Partial-State-Constrained Adaptive Intelligent Tracking Control of Nonlinear Nonstrict-Feedback Systems with Unmodeled Dynamics and Its Application

In this paper, an adaptive intelligent control scheme is presented to investigate the problem of adaptive tracking control for a class of nonstrict-feedback nonlinear systems with constrained states and unmodeled dynamics. By approximating the unknown nonlinear uncertainties, utilizing Barrier Lyapunov functions (BLFs), and designing a dynamic signal to deal with the constrained states and the unmodeled dynamics, respectively, an adaptive neural network (NN) controller is developed in the frame of the backstepping design. In order to simplify the design process, the nonstrict-feedback form is treated by using the special properties of Gaussian functions. -e proposed adaptive control scheme ensures that all variables involved in the closed-loop system are bounded, the corresponding state constraints are not violated. Meanwhile, the tracking error converges to a small neighborhood of the origin. In the end, the proposed intelligent design algorithm is applied to one-link manipulator to demonstrate the effectiveness of the obtained method.


Introduction
Over the past few decades, nonlinear control systems, which can be employed to model numerous applications such as biological systems, chemical processes, and aerospace vehicles, have aroused a wide range of concerns among researchers. In this area, adaptive control of nonlinear systems with uncertainties is a very active research subject. e backstepping method, as we all know, has been proposed in [1] as an effective method to solve the adaptive control problem of nonlinear strict-feedback systems with mismatched uncertainties. With the rapid development of adaptive control theory, the backstepping method has been widely used in the control design of different complex nonlinear systems such as interconnected large-scale systems, MIMO systems, and unmodeled dynamic systems [2][3][4].
On the other hand, unmodeled dynamics are common phenomenon in practical applications, which are mainly caused by modeling errors and external disturbances. e existence of unmodeled dynamics usually degrades control performance or leads to the instability for a control system, and thus dealing with unmodeled dynamics has drawn considerable attention from many scholars in the control field. In recent years, adaptive control of uncertain nonlinear systems with unmodeled dynamics has become a research hot spot, and many related achievements have been reported [5][6][7][8][9]. To just name a few, a robust controller is designed in [5] for a class of nonlinear systems with unmodeled dynamics by using the method of adaptive backstepping and introducing a dynamic signal; and an adaptive output feedback controller is designed in [6] for a class of stochastic nonlinear systems with output unmodeled dynamics by using a stochastic small-gain theorem. Obviously, all the above-mentioned references are nonlinear strict-feedback systems with unmodeled dynamics, and there are few results on nonlinear nonstrict-feedback systems with unmodeled dynamics at present. erefore, how to deal with the nonstrict-feedback form is one of the hardest issues in the research field of nonlinear systems.
It should be pointed out that none of the above related adaptive results can be used to deal with completely unknown nonlinearities of control systems. To solve this problem, some elegant intelligent adaptive control algorithms are proposed by using NN or fuzzy logic systems . Up to now, great progress has been made in the area of adaptive intelligent control for uncertain nonlinear systems with unmodeled dynamics, and a large number of valuable results are presented in [33][34][35][36][37][38][39][40][41][42][43][44]. For example, several adaptive intelligent control schemes are proposed in [33,44] for several classes of nonlinear systems made up of unmodeled dynamics and strict-feedback form by using fuzzy logic and NN compensators, respectively. A suitable learning controller is proposed in [39] to overcome the disadvantages caused by parameter uncertainties and unmodeled dynamics for a class of multi-input and multioutput nonlinear systems. Meanwhile, the corresponding results have been obtained for interconnected nonlinear systems with unmodeled dynamics in [40]. However, the problem of constraints inevitably appears in various systems. e research in the field of handling complicated constraints has been paid more and more attention by researchers, and by utilizing the BLFs or the nonlinear mappings (NMs) to deal with state constraints or output constraints, a series of significant results have been obtained [45][46][47][48]. However, there exist a few intelligent control algorithms for nonstrictfeedback nonlinear systems with unmodeled dynamics to deal with the state constraints until now.
Motivated by the above research situation, this paper proposes an adaptive tracking control strategy for a class of nonstrict-feedback nonlinear systems with unmodeled dynamics and state constraints. e unknown functions are estimated by NN, then a dynamic signal is designed to handle the dynamic uncertainties to ensure that the considered system can be controlled effectively. Meanwhile, by using the BLFs to handle the state constraints, the proposed adaptive control approach can guarantee the boundedness of all the signals in the closed-loop system, and the tracking error converges to a small neighborhood of the origin. e main contributions of the proposed method are summarized as follows: (1) Compared with the variable partition technique in [36], this paper uses the essential property of Gaussian functions to deal with the nonstrict-feedback form, so that the controller design process is relatively simpler. (2) Barrier functions are applied in the design process to constrain state variables into the specified regions, despite the presence of unmodeled dynamics at the front end of the studied system. (3) is paper adopts a dynamic signal to handle the dynamic uncertainties to ensure the considered system can be controlled effectively so that the conservative assumption about unmodeled dynamics in [5] is not used.
Remark 1. Plant (1) has a nonstrict-feedback structure, where the diffusion terms g i (·), (i � 1, 2 . . . , n) are the functions of ξ � [ξ 1 , . . . , ξ i ] T , which is different from the strict-feedback structure in [49] and the semistrict-feedback structure in [50], because the functions g i (·) are relevant to all stats of ξ. We will establish an adaptive intelligent controller for system [1] so that the output η can track a given trajectory η d , the corresponding state constraints are not violated, and all the reference signals of the closed-loop system are bounded. erefore, we give the following assumptions.
Remark 2. Assumption 1 implies that the unknown functions f k i (χ i ) are strictly positive or negative. Further, let us assume that 0 < b m ≤ |f i (ξ i )| < ∞ for generality. Assumption 2 is required in many literatures on the tracking control problem such as [36,40], since we need to figure out its time derivatives up to n th in the design process. ere exists the similar assumption in [5], where φ i1 (·) and φ i2 (·) are assumed to be available. However, Assumption 3 does not need this restriction and is thus more relaxed. Assumption 4 is the key condition to ensure the stability of the unmodeled dynamics in (1).

