Almost Disturbance Decoupling for a Class of Fractional-Order Nonlinear Systems with Zero Dynamics

The problem of almost disturbance decoupling is addressed for fractional-order nonlinear systems. A new definition for the norm is proposed to describe the effect of disturbances on the output tracking error for fractional-order systems. Based on the Lyapunov stability theory and the backstepping design method, a tracking controller is constructed to make the output tracking error converge to zero without external disturbances and to attenuate the effect of disturbances on the tracking error at zero initial conditions. In order to validate these theoretical results, a numerical example and two practical examples are given.


Introduction
Recently, there has been a continuous growth in the research of fractional-order systems (FOSs).
anks to the great efforts contributed by researchers, a large number of research results have been published on FOSs, see [1,2]. It has been proved that fractional-order calculus can model systems more concisely than integer-order calculus [3]. erefore, FOSs have been applied in numerous areas. e output tracking control problem is one of the most important research topics. It is worth mentioning that the output tracking control has been widely applied in engineering systems including industry and economics [4][5][6]. As for FOSs, few research outcomes are published. An output tracking controller for a class of fractional-order positive switched systems is developed via an observer-based controller method [7]. In [8], a novel adaptive iterative learning sliding mode controller is proposed to control FOSs. What is more, the output tracking controllers can also be developed with the help of backstepping technique for FOSs. For instance, Wang et al. addressed an adaptive fuzzy output tracking control law for a class of uncertain FOSs with backstepping technique [9]. An output feedback neural network tracking controller for fractional-order nonlinear systems is constructed with the backstepping method [10].
Over the past decades, backstepping has become a widely used technique for adaptive nonlinear control. e technique was comprehensively addressed [11], and the basic idea of the backstepping scheme is to design a controller recursively by using a set of virtual controllers. e integer-order backstepping technique has been widely applied in many fields [12,13]. An adaptive finite-time fuzzy funnel controller is presented for nonaffine nonlinear systems via backstepping technique [14]. An event-triggered adaptive tracking controller based on a new funnel function for uncertain nonlinear systems is proposed by using backstepping technique [15]. Few research outcomes applying backstepping technique to FOSs are published. In [16], synchronization of two fractional-order chaotic Coullet systems is investigated based on the adaptive backstepping. A novel adaptive control technique named adaptive fractional-order backstepping is proposed for fractional-order nonlinear systems (FONSs) with uncertain constant parameters [17]. In [18], Ding et [20], and Wei et al. originally presented an adaptive output feedback control law for fractional-order systems by using the backstepping method [21]. Owing to most of the results which construct Lyapunov functions limited to the use of the frequency distribution model or linear matrix inequality (LMI), and the fractional-order backstepping techniques are still to be further studied.
It is well known that disturbances exist widely in industrial processes. e issue of disturbance decoupling has drawn great attention. e exact disturbance decoupling problem is to design a feedback controller so that the disturbance does not affect the output of the closed-loop system, which can be solved by the state or output feedback under some strict conditions [22][23][24]. However, for many practical systems, the exact disturbance decoupling problem is not solvable because the strict conditions are not satisfied. erefore, the almost disturbance decoupling (ADD) problem is proposed by Weiland and Willems [25], which is to design a feedback controller such that the closed-loop system is stable and the L 2 gain from the disturbance to the output is less than or equal to an arbitrary positive constant. Up to now, there have been many research outcomes for the ADD problem for both linear and nonlinear control systems. For example, the ADD problem for SISO nonlinear systems is proposed [26]. e ADD problem of MIMO nonlinear systems and its application to chemical processes are addressed in [27]. In [28], Lin et al. discussed the problem of ADD for nonlinear systems with unstable zero dynamics. Also, in [29], Lin et al. developed a global result about the ADD problem for nonlinear systems with unstable zero dynamics. In [30], the authors investigated the problem of nonlinear output feedback tracking with almost disturbance decoupling. What is more, the ADD problem for high-order nonlinear systems is addressed by [31]. Tracking and almost disturbance decoupling for nonlinear systems with uncertainties are solved by Chen et al. [32]. Chen et al. [33] developed a direct adaptive fuzzy almost disturbance decoupling control method for nonlinear MIMO systems with unknown nonlinearities. For more research results on the ADD problem, see [34][35][36][37][38]. However, the research studies mentioned above are conducted only for integerorder systems. To the best of our knowledge, there are no reports on the ADD problem for FOSs. e active disturbance rejection control (ADRC) problem, which utilizes an estimation and cancellation method to attenuate the negative impact of internal unmodelled dynamics and external disturbances, has been paid extensive attention recently. e ADRC problem has been addressed for nonlinear systems [39][40][41][42][43][44].
Motivated by the aforementioned reviews, this article develops an adaptive backstepping control algorithm to solve the problem of ADD for a class of fractional-order nonlinear systems. To this end, the following two challenges have to be overcome: (1) Aghababa [45] has proved that Lin et al. [46] mistakenly applied the integer-order integral to deal with the fractional-order differential, which causes that the proposed controller in [46] cannot guarantee the H ∞ performance of the closed-loop system, which implies that the L 2 -norm cannot be directly used to solve the ADD problem. To solve this problem, a new form of norm based on the definition of Caputo fractional integral is proposed. (2) When the backstepping design method is used, the fractional-order derivative of the virtual controller is required, which is a function of state variables. erefore, the fractional-order derivative of a composite function is necessary. Unlike the integerorder derivative, the fractional-order derivative of a composite function is so complicated that it cannot be used to the fractional-order derivative of the virtual controller. To avoid such a challenge, the nonlinear functions in all the subsystems but last one are assumed to be linear.
Compared with the existing results, the main contributions are listed as follows: (1) e problem of ADD is first addressed for a class of fractional-order nonlinear systems with external disturbances and zero dynamics. e remaining parts are arranged as follows. In Section 2, several definitions and useful lemmas relevant to the fractional-order calculus are shown. Section 3 proposes the problem formulation and gives the aim of this paper. In Section 4, the main results of this paper are presented. Section 5 gives a numerical example and two practical examples to further support the theoretical results. In the end, the conclusions are summarized in Section 6.

