Novel Active Disturbance Rejection Control Based on Nested Linear Extended State Observers

In this paper, a Novel Active Disturbance Rejection Control (N-ADRC) strategy is proposed that replaces the Linear Extended state observer (LESO) used in Conventional ADRC (C-ADRC) with a Nested LESO. In the nested LESO, the inner-loop LESO actively estimates and eliminates the generalized disturbance. Increasing the bandwidth improves the estimation accuracy which may tolerate noise and conflict with H/W limitations and the sampling frequency of the system. Therefore, an alternative scenario is offered without increasing the bandwidth of the inner-loop LESO provided that the rate of change of the generalized disturbance estimation error is upper bounded. This is achieved by the placing an outer-loop LESO in parallel with the inner one, it estimates and eliminates the remaining generalized disturbance that eluded from the inner-loop LESO due to bandwidth limitations. The stability of LESO and nested LESO is investigated using Lyapunov stability analysis. Simulations on uncertain nonlinear SISO system with time-varying exogenous disturbance revealed that the proposed nested LESO can successfully deal with a generalized disturbance in both noisy and noise-free environments, where the Integral Time Absolute Error (ITAE) of the tracking error for the nested LESO is reduced by 69.87% from that of the LESO.


Introduction
The performance of a control system is excessively affected by system uncertainties, such as exogenous disturbances, unmodelled dynamics, and parameter perturbations. Guaranteeing simultaneously disturbance rejection and good tracking performance in light of the existence of large uncertainties complicates the design of any controller that aims to address these objectives.
Accordingly, anti-disturbance methods with both external-loop controllers and internal-loop estimators have been comprehensively utilized. The precision of such controls mainly depends on the accuracy of the observer in the internal-loop, this type of controller is called "model-free controller" in contrast to other controllers that require the dynamics of the system, e.g. disturbance observers based control [1]. There have been various observer design philosophies posited, including fuzzy observers, sliding mode observers, unknown input observers, perturbation observers, equivalent input observers, extended state observers, and disturbance observers. Of these observers, the extended state observer (ESO) was originally suggested by Han [2]; it is often favoured because, in terms of design, it requires the minimum information from the system. It estimates the internal states of the system, system uncertainties, and exogenous disturbances, and it can also be used to design a state feedback controller. Based on this, an ESO is considered to be an essential part of the active disturbance rejection control paradigm. ESO-based control design has thus been widely examined in recent years [3,4]. The basic principle behind the operation of ESO is to augment the mathematical model of the nonlinear dynamical system with an additional virtual state that describes all the unwanted dynamics, uncertainties, and exogenous disturbances, which is termed "generalized disturbance". This virtual state, together with the states of the dynamic system, is observed in real-time using the ESO. This form of control design has been applied to a broad range of systems due to its model-independent operation. Initially, each ESO was constructed with nonlinear gains; however, it is more realistic to design and tune the ESO using tuneable linear gains, as proposed in [5]. Two signals, the input and the output of the nonlinear system, thus feed the ESO with information [6]. ESO-based control system design offers generally good performance due to the simplicity of design of ESO, which offers a need for minimum information, high precision of convergence, and fast-tracking capabilities [7]. In [8], ESO is tested on the nonlinear kinematic model of the differential drive mobile robot (DDMR). In [9], a general ESO-based control technique for non-chain integrator systems with mismatched disturbances was proposed. Recently, numerous control problems in various fields have also been effectively resolved by utilizing the ESO technique, including PMSM control [10], and attitude control of an aircraft [11]. The authors in [12] introduced an ESO-based dynamic sliding-mode control for high-order mismatched uncertainties with applications in motion control systems, and this also presented excellent tracking performance. In [13], an improved nonlinear ESO was proposed which achieved an outstanding performance in terms of smoothness in the control signal which leads to less control energy required to attain the desired performance. Techniques other than classical ones for dealing with measurement noise are proposed in the literature, e.g., authors of [14] and [15] have proposed a novel class of Adaptive ESOs (AESOs) with time-varying observer gains. As a result, the proposed AESO combines both the advantages of NESO and LESO and provided more extra design flexibility than LESO. Techniques different from ESO based estimation methods like time-delay estimators to estimate the generalized disturbance are proposed in [16,17].
The weak points of the aforementioned methods lie in the following: (1) for LESO to increase the estimation accuracy, the bandwidth of the LESO has to be increased, which tolerates noise and leads to hardware difficulties. Additionally, LESO suffers from peaking phenomenon due to large gain values.
(2) For nonlinear ESO, the performance will abruptly deteriorate when the amplitude or derivative of the generalized disturbance goes large to a certain degree [18]. Moreover, stability analysis and performance analysis is very complicated for nonlinear ESO.
(3) For other classes of observers like AESO, the parameter tuning process becomes more time consuming as the observer order goes higher.
In this paper, we offer a novel simple structure base on LESO, namely, the nested LESO, which combines the advantages of both linear and nonlinear ESOs. It consists of two LESOs connected in parallel sharing the same plant output. The proposed observer efficiently estimates the generalized disturbance without increasing the observer bandwidth and requires fewer computations for the parameters tuning; since it is built from LESOs which needs a single parameter to be tuned, i.e., the LESO bandwidth. Moreover, due to its linear structure, the proposed nested LESO inherits the simplicity of the LESO stability analysis, while the performance evaluation of the closed-loop system of an uncertain nonlinear SISO system is achieved very easily with the proposed nested LESO.
An outline of this paper's contents and organisation follows. Section II presents the problem statement and contribution of this work. Section III briefly presents the concepts behind active disturbance rejection control (ADRC). A description of the proposed nested LESO and the relevant stability tests are included in section IV. The numerical simulations verifying the validity of the proposed configuration are provided in section V. Finally, the conclusion is given in section VI, along with recommendations for future work.

