Disturbance Observer-Based Adaptive Neural Network Output Feedback Control for Uncertain Nonlinear Systems

This article is devoted to the output feedback control of nonlinear system subject to unknown control directions, unknown Bouc–Wen hysteresis and unknown disturbances. During the control design process, the design obstacles caused by unknown control directions and Bouc–Wen hysteresis are eliminated by introducing linear state transformation and a new coordinate transformation, which avoids using the Nussbaum function with high-frequency oscillation to deal with the issue. Besides, to settle the issue caused by the unknown disturbances, a novel nonlinear disturbance observer is designed, which has the characteristics of simple structure, low coupling, and easy implementation. Especially, a compensation item is constructed to offset the redundant items generated in the backstepping design process. Simultaneously, using the neural network and backstepping technology, an output feedback controller is devised. The controller ensures that all closed-loop signals are bounded, and the system output, state observation error, and disturbance observation error converge to a small neighborhood of the origin. Finally, to illustrate the effectiveness of the proposed scheme, simulation verification is carried out based on a numerical example and a Nomoto ship model.

of research on nonlinear systems was advanced [5]- [8].Generally speaking, the nonlinear function in the system is often unknown.Therefore, to develop adaptive control, neural networks or fuzzy rules are usually employed to estimate uncertain nonlinear functions.It has become a trend to integrate the two methods into adaptive backstepping control design [9]- [13].More details, by combining backstepping technology and neural network, the control research of a kind of helicopter with three degrees of freedom was implemented in [10].The difference is that [11]- [13] were aimed at the nonlinear system in the form of nonstrict feedback.It is well known that the system dominated by strict lower triangular form is called strict feedback system, which is a special case of nonstrict feedback system.Generally, the control method used to deal with strict feedback system is no longer suitable for nonstrict feedback system.Based on this, some solutions were proposed, such as variable separation technology, adding restrictive conditions for nonlinear functions, and using the structural characteristics of neural networks or fuzzy logic systems, and so on.Concretely, by employing the structural characteristics of fuzzy logic system, [12] settled the difficulty that nonstrict feedback form brings to the control design.
In some practical applications, the system state cannot be measured directly.Under the limitation of unmeasurable state, state feedback adaptive control cannot be carried out smoothly.The proposal of the state observer successfully eliminated the obstacles caused by the unmeasured state [14]- [18].A reduced order observer was built to realize the observation of partial unmeasurable states in [18].In addition, the control direction of the actual system is often unknown, which is another obstacle to the development of the control design.In [19], a technique called Nussbaum function was proposed for the first time, which can solve the issue of the unknown control directions.For example, in the context of ship dynamics, [20] studied the issue with unknown control directions through system modeling, and successfully constructed the controller to achieve the ideal tracking performance.In addition, some research background and results based on unknown control directions were developed in [21]- [27].For example, in [27], an adaptive consensus control scheme was proposed for a class of nonlinear multiagent systems with nonidentical partially unknown control directions.
The existence of time-varying disturbances will produce some unexpected effects, such as weakening the system performance and causing the system instability; it is more meaningful to carry out control research on unknown time-varying disturbances.Aiming at the issue of unknown disturbance, a series of research work has been developed.On the one hand, the influence brought by unknown disturbance can be solved by constructing robust term, but it is gradually replaced due to its disadvantages such as energy waste and weakened tracking performance.On the other hand, the nonlinear disturbance observer can effectively observe unknown disturbances and improve robustness.Therefore, some research results were developed, which construct different disturbance observers based on different design methods [28]- [33].In particular, based on the Nussbaum function, a Nussbaum disturbance observer was constructed in [30].
For power systems and piezoelectric actuators and other similar practical systems, their actuators all have an inherent characteristic called hysteresis.Hysteresis is an unconventional nonlinear phenomenon with the characteristics of nonsmoothness and multimapping.The appearance of actuator hysteresis not only easily causes the limitation of control accuracy, but also makes the system unstable.At present, hysteresis models mainly include the Backlash-like hysteresis model, Prandtl-Ishlinskii model, Bouc-Wen model, and so on.In recent years, taking into account the existence of actuator hysteresis, a lot of efforts have been made in the research of nonlinear systems [34]- [40].In [35], the tracking control of stochastic system was realized in consideration of the Backlash-like hysteresis of actuator.To our knowledge, there are very few research results that take into account the Bouc-Wen hysteresis of the actuator, the controlled system has unknown control directions, unknown disturbances, and nonstrict feedback structure at the same time.
Encouraged by the above analysis, we study the output feedback control of nonlinear systems subject to unknown control directions, Bouc-Wen hysteresis, and unknown disturbances.The main contributions are as follows.
1) This article studies a more general system, which has the characteristics of unknown control directions, Bouc-Wen type hysteresis and nonstrict feedback structure.At present, the existing methods cannot be directly applied to the model studied in this article.In addition, Nussbaum function is usually introduced to compensate the design obstacles caused by unknown control directions, such as [21] and [22].However, Nussbaum function is a function with the property of upper and lower bound integral tends to infinity, which has the characteristics of high-frequency oscillation.To avoid the use of Nussbaum function, a new coordinate transformation is introduced to carry out the control design.
In particular, compared with [38], the curve presented by our simulation results is smoother, which reflects that the oscillation phenomenon caused by using Nussbaum function is avoided.2) Compared with [28] and [29], the design process of the proposed disturbance observer is simpler, clearer, and easier to implement.In particular, different from the high coupling of the disturbance observer in [29], the designed disturbance observer has the characteristics of low coupling and high cohesion, which reflects better readability and sustainability.At the same time, the simulation results show that the designed disturbance observer achieves the desired observation effect.In particular, to remove the redundant terms generated by the constructed disturbance observer, a compensation term is introduced in the backstepping design.The next part mainly consists of the following aspects.Section II introduces the preparatory statement.Section III includes three aspects: observer design, control design process, and a theorem.Section IV carries out the simulation verification.Section V is the conclusion.

