On The Design of a Novel Finite-Time Nonlinear Extended State Observer for Class of Nonlinear Systems with Mismatch Disturbances and Uncertainties

In this paper, a novel finite - time Nonlinear Extended State Observer (NLESO) is proposed and employed in Active Disturbance Rejection Control (ADRC) to stabilize a nonlinear system against system's uncertainties and discontinuous disturbances using output feedback technique. The first task was to aggregate the uncertainties, disturbances, and any other undesired nonlinearities in the system into a single term called the"generalized disturbance". Consequently, the ESO estimates the generalized disturbance and cancel it from the input channel in an online fashion. A peaking phenomenon that existed in Linear ESO (LESO) has been reduced significantly by adopting a saturation - like nonlinear function in the proposed Nonlinear ESO (NLESO). Stability analysis of the NLEO is studied using finite - time Lyapunov theory, and the comparisons are presented over simulations on Permanent Magnet DC (PMDC) motor to confirm the effectiveness of the proposed observer concerning LESO.


Introduction
In most control industries, it is hard to establish accurate mathematical models to describe the systems precisely.Also, some terms are not explicitly known in mathematical equations and, on the other hand, some unknown external disturbances exist around the system environment.The uncertainty, which includes internal uncertainty and external disturbance, is ubiquitous in practical control systems [1].
To tackle the shortcomings of Passive Anti-Disturbance Control (PADC) strategies in treating the disturbances, Han in [2] has proposed the alleged Active Disturbance Rejection Control (ADRC) paradigm.Generally speaking, the basic idea behind the ADRC is to straightforwardly oppose disturbances by feedforward compensation control configuration given disturbance estimations /cancellation principle [3].This procedure has discovered an expansive number of practical applications [4,5].Extended State Observer (ESO) is the central part of the ADRC, which works by augmenting the mathematical model of the nonlinear dynamical system with an additional virtual state.It describes all the unwanted dynamics, uncertainties, and exogenous disturbances and is called the "generalized disturbance" or "total disturbance".Then this total disturbance is estimated by the ESO and feedback into the control input channel for cancelation [2,6].
During recent years, the literature has found many types of research for analysis, design, and implementation of the ESO.Different forms of modern control theory based observers have been proposed to meet this need where several surveys of various disturbance observers can be found in [7][8][9].In 1971, Johnson introduced the Unknown \Input Observer (UIO) [10] to estimate the unknown input of the system.The transfer function based Disturbance Observer (DOB) [11], proposed later on by Japanese researchers, can predict the disturbance as well.The Perturbation observer (POB) was offered by Kwon and Chung in 2002 [12] in the discrete form to estimate the perturbation acted on the system.The "unknown input", the "disturbance" and the "perturbation" are just different names for the external disturbance, and the above observers can only deal with it.The aforementioned DOBC techniques employ some plant information for disturbance observation and control design.
The ESO is the one that uses less information as only the system relative degree should be known (defined later in this paper).For that reason, the ESO has become very popular in recent years.ADRC can be understood as a combination of an extended state observer (ESO) and a state feedback controller, where the ESO is utilized to observe the generalized disturbance taken as an augmented/extended state.The state feedback controller is used to regulate the tracking error between the real output and a reference signal for the physical plant [13].It should be remembered that the ESO together with the nonlinear plant takes the form of chain-like integrators where any of the linear or nonlinear control design methods, like those presented in [14][15][16][17][18][19] can be applied for feedback stabilization and performance enhancement.
The rest of the paper is organized as follows: in Section 2, paper scope and contribution are summarized.Problem statement and preliminaries with some mathematical definitions are given in Section 3. The main results of the proposed NLESO with the stability studies are presented in section Section 4. Section 5 describes an enhanced Version of ADRC with the proposed NLESO as its central core.The simulations of the enhanced ADRC on the permanent magnet DC motor are shown in Section 6. conclusions and future work are mentioned in Section 7.

Paper Scope and Contribution
In this paper, a new class of nonlinear extended state observers(NLESO) are proposed to actively reject the generalized disturbance for a general uncertain nonlinear system according to the principle of estimation/cancelation.For this NLESO, a saturation-like nonlinear error function was suggested to attenuate the large observer error at the starting stage and/or at the time of a discontinuous disturbance is injected to the system, consequently alleviating the peaking phenomenon.From the stability analysis of the error dynamics of NLESO observer, the finite-time stability analysis based on Lyapunov principle and Self-Stable Region(SSR) were introduced an applied on NLESO.

