Design of a Disturbance Rejection Controller for a Class of Nonholonomic Systems with Uncertainties

Cao, Dianguo; Chen, Jiaqian

doi:https://doi.org/10.1155/2021/9941611

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9941611 | https://doi.org/10.1155/2021/9941611

Design of a Disturbance Rejection Controller for a Class of Nonholonomic Systems with Uncertainties

Dianguo Cao¹and Jiaqian Chen¹

Academic Editor: Antonio M. Lopes

Received29 Apr 2021

Revised05 Oct 2021

Accepted27 Oct 2021

Published18 Nov 2021

Abstract

This study investigates the global output feedback stabilization problem for one type of the nonholonomic system with nonvanishing external disturbances. An extended state observer (ESO) is constructed in order to estimate the external disturbance and unmeasurable system states, in which the external disturbance term is seen as a general state. Thus, a new generalized error dynamic system is obtained. Accordingly, a disturbance rejection controller is designed by making use of the backstepping technique. A control law is given to ensure that all the signals in the closed-loop system are globally bounded, while the system states converge to an equilibrium point. The simulation example is proposed to verify that the control algorithm is effective.

1. Introduction

Within recent decades, the control of nonholonomic systems has always been one of the most popular tasks in control fields since such systems can be frequently found in mechanical systems, for example, car-like vehicles, wheeled mobile robots, knife-edge, and so on. In the theoretical analysis of the nonholonomic system model, some nonlinear feedback controllers for these systems were put forward in the literature to ensure that the systems are asymptotically stable or exponentially regulatable, for example, the studies [1–7] and references therein. By using an input/state scaling technique and switching algorithm, a class of feedback control law was obtained for nonholonomic chained systems with uncertainties to realize exponential stabilization [6, 7], and a switching-based state scaling is designed for prescribed-time stabilization of nonholonomic systems with actuator dead-zones [8]. In practical applications, especially in the research of nonholonomic wheeled mobile robot control, the controller design method to realize the robust stabilization of the system is given [6, 9, 10]. Considering the limitations of the hardware and environment of the actual system, the design method of the controller with saturated input is given in [11, 12]. In order to overcome the external disturbances, the robust tracking control for the wheeled mobile robot is proposed based on the ESO [13, 14].

The measurement of full states is usually difficult and sometimes impossible. Moreover, in practical applications, the systems usually contain unknown disturbances, measurement noise, and modeling errors, which are called nonvanishing total disturbances. These disturbances in reality will influence the performance of closed-loop systems. Therefore, it is of great significance to study the output feedback stabilization of nonholonomic systems with nonlinear uncertainties and external disturbances. The output feedback stabilization for nonholonomic systems is more complex and difficult than using the general nonlinear state feedback. The output feedback problem towards asymptotic stability and exponential stability of nonholonomic systems has previously been put forward [15, 16]. In [17–20], the adaptive output feedback global stabilization of a class of nonholonomic systems with parametric uncertainties and strong nonlinear drifts are solved. However, none of the above work considers the existence of disturbance items that do not disappear from the system even though uncertainties or nonlinear drifts exist. This means that the proposed output feedback scheme may be unstable because of the external disturbances. To reject the external disturbances, an output feedback controller has been proposed for nonholonomic systems with nonlinear uncertainties [21] and nonvanishing external disturbances [22–24]. In [23], the external disturbances are considered a generalized system state, and an ESO was constructed. By utilizing the so-called ESO, [25] further investigated the output regulation control problem towards one type of cascade nonlinear systems with the external disturbance, and the output feedback adaptive regulation problem was solved by the time-varying Kalman observer [25]. However, the output regulation controller in [22, 23, 25] requires that the nonlinear uncertainties in the systems are only related to the output of the systems.

The ESO in pioneering work [26] is the key creative advancement towards active disturbance rejection control (ADRC). The ESO has the capability for state observation and real-time estimation of generalized disturbances between the controlled object and the model of the controlled system [27, 28]. By using the ESO, this study addresses robust output feedback adaptive control towards one type of nonholonomic chained form systems that have nonvanishing external disturbances in the input channel and uncertain nonlinearity drift. Different from references [22, 23], in the model studied in this study, the upper bound function of nonlinear uncertainties depend not only on the output variables but also on the system state variables, in which such uncertain nonlinearities meet a linearly growing triangular condition.

The main contribution of this study is that the extended state observer (ESO) and gain scaling technique [29] are constructed. In order overcome unknown system states and the external disturbance, we reconstruct the system state, and the disturbance is regarded as an extended state. The ESO with dynamic gain is put forward, and the disturbance rejection controller based on an observer is developed by designing a variable observer gain to overcome the uncertainty. The controller design is carried out for one type of the nonholonomic system with nonvanishing external disturbances and uncertain nonlinearities satisfying a linearly growing triangular condition. This approach allows the external disturbances to be a larger class of signals.

2. Problem Formulation

In this study, we consider the following nonholonomic system with nonlinear uncertainties and nonvanishing external disturbance:where , are the system states, and the initial values are , , with as the initial moment of the system; is the control input, and is the system output. The functions are the uncertainties which represent possible modeling errors and neglected dynamics; , , , and are the uncertainties and bounded, where is the nonvanishing external disturbance and satisfies that . The assumptions and lemmas used in this article are listed as follows.

Assumption 1. For every , the following inequality holds:where the nonnegative smooth functions and are known.

