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The tracking problem for continuous-time systems is investigated. It is assumed that the states
of the systems are not available. An observer is firstly designed to estimate the states by using the

The tracking control is a fundamental and also the most important control problem no matter from the control theory and from the practical applications [

For those newly emerging advanced control theories and practical design techniques, it is obvious that there are some advantages over the traditional PID control. However, PID controllers have simple structures but can provide good tracking performance for the majority of industrial plants, such as chemical processes, motor drives, automotive, bio-mechanical systems, hydraulic systems, and flight vehicles. It is also necessary to mention that it is difficult to tune the PID gains (no theoretical optimal solution). Moreover, there is not an effective algorithm to design the PID controller for multi-input-multi-output systems. However, if we analyze the system in the state-space model, there is not a significant difference between the single-input-single-output systems and multi-input-multi-output systems [

On another research frontier, the robust control has attracted a lot of attention in the past decades [

In the robust control, there are mainly three control strategies:

For the sake of improving the transient response, only the feedback loop is insufficient. The feedforward loop is also necessary to contribute into the control law [

The tracking control scheme used in this paper is illustrated in Figure

Tracking control scheme for continues-time systems.

Consider the following continuous-time systems:

In the following, we will discuss our main assumptions. Based on these assumptions, we will present the controller design procedure in the following sections.

The matrix set

The determinant of the matrix

The output matrix

All the external excitations are energy bounded.

In the stabilization and the pole placement, we propose to use the observer-based state-feedback control. The dynamics of Luenberger observer can be represented by

To deal with another external input

To fulfill the proposed control scheme, the control law used is

In the tracking control, the tracking error is required to be as smaller as possible. Hence, we utilize the following cost function:

Here,

In summary, the closed-loop system with the controlled output has the following form:

It is important to emphasize that there are two external inputs

To design the observer and the feedback controller that the poles of the closed-loop system in (

To investigate the mixed

Before ending the section, a useful lemma named Schur complement is introduced.

Given a symmetric matrix

The pole placement in LMI regions with feedback control has attracted increasing attentions since it was originally proposed in [

A subset

For the closed-loop system in (

Note that there are external excitations in the closed-loop system. In order to evaluate the impact of the external excitations, we study the mixed

Given two positive scalars

The condition (

It is important to emphasize that the parameters to be determined are coupled with the positive-definite matrix

In order to deal with the challenge, an

It can be seen from (

Given a positive scalar

It follows from Theorem

Generally, we need to minimize the disturbance attenuation level

The minimum

Recalling the conditions in Theorem

Given two positive scalars

By using the Schur complement, the inequality (

Since

It is necessary to point out that there is one matrix equation in Theorem

It is necessary to show some examples on LMI regions. Generally, there are three types of regions are widely considered.

As shown in Figure

The disk is with the center at

The conic sector is with the center at the origin and with the inner angle

Illustration of LMI regions.

For a given

For a given

The minimum mixed

The design procedure of the controller is summarized as follows.

Derive the dynamics of the control plant or identify the system model of the control plant.

Augment the system to an augmented one in the form of (

Choose the weighting factor

Design the estimator gain

Design the gains

In this section, a numerical example is considered to show the effectiveness of the proposed design method.

Consider the continuous-time system in Figure

When the

By employing Corollary

The control problem for continuous-time systems under the framework of

This paper is supported by Natural Science Foundation of Zhejiang Proviance of China under Grant no. Y1080112.