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

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1 On The Design of a Novel Finite-Time Nonlinear Extended State Observer for Class of Nonlinear Systems with Mismatch Disturbances and Uncertainties Asst. Prof. Ibraheem Kasim Ibraheem, IEEE member University of Baghdad /College of Engineering/Electrical Engineering Department, 10001, Baghdad, Iraq ibraheemki@coeng.uobaghdad.edu.iq Abstract: 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. Moreover, in some nonlinear systems, the disturbances might not enter through the same channel of the control signal, thus leading to mismatched uncertainties and/or disturbances. Accordingly, a new procedure is proposed in this paper to transform the mismatched disturbance and uncertainties into the same input channel and converting the nonlinear uncertain system into a chain of integrals up to the relative degree of the system with the total disturbance relocated into the control input channel. 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 principles and the comparisons are given through simulations on Permanent Magnet DC (PMDC) motor to show the effectiveness of the proposed observer with respect to LESO. Keywords: extended state observers, bandwidth, estimation error, output feedback, tracking differentiator, nonlinear state error. I. INTRODUCTION In most control industries, it is hard to establish accurate mathematical models to describe the systems precisely. In addition, there are some terms that 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]. On one hand, Adaptive Control (AC) expects to deal with the undesired impacts caused by structure parameter variations. The possibility of AC is that the model parameters of the controlled plants are first identified on realtime manner, at that point the control parameters are tuned in light of the calculated model parameters to acquire fine response. The Robust Control (RC) centers around investigating the capacities of the controller against system perturbations, the control design of RC conservatively assumes the worse-case of the plant variations. Thus, the validity of RC is generally acquired at the cost of relinquishing the transient performance of other distinguished points uncertainties [2]. 1

2 On the other hand, the Sliding Mode Control (SMC) has fine capabilities in restraining the impacts of parameter variations and exogenous disturbances. Notwithstanding, the discontinuous switching of the SMC exhibits high fluctuations (chattering) of mechanical systems [3]. To this end, these control approaches are commonly classified as Passive Anti-Disturbance Control (PADC) strategies. To tackle the shortcomings of PADC strategies in treating the disturbances, Han in [4] 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 in view of disturbance estimations /cancellation principle [5]. This procedure has discovered an expansive number of practical applications [6]. 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 [4], [6], [7]. During the 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 [8] [10]. In 1971, Johnson proposed the Unknown \input Observer (UIO) [11] to estimate the unknown input of the system. The transfer function based DOB [12], proposed later on by Japanese researchers, can estimate the disturbance as well. The POB was proposed by Kwon and Chung in 2002 [13] in the discrete form to estimate the perturbation acted on the system. Above 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 kind of plant information for disturbance observation and control design. The ESO is the one that uses the 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. The observer design took a leap in 1995, when Han proposed the unique ESO[14], [15] to address not only the external disturbances but also the internal dynamic uncertainties, the whole effect of which is considered as the total disturbance and estimated by the ESO. The ESO can 2

3 estimate conventional state variables of the system, which may be advantageous during controller design. The ESO owes its popularity to the natural connection to the well-established tools of control theory, including state space representation and observer design. 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, which is also taken as an augmented/extended state, and the state feedback controller is used to regulate the tracking error between the real output and a reference signal for the physical plant [16]. The original work on ADRC proposed a nonlinear version of the ESO [4], but it is the simple, linear, Luenberger-like observer [17], [18], which is now the most common reconstructing tool seen in the literature on ADRC, due to its effectiveness and simplicity. One can now notice many works in which the standard types of observers are augmented with the idea of the extended state (representing the total disturbance), What's more, an idea of bandwidth parameterization is suggested in [16] to limit the number of tuning parameters of ADRC. Utilizing this idea, ADRC just has two tuning parameters, of which one is for the controller, and the other is for the ESO. The two tuning parameters specifically mirror the performance speed of the ESO and the feedback system respectively. Additionally, fewer tuning parameters make the practicality of ADRC easier. The guide on how to choose the tuning parameters for ADRC are given in [19]. At the beginning of the research of ADRC, time-domain analysis of the controller was the main focus. In recent times, a transfer function representation of ADRC was initially given in [20], where frequency-domain analysis have been positively carried out on a second-order system. The feedback system with ADRC can be represented by a unity feedback loop. In the performance analysis in [20], the Bode diagram and the stability margins of the closed-loop system have been obtained. The unchanged values of the margins against the variations of system parameters demonstrate the distinguished power of ADRC against system s parameter variations. It should be remembered that the ESO together with the nonlinear plant take the form of chain-like integrators where any of the linear or nonlinear control design methods can be applied for feedback stabilization and performance enhancement, see for example [21] [26] and the references therein. The rest of the paper is organized as follows: in section II, paper scope and contribution are summarized. Problem statement and preliminaries with some mathematical definitions are given in section III. The main results of the proposed NLESO with the stability studies are 3