Problem Statement and Preliminaries
Assume an n-dimensional SISO nonlinear system which is expressed by [20], which can be rewritten as a chain of integrals with nonlinear uncertainties appearing in the n-th equation, where () ∈ (ℝ, ℝ) is the control input, () the measured output, (•) ∈ (ℝ  , ℝ) an unknown system function, () ∈ (ℝ, ℝ) the uncertain exogenous disturbance, () = ( 1 (),  2 (), … ,   ())  the state vector of the nonlinear system and (0) = ( 10 ,  20 , … ,  0 ) the initial state, () = (,  1 (),  2 (), … ,   ()) + () is called ''generalized disturbance'' [21].By adding the extended state  +1 () ≝ () = (, .) + (), the system (1) can be written as, The following Linear ESO (LESO) needs to be designed to estimate the states of the nonlinear system as well as the generalized disturbance () and is described as [22], where  ̂+1 () =  ̂=  ̂(,  1 (),  2 (), … ,   ()) +  ̂(),   is a constant observer gain to be tuned, , where   ,  ∈ {1,2, … , ,  + 1} are pertinent constants, and  is the constant gain or the reciprocal of observer's bandwidth,  is the relative degree of the nonlinear system [20].The observer gain is directly proportional to the observer bandwidth.Selecting a bandwidth that is too large will lead to a drop in the estimation error within an acceptable bound [23].Therefore, the observer bandwidth is chosen to be sufficiently larger than the disturbance frequency and smaller than the frequency of unmodelled dynamics [24].On the other hand, the performance of the ESO will be deteriorated if the bandwidth of the ESO is selected as either too low or too high.The side effects of adopting a large value for bandwidth can be summarised as, the measurement noise causes a degradation on the output tracking and the control signal.Some unmodelled high frequencies dynamics may be activated beyond a certain frequency causing inconsistency in the closed-loop system.A dynamical system whose states approaches as accurate as possible the states of (3) through time and coincides with them as  → ∞.
Motivated by the above reasons, the aim is to design a nonlinear ESO(NLESO) is required to be designed whose Functions are as follows: 1. Its states approach as accurate as possible the states of (3) through time and coincide with them as  → ∞.
If the nonlinear functions   : ℝ → ℝ,  ∈ {1,2, … ,  + 1} were selected appropriately, the state variables of the nonlinear system could track the state variables of the original system and generalized disturbance.A nonlinear function is the mathematical fitting of "big error, small gain or small error, big gain" [25].This function is generally selected as a nonlinear combination power function and is given as [26,27] where  is a small number which is used to express the length of the linear part [28].The (•) is a piecewise continuous, nonlinear, saturation, a monotonous increasing function.The following assumptions are made for the next sections [29,30].

Assumption (A1
∎ Assumption (A4).The solution   of (2) satisfy |  ()| ≤  for some constant B > 0 for all i = 1, 2,…., n, and t ≥ 0. ∎ Next, the convergence analysis of LESO for uncertain nonlinear single-input single-output is considered.Definition 2 [31]: Assuming that S is a region in the state space, which contains the origin.If it satisfies the condition that any system's trajectory, which remains in it after a particular time, will eventually converge to the origin, then S is called the self-stable region (SSR) of the system.∎ In the next, we will derive a formula based on Lyapunov function which describes the convergence of the proposed NLESO.The assumptions given below are related to finite-time stability analysis of both the ESO and the closed-loop system.Lemma 1 [32,33]: the system is globally finite-time stable where k > 0,  ∈ (0, 1).For any initial value of e(t) at t = to, i.e. ,   , it is easily obtained that the solution trajectory of (8) will reach e = 0 in finite time Theorem 1 [33,34]: Consider the nonlinear system ̇= () with f(0) = 0. Suppose there exists a continuous function :  → ℝ on an open neighborhood  ⊆ ℝ n of the origin such that the following conditions hold, 1. V(x) is positive definite.

