Observer-Based Adaptive Fuzzy Predefined Performance Control of a Class of Nonlinear Pure-Feedback Systems with Input Delay

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Introduction
In the past years, the adaptive nonlinear systems based on the backstepping method has matured increasingly and received widespread attention in [1,2]. At an earlier time, there was an unmodeled nonlinear problem in the above system, which greatly limited the application of this technology. To solve the above problems, fuzzy logic systems (FLSs) and neural networks (NNs) were applied extensively to approximate unknown nonlinear function in [3][4][5][6][7]. However, the characteristic of the backstepping method is a class of recursive design procedures coupled with Lyapunov function candidates; hence, the repeated differentiation of virtual controller leads to the complexity explosion problem. Afterwards, the dynamic surface control (DSC) technology was integrated into the backstepping method to solve this problem in [8][9][10]. In addition, since the unmeasurable state in the application has a great restriction, the state observer was employed to estimate the unmeasured state in [11][12][13][14][15]. Among them, an equivalent output injection sliding mode observer was proposed in [12], which could estimate the status of each follower and its neighbor. And high gain observer was used to estimate the position, course, and speed of the vessel in [13]. In recent years, observer-based adaptive fuzzy control with the DSC technology was investigated in [16][17][18].
It is well known that different from strict-feedback systems, pure-feedback systems have nonaffine structure of the variables, which presents more challenges to the controller design. Fortunately, the mean value theorem was proposed to solve the variables coupling problem of nonaffine structures in [19,20]. Moreover, pure-feedback systems usually have the problem of unknown input control direction, which could be solved by Nussbaum functions in [21][22][23][24]. In addition, the input and output of the control systems have many restrictions, such as input saturation, dead zone, and input delay in [25][26][27][28][29][30][31][32][33]. It is worth mentioning that an adaptive predictor incorporated with a high-order neural network observer was proposed to obtain the predictions of the future system states in pure-feedback systems in [28], which were applied in the control design to avoid the input delay and nonlinearities. Subsequently, the input delay was solved by Pade approximation technique and intermediate variables in the strict-feedback systems in [29][30][31], which simplified the controller design. However, to the best of the author's knowledge, the combination of input delay and pure-feedback systems was rarely considered. erefore, the controller design of pure-feedback systems with input delay is complicated, which needs to be further developed.
On the other side, the predefined control performance is better able to achieve the desired performance, such as overshoot, convergence rate, and convergence accuracy. erefore, the prescribed performance control was proposed, which can satisfy preset transient and steady-state tracking performance in [34][35][36]. In particular, an adjustable finite-time prescribed performance function with fast convergence speed was adopted in [37,38], which ensures realtime adjustment of controller parameters cased by the tracking error. Although the research of the prescribed performance control method is approaching maturity, there was still limitation of unknown initial values. Fortunately, a predefined performance function with time-varying design parameters was proposed to reduce the impact of unknown initial tracking error in [39]. However, it is not applied to the unknown nonlinear pure-feedback systems. In summary, the existing predefined performance control methods are insufficient to deal with a class of nonlinear pure-feedback systems with input delay. erefore, the controller design for the above conditions needs to be developed.
Based on the above discussion, this article presents a method for observer-based adaptive fuzzy predefined performance control of a class of nonlinear pure-feedback systems with unknown control direction and input delay. State observer and FLSs are proposed to solve the problem of approximate unmeasurable state and unknown nonlinear functions, respectively. Compared with the existing literature, the main contributions of this paper are as follows: (1) In the existing literatures [34,37], the initial values in predefined performance control are assumed to be known. In order to relax that assumption, a novel predefined performance control method is proposed, which is a variable-parameter scheme independent of the initial error. erefore, the restriction of the unknown initial error in the predefined performance control is solved. (2) Compared with [36], the input delay is introduced into the pure-feedback systems. It is difficult to design the controllers due to input delay and nonaffine properties of the pure-feedback systems, which can be simplified by Pade approximation and mean value theorem, respectively. e framework of this article is as follows. In Section 2, preliminaries and problem formulation are presented. In Section 3, an observer-based adaptive fuzzy predefined performance controller is designed of a class of nonlinear pure-feedback systems with unknown control direction and input delay, and the stability analysis is given. In Section 4, an example simulation is given to verify the feasibility of the proposed method. Finally, the Section 5 is the conclusions and the prospect of the future work.
Notations: R denotes the set of real numbers, R i denotes the i-dimensional vector space, and R + is the set of all nonnegative real numbers. ‖ · ‖ indicates the Euclidean norm of vectors or matrix. For a matrix X, X T indicates its transpose and X − 1 indicates its inverse. For a matrix Q, λ min (Q) stands for the smallest eigenvalue of Q and λ max (Q) stands for the largest eigenvalue of Q.