A. Problem Formulation
The considered nonlinear system is sketched as follows: where xi = [x 1 , . . ., x i ] T ∈ R i , i = 1, . . ., n, y ∈ R, and u ∈ R are state vector, output, and control input, respectively.h i = 0 is a constant, which is called the unknown control coefficient and its sign is unknown.f i ( xi ) is completely unknown smooth nonlinear functions with f i (0) = 0. d i (t) is unknown disturbance.Suppose that the states x 2 , . . ., x n are unknown except that the output y is measurable.Concurrently, the actuator with Bouc-Wen-type hysteresis fault is considered, which can be written as where ς 1 and ς 2 are the unknown hysteresis parameters, and ν denotes the input of the hysteresis. is expressed as where ι, κ, and represent shape, amplitude of the model, and smoothness parameter that determine the transition from initial slope to asymptote slope, respectively.The above unknown parameters satisfy ι > |κ| and > 1.By analyzing (3), [40] indicated that the variable is bounded, which is expressed as . The variables and hysteresis are described in Figs. 1 and 2, respectively.The parameters are given as ς 1 = 3, ς 2 = 5, = 2, ι = 1, κ = 0.5, (t 0 ) = 0, and ν = 4 sin(2t).Remark 1: When the actuator has a hysteresis fault, it will weaken the system performance and reduce the control accuracy.The hysteresis with Bouc-Wen model has the following characteristics.First, the Bouc-Wen model has multiple mappings, and the shape of the hysteresis loop changes with the change of .Second, the Bouc-Wen model is nonsmooth.When the direction of the input ν changes, the turning point is not differentiable, which leads to the fact that the point does not meet the conditions for the controller design.Thirdly, the Bouc-Wen model is based on the form of (3), but the general solution of differential ( 3) is still unknown.To overcome these problems, [40] obtained the boundedness of the intermediate Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.variable by analyzing (3).Further, the control design is carried out according to the characteristics of Bouc-Wen hysteresis combined with backstepping technology in this article.
The existence of the control parameters h i , whose value and symbol are unknown, adds to the challenge of control design.For this reason, a linear state transformation is introduced for (1).Let where Obviously, the system (1) is transformed into an uncertain nonlinear system with unmeasurable states.Further, the output feedback control design is carried out on the system (4).
The objective of this article is to construct a control input ν, which makes all the signals in the closed-loop system bounded and system output, state observation error and disturbance observation error converged to near the origin.To reach the aim, some prior knowledge is stated as follows.