4 presented in section IV. Section V 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 VI. conclusions and future work are mentioned in section VII. II. 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. To deal with mismatch disturbances and uncertainties, a new method with its validation was proposed to deal with any mismatched nonlinear model and converting it into a chain of integrals form up to the relative degree of the system with the total disturbance term is relocated to the control input channel. Remark 1: Without loss of generality and for the rest of the paper, the vectors a and a(t) will be used interchangeably to mean vector with appropriate dimension as a function of time. Timeindexed vectors will be used in certain contexts to emphasize a specific meaning. n-th equation, III. PROBLEM STATEMENT AND PRELIMINARIES Assume an n-dimensional SISO nonlinear system which is expressed by [27]: y n (t) = f(t, x 1 (t), x 2 (t),, x n (t)) + w(t) + bu(t) (1) Which can be rewritten as a chain of integrals with nonlinear uncertainties appearing in the x 1(t) = x 2 (t), x 1 (0) = x 10, x 2(t) = x 3 (t), x 2 (0) = x 20, x n(t) = f(t, x 1 (t), x 2 (t),, x n (t)) + w(t) + bu(t), x n (t) = x n0, { y(t) = x 1 (t). Where u(t) C(R, R) is the control input, y(t) the measured output, f( ) C(R n, R) an unknown system function, w(t) C(R, R) the uncertain exogenous disturbance, x(t) = (2) 4

5 (x 1 (t), x 2 (t),, x n (t)) T the state vector of the nonlinear system and x(0) = (x 10, x 20,, x n0 ) the initial state, L(t) = f(t, x 1 (t), x 2 (t),, x n (t)) + w(t) is called generalized disturbance [28]. By adding the extended state x n+1 (t) L(t) = f(t,. ) + w(t), the system (1) can be written as: x 1(t) = x 2 (t), x 1 (0) = x 10, x 2(t) = x 3 (t), x 2 (0) = x 20, x n(t) = x n+1 (t) + bu(t), x(t) = x n0 x n+1(t) = Δ(t) = f (t, x 1 (t), x 2 (t),, x n (t)) + w (t), x n+1 (0) = x n+1,0, { y(t) = x 1 (t) Definition 1 [29]: Given the nonlinear system below x = f(x) + g(x) u y = h(x) where f, g, and h are sufficiently smooth in a domain D, f : D R n and g : D R n are called vector fields on D. The system has relative degree ρ, 1 ρ n, in D 0 D if x D 0 L g L f i 1 = 0, i = 1, 2,, ρ 1; L g L f ρ 1 0, where is the Lie Derivative of h with respect to f (3) or along f. The aim is to design a dynamical system whose states approaches as accurate as possible the states of (3) through time and coincides with them as t. The following Linear ESO needs to be designed to estimate the states of the nonlinear system as well as the generalized disturbance L(t) and is described as [30]: { x 1(t) = x 2(t) + β 1 (y(t) x 1(t)), x 2(t) = x 3(t) + β 2 (y(t) x 1(t)), x ρ(t) = x ρ+1 (t) + bu(t) + β ρ (y(t) x 1(t)), x ρ+1(t) = β ρ+1 (y(t) x 1(t)), Where x n+1 (t) = L = f (t, x 1 (t), x 2 (t),, x ρ (t)) + w (t), β i is a constant observer gain to be tuned, i {1,2,, ρ, ρ + 1}. With β i = a i ε i, where a i, i {1,2,, ρ, ρ + 1} are pertinent constants, and ε the constant gain or the reciprocal of observer s bandwidth [27]. 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 [31]. Therefore, the observer 5 (4)

6 bandwidth is chosen to be sufficiently larger than the disturbance frequency and smaller than the frequency of unmodelled dynamics [32]. 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 1) the measurement noise causes a degradation on the output tracking, introduces a chattering on the control signal [33], 2) the transient response of the ESO is diminished since large values of bandwidth lead to high gain observers [34], and 3) some unmodelled high frequencies dynamics may be activated beyond a certain frequency causing inconsistency in the closed-loop system. For these reasons, it is required to modify the conventional ESO in order to ensure high estimation accuracy and reduces the chattering phenomenon, which will be our main focus in this paper. Based on the aforementioned reasons, a nonlinear ESO(NLESO) is required to be designed to reduce peaking phenomenon as well as to avoid large transient behaviors [35]. A nonlinear observer is also applied to guarantee fast-convergence and robustness with respect to the noise [36]. A general nonlinear extended state observer is given by: x 1(t) = x 2(t) + g 1 (y(t) x 1(t)), x 2(t) = x 3(t) + g 2 (y(t) x 1(t)), x ρ(t) = x ρ+1 (t) + bu(t) + g ρ (y(t) x 1(t)), { x ρ+1(t) = g ρ+1 (y(t) x 1(t)). If the nonlinear functions g i : R R, i {1,2,, ρ + 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 [37]. This function is generally selected as a nonlinear combination power function and can be shown as follows [38][39]: fal(e, α, δ) = { δ 1 α e α sgn(e) e e δ e > δ where δ is a small number which is used to express the length of the linear part [40]. The fal( ) is a piecewise continuous, nonlinear, saturation, a monotonous increasing function. The following assumptions are made for the next sections [34], [41]: (5) (6) 6