𝑉 ̇(𝑥) + 𝑐𝑉 𝛼 ≤ 0.
Then the origin is finite-time stable, and the settling time tf depending on the initial conditions x(0) = xo is given as, for all xo in some open neighborhood of the origin, where c > 1, 0 <  < 1.

Main Results
In this section, the proposed NLESO will be presented followed by the stability analysis of its finite-time convergence based on self-stable region technique.The conversion of the mismatched disturbance into matched one will also be introduced.

The Proposed NLESO
The proposed NLESO for the general uncertain nonlinear system of ( 2) is designed as, where  ̂+1 () =  ̂,   =    0 −1 ,  0 > 0 is the ESO bandwidth and   , i = 1, 2,…..,  +1, are selected according to, such that the characteristic equation is Hurwitz, where  is the relative degree of the nonlinear system.The nonlinear function ℊ  : ℝ → ℝ is designed as: where   ,   , ,   and  are the positive design parameters,   's are used to help further reduce peaking phenomenon, a coherent problem with LESO, they are chosen such that  1 >  1 > ⋯ . .>  +1 .It should be noted that the proposed NLESO estimates the states of the uncertain nonlinear system up to relative degree  of the nonlinear system.For the chain of integrals characterized by ( 2) or (3), the relative degree is  = .So, the NLESO will estimate up to n-th states of (2) in addition to the generalized disturbance defined by  +1 ().
The proposed nonlinear function has a saturation-like profile which obeys the principle of "small error, large gain, and large error, small gain", it is an odd function in terms of the error e.It has the following additional features:   (0) = 0   (e) = k(e).e , where   ≤ () < ∞.

Remark 1:
The proof of the first part of Theorem 3 is based on the Filippov sense, where any discontinuous differential equation ̇= () ,  ∈ ℝ  and v is a locally bounded measurable vector function, is replaced by an equivalent differential inclusion ̇∈ () (see [36] ).While the second part was proved by self-stable region approach defined in definition 1, see [37] and the references therein.∎ The dynamics of the proposed NLESO given by ( 10) can be represented in terms of ( 14 The dynamics (20) tells us that the states of the NLESO suffer from observer error dynamics (14).

Mismatched Disturbance and System of Integrals Chain
The original ESO in [2], [38] assumes that the plant is expressed in the Integral Chain Form (ICF) satisfying the matched condition (Brunovsky form) [39].Therefore, its applicability is restricted to systems which, directly or using a change of variable, can be expressed in the ICF.Performing such transformation is not always easy as it is mentioned in [2], [40], especially if the system has zerodynamics.Furthermore, in certain nonlinear systems, the disturbances appear in the system in a different channel of the control input and hence does not satisfy the matching condition.Consequently, the standard manipulation of ADRC for this mismatched disturbance is no longer available.For example, assume the following uncertain nonlinear system, where xi = [x1, x2, … , xn] T ∈ ℝ n , u ∈ ℝ, y are the states of the system, the control input, and the system output, respectively.fi, are smooth function and are differentiable and fi(⋅) ≠ 0 for i = 1, 2, … , n-1, di(t) ∈ ℝ represents the external mismatched disturbance,   is the matched disturbance.Therefore, motivated by the successful results of the ESO, it was recently pointed out in [41] that it is imperative to develop ESO-based control techniques for systems without assuming the ICF and satisfying the matching condition( the disturbance must appear on the same channel of the control input, i.e., Brunovsky form).The next theorem is proposed to deal with mismatch disturbances and uncertainties assuming n = 2 for simplicity.
Theorem 4: Consider the 2 nd order affine nonlinear dynamical system with mismatched disturbance satisfying assumption A2 represented by: The above system can be transformed into the nonlinear model satisfying the matching condition (Brunovsky form) with the state-space given by: where  ̂( ) Proof: differentiate the first equation of ( 23) w.r.t t, one gets: Substitute ( 22) into (24) to get Rearrange ( 25), then, )) Then ( 26) is reduced to What remained is just  2 , and one can find an expression of  2 from the first equation of ( 22) and substitute this expression in (27) to get a matched nonlinear state-space equation in terms of the new coordinate system  ̃1,  ̃2.Finally, ( 27) is called the canonical form of ADRC ( Brunovsky form) [1].∎ To illustrate the aforementioned transformation, let apply it on the numerical example given by [42], Then Sub. ( 28) in (32) to have: and sub.( 29) in (33), Rearrange (34), results in Finally,  ̂( ̃1,  ̃2) = −2 ̃1 −  2 +  2 ̃1 +  2   ̃1 ,  ̂=   ̃1  +  ̇. Alternatively, Let  ̃1 =  1 ,  ̃2 =  ̃1 , then  (22) in the sense that after being transformed into the same channel of the control input, it is expressed in terms of the dynamic states of the nonlinear system and derivative of the original mismatched disturbance.In effect, the proposed NLESO will in real-time manner estimate and cancel  ̂( ̃1,  ̃2,  2 ) +  ̂ and depending on how well the NLESO estimate the dynamic states of the nonlinear system and the generalized disturbance, the nonlinear system together with the NLESO will look like a chain of integrals up to the relative degree  of the original uncertain nonlinear system.