Complexity
For simplicity in notations, t 0 D α t is denoted as D α for t 0 � 0.
Definition 2 (Caputo fractional integral [3]). Assume that f(t) is an integrable function. Its Caputo fractional integral of order α is defined by where α > 0 and t 0 is the lower terminal.
Properties of the Caputo fractional calculus, which are used in the paper, are shown as Lemmas 1 and 2.
Lemma 1 (an inequality of the Caputo fractional derivative [2]). For a differentiable function x(t) ∈ R n , a positivedefinite matrix P ∈ R n×n , and 0 < α < 1, the inequality is true for any t ≥ 0.
In particular, for a continuous function f(t) and 0 < α ≤ 1, the following holds:

Stability of Fractional-Order Systems. Consider the following FOS:
where t stands for the time and 0 < α ≤ 1.
Li et al. [48] extended the Lyapunov direct method to FOSs, which is shown as Lemma 3.

Problem Formulation
Consider a fractional-order strict-feedback nonlinear system with unknown disturbances and zero dynamics defined by where 0 < α < 1, z(t) denotes the zero dynamics of (10), and f 0 (z(t)) is a known nonlinear function with respect to z(t).
being the state of the system, while y(t) ∈ R and u(t) ∈ R are the system output and input, respectively. ω(t) ∈ R r represents the disturbance signal. n) with a i being a known vector. f(·) and g(·) are assumed to be nonlinear and known functions. P i ∈ R 1×r is a known vector for x i , and z, respectively. e aim is to construct a state feedback controller to drive the output y(t) to track a desired reference signal y r (t), i.e., lim t⟶∞ (y(t) − y r (t)) � 0 when ω(t) � 0 and to have an H ∞ tracking performance for the zero initial state, i.e., where ‖·‖ α is defined later and c > 0.

4.
1. e L α Norm. In this section, a new form of norm named L α norm based on the definition of Caputo fractional integral is presented.

Lemma 5.
e L α norm satisfies the following properties: (i) Nonnegative: (ii) Homogeneity: (iii) Triangle inequality: Proof (i) According to Definition (12), it can be easily verified that (13) is true. (ii) Clearly, (iii) For convenience, denote ω(t) as ω. It can be seen that the proof of condition (15) is equivalent to verify On the one hand, On the other hand, It is a simple exercise to show that the proof of (17) is equivalent to prove In order to verify (20), we construct a quadratic function of u as follows: After a little manipulation, one can get the following result: It is easily seen from (21) and (22) that the quadratic function intersects the u − axis at most one point, which implies that which is equivalent to (20).