7 Assumption (A1). The function f and w(t) is continuously differentiable for all (t, x(t)) R R n. u + f + w + f t + f n c x o + c j x j k i for some positive constants c j, j = 0, 1,.., n and positive integer k. Assumption (A2). The generalized disturbance L(t) is bounded and belongs to a known compact set L R, i.e, sup L(t) t Assumption (A3). There is a positive constant M such that Δ(t) M for t 0. Where Δ(t) = L (t) = f (t, x 1 (t), x 2 (t),, x n (t)) + w (t) = d f(t,x(t)) [f(t, x(t)) + w(t)] = + dt t n 1 i=1 x i+1 (t) f(t,x(t)) x i j=1 + [f(t, x 1 (t), x 2 (t),, x n (t)) + w(t) + bu(t)] f(t,x(t),w(t) x n + w (t) Remark 2 : in the next, Lemmas and theorems mentioned with numbers are taken from their own references, the rest are proposed and proved by us. Assumption (A4). The solution x i of (2) satisfy x i (t) B for some constant B > 0 for all i = 1, 2,., n, and t 0. is considered. Next, the convergence analysis of LESO for uncertain nonlinear single-input single-output Definition 2 [42]: 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 certain time, will eventually converge to the origin, then G is called the self-stable region (SSR) of the system. Theorem 1 [34]. Consider the system given in (2), and the linear extended state observer (3), then: x i (t) x i(t) 1 ω n+1 i (2Mλ2 max (P) 0 ω 0 λ min (P) lim x i(t) x i(t) = 1 2Mλ 2 max (P) t ω n+2 i 0 λ min (P) And leads to t lim x i (t) x i(t) =0. ω 0 (1 e 2λmax (P) ) + V(η(0)) ω0t ω0t λ min (P) e 2λmax (P) ) (7) = O ( ) (8) ω n+2 i 0 where x i (t), and x i(t) denote the solutions of (2) and (4) respectively, i {1,2, n + 1}. Proof: refer to [1], [43]. 1 7

8 It can be seen that the estimation error is inversely proportional to the observer s bandwidth ω 0 and (10) tells that to get zero estimation error at steady-state, one needs to increase ω 0 extremely very large (ω 0 ). 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 [44], [45]: the system e = k sgn(e) e α (9) is globally finite-time stable where k > 0, α (0, 1). For any initial value of e(t) at t = to, i.e., e o, it is easily obtained that the solution trajectory of (9) will reach e = 0 in finite time t f = e o 1 α (1 α)k. Theorem 2 [45],[46]: Consider the nonlinear system x = f(x) with f(0) = 0. Suppose there exists a continuous function V: D R on an open neighborhood D R n of the origin such that the following conditions hold: 1. V(x) is positive definite. 2. 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: t f V(x o) 1 α c(1 α) for all xo in some open neighborhood of the origin, where c > 1, 0 < α < 1. (10) A. The Proposed NLESO IV. MAIN RESULTS The proposed NLESO for the general uncertain nonlinear system of (2) is designed as: x 1(t) = x 2(t) + β 1 g 1 (ω 0 (y(t) x 1(t))), x 2(t) = x 3(t) + β 2 g 2 (ω 0 (y(t) x 1(t))), x ρ(t) = x ρ+1 (t) + bu(t) + β ρ g ρ (ω 0 (y(t) x 1(t))), x ρ+1(t) = β ρ+1 g ρ+1 (ω 0 (y(t) x 1(t))) { Where x n+1 (t) = L, β i = a i ω 0 i 1, ω 0 > 0 is the ESO bandwidth and a i, i = 1, 2,.., ρ +1, are selected according to: a i = (ρ+1)! i!(ρ+1 i)! (11) 8

9 such that the characteristic equation s ρ+1 + a 1 s ρ + + a ρ s + a ρ+1 = (s + 1) ρ+1 Is Hurwitz. The nonlinear function g i : R R is designed as: g i (ω 0 e) = c i (K α ω 0 e α sgn(ω 0 e) + K β ω 0 e β. ω 0 e) (12) where K α, K β, α, c i and β are the positive design parameters (see Figure 1), c i s are used to help further reduce peaking phenomenon, a coherent problem with LESO, they are chosen such that c 1 > c 1 >.. > c ρ+1. It should be noted that the proposed NLESO estimates the states of the uncertain nonlinear system up to relative degree ρ of the system. For the chain of integrals characterized by (2) or (3), the relative degree is ρ = n. So, the NLESO will estimate up to n-th states of (2) in addition to the generalized disturbance defined by x n+1 (t) g(e) Figure 1. The curve of g function when α = 0.533, β = 0.334, k α = and k β = 0.543, c i = 1. As can be seen, 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 (e). It has the following features: g (0) = g (e) = k(e). e, where k min k(l) <. 9 e

10 B. Stability Study of the proposed NLESO The convergence of the proposed NLESO is studied based on how well it estimates the states of the uncertain nonlinear system and the generalized disturbance, theorem 3 below shows the convergence analysis in terms of the estimation error dynamics and finite-time stability. Theorem 3: Consider the nonlinear system of (2) and assumptions 1-4 are satisfied, then proposed NLESO described by (11) is globally asymptotically stable, it is finite-time convergent to (2) with tf > to such that e i = 0, i = 1,2,., n + 1 for all t > tf. Proof: By introducing the augmented state x n+1 = L into (2), we obtained (3). Moreover, set { e i(t) = x i (t) x i(t) i {1, 2, n} e n+1 (t) = x n+1 (t) x n+1 (t) i = n + 1 (13) It should be noted that in (13) that x n+1 (t) x n+1 (t) = L L, where L and L are the generalized disturbances and the estimated generalized disturbance respectively. A direct computation shows that the estimation error dynamics of (13) satisfies: x 1(t) x 1(t) = x 2 (t) (x 2(t) + β 1 g 1 (ω 0 (y(t) x 1(t))) x 2(t) x 2(t) = x 3 (t) (x 3(t) + β 2 g 2 (ω 0 (y(t) x 1(t))) x n(t) x n(t) = x n+1 (t) + bu(t) (x n+1 (t) + bu(t) + β n g n (ω 0 (y(t) x 1(t))) (14) { x n+1(t) x n+1(t) = Δ(t) β n+1 g n+1 (ω 0 (y(t) x 1(t))) Substituting (12) in (14), gives e 1(t) = e 2 (t) β 1 c 1 K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β 1 c 1 K β ω 0 e 1 (t) β. ω 0 e 1 (t) e 2(t) = e 3 (t) β 2 c 2 K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β 2 c 2 K β ω 0 e 1 (t) β. ω 0 e 1 (t) e n(t) = e n+1 (t) β n c n K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β n c n K β ω 0 e 1 (t) β. ω 0 e 1 (t) { e n+1(t) = Δ(t) β n+1 c n+1 K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β n+1 c n+1 K β ω 0 e 1 (t) β. ω 0 e 1 (t) Where Δ(t) is the derivative of the generalized disturbance L and is given by (7). According to Lemma 1 and theorems 5-7 of [47], and if assumption A3 holds true, then the error dynamics of (17) are asymptotically stable, i.e., the error e 1 (t) in the first equation approaches zero, so do e 2 (t), e 3 (t),, e n+1 (t) go to zero. Moreover, the error dynamics of (15) are finite-time stable, i.e., it converges to (2) within tf > to such that e i = 0, i = 0, 1,., n + 1 For all t > tf. To prove it is time-finite convergent, we use the self-stable region (SSR) approach in [42] to accomplish this task, and we proceed as follows, (15) 10