Application of The Proposed NLESO in ADRC
The classical Active Disturbance Rejection Control ( ADRC) proposed by J. Han [2] is built by combining the tracking differentiator (TD), the nonlinear state error combination (NLSEF), and the linear extended state observer (LESO).In this work, an enhanced version of the ADRC(EADRC) which is called EADRC-NLESO is illustrated to emphasize that the proposed NLESO is employed in the design.It consists of a Second Order Nonlinear Differentiator (SOND) [43], an Improved Nonlinear State Error Feedback (INLSEF) controller [44], and the proposed NLESO.In the Improved INLSEF controller, the algorithm uses the sign(.)together and the exponential function which are integrated as follows,   =   = () = ()  () +   , Where  ∈ ℝ n is the vector of the state error, defined as  = [  (0) … . () … . (−1) ]  .In this regard, e (i) is the i-th derivative of the state error defined as,  () =  () −  () .The function k(e) and the function f(e) are defined in [44].It must be mentioned that   = 0 in our work, where integral action in ADRC is almost achieved by the ESO.Supposedly, the ESO will estimate and cancel online all the errors caused by any kind of discrepancy in the nonlinear system including the external disturbances.The NLESO (for n = 2) has the following state-space representation, where  ̂() = [ ̂1(),  ̂2(),  ̂3()]  ∈ ℝ 3 , is a vector that includes the predictable states of the plant and the total-disturbance.The coefficients  1 = 3,  2 = 3  ,  3 =   2 ,   , ,   ,  1 ,  2 ,  3 and  ∈ ℝ + are NLESO design parameters.Another structure of ADRC was designed to compare the performance of the proposed EADRC-NLESO with it.It has the same configuration of the EADRC-NLESO, but with LESO instead of NLESO, throughout the simulations, it is referred to as EADRC-LESO.Based on the above, the control signal which actuates the nonlinear system in the ADRC paradigm is given by  =   −  ̂3()  (37)

