Adaptive Nonlinear Output Tracking Control With Rejection of Unmatched Biased Sinusoidal Disturbances for Nonlinear Systems

An adaptive nonlinear output tracking control method is proposed to reject unmatched biased sinusoidal disturbances for nonlinear systems that are in the strict feedback form. The proposed method consists of a disturbance observer (DOB) and an adaptive nonlinear controller. The DOB is in the form of a high-pass filter, and it estimates biased sinusoidal disturbances with unknown frequencies. The performance of the DOB is investigated in the time and frequency domains. As the DOB cannot accurately estimate biased sinusoidal disturbances with unknown frequencies, an adaptive nonlinear controller is designed to compensate for the disturbance estimation errors and ensure output tracking. The adaptive nonlinear controller is developed via the backstepping procedure. Thus, matching conditions are not significant in this design. The discontinuous “sgn” function is applied to the controller to compensate for the disturbance estimation errors. An update law of the control gain of the discontinuous “sgn” function is proposed to suppress the disturbance estimation errors without the knowledge of their upper bounds. Low-pass filters are embedded in the controller to smooth the derivative of the discontinuous “sgn” function. Consequently, multiple unmatched biased sinusoidal disturbances are compensated by the proposed method without any knowledge of the disturbances. The proposed method can reject multiple biased sinusoidal disturbances while it guarantees the constraint of the tracking error in steady-state response. Furthermore, the proposed method can reject multiple biased sinusoidal disturbances.


I. INTRODUCTION
The rejection of sinusoidal disturbances is of particular interest in control systems because these disturbances commonly occur in practice [1]. Sinusoidal disturbances occur in a large number of applications, such as rotating magnetic bearings [2], precise piezoactuated nanopositioning [3], hard disk drives [4], optical disk drives [5], rotating pumps in cryocooler expanders [6], helicopter rotor blades [7], and aircraft landing on oscillating carriers [8].
The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen .
To address this problem, the internal model principle (IMP) approach was developed to cancel sinusoidal disturbances using disturbance dynamics [9]- [12]. In this approach, a pair of poles that matches the frequency of a sinusoidal disturbance is requiblack to perfectly cancel the disturbance because the sensitivity function from the disturbance to an output is in the form of a notch filter. However, it is difficult to exactly determine the disturbance frequency because it may vary or may be unknown during system operation. Therefore, various adaptive algorithms have been proposed to estimate the disturbance frequency [5], [13], [14]. Even though these algorithms solve general problems, control laws are difficult to implement in practice and may easily destabilize a system owing to the presence of unmodeled dynamics [15]. Other approaches for canceling sinusoidal disturbances in nonlinear systems are global stabilization and disturbance suppression using output feedback [16]. In addition, a global output feedback compensator was designed to reject unknown sinusoidal disturbances for a class of nonlinear minimum phase systems [17]. These methods effectively cancel sinusoidal disturbances with unknown frequencies when systems are in the normal form or output feedback form and satisfy matching conditions. In practice, a disturbance is not a pure sinusoidal signal but a biased sinusoidal signal owing to the asymmetry of systems. Furthermore, in IMP-based methods, n periodic generators are requiblack to reject n biased sinusoidal disturbances.
Observer based control methods were used to estimate biased sinusoidal disturbances [24]- [29]. A disturbance observers (DOBs)-based controller was designed to cancel disturbances in the frequency domain [24]. A DOB designed in the time domain was generalized to estimate higher-order disturbances in a time-series expansion [25]. DOB-based tracking control algorithms were presented in [26], [27]. The disturbance estimation error cannot converge to zero because a biased sinusoidal disturbance varies with time. Variable structure control methods were applied to tracking controllers to suppress the disturbance estimation error [28], [29]. However, these methods require the upper bound of the disturbance estimation error, which is difficult to determine. Furthermore, the use of the discontinuous function ''sgn'' may result in the chattering in the control input. Extended observer based control methods were proposed [18], [19]. In [18], only matched disturbance was consideblack and in [19], it may be difficult to satisfy the constraint of the tracking error in steady-state response.
Recently, fuzzy and neural network based on control methods were proposed to compensate for disturbances [20], [21]. These method can effectively compensate for the disturbance, but, they require the upper bounds of the absolute values of the disturbances. An adaptive asymptotic tracking control was developed for uncertain nonlinear time-delay systems depended on delay estimation information [22]. In this paper, external disturbances were not consideblack. In [23], an adaptive exact sliding tracking control was proposed for high-order strict-feedback systems with mismatched nonlinearities and external disturbances, but, its control input had ripples.
In this paper, we propose adaptive nonlinear output tracking control to reject unmatched biased sinusoidal disturbances for the nonlinear systems that are in the strict feedback form. The proposed method consists of a DOB and an adaptive nonlinear controller. The DOB is designed to estimate the biased sinusoidal disturbances using the full state feedback and system information. The DOB is in the form of a high-pass filter, and it estimates biased sinusoidal disturbances with unknown frequencies. The performance of the DOB is studied in the time and frequency domains. As the DOB cannot accurately estimate biased sinusoidal disturbances with unknown frequencies, an adaptive nonlinear controller is designed to compensate for the disturbance estimation error and ensure global output tracking. The adaptive nonlinear controller is developed via the backstepping procedure. Thus, matching conditions are not significant in this design. The discontinuous ''sgn'' function is applied to the controller to compensate for the disturbance estimation error. An update law of the control gain of the discontinuous ''sgn'' function is proposed to suppress the disturbance estimation error without the knowledge of its upper bound. Low-pass filters are embedded in the controller to smooth the derivative of the discontinuous ''sgn'' function. The combination of the backstepping, the adaptive gain, and the discontinuous ''sgn'' function can blackuce the ripple in the control input while it can suppress the disturbance estimation error. Consequently, unmatched biased sinusoidal disturbances are compensated by the proposed method without any knowledge of the disturbances. The proposed method can reject multiple biased sinusoidal disturbances while it guarantees the constraint of the tracking error in steady-state response. Closed-loop stability is mathematically proven. The performance of the proposed method is validated via simulations.