Paper Contribution
In this paper, a novel ADRC is constructed by connecting a second LESO in parallel with an original LESO (the inner LESO), to construct a nested LESO. The advantage of this configuration is that the second LESO estimates and eliminates the remaining generalized disturbance that eluded from the inner LESO due to bandwidth limitations. Its excellent performance becomes very evident when considered in terms of measurement noise. The proposed observer with nested structure is differed from the state observer with a cascade structure, where the latter is just a state observer and used in special applications with delayed measurements such as the presence of an arbitrarily long delay in the output [20] or for position and attitude estimation of Unmanned Air Vehicles (UAVs) [21]. It should be emphasized that our main contribution is proposing a new structure to build a modified linear extended state observer by nesting two LESOs, sharing the same output rather than modifying the internal structure of the LESO. To the best of the authors' knowledge, using double LESOs within the same ADRC structure, with applications in highly uncertain nonlinear systems, has not previously appeared in the literature.

Conventional Active Disturbance Rejection Control Problem
In ADRC, the model of the nonlinear system is extended with an additional virtual state variable, which lumps all of the unwanted dynamics, uncertainties, and disturbances that remain unobserved in the standard system into a single term known as "generalized disturbance". In addition to estimating the states of the nonlinear system, the ESO performs online estimation and cancellation of this virtual state. In this scenario, the nonlinear system is converted into a chain of integrators, which allows the control system design to be simpler. Fig. 1 demonstrations the structure of a Conventional ADRC, (C-ADRC) which contains three key parts: the Tracking Differentiator (TD), an Extended State Observer (ESO), and Non-linear state error feedback (NLSEF) [22]. The tracking differentiator generates the required signal profile, which is the signal itself, free from noise, and a set of signal derivatives (1 st derivative, 2 nd derivative, …). The NLSEF acts as a nonlinear combination of the error profile. The ESO function is as discussed in the introduction section [23].

Tracking Differentiator (TD)
In the tracking differentiator, the output profile of the nonlinear system, in Brunovsky canonical form [24], must track the transient profile of the reference signal to resolve the problem of set-point jump in the traditional PID controller as stated in the seminal work [2]. In this manner, when a rapid change occurs in the set-point for any reason, the output signal of the plant will follow the output of the TD and will change gradually to reach the desired set-point [25]. TD can be represented as where r1 is the tracking signal of the input r, and r2 is the tracking signal of the derivative of the input r. To speed up or slow down the system during transient effects, the coefficient R is adapted, making it application dependent [25]. Other versions of enhanced TD are proposed in [26][27][28][29].

Non-Linear state error feedback
The linear weighting sum of the PID control is another limitation, involving as it does only the present, predictive, and accumulative errors, omitting other important parameters that could enhance its performance [25]. In the seminal work [2], the following nonlinear control law was suggested [2]: where α is a tuning parameter. The error signal, e, can thus reach zero more rapidly where α ˂ 1 [25]. Other forms of nonlinear control laws are suggested in [30][31][32][33].