System Descriptions and Assumptions.
A class of nonlinear pure-feedback systems with input delay is considered as ) are unknown smooth functions, d i (t) means unknown and bounded external disturbance inputs, and δ denotes the input delay, which is an small unknown positive constant caused by network delay. Moreover, the output y is measurable. Because of the coupling between states x i+1 and u(t − δ) in smooth functions f i (x i , x i+1 ) and f n (x n , u(t − δ)), which makes the desired control objectives difficult to design, the mean value theorem is used as i+1 is certain point between zero and x i+1 , and u 0 (t− δ) is certain point between zero and u(t − δ).
Substituting (2) into system (1), one can obtain as x n � f n x n + g n u(t − δ) + d n (t),

Complexity
To get the actual control input u by removing the effect of input delay δ, the Pade approximation method and the delay theorem of Laplace transform can be used, which solve the analysis complexity problem caused by time delay, and it follows that where s represents the Laplace variable and ℓ u(t) { } is the Laplace transform of u(t).
Define the intermediate variable x n+1 as By transforming formula (5), one can be given as By taking the inverse Laplace transform, where β � 2/δ is a variable.

Remark 1.
Pade approximation has been used in [31]. In this article, since the Pade approximation is applied to solve a class of small time delay problems, e − δs is approximately equal to 1 − δs/2/(1 + δs/2) when the time delay is very small. And the intermediate variable x n+1 is not a real variable of system (1), which can be viewed as an error variable. And this has been verified in the simulation in [31]. By using the above methods, (3) can be further written as Assumption 1 (see [9]). e expected signal y d and its derivatives _ y d and € y d are all known and bounded, which is Assumption 2 (see [20]). e sign of g i is unknown, but g i has the same sign and its public super bound is known, which is 0 < |g i | < g * .
Assumption 4 (see [27]). ere is a known constant s i that . , x i ] T , and ‖X‖ represents the 2 norm of the vector X.

Fuzzy Logic Systems.
Because the nonlinear function is unknown, FLSs is proposed. Build FLSs with the if-then rules. R q : if x 1 is F q 1 and x 2 is F q 2 and . . . and x n is F q n . en, y is B q , q � 1, 2, . . . , a. Here, x � [x 1 , . . . , x n ] T and y are the FLS input and output, respectively. Fuzzy sets F q i and B q , associated with the fuzzy functions μ F q i (x i ) and μ B q (y), respectively. a is the rules number. us, FLS can be calculated by formula where Lemma 1 (see [40]). Let f(x) be a continuous function defined on a compact set Ω. en, for any constant ε > 0, there exists an FLS such as Define the idealized parameter vector θ * i as By Lemma 1, the nonlinear functions can be approximated by the following FLSs: The fuzzy minimum approximation errors can be defined as Assumption 5. e approximation error ε i is bounded, and there is a constant ε * i that satisfies |ε i | ≤ ε * i . From (11), system (8) can be expressed as where Rewriting (12) in the following formula,

Nussbaum-Type Function. A continuous function N(μ)
is called the Nussbaum function if it has the following properties: where m is the integral upper boundary. For instance, the frequently used continuous Nussbaum-type functions contain μ 2 cos(μ), μ 2 sin(μ), e μ 2 cos(μ), and so on. In this work, the continuous Nussbaum-type function N(μ) � μ 2 cos(μ) is utilized.

