Finite-Time Tracking Control for Nonstrict-Feedback State-Delayed Nonlinear Systems with Full-State Constraints and Unmodeled Dynamics

.e problem of finite-time tracking control is discussed for a class of uncertain nonstrict-feedback time-varying state delay nonlinear systems with full-state constraints and unmodeled dynamics. Different from traditional finite-control methods, a C1 smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback, so that the desired fast finite-time control performance can be guaranteed. By constructing appropriate Lyapunov-Krasovskii functionals, the uncertain terms produced by time-varying state delays are compensated for and unmodeled dynamics is coped with by introducing a dynamical signal. In order to avoid the inherent problem of “complexity of explosion” in the backstepping-design process, the DSC technology with a novel nonlinear filter is introduced to simplify the structure of the controller. Furthermore, the results show that all the internal error signals are driven to converge into small regions in a finite time, and the full-state constraints are not violated. Simulation results verify the effectiveness of the proposed method.

It is known to all that many practical systems encounter the effect of the constraints, such as the temperature of chemical reactor and physical stoppages. us, the research about the systems with state constraints is very meaningful and necessary on account of the existence of state constraints which may undermine the stability of the system. In order to tackle the problem of state constraints, some effective control techniques (e.g., model predictive control (MPC) [18,19], reference governors (RGs) [20], one-to-one nonlinear mapping (NM) [21][22][23], and barrier Lyapunov functions (BLFs) [24][25][26][27][28]) have been presented. Due to the fact that MPC and RGs require strong online computing capability to guarantee constraints, this requirement restricts their applications in engineering design. erefore, one-to-one NM and the BLFs-based methods become the main methods to deal with the constrained nonlinear systems. ere exist many significant results which focus on lower-triangular structure nonlinear systems with different constraints (e.g., input constraints [3], output constraints [24], partial-state constraints [25], and full-state constraints [21-23, 26, 27]). In addition, the rate of convergence is also an essential consideration for most practical systems. e works mentioned above only obtain asymptotic or exponential stability with infinite time, which cannot meet the requirement of finite-time control in most practical control systems. As a consequence, a considerable number of meaningful researches (e.g., see [28][29][30][31][32][33]) have been proposed on finite-time control for nonlinear systems. However, most of the works are to present C 0 finite-time controller by using a backstepping technique together with a nonsmooth fractional feedback design method. In order to achieve a faster convergence rate, the authors in [34] originally proposed a C 1 smooth finite-time adaptive NN controller by using a smooth switch between the fractional and cubic form state feedback. Moreover, there are other significant results presented in [35][36][37][38][39][40][41], such that two globally stable adaptive controllers were proposed in [35,36]. To obtain the tracking accuracy, a practical adaptive fuzzy tracking controller for a class of perturbed nonlinear systems with backlash nonlinearity has been designed in [37]. An adaptive fuzzy output-feedback tracking control technique for switched stochastic pure-feedback nonlinear systems has been presented in [38]. e authors in [39] proposed an observed-based adaptive finite-time tracking control technique for a class of nonstrict-feedback nonlinear systems with input saturation. An adaptive finite-time output-feedback controller for switched pure-feedback nonlinear systems with average dwell time has been given in [40]. A decentralized event-triggered controller for interconnected systems with unknown disturbances has been proposed in [41].
In addition, time delays frequently occur in some practical engineering systems. As stated in [48], their existence can deteriorate the transient performance and even can destroy the stability of the control systems. us, the research on nonlinear time-delay systems has become one of the hot topics and some meaningful results have been achieved during the past decades [49][50][51][52][53]. For uncertain nonlinear time-delay systems, the effective controller was developed originally in [50] by combining the backstepping technique with Lyapunov-Krasovskii functionals. Soon afterward, this method was extended to nonlinear strictfeedback time-delay system with unknown control gain functions [51] and uncertain multi-input/multi-output nonlinear systems with time delays [52]. Later, some improved control schemes based on [50] were proposed (e.g., see [35,53,54]).
Although many significant research results on adaptive neural network control for uncertain nonstrict-feedback systems have been obtained in [11][12][13][14][15][16][17], their considered systems did not include unmodeled dynamics or full-state constraints. In [21][22][23][24][25][26][27][28], the effective controllers have been designed for the lower-triangular structure nonlinear systems with state constraints and unmodeled dynamics, but their considered systems did not include state delay and their control methods may be invalid to nonstrict-feedback systems on account of subsystem function which contains the whole state variables. Furthermore, the above-mentioned control methods only obtain asymptotic or exponential stability with infinite time. To the best knowledge of the authors, finite-time tracking control for a class of uncertain nonstrict-feedback time-varying state-delayed nonlinear systems with full-state constraints and unmodeled dynamics has not been fully discussed in the literature, which is still open and remains unsolved. In this paper, we are committed to solving the problem mentioned above. e main contributions of the paper are summarized as follows: (i) In contrast to the existing results reported in [21][22][23][24][25][26][27][28]47] where the control methods have been proposed for nonlinear strict-feedback or purefeedback systems with state or output constraints and unmodeled dynamics, a generalization of the results is proposed for a class of nonstrict-feedback state delay systems with state constraints and unmodeled dynamics of which the subsystem function contains the whole state variables. To the best of authors' knowledge, it is the first time to develop an adaptive DSC method for uncertain nonstrict-feedback state delay systems with state constraints and unmodeled dynamics. (ii) Different from the finite-control methods in [31][32][33], a C 1 smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback reported in [34], so that the desired fast finite-time control performance can be guaranteed. Moreover, unmodeled dynamics is coped with by introducing a dynamical signal and the uncertain terms produced by time-varying state delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. e results show that all the error signals are driven to converge into small regions in a finite time, and the full-state constraints are never violated. e remainder of this paper is organized as follows. In Section 2, the problem formulation and preliminaries are presented. Adaptive DSC design and stability analysis are given in Section 3. Simulation results verify the effectiveness of the proposed control approach in Section 4, followed by Section 5, which concludes this paper.
Notation. In this paper, R denotes a set of real numbers, R + denotes a set of nonnegative real numbers, R m×n denotes a set of m × n real matrices, R n denotes a set of n-dimensional real vectors, sup(·) denotes the least upper bound, ‖·‖ denotes 2-norm of a vector or matrix, |·| denotes an absolute value of a real number ·, exp(·) denotes an exponential function of ·, and log(·) denotes the natural logarithm of ·.

