Asymptotic Tracking Control for Mismatched Uncertain Systems with Active Disturbance Rejection

: By introducing a set of exact disturbance estimators, a continuously tracking controller for a class of mismatched uncertain systems with exogenous disturbances will be proposed. The most appealing superiority is that the proposed exact disturbance estimators can not only estimate the external disturbances but also achieve an asymptotic estimation performance. Furthermore, with the help of a set of first-order asymptotic filters and an auxiliary system, the developed control al-gorithm is able to compensate for these total disturbances feedforwardly. Consequently, the whole closed-loop stability with an asymptotic tracking performance is strictly analyzed, and meanwhile applications are conducted to indicate the effectiveness of the proposed controller


Introduction
Disturbances extensively exist in all practical systems, which may cause critical control performance degradation and even instability in developing high-performance closed-loop controllers [1][2][3].Over the past decades, many advanced control algorithms such as adaptive robust control [4], robust adaptive control [5,6], sliding mode control [7,8] and so on have been proposed for various nonlinear systems to cope with modeling uncertainties.Additionally, many studies focus on rejecting disturbances by combining with disturbance observers.
Currently, there has been a growing interest in disturbance-observer-based control strategies with an active disturbance rejection ability for uncertain nonlinear systems [9][10][11][12][13][14][15][16][17][18].And the main concept of these control strategies is to estimate the disturbances via different disturbance observers and thus compensate for them feedforwardly in developing the closed-loop controllers.Typically, Chen et al., proposed a nonlinear disturbance observer (NDOB) for nonlinear systems with disturbances governed by the exogenous system to estimate the total disturbances in an exponentially convergent rate [19].It is worth noting that NDOB has been successfully applied to various practical systems [20,21].Moreover, Won et al., have proposed a high-gain-disturbance-observer-based controller for hydraulic systems to improve the output tracking performance and meanwhile constrain the output tracking error [14].Furthermore, Han has developed an active disturbance rejection controller (ADRC) for uncertain systems [22].And the main support for ADRC is the extended state observer (ESO) [12,23,24].Moreover, there are still some other disturbance observers [25][26][27] have been proposed.Notably, the aforementioned disturbance observers can only achieve a bounded estimation performance.How to develop exact disturbance estimators and meanwhile acquire an asymptotic tracking performance, especially for mismatched nonlinear systems, is extremely important and challenging in designing high-performance closed-loop controllers.
Inspired by the above discussions, we will propose an asymptotic tracking controller for systems with matched and mismatched exogenous disturbances, which is of great significance both in theory and practice.Especially, the main contributions of this paper are shown as follows: (1) A set of exact disturbance estimators (EDEs) with optimized design parameters which can acquire an asymptotic estimation performance is proposed; (2) Both mismatched and matched exogenous disturbances can be effectively compensated, and meanwhile an asymptotic tracking performance can be acquired.
being the estimation error; and sgn(•) is the signum function.In addition, the variable i = 1,…, n with n being the system order; the variable j = 1,…, n-1; and the variable θ = 2,…, n-1.

Problem Formulation
A class of nonlinear systems is employed as where  (1), which means that it can be applied to many practical systems, such as motor servo systems [28,29] and so on.

Exact Disturbance Estimator
Inspired by the ESO, we extend the external disturbances fi(t) as new state variables φfi.According to Assumption 2, we can define ( ) ( ) and then transform the system (1) as Stimulated by [30,31], a set of novel EDEs can be proposed as [32] where Li1 and Li2 are adjustable positive design parameters; in addition, ˆi α are the esti- mates of αi which satisfy (23) which will be introduced later.
As conducted in [23], we can parameterize Li1 and Li2 as 2βoi and 2 oi β , respectively, with βoi being adjustable positive design parameters.Therefore, (3) can be rearranged as Especially, ˆi α can be updated via 1 ˆsgn( ) where λi are positive design parameters and ηi Due to incalculable variables existing in (5), we can obtain ˆi α via the following method: Noting ( 2) and ( 4), we can obtain After introducing new vectors as ηi where As Bo is Hurwitz, there is a positive definite matrix Fo guaranteeing