Remark 2.
e parameters g i and g n are time-varying parameters and their signs are unknown. If the control direction changes rapidly, it is difficult to effectively guarantee the stability of the closed-loop system under the selfadaptive condition. erefore, compared with the control direction which is assumed to be known, Assumption2 is more flexible in the application. In addition, similar to [20], this paper uses the Nussbaum functions to solve the control direction problem, which relaxes the prior knowledge.

Fuzzy State Observer Design.
To estimate the unmeasurable states of the system, the corresponding fuzzy observer is designed as Rewriting (16) in the following formula, , 0] T . e observer gain matrix K is given such that A 0 is a Hurwitz matrix. erefore, for any chosen positive definite matrix Q � Q T > 0, there is a positive definite matrix P � P T > 0 that satisfies e observer errors can be obtained as From (12), (16), and (17), the observer error is where Consider the Lyapunov function candidate as e time derivative of V 0 with (20) is By using Young's inequality and Assumptions 3-5, the inequalities can be obtained as 4 Complexity where r 0 � 1 + ‖P‖ 2 n j�2 L 2 j . Substituting (23)- (25) into (22) yields

Tracking Error Transformation.
A new variable parameter independent of the initial error is proposed, which satisfies the expected variable-parameter scheme tracking performance constraints. en, the tracking error is defined as E 1 � y − y d . Predefined performance control constraint will be obtained as inequality holds for all t ≥ 0: where smooth function ι d (t) and ι u (t) satisfies the follow properties.
In this article, ι d (t) and ι u (t) can be chosen as where λ d , λ u , h d , and h u are positive constants. And ρ(t) is an appropriate performance boundary function and is defined as ρ(t) � (ρ(0) − ρ ∞ )e − λt + ρ ∞ , which satisfies the following. (1) ρ(t) is positive and strictly decreasing.
Integrating equality (28) over [0, t), we have In the same way, By the above analysis, ι d (t) and ι u (t) converge exponentially to constants h d /λ d and h u /λ u , and the convergence rate can be improved by adding λ d and λ u . When ι d (t) and ι u (t) converge to constant values, inequality constraint (27) is degenerated as follows: According to (31), when the system is stable, the upper bound of the steady-state error is max (h d /λ d ), (h u /h u )}ρ(∞), and the error convergence speed and the maximum overpass can be adjusted by the coefficient λ d , λ u , h d , h u , and ρ(t). e inequality constraint is transformed into equality constant, and the error transformation function ϕ(z, ι d , ι u ) is defined as where z is transform error, and the continuous function ϕ(z, ι down , ι up ) satisfies the following properties. (1) ϕ(z, ι d , ι u ) is smooth and strictly increasing; By the properties of function ϕ(z, ι d , ι u ), the inverse transformation is e error transformation function is defined as erefore, by (32), the boundless of the error transformation function z 1 results in the prescribed performance of the tracking error E 1 .
Differentiating (33), it yields Combining (35) and (12), one can obtain as where ϖ � (zϕ − 1 )/(z(E 1 /ρ))1/ρ > 0 and D � (− zϕ − 1 )/(z (E 1 /ρ))1/ρ(y d . + E 1 _ ρ/ρ). By derivation, Remark 3. Note that the functions ι d and ι u are set to solve the problem of unknown initial error E(0). In the predefined performance control of [34], the initial error is assumed to be known, but in the actual control, the exact value of the initial error is often not obtained, which limits the use of the predefined performance control. e reason why the initial values of ι d (t) and ι u (t) are set to infinity is mainly to guarantee E(0) ∈ (− ι d , ι u ); however, ι d (0) and ι u (0) just need to be set to a sufficiently large constant in the actual design. e facts justify this treatment.