11 First, for simplicity assume n = 1, then the error dynamics of the NLESO is expressed as, { e 1(t) = e 2 (t) β 1 c 1 K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β 1 c 1 K β ω 0 e 1 (t) β. ω 0 e 1 (t) e 2(t) = Δ(t) β 2 c 2 K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β 2 c 2 K β ω 0 e 1 (t) β. ω 0 e 1 (t) (16) Define m 2 (e 1, e 2 ) = e 2 β 1 c 1 K α ω 0 e 1 α sgn(e 1 ) β 1 c 1 K β ω 0 e 1 β ω 0 e 1 + kq 1 (e 1 ) sgn(e 1 ), where q 1 (e 1 (t)) is positive definite continuous function, i.e., q 1 (0) = 0, k > 1. Also, define S = {e 1, e 2 : m 2 (e 1, e 2 ) q 1 (e 1 )}, assume that there exists (e 1, e 2 ) S, t > T. According to the structure of S or e 2 β 1 c 1 K α ω 0 e 1 α sgn(ω 0 e 1 ) β 1 c 1 K β ω 0 e 1 β ω 0 e 1 + kq 1 (e 1 ) sgn(e 1 ) q 1 (e 1 ) e 2 β 1 c 1 K α ω 0 e 1 α sgn(ω 0 e 1 ) β 1 c 1 K β ω 0 e 1 β ω 0 e 1 q 1 (e 1 ) kq 1 (e 1 ) sgn(e 1 ) (17) Choose a Lyapunov candidate function V(e 1 (t)) as Then, e 1 (t) V(e 1 (t)) = e 1 (t)de 1 (t) 0 V (e 1 (t)) = e 1 de 1 (t) dt = e 1 (e 2 (t) β 1 c 1 K α ω 0 e 1 (t) α sgn(ω 0 e 1 (t)) β 1 c 1 K β ω 0 e 1 (t) β. ω 0 e 1 (t)) (18) Sub (17) in (18), we get V (e 1 (t)) e 1 (q 1 (e 1 ) kq 1 (e 1 ) ) (k 1) e 1 q 1 (e 1 ) (19) From the positive definiteness definition of q 1 (e 1 ) and since k > 1, we conclude that V (e 1 (t)) is negative definite. Hence, and based on the structure of S t e 1 (t) 0 t e 2 (t) 0 Let q 1 (e 1 ) = k 1 e 1 (t) α, with k 1 > 0 and 0 < α <1. With this choice of q 1 (e 1 ), V (e 1 (t)) becomes From (18), V = e 1 2 V (e 1 (t)) r e 1 1+α 2 or e 1 = (2V) 1 2, and V (e 1 (t)) gets its final form as V (e 1 (t)) r(2v) 1+α 2 (20) With r = (k 1) k 1. According to Theorem 2, the error dynamics of (16) is finite-time convergent with 11

12 t f V(e 1 (t 0 ))1 α c(1 α ) Where α = 1+α 2. Remark 3: the proof of the first part of Theorem 3 is based on the Filippov sense, where any discontinuous differential equation x = v(x), x R n and v is a locally bounded measurable vector function, is replaced by an equivalent differential inclusion x V(x) (see[48] ). While the second part was proved by self-stable region approach defined in definition 1inherent property that second order systems, see [49] and the references therein. The dynamics of the proposed NLESO given by (11) can be represented in terms of (15) as, x 1(t) = x 2(t) + e 2 (t) e 1(t), x i(t) = x i+1 (t) + e i+1 (t) e i(t), i = 1,.., n 1 x n(t) = x n+1 (t) + e n+1 (t) e n(t) + u(t) { x n+1(t) = Δ(t) e n+1(t) The dynamics (23) tells us that the states of the NLESO suffer from the observer error dynamics (15). C. Mismatched disturbance and System of Integrals chain The original ESO in [4], [50] assumes that the plant is expressed in the Integral Chain Form (ICF) satisfying the matched condition [51], [6]. Therefore, its applicability is restricted to systems which, directly or by means of a change of variable, can be expressed in the ICF. Performing such transformation is not always easy as it is mentioned in [4], [52], 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, x 1 = f 1 (x 1,., x n ) + b 1 d 1 x i = f i (x 1,., x n ) + b i d i x n = f i (x 1,.., x n ) + b n u+b n d n { y = x 1 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 (21) (22) 12