Problem Formulation and Preliminaries
2.1. Problem Statement. Consider a class of uncertain nonstrict-feedback state-delayed nonlinear systems with unmodeled dynamics for i � 1, 2, . . . , n − 1 in the following form: . , x n ] T ∈ R n is the state vector, ξ ∈ R n 0 is the unmodeled dynamics, and u, y, T i (t) denote the system input, the system output, and the unknown timevarying delays, respectively.
. , x i ] T and δ i (ξ, x, t) be the unknown uncertain disturbances. All the states x i are required to remain in the sets Ω x i � x i : |x i | < k c i , where k c i are positive constants. (1) is called a nonstrict-feedback form in which the system function f i (·) and its bounding function contain all the state variables [10]. Apparently, strict-feedback and pure-feedback structures are the special cases of system (1). e methods proposed in [21-28, 31-33, 47] cannot be directly applied to system (1) on account of its nonstrict-feedback structure. e control objective of this paper is to construct an adaptive NN controller u(t) to make sure that the output y follows the desired trajectory y r in a finite time, while every state x i ∈ Ω x i is never violated.

RBFNN Approximation.
In this paper, for i � 1, . . . , n, the unknown smooth nonlinear functions � F i (Z i ): R m ⟶ R will be approximated on a compact set Ω i ⊂ R m by the following RBFNN: where Z i , W i , l denote input vectors, weight vectors, and NN node number, respectively. ε i (Z i ) are the NN inherent approximation errors which are bounded over the compact sets; that is, ε i (Z i ) ≤ ε i , where ϵ i are unknown constants and known smooth vector functions with s q (Z i ) being chosen as the commonly used Gaussian functions, which have the form where μ q � [μ q1 , . . . , μ qm ] T is the center vector and η q is the spreads of the Gaussian function. e optimal weight vector W i is defined as where W i is the estimate of W i .

Key Definition and Lemmas
Definition 1 (see [21]). e unmodeled dynamics ξ is said to be exponentially input-state-practically stable (exp-ISpS), that is, for system _ ξ � q(ξ, x, t), if there exist functions � α 1 , � α 2 of class K ∞ and a Lyapunov function V(ξ), such that and there exist two constants c > 0, d ≥ 0 and a class K ∞ function c, such that where c and d are known positive constants and c(·) is a known function of class K ∞ .

and a signal described by
such that Lemma 2 (see [11]).
Lemma 6 (see [57]). For To obtain the control objective, the following assumptions are needed.