Controller Design
Firstly, we introduce a set of error variables ei(t) and compensation signals εi(t) as [33] where e1 = φ1-φ1d(t) is the tracking error; ζi represent the auxiliary variables; and vjf indicate the filtered values of the virtual control laws vj to be synthesized later, which can be produced via the following filters [34]., (0): (0) ˆ( ) where Lcj are the positive parameters; ξj = vj -vjf indicate the filtering errors; and δj(t) > 0 and satisfy 0 ( )


, t ∀ ≥ 0, with ϑi being some positive constants [34]; especially, σj represent the upper bounds of j v  , which can be updated via with rj being the adjustable positive gains.
An auxiliary system is introduced as [33] 1 ( ) where ki are the positive gains.
Proof.See Appendix B. □ Remark 2. Notably, some advanced control strategies [35,36] have been proposed.However, compared with these controllers, we have constructed a set of novel EDEs via the traditional ESO which can estimate the states and disturbances asymptotically.Additionally, to prevent over-parameterization of ˆi α and ˆj σ , we can constrain their adaptive laws through the projection map- ping function in [1].

Illustration Example
A one-link robot arm driven by the permanent magnet direct-current motor will be employed to verify the performance of the designed controller.Deniting state variables φ1 = ym, φ2 m y =  and φ3 = KmIm/Jm, the considered system can be presented as follows where ym and Im are the angular displacement of the load and the electric current, respectively; 1 1 ( ) ) ( ) The definitions and physical val- ues of the system parameters are given in Table 1.For ( 25), the following four controllers are employed to track trajectory φ1d = 0.1sin(πt)[1-exp(-0.01t 3 )]rad.
(1) C1: This is the developed controller in Section 3.And its parameters are tuned as Table 2.For fairness, all design parameters of C2, C3 and C4 are chosen as same as that of C1.The contrastive tracking errors are plotted in Figure 1.It can be clearly discovered that C1 performs the best tracking performance in terms of both transient and final tracking errors, which verifies the effectiveness of the compensation performance for mismatched and matched external disturbances.Moreover, it follows from Figure 1 that the tracking error of C1 gradually approaches zero, which demonstrates the achievable asymptotic output tracking performance.Furthermore, it also means that mismatched and matched external disturbances can be exactly estimated by the introduced observer.To support this claim, the estimation performance of the system states and modeling uncertainties with the proposed observer (5) are exhibited in Figures 2 and 3, respectively.In addition, Figure 4 plots the control input of C1, which demonstrates that the resulting control law is smooth and meanwhile bounded.

Conclusions
A novel asymptotic tracking controller for a class of nonlinear systems with mismatched and matched exogenous disturbances has been proposed.Especially, a set of novel exact disturbance estimators with nonlinear robust terms to further reject disturbances has been creatively constructed to estimate the total disturbances in real time.Meanwhile, the exact disturbance estimations have been exploited in designing the resulting control scheme to eliminate the effects of disturbances.Especially, asymptotic tracking performance and asymptotic disturbance estimation performance have been demonstrated via strict theoretical analysis.In addition, the application on a one-link robotic arm driven by a direct-current servo motor has been conducted to verify the achievable results.
Therefore, we have This proves Proposition 1. □

Appendix B
Proof of Theorem 1.A set of auxiliary functions Wi is defined as: 0 : ( ) It follows from Proposition 1 that Wi ≥ 0. Therefore, a Lyapunov candidate VL1 can be employed as Based on (10), the filtering error dynamics can be arranged as After substituting ( 12), ( 15), ( 18), ( 21) and (A5) into the time derivative of VL1, we have Noting ( 5), ( 10) and ( 11), we have where ςi are some positive constants.

5 ( 2 )
C2: It is same as C1 but without compensation of the mismatched external disturbances.(3) C3: It is same as C1 but without compensation of the matched external disturbances.(4) C4: It is same as C1 but without compensation of the mismatched and matched external disturbances simultaneously.

Figure 2 .
Figure 2. Estimation performance of φ2 and φf2 with the constructed observer.

Figure 3 .
Figure 3. Estimation performance of φ3 and φf3 with the constructed observer.

Figure 4 .
Figure 4.The control input of C1.

Table 1 .
The physical parameters of the system.

Table 2 .
The controller parameters of C1.