Simulations Results
As an application of the EADRC, the following numerical simulations include the nonlinear control of Permanent Magnet DC (PMDC) motor shown in Figure 1 with Coulomb friction force using EADRC.The nonlinear model of the PMDC motor including the external disturbance is of mismatched type, see (25).Applying Newton's law and Kirchoff's law, we get the following equations, where  is the input voltage applied to the motor (Volt) ,   is electromotive force constant constant (Volt / rad/s),   is the torque constant (N.m/A) ,  is the electric resistance constant(Ohm),  is the electric self-inductance (Henry),   is the total-equivalent moment of inertia(kg.m 2 ),   =Jm+ JL/N 2 , where JL is the load moment of inertia (kg.m 2 ), Jm is the motor armature moment of inertia(kg.m 2 ),   is the total-equivalent viscous damping of the combined motor rotor and load (N.m/rad.s),  =bm+ bL/N 2 , bm is the motor's rotor damping (N.m/rad.s),and bL is the load viscous damping (N.m/rad.s),N is the gearbox ratio,   (N.m) is the load torque applied at the motor side.
where   is the Coulomb friction force [45], it is nonlinear function of  1 , that is why the system of ( 26) is nonlinear,   is the external disturbance torque, usually of discontinuous type.The values of the parameters for PMDC motor are [46]: The PMDC controlled by both EADRC-NLESO and EADRC-LESO is tested by applying a reference angular-velocity equals to 1 rad per second at t = 0 and for Tf = 10 sec.To verify "Peaking Phenomenon", the initial conditions of both NLESO and LESO were set to  ̂1(0) = 0.5,  ̂2(0) =  ̂3(0) = 0, and that of the PMDC motor were,  1 (0) =  2 (0) =  3 (0) = 0.The results are shown in Figures 2 and 3   For   = 35, the output response of the PMDC motor system of (40) using EADRC-NLESO are plotted in Figure 2. The angular velocity  1 () and its estimation  ̂1() are drawn in Figure 2 (a), while the angular acceleration  2 () and its estimation  ̂2() are plotted in Figure 2 (b).The generalized disturbance  3 () together with its estimation are depicted in figures 2 (c).It can be seen that the estimation using NLESO is almost guaranteed.For the same   , the numerical results of the PMDC motor of (39) are redrawn in Figure 3, but using EADRC-LESO.It was clear that the LESO satisfactorily achieve state and generalized disturbance estimation, but suffers from "Peaking phenomenon", where  ̂1() peaks to -0.225,  ̂2() to -13.3 and  3 () to -139.6, compared to NLESO where  ̂1() peaks to -0.026,  ̂2() to -3.27 and  3 () did not peak.The under estimation in generalized disturbance  ̂3() (i.e. 3 () does not exactly follow  3 () ) can be treated successfully by increasing   , but on the account of noise filtration.The control signal in EADRC-LESO peaks down to -16 volts and up to 26 volts, whereas it peaks just to about 16.5 volts in EADRC-NLESO.It obvious that peaking in EADRC-NLESO is much smaller than that of EADRC -LESO.
To investigate the performance of the proposed NLESO to an exogenous disturbances, an experiment was conducted with an external torque acting as a step disturbance equal to 2 N.m ((2N.m /3)=0.666N.m seen from the rotor side) is applied after the gearbox during the simulation at t = 5 sec using MATLAB Simulink environment.The numerical results are shown in Figure 4. From this figure, it is easy to verify that both methods cancel the effect of the disturbance on the angular velocity efficiently with the EADRC-NLESO exhibits an undershoot larger than that of the EADRC-LESO (see Figure 3 (a), at t = 5).The Integration-Time-Absolute-Error (ITAE) performance measure defined as, where y is motor angular velocity output, and r is the reference signal is used to measure the performance of both NLESO and LESO at steady-state.Its value in EADRC-LESO is 2.238968, and the ITAE value using EADRC-NLESO is 0.485433.It is clear that EADRC-NLESO outperforms EADRC-LESO significantly.Also, the control signal in the EADRC-LESO had a high peak at the starting and fluctuated after that until it reached the time of disturbance occurrence, again it overshoots to 9.3 volts.On the other hand, the control signal in EADRC-NLESO overshoots with positive values only and to the half of that in EADRC-LESO.This leads to an increase of the energy required in the EADRC-LESO case, where an energy index defined as the integral square of the control signal (u) denoted as ISU is used to measure how much energy the control scheme requires, i.e.Based on ISU index, the control energy in EADRC-NLESO is 161.60068 and 172.92265 in EADRC-LESO.It must be mentioned that a limiter of ∓12 volts has been placed before the PMDC motor to limit the control signal input within the safe bounds.