Assumption 2.
ere exist unknown nonnegative continuous functions φ i1 and nondecreasing continuous functions φ i2 such that where

Remark 2. From Definition 1 and Assumption 1, we have
is inequality will be used to cope with the uncertain terms in the following controller design.

Assumption 3.
e sign of g i (� x i ) is known, and there exist some unknown positive constants a i and b i such that Without loss of generality, this paper assumes that g i (� x i ) > 0.

Assumption 4.
e reference trajectory y r (t) and its derivatives about time _ y r and € y r are in a bounded region Ω d , and there exists a known constant A 0 , such that |y r | ≤ A 0 < k c1 .

Assumption 5.
e unknown continuous functions ) satisfy the following inequality: and the time-varying state delays T i (t) satisfy the in- ) are unknown positive smooth functions and T max and � T max are unknown constants.

Adaptive DSC Design and Stability Analysis
3.1. Adaptive DSC Design. Similar to traditional backstepping, the backstepping-design procedure with n steps is developed to construct the adaptive neural controller in this part. By using the backstepping technique, the proposed adaptive DSC scheme contains n steps as follows.
Step 1. Define the first surface error z 1 � x 1 − y r ; the time derivative of z 1 is defined as e virtual control law α 1 and the update law for _ ϖ 1 are designed as where c 1 , μ 1 , k b1 , κ 1 , ρ 1 , σ 11 , σ 12 , l 1 are positive design parameters, ϖ 1 is an estimate of ϖ 1 , where 0 < h � (h 1 /h 2 ) < 1, h 1 and h 2 are the positive odd integers, Consider the BLF candidate V z 1 as Obviously, V z 1 is positive definite and continuously differentiable. Based on Assumptions 2 and 5 and Young's inequality, we obtain the time derivative of V z 1 as follows: 4 Complexity where Note that � F 1 (Z 1 ) is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds: where y r ] T , and ε 1 > 0 is any given. By using Young's inequality and Lemma 2, one has Substituting (13), (14), and (20) into (17), we can obtain By utilizing Young's inequality, the following inequalities can be obtained: erefore, we have According to the inequality 2b 1 ϖ 1 ϖ 1 ≤ ϖ 2 1 − ϖ 2 1 and Lemma 4, one as where ζ 1 and ζ 2 are defined in Lemma 4.
To deal with the time delay in equation (24), define the Lyapunov-Krasovskii functional as follows: where c > 0 is a positive constant. Using Assumption 5, we obtain that the derivative of V U 1 is 6 Complexity From equations (24) and (26), we have where Φ 1 � (e cT max /2(1 − � T max ))ρ 2 11 (x 1 (t)). To move on, introduce the coordinate transformation where z i , α i− 1 , and y i denote the tracking error, the virtual control input, and the boundary layer error for i � 2, 3, . . . , n, respectively. w i is the output of the following first-order filter: where τ i1 and τ i2 are the positive design constants and h is defined in (15). (29), it can be seen that the proposed filter involves both the linear and fractional terms. In particular, when τ i1 � 0 or τ i2 � 0, filter (29) degrades into the fractional filter used in [58] and the linear filter as widely used in the literature [21][22][23], respectively. It is the key to ensure the fast finite-time stability of the closed-loop system, which will be detailed in the following analysis.

Remark 3. From
Remark 4. As mentioned in [34], by designing ι 11 and ι 12 properly, both the virtual control input α 1 and its derivative _ α 1 are ensured to be inherently continuous in the set Ω x i . It means that the virtual control input α 1 defined in (13) is C 1 continuous in the set Ω x i . From (13), it is not hard to see that α 1 and its derivative _ α 1 are the functions of the variables z 1 , ϖ 1 , y r . and z 1 , z 2 , _ ϖ 1 , y 2 , _ y r , € y r , respectively. Combining the continuity of _ α 1 and (28) and (29), it can be seen that there exists a continuous function λ 2 (z 1 , z 2 , _ ϖ 1 , y 2 , _ y r , € y r ) which satisfies Step 2. (i � 2, 3, . . . , n − 1) Define the i th surface error z i � x i − w i ; the time derivative of z i is defined as e virtual control law α i and the update law _ ϖ i are designed as where where h is defined in (15), , and τ i <k bi is a small positive constant.
Consider the BLF candidate V z i as where V z i is also positive definite and continuously differentiable in the set |z i | < k bi . Similar to (17), the time derivative of V z i is where Complexity 7 Note that � F i (Z i ) is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds: where w i ] T , and ε i > 0 is any given. By using Young's inequality and Lemma 2, one has Substituting (32), (33), and (39) into (36), we can obtain