B. Preparatory Knowledge Assumption 1: [29] There exist unknown positive constants d *
i and di such that Obviously, from the above assumption, it can be deduced that there are unknown constants such that the compound disturbance is largely due to exogenous effects, which have limited energy.Some practical disturbances such as wind disturbance, wave disturbance, and friction are usually assumed to satisfy Assumption 1.The same or similar description of Assumption 1 is widely used, such as [28] and [29], and so on.
Lemma 1: [35] In the case of given bounded initial conditions, if there exists a C 1 continuous and positive Lyapunov function then the solution x(t) is uniformly bounded.As mentioned above, C 1 is defined as a set of functions with continuous 1th partial derivatives.β 0 , β 1 : R n → R are K class functions and o 1 , o 2 are positive constants.
The radial basis function neural networks is introduced to estimate uncertain nonlinear functions, which can be described as f nn (z) = W T S(z), where zin z ⊂ R q is input vector and l > 1, W = [w 1 , . . ., w l ] T ∈ R l is weight vector, and S(z) = [s 1 (z), . . ., s l (z)] T stands for the basis function vector, where is called the center of the receptive field and η is the width of the Gaussian function.It should be noted that the basis vector S meets the requirement of 0 < S T S ≤ s * i , where s * i is the dimension of S. The following lemma states that continuous function f (z) : R n → R can be approximated by neural networks.

A. State Observer and Disturbance Observer
Since the state of system (4) is unmeasurable, it is necessary to introduce a state observer to estimate unknown states.Let G i (ẑ) = g i (z) − G i (ẑ); the system (4) can be rephrased as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Next, neural network is used to approximate where k i > 0 is a designed parameter.Equations from ( 6) to ( 7) can be rewritten as Based on the above discussion, consider the following state observer for (8): where ẑi represents the estimate of z i , zi = [ẑ 1 , ẑ2 , . . ., ẑi ] T and zn = ẑ.D i (t) is abbreviated as D i .Di represents the estimate of D i , the disturbance observation error Di = D i − Di , i = 1, 2, . . ., n.By choosing the appropriate positive parameter k i , it can be obtained that is a Hurwitz matrix.Then, there exists positive definite matrix P = P T and Q = Q T such that A T P + P A = −Q.The observer error is defined as z = z − ẑ.Then, one has where Based on the above analysis, the complete system is expressed as To design nonlinear disturbance observer, an auxiliary variable is introduced.
According to (11) and ( 12), the following can be obtained: Remark 3: Equations ( 12) and ( 13) show the design process of disturbance observer.Compared with literature [28] and [29], the design process of this article is simpler, clearer, and easier to implement.Especially, the disturbance observer constructed in [29] involves auxiliary variables, intermediate variables, and estimates errors, disturbance errors, etc., in which auxiliary variables and intermediate variables are still coupled to a certain extent.However, (13) shows that the designed disturbance observer Ḋi only depends on the transformed state estimate ẑ1 and the current adaptive parameter Di .It can be seen that the designed disturbance observer has the characteristics of low coupling and high cohesion, which reflects better readability and sustainability.In addition, the simulation results also show the disturbance observer (13) achieves the desired observation effect.
Furthermore, the change of coordinates is given as follows: In the process of control design, the virtual control law α i , adaptive law θi and Ẇi are constructed as below where i = 1, 2, . . ., n, d i , p i , l i , b i , m i denote positive design parameter and X 11 = ẑ1 , 2 and M i are the ideal constant weight vectors.Ŵi and θi are recorded as the estimation of W i and θ i , respectively.Wi = W i − Ŵi and θi = θ i − θi .Remark 4: Nussbaum function is a function with the property of upper and lower bound integral tending to infinity, which is generally used to deal with unknown control directions, such as [21], [22] and [38].However, the Nussbaum Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
function is characterized by high-frequency oscillation.In literatures [22] and [38], in order to facilitate control design, linear state transformation is introduced to overcome the influence of unknown control coefficients.Similarly, it can be seen that we borrow the same approach to simplify control design.The difference is that Nussbaum functions still appear in controllers designed in the above literatures.In this article, a new coordinate transformation ( 14) is introduced to avoid using Nussbaum function to solve the issue of unknown control directions.Furthermore, compared with [38], the curve presented by our simulation results is smoother, which reflects that the oscillation phenomenon caused by using Nussbaum function is avoided.
Next, combined with backstepping technology, adaptive output feedback control design will be carried out in Section III-B.
Choosing candidate Lyapunov function as where a i and τ i are positive parameters.From (34), one has By utilizing Young's inequality, one further has Substituting ( 13), (38), and ( 39) into (37) yields where where approximation error δ i (X i ) satisfies ||δ i (X i )|| ≤ ε i and ε i > 0. In the same way, one has Considering ( 15)-( 17), one yields where Step n: According to ( 11) and ( 14), one can obtain that where Defining the Lyapunov candidate function as by using (44), one has The following is obtained from Young's inequality: Substituting ( 13) and ( 48) into (47), the following can be obtained: where Using the same treatment method as ( 31)- (32), one has Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where ||δ n (X n )|| ≤ ε n , ε n > 0, and X n = X nn .Let the input of the hysteresis ν = α n , adaptive law θn and Ẇn are constructed as the same as ( 15)-( 17); it can be deduced that By considering Assumption 1 and applying Young's inequality, the following holds: and according to the definition of , the following can be obtained: Taking ( 52)-( 56) into (51) results in where Remark 5: It was noticed that the existence of the last term in (51) caused the Lyapunov function to fail to meet the conditions stated in Lemma 1.Therefore, to eliminate the redundant term generated by the constructed disturbance observer, a compensation term is introduced by means of neural network in the first step of backstepping design.Considering (55) and (56), the redundant item is eliminated.