Another experiment has been conducted to test the proposed NLESO against measurement noise.Assume that y(t) has been contaminated by n(t) usually (Gaussian) random signal where n(t) is usually (Gaussian) distributed random signal with 36 × 10 −6 and zero mean and is added using MATLAB Simulink block called random number.With the same values of the parameters in the above simulations including the bandwidth (  ), the results are presented in Figure 5.It can be seen that the noise is still exist in the output response of the EADRC-LESO, while EADRC-NLESO produces a smoother response and suppresses noise evidently.
Finally, we end our simulations by subjecting the PMDC motor of (40) to parameter uncertainties and test the efficiency of EADRC-LESO, EADRC-NLESO, and compared to each other.Let the parameters to be varied are defined in allowable range as follows, where  ̅  = 0.2752 ,  ̅  = 0.3922, and  ̅ = 0.1557 are called the nominal values of   ,   , and ∆  , ∆  , and ∆  are the possible relative changes in their respective parameters.Assume that   =   =   = −1 and let ∆  = 0.2, ∆  = 0.4, and   = 0.5, the angular velocity of the PMDC motor of (39) using both EADRC-LESO and EADRC-NLESO are graphed in Figure 6.
To end this section, let us discuss the advantages and disadvantages of using NLESO over LESO in ADRC configuration.At steady state, the error dynamics in (14) or (15) will be zero, i.e. ̇1() = ̇2() = ̇3() = 0, then , one can find a relationship between error  3 () in the last equation of (15) in terms of ∆, β3, c3, and other parameters,  3 () = ( ∆,  3 ,  3 , . .), where (. ) is a nonlinear function.The same can be done for  2 () and  1 (), i.e  2 () = ( ∆,  2 , … .),  3 () = ( ∆,  1 , … .), and this makes the errors of the NLESO are sensitive to ∆, which is the rate the generalized disturbance (), while in LESO, the errors are linearly depending on ∆.That explains the jumps in the generalized disturbance using EADRC-NLESO at the points in the times where () exhibits a sudden increase or decrease (e.g., sudden external disturbance) or parameter variations in the nonlinear system of the PMDC motor described in (39).Figures 4(a Next, we comment on the peaking phenomenon in both EADRC-LESO and EADRC-NLESO scenarios; it occurs at the starting when there is an initial condition  ̂1(0) for  ̂1() different from that of  1 ()(.. ̂1(0) ≠  1 (0)), this makes the terms in the equations of ( 14) that depends on   , i.e.    0 −1 ℊ  ( 0 (() −  ̂1())) very large for some ℊ  's and with large  0 .This term with a large value in the right hand side makes the ESO to produce large fluctuation on its output channels.For example when ℊ  () = , that means the ESO acts as a LESO, then the term    0 −1 ℊ  ( 0 ((0) −  ̂1(0))) will have a high value for (0) −  ̂1(0) and large  0 .The nonlinear function of (12) in our proposed NLESO (11) has a saturation-like behaviour for large e1, in addition to the attenuating factors   ( 1 >  1 > ⋯ . .>  +1 ), that explains why our proposed NLESO has a smaller peaking than LESO for the same  0 .for example the  0 -dependent term in the i-th equation of ( 14) can be expressed as  The reduction in the peaking with the proposed saturation-like function ℊ  (.) defined in (11) in our proposed NLESO of ( 11) is noteworthy .
The same reasoning can be extended to illustrate why the noise is attenuated using our proposed NLESO of (11) with the saturation-like function ℊ  (.) defined in (11), where the noise is magnified to ( 0 ) 3 .() in the i-th equation of the LESO of (4).While with our proposed saturation-like function ℊ  (.) defined in (11), the magnification of the noise () in our proposed NLESO is        0 +−1 |((0) −  ̂1(0))|  which is proven to be less than ( 0 ) 3 .The results of Figure 5 illustrate this justification.

Conclusions
In this paper, a novel saturation-like function has been proposed and employed in the design of a nonlinear extended state observer used to estimate the states and the generalized disturbance of any uncertain nonlinear system with mismatch disturbance.Stability analysis based on Lyapunov principles has shown the asymptotic convergence of the proposed ESO and finite-time stability is always guaranteed provided that the generalized disturbance is bounded.The advantage of the proposed ESO is that it produced smaller peaking and had immunity against measurement noise and parameter variations.All the mathematical investigations and the conducted experiments included in this paper proved that the proposed ESO presented better performance than the linear ESO for the mentioned reasons.Employing the proposed ESO in the ADRC configuration provided an excellent tool to control any uncertain nonlinear system and to counteract the generalized disturbance.As a future direction, this work can be extended to a general MIMO uncertain nonlinear system and apply the proposed ESO to non-affine control system like ball-and-beam system.

Figure 2 .
Results of the numerical simulations of the PMDC motor by EADRC-NLESO.

Figure 4 .
Angular velocity due to an external step disturbance of 2 N.m.

Figure 5 .Figure 6 .
) and 6 clarify this reasoning.The numerical results for PMDC motor of (39) with measurement noise.The angular velocity of PMDC motor of (39) with parameter uncertainties.