Lemma 1 (see [30]). For any , any scalar , and any positive definite matrix , the following inequality holds:

Lemma 2 (see [31, 32]). For any , there exist positive real numbers and , positive definite matrix , and positive constants , such that the following inequality is satisfied:where is the identity matrix of order , and and are the matrices denoted as

3. Controller Design and Stability Analysis

3.1. Output Feedback Controller Design

Lemma 3 (see [33]). For the first subsystem of (1), if the first control law is chosen aswhere is regarded as the initial condition, , then as the corresponding solution, exists and , . is designed as a positive constant parameter. Furthermore, .

Proof. Substituting (6) into the first formula of system (1), we can obtainIntegrating this nonlinear equation, the solution is This shows that at any time if , and thus, . Choosing , it can be obtained from (6) thatThus, asymptotically approaches zero. For any bounded because are smooth functions, there is a positive constant for , . This yields thatIntegrating both sides of this equation, it can be obtained thatThis means that converges to zero, but at any given moment, so that .
We introduce the following input state scaling:Unknown nonvanishing external disturbance is treated as a generalized state. To realize symbol consistency, it is defined asIn the new state , system (1) is converted to

Lemma 4. For any given in (6), there is a known nonnegative smooth function , such that , .

Proof. The following calculation is completed:This completes the proof of the lemma.
We know from Assumption 1 that there is a nonnegative smooth function :Considering that are unmeasurable signals that cannot be used in feedback control, the dynamic observer for (13) is denoted as follows:where is the dynamic gain, which will be designed later according to the requirements. The observer error dynamics is defined asWe can determine from (13), (16), and (17) thatIntroducing dynamic gain scaling,and defining , we arrive atDenoting , , , and , we haveNow, using Assumption 1 and (15), it follows thatChoosing , it is then obtained thatIntroducing the following transformation,where is a constant number that will be given later, because , and , we haveNow, using Lemma 1, it follows that inequality,holds, where and are the nonnegative smooth functions. On the other hand, by using Young’s inequality, one hasCorrespondingly, we can obtain that Step 1. Choosing , we have By using Young’s inequality, we have Where are the constants. Choosing , we then obtain Step 2. Choosing , where is a designed positive constant,From Lemma 1, we can derive the following inequalities:where are the constants. Using the relations and , we haveDefining , it is derived thatwhere . Since , this implies thatLetting , it can be concluded thatwhere are the constants. Furthermore, we haveLet us denotethus, it is obtained thatFor simplicity, let and ; then,Step i . Assume that in Step , we haveLetting the i^th candidate Lyapunov function be and defining , where are the designed positive constants,By applying Young’s inequality, one haswhere are the constants. Using the relationsand , it follows thatSince , this implies thatDefining ,where are the constants.
Denotingwe haveChoosing and using the formulawe again obtainStep n: for the last step, we choose , where is a designed positive constant, and by using (30), (32), and (43), we haveThis is similar to step , in thatwhere are the constants. According to the relationsand , we haveBecause , it is obtainedDefining ,where are the constants.
Denotingwe haveFor simplicity, we define , and then,where are the constants. Now, let us choose the final control signal asFinally, it is derived that

3.2. Stability Analysis

Choosing , we can obtain that

Select parameterswhere denote , such that

Denoting and , then

Defining , we can obtain that

are bounded. Now, we shall prove by contradiction that is bounded. Assume is unbounded in . We notice that , so , and thus, there exists a finite time , such that , and we have

Integrating both sides of the top equation, when , we can obtain

Since is bounded in , is bounded, , which is a contradiction. Thus, is bounded in . When , one has

These two equations contradict each other. Thus, is bounded in . Integrating , we havenamely,

Because . Because , it can be obtained from (19) and (20) that is bounded. According to Barbalat’s lemma, , and .

4. Simulation Results

In this section, we consider a simulation example to prove that the controller design in this study is effective. Consider the following three-dimensional uncertain nonholonomic system with nonvanishing external disturbance:

This example shows that the nonlinear uncertainty in third equation is related to the unknown system state , so that the assumption condition in [23] is not satisfied, and thus, the given method in that of article is not available to deal with this model. However, this example satisfies the given Assumption 1. By designing the controller, execution simulation algorithm, and choosing parameters , we havewhere the initial states are , and , and dynamic gain is selected as . In Figure 1, the simulation results are shown. This study presents an output feedback control scheme that realizes stability control, and the control inputs and are bounded, as shown in Figure 2.

5. Conclusion

This study solves the problems of output feedback control for one type of the nonholonomic system with nonvanishing external disturbances and nonlinear uncertainties for which the strong uncertainties are restricted by a generalized lower triangular linearly growing condition. The system is reconstructed by introducing a new extended state observer. The external disturbance is viewed as a general state. An adjustable varying gain scaling transformation and the extended state observer are used to carry out output feedback control and overcome the uncertainties and disturbances. The output of the system and states of the system go to zero, and all signals of the closed-loop system are guaranteed to be bounded. Simulation examples show that the control algorithm is effective. How to reduce the uncertainty and external disturbance assumptions of the model (1) and make the types of the models more extensive will be further considered.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (62073187), the Major Scientific and Technological Innovation Projects in Shandong Province (2019JZZY011111), Rizhao Science and Technology Innovation Special Plan (2019cxzx2212), and Funding for the Key R&D Program of Jining (2020JKNS004).

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Copyright © 2021 Dianguo Cao and Jiaqian Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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