Complexity
By utilizing Young's inequality, the following inequalities can be obtained: erefore, we have According to the inequality 2b i ϖ i ϖ i ≤ ϖ 2 i − ϖ 2 i and Lemma 4, one has where ζ 1 and ζ 2 are defined in Lemma 4. To handle the time delay, define the Lyapunov-Krasovskii functional as follows: where c > 0 is a positive constant. By using Assumption 5, we obtain that the derivative of V Ui is From in equations (43) and (45), we have

Similar to the analysis in Remark 4, there exists a continuous function
Step 3. Define the n th surface error z n � x n − w n ; the time derivative of z n is defined as e actual control law u and the update law _ ϖ n are designed as where c n , μ n , k bn , κ n , ρ n , σ n1 , σ n2 , l n are positive design parameters, ϖ n is an estimate of ϖ n , ϖ n � ϖ n − b n ϖ n , ϖ n � ‖W n ‖ 2 , β n (z n ) is defined as , and τ n <k bn is a small positive constant.
Consider the BLF candidate V z n as Similar to (17) and (36), we can obtain the time derivative of V z n as follows: Note that � F n (Z n ) is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds: where W T n S n (Z n ) is an NN, |ε n (Z n )| ≤ ε n , Z n � [� x n , z n− 1 , z n , r, w n− 1 , _ w n ] T , and ε n > 0 is any given. By using Young's inequality and Lemma 2, one has z n k 2 bn − z 2 n � F n Z n � z n k 2 bn − z 2 n W T n S n Z n + ε n Z n , where Ξ n � Z n � [� x n , z n− 1 , z n , r, w n− 1 , _ w n ] T . Substituting (49), (50), and (56) into (53), we can obtain By utilizing Young's inequality, the following inequality can be obtained: erefore, we have According to the inequality 2b n ϖ n ϖ n ≤ ϖ 2 n − ϖ 2 n and Lemma 4, one has where ζ 1 and ζ 2 are defined in Lemma 4. To handle the time delay, define the Lyapunov-Krasovskii functional as follows: where c > 0 is a positive constant. By using Assumption 5, we obtain that the derivative of V Un is From equations (60) and (62), we have where Φ n � n j�1 (e cT max /(2(1 − � T max )))ρ 2 nj (x j (t)).

Stability Analysis.
In this subsection, we present the stability analysis of the resulting closed-loop system. e main results are presented by the following theorem.
Theorem 1 Consider the nonlinear system (1) with Assumptions 1-5. Let the actual control input and the NN adaptive law be designed as (49) and (50), respectively. If the initial conditions satisfy V(0) ≤ Δ, |z i (0)| ≤ k bi , in which Δ > k bi is any positive constant for i � 1, 2, . . . , n and k bi are properly chosen, such that k c1 > k b1 + A 0 and k ci > � w i + k bi with � w i � sup w i for i � 2, 3, . . . , n, one has that all internal signals z i , ϖ i and y i+1 in the closed-loop system are semiglobally uniformly ultimately bounded and the tracking error will converge into the arbitrarily small regions in a finite time. Meanwhile, each state x i will remain in the set Ω x i ; that is, the full-state constraints are never violated.
Proof. Construct the overall Lyapunov function candidate where V y i � (a 2 i /2b i )y 2 i+1 and V z i , V U i are defined in (35) and (44), respectively.

Conclusions
e problem of finite-time tracking control for a class of uncertain nonstrict-feedback state-delayed nonlinear systems with full-state constraints and unmodeled dynamics has been proposed in this paper. Unmodeled dynamics is dealt with by introducing a dynamical signal and the uncertain terms produced by time-varying state delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. By utilizing a smooth switch between the fractional and cubic form state feedback, novel C 1 smooth finite-time NN control laws have been provided for nonlinear systems with full-state constraints. Based on a modified DSC method and adaptive NN control, together with the BLFs, the fast finite-time control performance of the closed-loop nonlinear systems can be ensured, while the full-state constraints are never violated. eoretical proofs and experimental simulation show that all the internal signals in the closed-loop system are uniformly bounded, and the tracking error signals can converge into compact sets in a finite time with sufficient accuracy, respectively. To extend this control scheme to solve the finite-time tracking control problem for some more complicated systems, such as MIMO nonlinear systems, switched nonlinear systems are also the direction of our future efforts.

Data Availability
is paper is a theoretical study and no data were used to support this study.

Conflicts of Interest
e authors declare that they do not have any financial or nonfinancial conflicts of interest.  16 Complexity