Control Laws for H ∞ Tracking Performance
Assumption 1. For the zero dynamics D α z � f 0 (z) in system (10), there exists a positive-definite differentiable function V 0 so that its Caputo fractional derivative with order 0 < α < 1 satisfies the following inequality: 4 Complexity

Assumption 2.
e known nonlinear function g(·) satisfies the following inequality: Assumption 3. e desired reference signal y r (t) and D i are bounded and piecewise continuous, Define the control law by and the virtual control laws by α 0 � y r (t), where and (26), and feedback controller (25), the closed-loop system corresponding to (10) has the following properties:
Proof. e proof is given by constructing virtual control laws and a control law based on backstepping design approach.
Step 1: select the following Lyapunov function: en, the fractional derivative of V 1 is given by where the first inequality is generated from (3). According to Lemma 4, let p � q � 2 and ε � 1/ �� 2c; then, (34) can be estimated by en, choose a virtual control law as where k 1 and c 1 are positive constants and can be designed by us. Furthermore, substituting (36) into (35) results in Complexity 5 Step 2: Define a Lyapunov function as It follows from (28) and (27) that with Substituting (40) into (39) yields Similar to (35), applying Lemma 4 to P 2 ω, (41) can be estimated by en, the fractional derivative of V 2 is given by where the first inequality is based on (37) and (3), respectively.

Similar to
Step 1, the virtual control law is chosen as where k 2 is a positive constant. With virtual controller (43), (44) can be expressed as Step i (i � 3, . . ., n − 1): In this step, we repeat the procedure in a recursive way. It follows from (27) and (28) that with Substituting (47) into (46) yields where the inequality is based on Lemma 4 with p � q � 2 and ε � 1/ �� 2c. With (48), it can be derived that the Lyapunov function has the following fractional derivative: e virtual control law is chosen as where k i is a positive constant. Next, substituting (51) into (50) results in Step n: In the final step, select a Lyapunov function as It follows from (27) and (28) that with Substituting (55) into (54) yields where the inequality is based on Lemma 4 with p � q � 2 and ε � 1/ �� 2c. en, the fractional-order derivative of V n can be estimated by Introduce the control law as follows: where k i is a positive constant. Next, substituting (58) into (57) results in (1) When ω(t) � 0, D α V n ≤ 0 can be guaranteed. en, the output signal y(t) asymptotically tracks the desired reference signal y r (t).
(2) When ω(t) ≠ 0, (59) can be estimated by where c � n s�1 c s . To simplify the notation, V n is abbreviated to be V. It follows from (53) that V ≥ 0. For the H ∞ performance, assume the initial condition is zero. en, V(0) � 0 is true. Calculate the Caputo fractional-order integral with order α of (60): It means that i.e., en,   Complexity i.e., with c � ���� c/k 1 .
It means that the effect of disturbances on the output tracking error is attenuated to a given degree c. erefore, the H ∞ tracking performance can be achieved.

Simulation Results
In order to verify the excellent properties of the proposed tracking controller, a numerical example is presented as Example 1, and two practical examples are given as Examples 2 and 3.

Example 1.
Consider system (10) with α � 0.93,    Complexity e virtual laws are chosen as e adaptive control law is determined by where the design parameters are set to k 1 � 12, k 2 � 10, k 3 � 10, c 1 � 0.01, c 2 � 0.2, and c 3 � 4. Simulation results of Example 1 are shown as follows. Figure 1 gives the tracking performance, from which it can be concluded that the asymptotic tracking can be realized. It is obviously seen that the state variables are influenced by external disturbances in Figure 2. Figure 3 indicates the trajectories of the control input.

Example 2.
Consider the fractional-order model of the magnetic leviation (FO-MAGLEV) suspension system in [49]: with α � 0.93, P 1 � 0.1, P 2 � 0, P 3 � −65.17 * 0.      Figure 4. e state responses of Example 2 are shown in Figure 5. Figure 6 presents the control input of Example 2. It is seen that the tracking controller succeeds to make the output tracking error converge to zero without external disturbances and to attenuate the effect of disturbances on the tracking error.

Example 3.
Consider the fractional-order model of the horizontal platform system (FO-HPS) given in [50]:    Complexity e virtual laws and the adaptive control law are determined by (27) and (26), where the design parameters are chosen as k 1 � 10, k 2 � 10, c 1 � 0.01, and c 2 � 0.05. e simulation results of Example 3 are presented in Figures 7-9. e tracking performance of Example 3 is shown in Figure 7. e state responses of Example 3 are given in Figure 8. Figure 9 presents the control input of Example 3. It is observed that the reference signal can be tracked closely by the output signal and the impact of the external disturbances on the output tracking error can be reduced to a certain extent.

Conclusion
e ADD problem has been addressed for a type of nonlinear FOSs with time-varying disturbances and zero dynamics. With the help of the backstepping method, a stabilising state feedback control law has been constructed, which guarantees that asymptotic tracking is achieved without external disturbances and the effect of the external disturbances on the output tracking error can be reduced to a certain extent. In addition, a numerical example and two practical examples are given to validate the theoretical results.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.