sensors to measure the output of a system under control. In short, all the state variables are assumed to be available outputs. However, for many control systems, state variables are not accessible to direct measurement, or the number of measurement devices is limited, possibly due to cost considerations. In order to effectively use state feedback, developing an approach to estimating the system states is inevitable. As a result, an important problem regarding linear multivariable control systems is designing an observer for a given dynamic system. Therefore, many researchers have proposed various methods for designing observers ([

For delay-free systems, some research has been devoted to considering the observer design issue of reducing the large error during the transient period of observation [

The remainder of the paper is organized as follows. Section

Consider the following time-delay system:

For estimating state vector

The

It can be shown that

A quadratic performance measurement of estimation error is defined as

The optimal observer design problem for the time-delay system is to directly find the observer gain matrix

For the state observer in (

The state estimation error vector

Let the Lyapunov function candidate for the error dynamic system described by (

By using the Schur complete formula [

By using the asymptotic stability criterion of (

Check the LMI-based constraint condition of the state estimation error vector asymptotically converging to zero in (

Minimize the quadratic performance measurement of estimation error in (

In order to help design the observer gain matrix, the orthogonal functions approach (OFA) is employed in this paper. The OFA has been successfully applied to investigate various problems of systems and control [

For

Using the Kronecker product, the solution

For

Applying the Kronecker product, the solution

Now, substituting the truncated OF representations of the estimation error vectors

From the aforementioned procedures for solving

Input is as follows: population size

First, randomly generate the initial population with the chromosomes of form

By integrating (

If yes, go to Step 5. Otherwise, go back to Step 3.

The optimal gain matrix

For studying the stability of time-delay systems, both delay-independent and delay-dependent criteria have been proposed in the literature. The purpose of both delay-independent and delay-dependent criteria is to guarantee that the steady-state errors are reduced. But the purpose of the proposed observer design method with the quadratic performance measurement of (

To conquer the inherent difficulty of giving a priori proper bound of parameters in which the optimal solution is placed, it is desirable to allow the HTGA to dynamically expand the search space. To this end, the search-space expansion schemes developed in [

One example is given in this section to verify the efficiency of the proposed method for designing the optimal observer gain matrix

Consider the time-delay system described by (

To the authors’ best knowledge, for the observer design of time-delay systems, there is no extant literature investigating the issue of minimizing estimation errors during the transient period of observation. That is, for state delay systems, the extant approaches do not consider to minimizing estimation errors during the transient period of observation. So, no existing methods can be used to compare the proposed optimal design approach. Thus, only the comparisons of the state estimation errors of optimal estimation and nonoptimal estimation are given in the simulation. Letting

A comparison of state estimation errors

A comparison of state estimation errors

A new approach has been proposed in this paper to design the observer gain matrix

No data were used to support this study.

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