C. Stability Analysis
Theorem 1: Consider the system (1), the state observer (9), disturbance observer (13), virtual controller signals (15), and adaptive laws ( 16) and (17).Under Assumption 1, considering Lemma 1, we can draw a conclusion that all signals in the closed-loop system are bounded, and system output, state observation error, and disturbance observation error converge to near zero.
The simulation results are depicted in Figs.3-7.It can be concluded from Fig. 3 that the trajectory responses of x 1 and x 2 converging to near zero under the proposed control method.The state observer errors z1 and z2 and the disturbance observer errors D1 (t) and D2 (t) converge to near zero, which are drawn in Figs. 4 and 5, respectively.The above images verify the fact that the designed state observer and disturbance observer can both achieve better observations.Fig. 6 describes the trajectories of the adaptive parameters θ1 and θ2 , which   indicates that θ1 and θ2 are bounded.Fig. 7 shows the responses of the controller input ν and actuator output u.
Example 2: The issue of a linear first-order Nomoto ship model given by [24] where ψ is the heading angle of the ship, ψ is the ship's heading angular rate, and u is the rudder angle.K index and T index are widely used in ship maneuverability evaluation.K is the ratio of the torque generated by the rudder to the damping moment, and T is the ratio of the moment of inertia to the damping moment.Parameter T > 0 when a ship is line movement stable, whereas T < 0. In addition, the ship motion Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In this article, we do not need to know the prior knowledge of the sign of the control coefficient h 2 .The unknown  Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The simulation results are depicted in Figs.8-12.Under the designed control scheme, the trajectory responses of states x 1 and x 2 converging to near zero are shown in Fig. 8.The state observer errors z1 and z2 and the disturbance observation errors D1 (t) and D2 (t) are plotted in Figs. 9 and 10, respectively.The trajectories in the above figures imply that all observation errors converge to near zero.Fig. 11 portrays the boundedness of the adaptive law parameters θ1 and θ2 .Fig. 12 shows the responses of the controller input ν and actuator output u.

V. CONCLUSION
This article discusses the output feedback control of uncertain nonlinear systems.In the process of backstepping design, with the help of a linear state transformation and a new coordinate transformation, the design obstacles caused by unknown directions and Bouc-Wen hysteresis are removed, and the use of Nussbaum function is avoided.In addition, state observer and disturbance observer are devised.The disturbance observer has simpler structure and lower coupling, which is not only easy to implement but also reflects better readability and sustainability.In particular, a compensation term is introduced to eliminate the redundant term generated in the process of backstepping design.Meanwhile, the neural network is integrated into the backstepping technology to construct the output feedback controller, which achieves the control goal.The last numerical example and practical example verify the effectiveness of the proposed method.