13 and fi( ) 0 for i = 1, 2,, n-1. di(t) R represents the external mismatched disturbance, d n is the matched disturbance. Therefore, motivated by the successful results of the ESO, it was recently pointed out in [53] 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). The next theorem is proposed to deal with mismatch disturbances and uncertainties assuming n = 2 for simplicity. Theorem 4: Consider the 2 nd disturbance satisfying assumption A2 represented by: x 1 = f 1 (x 1, x 2 ) + b 1 d { x 2 = f 2 (x 1, x 2 ) + b 2 u y = x 1 order affine nonlinear dynamical system with mismatched The above system can be transformed into the nonlinear model satisfying the matching condition with the state-space given by: x 1 = x 2 { x 2 = f (x 1, x 2, x 2 ) + b (u + d ) y = x 1 Where f (x 1, x 2 ) = f 1 (x 1,x 2 ) x 1 f 1 (x 1, x 2 ) + f 1 (x 1,x 2 ) x 2 f 2 (x 1, x 2 ), b = b 2 f 1 (x 1,x 2 ) x 2, d = (b 1 f 1 (x 1,x 2 ) x 1 d + b 1 d )/(b 2 f 1 (x 1,x 2 ) x 2 ) Proof: differentiate the first equation of (23) with respect to t, one gets: (23) (24) x 1 = f 1(x 1,x 2 ) x 1 x 1 + f 1(x 1,x 2 ) x 2 x 2 + b 1 d (25) Substitute (23) into (25) to get x 1 = f 1 (x 1,x 2 ) x 1 (f 1 (x 1, x 2 ) + b 1 d) + f 1 (x 1,x 2 ) x 2 (f 2 (x 1, x 2 ) + b 2 u) + b 1 d (26) Rearrange (26), then, x 1 = f 1 (x 1,x 2 ) x 1 f 1 (x 1, x 2 ) + f 1 (x 1,x 2 ) x 2 f 2 (x 1, x 2 ) + b 2 f 1 (x 1,x 2 ) x 2 (u + ( b 1d +b 1 Then (27) is reduced to x 1 = f (x 1, x 2 ) + b (u + d ) f1 (x 1,x2 ) x1 b 2 f1 (x 1,x2 ) x2 d )) (27) 13

14 Let, x 1 = x 1 x 2 = x 1. Then x 1 = x 2 { x 2 = f (x 1, x 2, x 2 ) + b (u + d ) y = x 1 What remained is just x 2, and one can find an expression of x 2 from the first equation of (23) and substitute this expression in (28) to get a matched nonlinear state-space equation in terms of the new coordinate system x 1, x 2. Finally, (28) is called the canonical form of ADRC[1]. To illustrate the aforementioned transformation, let apply it on the numerical example given by [54], x 1 = x 2 + e x 1 + d { x 2 = 2x 1 x 2 + u y = x 1 It is clear that f 1 (x 1, x 2 ) = x 2 + e x 1, b 1 = 1, f 2 (x 1, x 2 ) = 2x 1 x 2, b 2 = 1. Let Then x 1 = x 1 (30) x 2 = f 1 (x 1, x 2 ) + d = x 2 + e x 1 + d (31) x 1 = x 2 (32) x 2 = x 1 = x 2 + e x 1x 1 + d (33) Sub. (28) in (32) to have: x 2 = 2x 1 x 2 + u + e x 1(x 2 + e x 1 + d) + d (34) and sub. (29) in (34), Rearrange (34), results in Finally, Alternatively, x 2 = 2x 1 x 2 + u + e x 1x 2 + e 2x 1 + e x 1d + d (35) f (x 1, x 2) = 2x 1 x 2 + e 2x 1 + x 2 e x 1, d = e x 1d + d. b = b 2 f 1 (x 1,x 2 ) x 2 = 1. 1 = 1 f 1 (x 1, x 2 ) f 1 (x 1, x 2 ) d = (b 1 d + b x 1 d )/(b 2 ) = 1. ex1d + d = 1. e x 1d + d 1 x 2 1 f (x 1, x 2 ) = f 1(x 1, x 2 ) x 1 f 1 (x 1, x 2 ) + f 1(x 1, x 2 ) x 2 f 2 (x 1, x 2 ) = e x 1(x 2 + e x 1) + 1. ( 2x 1 x 2 ) (28) (29) 14

15 Let x 1 = x 1, x 2 = x 1, then x 1 = x 2 { x 2 = f (x 1, x 2, x 2 ) + b (u + d ) y = x 1 With f (x 1, x 2, x 2 ) = x 2 e x 1 + e 2x 1 2x 1 x 2, b = 1, d = e x 1d + d One can further eliminate x 2 from (35) by substituting x 2 = x 2 e x 1 d in (35) to get x 2 = 2x 1 (x 2 e x 1 d) + u + e x 1(x 2 e x 1 d) + e 2x 1 + e x 1d + d = 2x 1 x 2 + e x 1 + d + u + x 2e x 1 e 2x 1 d + e 2x 1 + e x 1d + d = 2x 1 x 2 + e x 1 + u + x 2e x 1 + e x 1d + d Which with the same b = 1, leads to, (36) f new (x 1, x 2) = 2x 1 x 2 + e x 1 + x 2e x 1, d new = e x 1d + d I t must be noted that the matched disturbance d or d new is different from the original mismatched disturbance d of (23) 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 f (x 1, x 2, x 2 ) + d 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 chain of integrals up to the relative degree ρ of the original uncertain nonlinear system. V. Application of The Proposed NLESO in ADRC The classical Active Disturbance Rejection Control ( ADRC) proposed by J. Han [4] is built by combining the tracking differentiator (TD), the nonlinear state error combination (NLSEF), and the linear extended state observer (LESO) [22]. In Figure 2, 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. The second order nonlinear differentiator (SOND) proposed in our previous work [55] adopting the hyperbolic tangent function is given as, 15