II. PROBLEM FORMULATION
We consider the class of the systems that are is in the strict feedback form [30] as follows: . .
x n−1 = f n−1 (x 1 , . . . , x n−1 ) + g n−1 (x 1 , . . . , x n−1 )x n + d n−1 x n = f n (x 1 , . . . , x n ) + g n (x 1 , . . . , x n )u + d n where x ∈ R n , y ∈ R, and u ∈ R are the state, output and the input of systems, respectively. d i for i ∈ [1, n] denotes biased sinusoidal disturbances with unknown frequencies. f i and g i for i ∈ [1, n] are smooth and always positive and bounded for all x. f i is zero at the origin for all i and x, and g i is always positive for all i and x. The disturbance d i is biased sinusoidal signal given by The main goal of the controller design is to make the system output, y = x 1 , track the desiblack output, y d = x 1 d , under the disturbances.

III. DISTURBANCE OBSERVER DESIGN
To estimate multiple biased sinusoidal disturbances, the dynamics ofx are defined as aṡ x n = f n (x 1 , . . . , x n ) + g n (x 1 , . . . , x n )u +d n wherex i andd i are the estimations of x i and d i , respectively.
where k P i and k I i for i ∈ [1, n] are positive constants to make the polynomial s 2 + k P i s + k I i a Hurwitz polynomial, and d i (0) = 0 for all i. The disturbance estimation error is defined asd Theorem 1: Suppose that the DOBs (2) and (3) are applied to the nonlinear system (1). The high pass-filter H i (s) is If the cutoff frequency ω c i of H i (s) is above the maximum frequency ω i max of the disturbance, then the disturbance estimation errord i , ∀i = [1, n] is bounded and T i exists such that The boundedness of the estimation error is proven in both transient and steady-state responses. First, the boundedness of the estimation error is proven in the transient response. Becauseḋ i =ḋ −ḋ i , from (1), (2), and (4), the derivative of (3) provideṡ d I i is defined as t 0d i ds. Then we obtain As k P i and k I i are positive constants to make the polynomial (8), e d i can be obtained [31] as As Thus,d i is bounded in the transient response. The boundedness depends on k I i and k P i .d i is studied in the frequency domain to analyze its upper bound in the steady-state response. Differentiating both sides of (3) twice provideŝ With (1) and (2), (12) becomes From (13), we obtain the error transfer functions H (s) as where s is the Laplace operator. In (14), H i (s) is a typical 2 nd order high pass-filter. The disturbances are biased sinusoidal signals, thus they are bounded. In (18), the transfers function from the disturbances to the disturbance estimation errors are in the form of the high pass filter. Consequently, the disturbance estimation errors are also bounded with the bounded disturbances. In H i (s), the cutoff frequency ω c i depends on observer parameters k P i and k I i . If ω c i is higher than the maximum frequency ω i max of a disturbance, then the estimation errord i is suppressed by the high pass filter (14). Thus there exists T 1 i such that for t > T 1 i . In the observer gain tuning, it is recommended that the observer gains are chosen such that ω c i is higher than the expected frequency of a main component of a disturbance. Even though DOBs can accurately estimate biased sinusoidal disturbances, the disturbance estimation errors cannot converge to zero because the derivatives of biased sinusoidal disturbances are not zero. The next section describes the design of the adaptive nonlinear controller to compensate for the disturbance estimation errors in tracking.

