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A novel method of robust trajectory linearization control for a class of nonlinear systems with uncertainties based on disturbance rejection is proposed. Firstly, on the basis of trajectory linearization control (TLC) method, a feedback linearization based control law is designed to transform the original tracking error dynamics to the canonical integral-chain form. To address the issue of reducing the influence made by uncertainties, with tracking error as input, linear extended state observer (LESO) is constructed to estimate the tracking error vector, as well as the uncertainties in an integrated manner. Meanwhile, the boundedness of the estimated error is investigated by theoretical analysis. In addition, decoupled controller (which has the characteristic of well-tuning and simple form) based on LESO is synthesized to realize the output tracking for closed-loop system. The closed-loop stability of the system under the proposed LESO-based control structure is established. Also, simulation results are presented to illustrate the effectiveness of the control strategy.

Trajectory linearization control (TLC) is a novel nonlinear tracking and decoupling control method, which combines an open-loop nonlinear dynamic inversion and a linear time-varying (LTV) feedback stabilization, which guarantees that TLC’s output achieves exponential stability along the nominal trajectory. Therefore, owing to the specific structure, it provides a certain extent of robust stability and can be capable of rejecting disturbance in nature, for which TLC has been successfully applied to missile and reusable launch vehicle flight control systems [

However, in [

It is not difficult to recognize that the focal point of [

Above all, the essence of this problem is really disturbance rejection, with the notion of disturbance generalized to symbolize the uncertainties, both internal and external to the plant [

This is the first paper that employs LESO to improve the robustness and capability in disturbance rejection for TLC. Compared with methods proposed in [

Unlike the conclusions on stability made by [

Compared with [

The paper is organized as follows. The review of TLC and controller design procedure based on LESO are presented in Sections

Consider a multi-input multi-output (MIMO) nonlinear system:

Firstly, without consideration of disturbance described by

Let

Since

Consider the LTV system derived by Taylor expansion at the equilibrium point (

Assume that systems (

Let

The system matrices pair

According to Assumption

With the consideration of control quality for closed-loop system, the augmented tracking error in forms of PI can be written in the following state space form:

The state vector

Let

It is obvious that, with the relative order and system order of (

Meanwhile, define

For subsystem (

However, the control law cannot be synthesized unless d is estimated by observers. To deal with the estimation issue in (

For simplicity, let

So far, by adopting direct feedback linearization, the original tracking error dynamics which take the form of linear time-varying have been transformed to canonical integral-chain form. Consequently, for (

Although

Furthermore, define

For simplicity, let

Assuming

If there exist three different negative real eigenvalues for

Note that

When

Hence, we have

From Theorem

With respect to

Suppose that the estimated errors of LESO satisfy

The controller of the LTV system stated above satisfies

With virtual control variable designed as

Note that

Substituting (

It is obvious that the relationship between the output

Here, without loss of generality, the gain parameters

Next, we mainly prove the conclusion (2).

Let the tracking error of

Since

When

Suppose that there exist two different real eigenvalues for

Similar to Theorem

Hence, we have

It can be seen that

Therefore, there exists

Let

Since

Since the LTV subsystems are decoupled with each other, the tracking errors of closed-loop system satisfy the following:

It is worth pointing out that conclusion (2) of Theorem

For the tracking error dynamics described by (

From Theorem

The solution of (

Similar to Theorem

Substituting the above inequality into (

It is usually desirable in observer design that

From Theorem

To demonstrate the effectiveness of the proposed approach, a numerical example is considered, which is described by Changsheng et al. [

According to the design procedure of the TLC method, the nominal input can be obtained:

To maintain causality, the derivative of

According to the design framework of TLC [

Correspondingly, the original system (

By linearizing (

The tracking and disturbance rejection performance of TLC combined with LESO are tested under the following different scenarios.

There exist no unmodeled dynamics and disturbances.

The unmodeled dynamics exist in the system described as

Both unmodeled dynamics and external disturbances exist in the system described as

Thus, the system (

Suppose that the tracking error

In this simulation, the tuning parameters are

Above all, the overall controller of the closed-loop system can be synthesized as follows:

In order to compare conveniently, here, the control law of [

Firstly, we suppose that the reference command is the same with [

The tracking performance of original TLC method tested under the aforementioned scenarios is shown in Figure

Simulation results for original TLC method.

From the simulation in Figure

The performance for the proposed method and control scheme presented in [

Simulation results for proposed method under Case

Simulation results for proposed method under Case

Simulation results for the method in [

Simulation results for the method in [

Tracking errors for proposed method and the method in [

From the simulation in Figures

To further demonstrate the relationship between the tracking error and the bandwidth, Figure

Tracking error for proposed method with different design parameters.

Estimated error for proposed method with different design parameters.

Next, in order to illustrate the control strategy can also work well when the desired trajectory proceeds with abrupt disturbance, we suppose a step disturbance with the amplitude of 3 at

Tracking response for proposed method and the method in [

Above all, compared with [

The main result in this paper is the validation of proposed method through theoretical analysis and simulation. The BIBO stability and ultimate tracking error bound are rigorously analyzed based on the proposed robust TLC’s specific structure. It is shown that the ultimate upper bound of closed-loop tracking error monotonously decreases with the product of LESO’s and controller’s bandwidth. Thus, the analysis provides a guideline to select the two tuning parameters. The theoretical study is further supported by the simulation results. Both stability analysis and simulation results validate the effectiveness of the proposed method.

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

This research has been funded in part by the National Natural Science Foundation of China under Grant 61175084/F030601 and in part by Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT 13004.