16 Fig. 2. The EADRC-NLESO Configuration. x 1(t) = x 2 (t) { x 2(t) = σ 2 tanh ( bx 1 (t) (1 a)r ) σx c 2 (t) Where x 1 (t) is an estimation of the actual input r, and x 2 (t) is an estimation of the derivative of the actual input. the coefficients a, b, c, and σ are design factors, where 0 < a < 1, b > 0, c > 0, and σ > 0. In the Improved Nonlinear State Error Feedback (INLSEF) controller [56], the algorithm uses the sign(.) together with the exponential function which are integrated as follows, u INLSEF = u o = Ψ(e) = k(e) T f(e) + u int, Where e R n is the vector of the state error, defined as e = [e (0). e (i). e (n 1) ] T [23]. In this regard, e (i) is the i-th derivative of the state error defined as, e (i) = x (i) z (i). The function k(e) is the nonlinear gain, defined as (n = 2): k(e) = [ k 1(e) k 2 (e) ] = [ (k 11 + (k 21 + k 12 1+exp(μ 1 (e (0) ) 2 ) ) (37) ] (38) k 22 ) 1+exp(μ 2 (e (1) ) 2 ) The coefficients k i1, k i2,, μ i R +, i = 1,2, are the controller design parameters. The function f(e) is the error function, defined as: f(e) = [ e (0) α1 sign(e) e (1) α2 sign(e (1) ) ] T It must be mentioned that u int = 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: 16

17 { wshere x x 1 = x 2(t) + β 1 c 1 [K α ω 0 (y x 1(t)) α sign(ω 0 (y x 1(t))) + K β ω 0 (y x 1(t)) β (ω 0 (y x 1(t)))] x 2 = x 3(t) + bu + β 2 c 2 [K α ω 0 (y x 1(t)) α sign(ω 0 (y x 1(t))) + K β ω 0 (y x 1(t)) β (ω 0 (y x 1(t)))] x 3 = β 3 c 3 [K α ω 0 (y x 1(t)) α sign(ω 0 (y x 1(t))) + K β ω 0 (y x 1(t)) β (ω 0 (y x 1(t)))] (t) = [x 1(t), x 2(t), x 3(t)] T R 3, is a vector that includes the predictable states of the plant and the total-disturbance. The coefficients β 1 = 3, β 2 = 3ω o, β 3 = ω o 2, K α, α, K β, c 1, c 2, c 3 and β 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 Figure 2, but with LESO, throughout the simulations, it is referred to as EADRC-LESO. Based on the above, the control signal which actuates the nonlinear system in ADRC paradigm is given by v = u o x 3 (t) b VI. Simulations Results As an application of the EADRC, the following numerical simulations include the control of Permanent Magnet DC (PMDC) motor shown in Figure 3 with Coulomb friction force using EADRC. The nonlinear model of the PMDC motor including the external disturbance is of mismatched type, see (41) (39) (40) Figure 3. Schematic diagram of PMDC motor. Applying Newton s law and Kirchoff s law, we get the following equations, { J d 2 θ eq = T T dθ dt 2 L b eq dt L di { = Ri + v e dt d 2 θ = 1 dθ (K dt 2 J t i T L b eq ) eq dt di dt = 1 L ( Ri + v K b 17 (41) dθ ) dt

18 where v is the input voltage applied to motor (Volt), K b is electromotive force constant constant (Volt / rad/s), K t is the torque constant (N.m/A), R is the electric resistance constant(ohm), L is the electric self-inductance (Henry), J eq is the total-equivalent moment of inertia(kg.m 2 ), J eq =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 ), b eq is the total-equivalent viscous damping of the combined motor rotor and load (N.m/rad.s), b eq =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. Applying Theorem 4 (See Appendix A for the detailed derivation) we get the simplified ICF nonlinear state-space representation of the PMDC motor which is given by [5]: { x 1 = x 2 x 2 = R b eq+k t K b x L J 1 (L b eq+r J eq ) x eq L J 2 + eq y = x 1 /N K t L J eq (v + d) As can be seen, the relative degree ρ = n. The state x 1 is the angular velocity (rad/s) of the PMDC motor before dividing the actual angular velocity of the motor shaft dθ by the gear ratio N, while the output y is measured after the gearbox, i.e., y = 1 N x 1. The state x 2 is the angular acceleration (rad/s 2 ). The equivalent disturbance at the input d = L K t T L R K t T L and T L (N.m) is the load torque applied at the motor side. The load torque T L = 1 N (T ext + F c sgn(x 1 )), where F c is the Coulomb friction force [57], it is nonlinear function of x 1, that is why the system of (42) is nonlinear. The values of the parameters for PMDC motor are [6], [58]: Ra = , La = 0.82, Kb = 1.185, Kt = , n = 3.0, Jeq = , and beq = , F c = 1. The parameters of the proposed EADRC-NLESO are as follows, INLSEF: k11 = , k12 = , k21= , k22 = , μ1 = , μ2 = , α1 = , α2 = The SOND: a = , b = , c = , σ = The NLESO: ω o = 35, Kα = , α = , Kβ = 0.38, β = , β1 = 3, β2 =105, β3 = 1225, c1 = 0.5, c2 = 0.125, c3 = While the parameters of the EADRC-LESO are, INLSEF: k11 = , k12 = , k21= , k22 = , μ1 = , μ2 = , α1 = , α2 = The SOND: a = , b = , c = , σ = The LESO: ω o = 35, β1 = 105, β2 =3675, β3 = 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 dt (42) 18