IV. ADAPTIVE NONLINEAR CONTROLLER DESIGN
The output tracking error is defined as e 1 = x 1 − x 1 d . Then tracking errors e = e 1 e 2 . . . e n T can be written as The tracking error dynamics arė e n =ẋ n −ẋ n d = f n + g n u + d n −ẋ n d . As may be written aṡ 216212 VOLUME 8, 2020 The adaptive nonlinear controller is designed via the backstepping procedure as where control gains k i > 0, ∀i ∈ [1, n].ˆ i ∀i ∈ [1, n] is the adaptive gain to suppress the disturbance estimation error in the steady-state response. i max should be known to sufficiently suppress the disturbance estimation error in the steady-state response. However, it is difficult to determine i max even though the disturbance estimation errors are bounded. We define the desiblack adaptive gain as i d > i max . Note that i d is also unknown. To makeˆ i converge to i d , the update law forˆ i ∀i ∈ [1, n] is designed aṡ whereˆ i (0) > 0 and c i > 1.ˆ i increases with nonzero e i . Then,ˆ i ≥ i max in finite time T 2 i where T 2 i is yet to be defined. e i becomes zero after t ≥ T 2 i . This yields thatˆ i admits a bounded value, which yields that there always exists a positive constant i d > i max for ∀t and i d > i max . The convergence of e i to zero is proven later. Theorem 2: Suppose that the DOBs (2) (3), nonlinear controller (19), and adaptive gain update law (20) are applied to the tracking error dynamics (18). Then, the origin of the tracking error dynamics (18) is globally exponentially stable for t > T max i where T max i = max i T 2 i andˆ i ∀i ∈ [1, n] converges to i d .
The tracking error dynamics (21) arw input-to-state stable (ISS). In the adaptive gain update law (20), the derivative ofˆ i ∀i ∈ [1, n] is always positive and nonzero. Thus,ˆ i increases with nonzero e i . Thenˆ i = i max in finite time T 2 i . After t > T 2 i ,ˆ i > i max . Hence, the term −ˆ i sgn(e 1 ) can sufficiently suppress the disturbance estimation errord i . A Lyapunov candidate function V e is defined as Then we obtainV where K = diag(k 1 , . . . , k n ), = [ˆ 1 sgn(e 1 ) − d 1 , . . . ,ˆ n sgn(e n )−d n ] T , andˆ i sgn(e i )−d i ≥ 0, ∀i ∈ [1, n]. Thus, after t > T 2 max where T 2 max = max i T 2 i , e converges to zero in finite time T 3 . After t > T 3 ,ˆ i admits a bounded value i d . A Lyapunov candidate function, V , is defined for the stability analysis of the controller (19) and adaptive gain update law (20) as follows: Then we obtaiṅ With the adaptive gain update law (20), (25) becomeṡ As (c i − 1) > 0, the origin of the tracking error dynamics (18) is globally exponentially stable for t > T max i where The backstepping controller (19) can ensure exponential stability. However,ˆ i continuously increases with nonzero errors; thus, adaptive gain may become extremely large in a few cases. To prevent this problem, the update law ofˆ i (20) is modified as˙ where ε i is the constraint for e i . With the modified update law (27), the backstepping controller (19) ensures the boundedness of the tracking errors because the tracking error dynamics (21) are ISS. The backstepping controller (19) involves the derivatives of the discontinuous ''sgn'' function, which is bounded in the digital implementation. This problem has been resolved in an ad hoc manner via numerical differentiation, i.e., in other words,ẋ where T is the sampling time. This ad hoc approach has worked well in numerous experimental applications [34], [35]. However, even thoughẋ i+1 d becomes bounded using of the ad hoc approach, this approach may face the problem of the explosion of terms. To overcome this problem, low pass filters are designed to smooth the signalẋ i+1 as follows: VOLUME 8, 2020 With the use ofx i d instead of x i d , the nonlinear controller (19) becomes where are the tracking errors. With the DOBs (2) (3), the backstepping controller (30), and the adaptive gain update law (20), the tracking error dynamics can be derived aṡ e 1 = −k 1 e 1 + g 1 e 2 + (d 1 −ˆ 1 sgn(e 1 )) e n = −k n e n − g n−1 e n−1 + (d n −ˆ n sgn(e n )).
♦ Proof: The tracking error dynamics (31) can be rewritten byė In Theorem 1, it was proven that the tracking error dynamics (21)   small-gain theorem [30] for the closed-loop system. Consequently, e 1 is ultimately uniformly bounded. Remark 1: The proposed method can be easily extended to the case of multiple biased sinusoidal disturbances. Disturbances can be expressed as the sums of biased sinusoidal signals as follows: These multiple biased sinusoidal disturbances (33) can be regarded as one biased sinusoidal disturbance. Consequently, as the proposed method does not require the frequencies of the disturbances, it can compensate multiple biased sinusoidal disturbances. In contrast, n periodic generators, such as the IMP and compensators, are requiblack to reject n biased sinusoidal disturbances. This is the main advantage of the proposed method.
♦ The block diagram of the control method is shown in Fig. 1. The DOBs (2) and (3) estimate the disturbances using the state x. Then, adaptive gains are updated using the tracking error e by (31). Based on the updated adaptive gains, the control input is generated using the state x, the tracking error e, and the estimated disturbancesd i by (34).
A. CASE 1 In Case 1, d 1 = 10 sin(5 × 2π t) + 2 and d 2 = 5 sin(2 × 2π t) + 2 sin(3 × 2π t) were used as the disturbances. VOLUME 8, 2020 The simulation results of the proposed method are shown in Figs. 2, 3 and 4. The adaptive nonlinear controller accurately tracked the desiblack states. The proposed DOB accurately estimated the disturbances. However, as the bandwidth of the DOB was not sufficiently larger than the disturbance frequencies, the estimation errors of d 1 and d 2 were 5% and 2%, respectively. The disturbance estimation errors caused the tracking errors, as shown in Figs. 2 and 3. To suppress these tracking errors, adaptive gainsˆ 1 andˆ 2 increased until |e 1 | = 0.005 and |e 1 | = 0.003, as shown in Fig. 4. After |e 1 | and |e 2 | settled below 0.005 and 0.003, respectively,ˆ 1 andˆ 2 did not increase. As shown in Fig. 4(c), the control input was smooth because the low-pass filter was embedded in the adaptive nonlinear controller. In the control (36) of the proposed method, ''sgn'' function that is the function of e 2 was used. As shown in Fig. 2 (d), the sign of e2 did not vary frequently. Thus, the chattering was not a problem in the proposed method.