ESO
Observers acquire data about the system states from its inputs and outputs progressively.
Luenberger first recommended the rule of observers in [34], where it was concluded that the state vector of the system can be estimated by observing the input and output of the system.
Subsequently, there have been numerous varieties of state observers outlined in the literature that rely upon the mathematical model of the system, including high gain observers and sliding mode observers [25]. The ESO was the first observer presented that was autonomous of the mathematical model and presented within the framework of ADRC. Furthermore, ESO has denoted estimators, which is considered a vital part of modern controls. The basic principle of the ESO is to observe the constituent parts of the generalized disturbance in real-time, including model discrepancy, exogenous disturbances, and the unmodelled dynamics of the nonlinear system. Additionally, it compensates for unpredicted disturbances in the control signal. ESO can be classified into two types. The first is linear ESO, which is an extension of the Luenberger observer [34,35], where the equations of the ESO contain only the linear correcting terms in order to simplify the calculations.
These terms manipulate the error between the actual states of the system and the estimated states in such a way that the error approaches zero. The second type is nonlinear ESO, where the error correcting terms include a nonlinear function of the error. These nonlinear functions have the advantage of enhancing the estimation error more rapidly and smoothly than the linear ESO.
Two approaches are common for ESO tuning: the pole-placement approach, and the bandwidth-based method. If the end goal is to reduce the number of parameters of the ESO, the parameters of the ESO can be expressed as a function of the bandwidth of the ESO, allowing only a single parameter of the ESO to be chosen or tuned. Selecting a bandwidth that is too large leads to a drop in the estimation error that nevertheless remains within an acceptable bound [36].
Observer bandwidth is chosen to be sufficiently larger than the disturbance frequency and smaller than the frequency of the unmodelled dynamics [37]. However, the performance of the ESO will deteriorate if the bandwidth of the ESO is selected to be too low or The authors in [14] designed a new class of adaptive ESO (AESO) in which the observer bandwidth varied with time to provide better performance than the LESO. The disadvantage of this method is that the parameter tuning may become more complex as AESO order increases [14].
To alleviate the peaking phenomenon caused by different initial values of the ESO, the small variable ε was designed as in [40]: The ESO parameters are tuned using Evolutionary Algorithm (EA) optimization techniques (BFO and PSO) rather than a manual process. Eventually, the ESO begins estimating these states.
Consequently, the effect of lumped disturbances is cancelled and the controller actively compensates for the disturbances in real time [35].

Main Results
The innovative ADRC is constructed by adding an extra LESO, which shares an output signal with the plant to be controlled with inner loop LESO. The structure of the novel ADRC is presented in Fig. 2. The inner LESO accomplishes the estimation of plant states and generalized disturbance.
In a situation where a suitably low bandwidth 0 is selected for the inner LESO to reduce noise, the estimation of the generalized disturbance through the augmented state is associated with a relatively large estimation error, this situation is deeply considered in [41]. The outer loop LESO will thus complete the rejection process by choosing an appropriate control law that depends on the estimated generalized disturbance ̂+ 1 .

Assumption[ (A1):
The function is continuously differentiable and there is a positive constant such that | ( )| ≤ 0 ≤ < ∞ This assumption represents wide range of fast and slow disturbances which exist in many realworld applications.
Letting ( ) = ‖ ‖ 2 , we can assume The stability of the proposed nested LESO is conducted in the following steps. Firstly, demonstrating the stability of the inner loop LESO by deriving the error dynamics of the system in (1) and proving its stability using Lyapunov function (Theorem1). Secondly, deriving the statespace equation of the nonlinear system combined with the inner loop LESO ( dotted square in Fig. 2) given by (26). Then, proving that the derivative of the generalized disturbance estimation error where ( ) , and ̂( ) denote the solutions of (2) and (3) Subtracting (3) Both (15) and (17) are substituted into (16) and the result is Solving ordinary differential equation (20) gives, From assumption (A3), ( )‖ ‖ 2 ≤ ( ). This leads to ‖ ‖ ≤ √  (20), Finally, Consider the control law described by Substituting (24) and (25) into (2) gives Adding an augmented state to the resultant system (26) thus creates  (2) and (3), From (23), lim From (29) and (30) ∎ Corollary 1. Consider the system given in (27), and the linear extended state observer (28). Here, , where ( ), and ̂( ) denote the solutions to (27) and (28) The stability of the closed-loop system with the N-ADRC is considered in the following theorem.