19 Peaking Phenomenon, the initial conditions of both NLESO and LESO were set to x 1(0) = 0.5, x 2(0) = x 3(0) = 0, and that of the PMDC motor were, x 1 (0) = x 2 (0) = x 3 (0) = 0. The results are shown in Figures 4 and 5. (a) (b) Figure 4. Continued 19

20 (c) Figure 4. Results of the numerical simulations of the PMDC motor by EADRC-NLESO. (d) 20

21 (a) (b) Figure 5. Continued 21

22 (c) (d) Figure 5. Results of the numerical simulations of the PMDC motor by EADRC-LESO. For ω o = 35, the output response of the PMDC motor system of (42) using EADRC- NLESO are plotted in Figure 4. The angular velocity x 1 (t) and its estimation x 1(t) are drawn in Figure 4 (a), while the angular acceleration x 2 (t) and its estimation x 2(t) are plotted in 4 (b). The 22

23 generalized disturbance x 3 (t) together with its estimation are depicted in figures 4 (c). It can be seen that the estimation using NLESO is almost guaranteed. For the same ω o, the numerical results of the PMDC motor of (42) are redrawn in Figure 5, but using EADRC-LESO. It was clear that the LESO satisfactorily achieve state and generalized disturbance estimation, but suffers from Peaking phenomenon, where x 1(t) peaks to , x 2(t) to and x 3 (t) to , compared to NLESO where x 1(t) peaks to , x 2(t) to and x 3 (t) did not peak. The under estimation in generalized disturbance x 3(t) (i.e. x 3 (t) does not exactly follow x 3 (t) ) can be treated successfully by increasing ω o, but on the account of noise filtration. The control signal in EADRC-LESO peaks 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, an experiment was conducted with an external torque acting as a step disturbance equal to 2 N.m ((2N.m /3)=0.666 N.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 6. (a) Figure 6. Continued 23

24 (b) Figure 6. Angular velocity due to an external step disturbance of 2 N.m. From Figure 6, 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 6 (a), at t = 5). The Integration-Time-Absolute-Error (ITAE) performance measure defined as, ITAE = 10 0 t r y dt is for the output signal Where y is the PMDC 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 and the ITAE value using EADRC-NLESO is 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. ISU = u 2 dt

25 Based on ISU index, the control energy in EADRC-NLESO is and in EADRC-LESO. It must be mentioned that a limiter of 12 volt 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 has been contaminated by n(t) normally (Gaussian) distributed random signal y n (t) = y(t) + n(t) where n(t) is normally (Gaussian) distributed random signal with 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 (ω o ), the results are presented in Figures 7. (a) Figure 7 Continued 25

26 (b) (c) Figure 7. Continued 26

27 (d) Figure 7. The numerical results for PMDC motor of (42) with Gaussian distributed measurement noise. From Figure 6, the noise is still exist in the output response of the EADRC-LESO, while EADRC-NLESO produces smoother response and suppresses noise evidently. Finally, we end our simulations by subjecting the PMDC motor of (42) 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 their allowable range as follows, J eq = J eq(1 + J δ J ), b eq = b eq (1 + b δ b ), R = R (1 + R δ R ) Where J eq = , b eq = , and R = are called the nominal values of J eq, b eq, and J, b, and R are the possible relative changes in their respective parameters. Assume that δ J = δ b = δ R = 1 and let J = 0.2, b = 0.4, and δ R = 0.5, the angular velocity of (42) using both EADRC-LESO and EADRC-NLESO are graphed in Figure 8. 27

28 (a) (b) Figure 8. The angular velocity of PMDC motor of (42) with parameter uncertainties. 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. e 1(t) = e 2(t) = e 3(t) = 0, then, one can find a relationship between error e 3 (t) in the last 28

29 equation of (15) in terms of, β3, c3, and other parameters, e 3 (t) = φ(, β 3, c 3,.. ), where φ(. ) is a nonlinear function. The same can be done for e 2 (t) and e 1 (t), i.e e 2 (t) = γ(, β 2,. ), e 3 (t) = θ(, β 1,. ), and this makes the errors of the NLESO are sensitive to, which is the rate the generalized disturbance L(t), 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 L(t) exhibits a sudden increase or decrease (e.g. sudden external disturbance) or parameter variations in the nonlinear system of (42). Figures 6(a) and 8 clarify this reasoning. 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 x 1(0) for x 1(t) different from that of x 1 (t)(i. e. x 1(0) x 1 (0)), this makes the terms in the equations of (11) that i 1 depends on ω o, i.e. a i ω 0 g i (ω 0 (y(t) x 1(t))) very large for some g i s and with large ω 0. This large term in the right hand side makes the ESO to produce large fluctuation on its output channels. For example when g i (e) = e, that means the ESO acts as a LESO, then the term i 1 a i ω 0 g i (ω 0 (y(0) x 1(0))) will have a high value for y(0) x 1(0) and large ω 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 c i (c 1 > c 1 >.. > c n+1 ), that explains why our proposed NLESO has a smaller peaking than LESO for the same ω 0. for example the ω 0 -dependent term in the i-th equation of (11) can be expressed as a i ω i 1 0 c i K α ω 0 (y(0) x 1(0)) α + a i ω i 1 0 c i K β ω 0 (y(0) x 1(0)) β. ω 0. e 1 (0) The K β has small value in most cases as compared to K α, so for the easiness of illustration, we neglect the second term and the ω 0 -dependent becomes a i c i K α ω α+i 1 0 (y(0) x 1(0)) α, and since α + i 1 < i, then ω α+i 1 i 0 < ω 0 in the LESO for big ω 0. For example assume ω 0 = 35, α = 0.3, n = 2, and (y(0) x 1(0)) = 1. In the third equation (i = 3), the ω 0 -dependent term is (35) = , whereas the corresponding term in the LESO is (ω 0 ) 3 = (35) 3 = The reduction in the peaking with the proposed saturation-like function g i (.) defined in (12) 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 i (.) defined in (12). When a noise is magnified 29