Control Design and Stability Analysis
In this section, by utilizing adaptive backstepping technique and Lyapunov stability theorem, an observer-based adaptive fuzzy predefined performance decentralized control approach is developed.
Complexity 5 e coordinate changes are as follows: where α i is the virtual control law and α i and ξ i are the filter signal and filter error of the first-order filter, respectively. Define a first-order filter as where ζ i is the design constant.
Step 2 (i (i � 2, . . . , n − 1)). e time derivative of E i is 6 Complexity (54) Consider the Lyapunov function candidate as where υ i > 0 is a design constant. e derivative of V i with time is where By using Young's inequality, From (57) and (58), one can obtain as Select the virtual control function α i and the adaptive laws θ i as Substituting (60)-(62) into (59) yields According to the definition of α 1 , α 2 , and α i , one can get . . , ξ i ) T : ϑ < ψ is a prefixed compact set, where the compact set C i can be made larger as needed. erefore, the maximum value of the continuous function η i is B i+1 on C i * Δ 0 based on Assumption 1 of the compact set Υ and the compact set C i . It is obvious that Utilising Young's inequality, where π i is the design constant. Substitute (65) and (66) into (63) as follows: Step 3 n. e time derivative of E n is Consider the Lyapunov function candidate as where υ n > 0 is a design constant. e derivative of V n with time along with (68) is By using Young's inequality, From (71), one can obtain Select the actual control input u and the adaptive laws θ n as u � N μ n ω n E n + 1 2 E n + θ T n φ n x n ) + k n e 1 − _ α n ], (73)  (31), and the observer errors converge within a small zero region.
Proof. Let where c will be positive by choosing appropriate parameters.

Simulation Example
Consider a class of nonlinear pure-feedback systems with unavailable states, unknown control direction, and input delay: where x 1 and x 2 are the system states and u and y are the system input and output, respectively. e smooth functions are used as δ)), and the external disturbances in this simulation are given as d 1 (t) � 0.02 cos(t) and d 2 (t) � 0.03 sin(t). e input delay is chosen as δ � 0.01, and the reference signal is given as y d � sin(t). e parameters in control functions and adaptive laws are given as ω 1 � 0.1, ω 2 � 40, υ 1 � υ 2 � 5, and σ 1 � σ 2 � 20. Parameter in a first-order filter is ζ 2 � 0.05. As for the state observer, the observer gains are selected as K � [k 1 , k 2 ] T � [10, 100] T . In predefined performance controls, e simulation results are shown in Figures 1-7, where the red and blue lines represent the approach proposed in this article, and the black line represents the removal of the variable-parameter predefined performance. Figure 1 shows the output trajectories of y and the expected output signal y d . e output tracking error and the prescribed performance boundaries are shown in  controller in this paper is obviously better in both the initial oscillation frequency and the tracking effect.
Remark 4. Compared with the existing literature, an adaptive fuzzy predefined performance controller is proposed in this paper, which makes the tracking error convergence in the preset range better. It can be seen from Figure 2 that different from the literatures [34,37], the novel predefined performance control is a variable-parameter scheme, which relaxes the limitations of known initial error in predefined performance. In addition, from the simulation data, it can be seen that the simultaneous consideration of input delay and pure-feedback system brings great difficulty to controller design. e control method proposed in this paper has excellent control performance, which is shown in Figure 1.

Conclusion
In this paper, an observer-based adaptive fuzzy predefined performance controller has been introduced of a class of nonlinear pure-feedback systems with unknown control direction and input delay. A novel predefined performance with variable-parameter scheme has been investigated, which solved the problem of unknown initial error. In order to overcome system complexity caused by input delay and pure-feedback systems, the Pade approximation and mean value theorem has been employed, respectively. In addition, Nussbaum functions have been used to deal with the unknown control direction and a first-order filter has applied to approximate repeated differentiations problem of the virtual controllers. State observer and FLSs have been proposed to estimate the unmeasured states and approximate the unknown nonlinear functions, respectively. erefore, it has been proved that stability of the entire closed-loop system is SGUUB in limitation of the predefined performance control. e tracking errors have converged within a predefined range, while the observer estimation errors have converged within a small zero region. Finally, the simulation results have verified the effectiveness of the proposed control method. In the future research, an observer-based adaptive fuzzy predefined performance controller will be considered in multiagent systems.

Data Availability
All data in this paper are from Simulink in MATLAB, and all data have been given in this paper.

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