Theorem 2:
Consider an -dimensional uncertain nonlinear SISO system given in (1). The system (1) is controlled by the Linearization Control Law (LCL) given by: where is given as, where : ℝ → ℝ + is an even nonlinear gain function.

Proof:
The tracking error between the reference trajectory and the corresponding plant estimated states is given as: With outer LESO and TD as in assumptions A4 and A5 respectively, the tracking error can be described as, For the system given in (1), the states are expressed in term of the plant output, Substitute (41) in (40), and the tracking error is given by Differentiating (42)  It follows from (35) and (45) that The tracking error dynamics given in (46) associated with the control law designed in (38) produces the following closed-loop error dynamics The dynamics given in (47) can by represented as and ̃= (̃1,̃2, … ,̃) . The characteristic polynomial of is given by The proposed Non-Linear state error feedback controller in this work is based on (•)function given in (7) which can be written in terms of (•) as follows, which is a positive even function. The design parameters ( , ) of (49) are selected to ensure that the roots of the characteristic polynomial (48) have strictly negative real parts i.e. Hurwitz (stable) polynomial.
(c) The TD is given as [11] : where 1 is tracking signal of the input , and 2 tracking signal of the derivative of the input .
The Novel-ADRC (N-ADRC) based on nested LESO was also implemented for the system (50) with the following configuration,  Both controllers and the suggested system were numerically simulated using MATLAB®/Simulink® ODE45 solver for models with continuous states. The reference input (r(t)) to the system was cos (0.5 ) applied at t = 0 sec. Two test conditions were considered for this work.
In the first case, the output of the proposed system did not include any measurement noise, while in the second test case, a Gaussian noise was applied with variance ( ) equal to 10 −4 and the mean It is worthy to mention that in our simulation we have set the bandwidth ( 0 ) of the ESO in the C-ADRC to 100 rad/sec while for our proposed structure, a bandwidth ( 0 ) for the inner ESO was set to 100 rad/sec, and a bandwidth ( 0 ′ ) for the outer ESO had a value of 22.83 rad/sec. It is clear that a big reduction in the bandwidth requirements in our proposed structure achieved a noticeable improvement in the performance in terms of both ITAE and ISU, especially in the noisy case.  The estimation error of the generalized disturbances for the inner LESO is described by 3 which is given in (12), and the generalized disturbance estimation error of the outer LESO is described by 3 which is given in (32); both of these are illustrated in fig. 4. The ITAE of 3 is 10.5769 and the ITAE of 3 is 5.8251, displaying a percentage reduction in the ITAE equal to 45%. Fig. 4 more clearly illustrates the reduction in 3 against 3 . As illustrated in Fig.5 the derivative of the generalized disturbance ∆( ) = is bounded during the transient period by 5.34 and at the steady-state by 0.3. Assumption A2 is already satisfied.

Conclusion
This paper presented a novel approach to the design of a new class of LESO achieved by nesting an additional LESO in parallel with the original to obtain an N-ADRC. The proposed N-ADRC was successfully applied to the hypothetical SISO and a highly uncertain nonlinear SISO system with exogenous disturbance as given in (50). It can be concluded that the N-ADRC outperforms the C-ADRC in terms of control effort, output tracking, and disturbance rejection, as well as, more obviously, in the case of measurement error. In contrast with the C-ADRC, when the order of the LESO increases, the issue of measurement noise could be challenging, where increasing bandwidth is the only option for obtaining better performance, the main outcome of this work was to show that an outer loop LESO connected in parallel with the inner loop LESO removes the need to increase the bandwidth of the inner loop LESO as has been shown through the numerical simulations of this work. Furthermore, the N-ADRC can converge to the states of the original system asymptotically. The N-ADRC reduced the ITAE dramatically for cases both with and without measurement noise. Due to its simplicity, N-ADRC is suitable to be implemented in realtime applications. In future work, this approach can be extended to nest more than two LESOs, and nonlinear ESOs could also be used and their performance investigated for MIMO systems.