30 to (ω 0 ) 3. n(t) in the i-th equation of the LESO of (4). While with our proposed saturation-like function g i (.) defined in (12), the magnification of the noise n(t) in our proposed NLESO is a i c i K α ω α+i 1 0 (y(0) x 1(0)) α which is proven to be less than (ω 0 ) 3. The results of Figure 6 illustrate this justification. VII. Conclusions and Future Work 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 have showed 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 general MIMO uncertain nonlinear system and apply the proposed ESO to non-affine control system like balland-beam system. Appendix A. Conversion of PMDC mismatched model into a canonical form of ADRC Let x 1 = dθ dt, x 2 = i, then, the mismatched nonlinear mathematical model of the PMDC motor can written as x 1 = 1 (K J t x 2 T L b eq x 1 ) { eq x 2 = 1 ( Rx L 2 + v K b x 1 ) (43) Let x 1 = x 1, 30

31 x 2 = x 1 = 1 J eq (K t x 2 T L b eq x 1 ) (44) Then, Sub. (43) into (45), we get x 1 = x 2 x 2 = 1 J eq (K t x 2 T L b eq x 1) (45) x 2 = 1 J eq [K t 1 L ( Rx 2 + v K b x 1) T L b eq 1 J eq (K t x 2 T L b eq x 1)] = x 2 [ K tr From (44), + b eqk t J eq L J eq ] + 2 x 1 [ beq 2 K tk b J2 eq ] + K t v + b eq T J eq L J eq L 2 L 1 J T L (46) eq To express (46) in the new coordinate system ( x 1, x 2), we need to eliminate x 2 from (46). J eq x 2 = 1 K t (J eq x 2 + T L + b eq x 1) (47) Sub. (47) in (46) to have x 2 = 1 K t (J eq x 2 + T L + b eq x 1) [ K tr = R L x 2 R J eq L T L b eqr J eq L x 1 b eq 1 J eq T L + b eqk t J eq L J eq J x 2 b eq eq J eq 2 ] + x 1 [ beq 2 T beq 2 2 L J eq 2 K tk b J eq = R L x 2 R J eq L T L b eqr J eq L x 1 b eq J eq x 2 K tk b J eq L x 1+ K t J eq L v 1 J eq T L = ( R + b eq ) L J x 2 ( b eqr + K tk b ) eq J eq L J eq L x 1 + K t (v + d) J eq L J eq L ] + K t v + b eq J eq L 2 T L 1 T J L (46) eq 2 J eq 2 x 1 + beq 2 x 1 K tk b x 1+ K t v + b eq J eq L J eq L 2 T L J eq J eq Where d = R K t T L L K t T L. Finally, 31

32 x 1 = x 2 { x 2 = ( R + b eq ) L J x 2 ( b eqr + K tk b eq J eq L J eq L ) x 1 + K t (v + d) (48) J eq L Which is exactly the model given by (42) and the load torque is given as: T L = T ext + F c sgn(x 1 ). Acknowledgment Author deeply thanks electrical engineering Department/College of Engineering / Baghdad University for providing us with the online library. Also, special thanks go to my Ph.D. student Wameedh R. Abdul- Adheem for the deep discussions on the topics of this paper. References [1] Bao-Zhu Guo and Zhi-Liang Zhao, Active Disturbance Rejection Control for Nonlinear Systems: An Introduction, 1 st ed., John Wiley & Sons: Singapore Pte. Ltd, [2] L. H. S. de Andrade and C. M. Agulhari, Robust mixed H2/H control of an nonlinear multivariable system 12 th (INDUSCON), 2016, pp IEEE International Conference on Industry Applications [3] Byeong Rim Jo, Hee Wook Ahn and Myung Joong Youn, Multi-Variable Sliding Mode Control of Quantum Boost SRC, IEEE Trans. Control Syst. Technol., vol. 2, no. 2, pp , [4] J. Han, From PID to Active Disturbance Rejection Control, IEEE Trans. Ind. Electron., Vol. 56, No. 3, pp , [5] Wameedh Riyadh Abdul-adheem and Ibraheem Kasim Ibraheem, Improved Sliding Mode Nonlinear Extended State Observer based Active Disturbance Rejection Control for Uncertain Systems with Unknown Total Disturbance, International Journal of Advanced Computer Science and Applications, Vol. 7, No. 12, pp , [6] Ibraheem Kasim Ibraheem and Wameedh Riyadh Abdul-Adheem, An Improved Active Disturbance Rejection Control for a Differential Drive Mobile Robot with Mismatched Disturbances and Uncertainties, 3 rd International Conference on Electrical and Electronic 32

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