B. CASE 2
In Case 2, to evaluate the performance of the proposed method with the disturbances those are suddenly changed,    2π t), 0 ≤ t < 2 10 sin(5 × 2π t) + 5 sin(5 × 4π t), t ≥ 2 The proposed method was compablack to the backstepping with the DOB as follows: where e 1 = x 1 − x 1 d and e 2 = x 2 −x 2 d , k 1 = 1, k 2 = 1, ρ 1 = 2, ρ 2 = 0.7. In (38), the disturbance estimated by the DOB 35 was used. The main difference between the proposed method (36) and the backstepping with the DOB (38) was the control gain of the discontinuous function ''sgn''. In the backstepping with the DOB (38), the constant gains ρ 1 and ρ 2 were used. For the backstepping with the DOB (38), we consideblack the case where the change of the disturbances were not expected. From the simulation results of Case 1, the sufficiently large gains ρ 1 = 2 and ρ 2 = 0.7 were used in the backstepping with the DOB (38). The simulation results of the proposed method are shown in Figs. 5, 6, and 7.
The estimated disturbances were shown in Fig. 5. The disturbances suddenly changed in t = 2. The tracking errors of the backstepping with the DOB (38) were smaller than those of the proposed method for t < 2 because the large gains were used. After the disturbances got larger in t = 2, the tracking errors of the backstepping with the DOB (38) increased due to the increased disturbances. In the proposed method, the tracking errors increased due to the increased disturbances in t = 2. However, the tracking errors were again blackuced until |e 1 | = 0.005 and |e 1 | = 0.003 because the adaptive gainsˆ 1 andˆ 2 increased until |e 1 | = 0.005 and |e 1 | = 0.003 as shown in Figs. 7 (c) and (d).

VI. CONCLUSION
Nonlinear output tracking control was developed to reject unmatched biased sinusoidal disturbances for nonlinear systems. A DOB was proposed to estimate biased sinusoidal disturbances with unknown frequencies. An adaptive nonlinear controller was designed via the backstepping procedure to compensate for the disturbance estimation errors and ensure global output tracking. The discontinuous ''sgn'' function with adaptive gain was applied to the controller to compensate for the disturbance estimation errors. Low-pass filters were embedded in the controller to smooth the derivative of the discontinuous ''sgn'' function. Through simulations, it was shown that the proposed method was effective for output tracking under multiple biased sinusoidal disturbances. Furthermore, it was observed that the proposed method was robust against the suddenly changed disturbances. In future work, we will design the controller to cancel general disturbances without the information of the nonlinear terms of the system and full state feedback.