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

*is 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. *us, 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. *e simulation example is proposed to verify that the control algorithm is effective.


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][2][3][4][5][6][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]. e 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. ese disturbances in reality will influence the performance of closed-loop systems. erefore, it is of great significance to study the output feedback stabilization of nonholonomic systems with nonlinear uncertainties and external disturbances. e output feedback stabilization for nonholonomic systems is more complex and difficult than using the general nonlinear state feedback. e output feedback problem towards asymptotic stability and exponential stability of nonholonomic systems has previously been put forward [15,16]. In [17][18][19][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.
is 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][23][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. e ESO in pioneering work [26] is the key creative advancement towards active disturbance rejection control (ADRC). e 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.
e 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. e 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. e 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.
is approach allows the external disturbances to be a larger class of signals.

Problem Formulation
In this study, we consider the following nonholonomic system with nonlinear uncertainties and nonvanishing external disturbance: . , x n ] T ∈ R n are the system states, and the initial values are x 0 (t 0 ), x(t 0 ), with t 0 as the initial moment of the system; u � [u 0 , u] T ∈ R 2 is the control input, and y ∈ R 2 is the system output. e functions ϕ d i (·), i � 1, 2, . . . , n are the uncertainties which represent possible modeling errors and neglected dynamics; w 0 (t), _ w 0 (t), w(t) ∈ R, and _ w(t) are the uncertainties and bounded, where w(t) ∈ R is the nonvanishing external disturbance and satisfies that _ w(t) ∈ L 2 . e assumptions and lemmas used in this article are listed as follows.
Lemma 1 (see [30]). For any x, y ∈ R, any scalar k > 0, and any positive definite matrix M ∈ R (n+1)×(n+1) , the following inequality holds: Lemma 2 (see [31,32]). For any μ > 0, there exist positive real numbers d 1 and d 2 , positive definite matrix P, and positive constants a i , such that the following inequality is satisfied: where I i is the identity matrix of order i, and A and D are the (n + 1) × (n + 1) matrices denoted as

Output Feedback Controller Design
Lemma 3 (see [33]). For the first subsystem of (1), if the first control law u 0 is chosen as where (t 0 , x 0 (t 0 )) is regarded as the initial condition, x 0 (t 0 ) ≠ 0, then as the corresponding solution, Proof. Substituting (6) into the first formula of system (1), we can obtain 2

Mathematical Problems in Engineering
Integrating this nonlinear equation, the solution is us, x 0 (t) asymptotically approaches zero. For any bounded Integrating both sides of this equation, it can be obtained that is means that x 0 (t) converges to zero, but x 0 (t) ≠ 0 at any given moment, so that |u 0 (t)| > 0.
We introduce the following input state scaling: Unknown nonvanishing external disturbance w(t) is treated as a generalized state. To realize symbol consistency, it is defined as In the new state ζ, system (1) is converted to □ Lemma 4. For any given u 0 in (6), there is a known non- Proof. e following calculation is completed: is completes the proof of the lemma. We know from Assumption 1 that there is a nonnegative smooth function α(x 0 ): Considering that (ζ 1 , . . . , ζ n+1 ) are unmeasurable signals that cannot be used in feedback control, the dynamic observer for (13) is denoted as follows: Now, using Assumption 1 and (15), it follows that Choosing V ε � ε T Pε, it is then obtained that Introducing the following transformation, where g i− 1 > 0 is a constant number that will be given later, because ε i � e i /c i− 1+μ , and e i � ζ i −ζ i , we have Now, using Lemma 1, it follows that inequality, holds, where φ 1 (x 0 ) and φ 2 (x 0 ) are the nonnegative smooth functions. On the other hand, by using Young's inequality, one has 2ε T Pbh ≤ cL 1 ‖ε‖ 2 + ch 2 , Correspondingly, we can obtain that Step 1. Choosing V 1 � 1/2z 2 1 � 1/2x 2 1 , we have 4 Mathematical Problems in Engineering By using Young's inequality, we have Where M > 0, β 1 � a 2 1 /4M 1 are the constants. Choosing α 1 � − g 1x1 � − (n + β 1 )x 1 , δ 1 � μ, we then obtain Step 2. Choosing V 2 � p 1 V 1 + 1/2x 2 2 , where p 1 is a designed positive constant, From Lemma 1, we can derive the following inequalities: (42) Step i (2 < i ≤ n − 1). Assume that in Step i − 1, we have Letting the i th candidate Lyapunov function be where p 2 , . . . , p i− 2 are the designed positive constants, By applying Young's inequality, one has are the constants. Using the relations Mathematical Problems in Engineering (48) (49) where ij are the constants. Denoting we have 8 Mathematical Problems in Engineering Choosing and using the formula we again obtain Step n: for the last step, we choose V n � p n− 1 V n− 1 + 1/2x 2 n + 1/2z 2 n+1 , where p n− 1 is a designed positive constant, and by using (30), (32), and (43), we have

Conclusion
is 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. e system is reconstructed by introducing a new extended state observer. e 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. e 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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.