Preliminaries.
To facilitate the design and analysis, the following Lemmas are given.
where the design parameters are κ i > 0 and μ i > 0.
For clarity, let us abbreviate the functions Now, let us start the design process. Step Next, choose the following Lyapunov function: where log(ϑ) stands for the natural logarithm of ϑ, θ 1 � θ 1 − θ 1 denotes the parameter error, and κ 0 and κ 1 are positive constants. In the set Ω x 1 , V 1 is continuous. us, according to Assumption 2, the derivative of V 1 along with (8) leads to (18). According to Lemma 2, the following inequality holds: where ) is a smooth function. e same method in [32] is repeated where (18), (19), and (20), it yields where (21) is discontinuous, and the NN cannot be directly modeled, so we introduce a hyperbolic tangent function tan h 2 (x 1 /r) and (21) becomes

Complexity
where r is the positive constant, and the unknown nonlinear function g 1 (X 1 ) is expressed as For where δ 1 (X 1 ) is the error of this model and T . Based on Lemma 3 and Lemma 5, the following inequality holds: where where x 2 � ξ 2 − α 1 . Next, using Assumption 1 and designing a virtual control signal α 1 in (12) when i � 1, en, we choose an adaptive law _ θ 1 from (15), when where κ 1 > 0 and μ 1 > 0 are constants. Substituting (28) into (27), it yields Under the action of we get where where en, construct a Lyapunov function V i as where κ i > 0 is a design parameter and θ i � θ i − θ i is the error. In the set Ω where and similar to the method in the Step 1, we can obtain where k j � c j b m > 0, d j (t) ≤ d j and (B j � (1/2)a 2 j + (1/2)ε 2 j + (μ j b m /2κ j )θ 2 j + λ j ′ + ℓ j ′ + d j ). (j � 1, 2, . . . , i − 1.). By using Assumption 3 and the absolute value inequality, it yields Using the same method as (19) and (20), we can get where 2 and d i (t) ≥ 0 for ∀t ≥ 0. en, substitute (36)-(39) into (35) to get 6 Complexity where For ∀ε i > 0, the unknown smooth function g i (X i ) is estimated by the RBF NN E T i H i (X i ) and we have where us, it yields where a i is the design parameter, . erefore, substitute (45) into (42) to get where x i+1 � ξ i+1 − α i . Next, designing a virtual control signal α i in (12) and an adaptive law θ i from (15), then, using the same method as (27)-(31), we can get where Step n: In this step, we design the real controller u from (13), so the derivative of _ x n is where _ α n− 1 is specified in (33) when i � n. en, choose the Lyapunov function as en, the dynamic equation of V n is where Ο n � Ο n − n− 1 j�1 (zα n− 1 /zξ j )Ο j and Ξ n− 1 has been defined in (38) with i � n.

Complexity 7
Using the same method as (38) to (40), we can get where φ n1 (ξ n , θ n− 1 , v) and φ n2 (ξ n , θ n− 1 , v) are defined in (47) and (48), respectively. In view of (51), (50) is written as where d n (t) ≤ d n and g n X n � 1 2 , there exists T such that for any t > T, it has |x 1 | ≤ Δ. As t ⟶ ∞, We can see that x 1 can be made arbitrarily small by selecting the design parameters appropriately.
is completes the proof.

Simulation Example
Example 1. In order to test the applicability of the proposed control method, the following one-link manipulator with motor dynamics and disturbances is considered: Complexity where q, _ q, and € q are the link angular position, velocity, and acceleration. τ is the torque, τ d � q 2 cos( _ qτ) denotes the current disturbance, and u is the control input representing the voltage. Take these parameters as D � 1 kgm 2 , B � 1 Nm, M m � 0.1 H, H m � 1.0 Ω, and K m � 0.2(Nm/A). Moreover, the sketch of the one-link manipulator is given in Figure 1.

Conclusions
e paper has investigated the problem of adaptive tracking control for a class of nonstrict-feedback nonlinear systems with partial-state-constraints and unmodeled dynamics. An adaptive neural tracking control method has been presented by using the adaptive backstepping technique. Based on the inherent properties of Gaussian functions, and the universal approximation ability of RBF NN, a new method is proposed to deal with the nonstrict-feedback form of the considered nonlinear system. e proposed control algorithm can guarantee the boundedness of all the resulting closed-loop signals, the tracking error to converge to a small neighborhood of the origin, and the corresponding state constraints are not violated. Finally, the effectiveness and practicability of the obtained result are shown by a practical simulation example.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest.