Barrier Lyapunov function-based fixed-time FTC for high-order nonlinear systems with predefined tracking accuracy

This article proposes a fixed-time adaptive fault-tolerant control methodology for a larger class of high-order (powers are positive odd integers) nonlinear systems subject to asymmetric time-varying state constraints and actuator faults. In contrast with the state-of-the-art control methodologies, the distinguishing features of this study lie in that: (a) high-order asymmetric time-varying tan-type barrier Lyapunov function (BLF) is devised such that the state variables can be convergent to the preassigned compact sets all the time provided their initial values remain therein, which not only preserves the constraints satisfaction, but warrants the validity of the adopted neural network approximator; (b) the proposed control design ensures the tracking errors converge to specified residual sets within fixed time and makes the size of the convergence regions of tracking errors adjustable a priori by means of a new BLF-based tuning function and a projection operator; (c) a variable-separable lemma is delicately embedded into the control design to extract the control terms in a “linear-like” fashion which not only overcomes the difficulty that virtual control signals appear in a non-affine manner, but also solves the problem of actuator faults. Comparative simulations results finally validate the effectiveness of the proposed scheme.

preserves the constraints satisfaction, but warrants the validity of the adopted neural network approximator; (b) the proposed control design ensures the tracking errors converge to specified residual sets within fixed time and makes the size of the convergence regions of tracking errors adjustable a priori by means of a new BLF-based tuning function and a projection operator; (c) a variable-separable lemma is delicately embedded into the control design to extract the control terms in a "linear-like" fashion which not only overcomes the difficulty that virtual control signals appear in a nonaffine manner, but also solves the problem of actuator faults. Comparative simulations results finally validate the effectiveness of the proposed scheme.
Keywords High-order nonlinear systems · Fixed-time stability · High-order tan-type BLF · Predefined tracking accuracy

Introduction
Many actual systems can be described as high-order nonlinear systems, such as connected vehicles dynamics (e.g., a tractor with several trailers, etc.), synchronous motors, or aircraft wing rock dynamics [1][2][3][4][5]. Compared with strict-feedback and pure-feedback systems [6][7][8][9][10][11], high-order nonlinear systems are more general in the sense that positive-odd-integer powers appear in the dynamics. It is well documented in the literature that high-order nonlinear systems are intrin-sically more challenging than strict-feedback and purefeedback systems, as feedback linearization and backstepping methods fail to work [12]. To handle such problem, the adding-one-power-integrator method was successfully proposed in [13] by introducing iteratively one high power integrator instead of a linear one. Although a number of control problems (i.e., global robust stabilization [14], practical tracking control [15], and asymptotic tracking control [16]) have been solved, the system nonlinearities considered in the above-mentioned results are required to satisfy the growth constraints, i.e., |ψ m (·)| ≤ (|χ 1 | r 1 + . . . + |χ m | r m )ρ m (χ 1 , . . . , χ m ) with ρ m (·)(1 ≤ m ≤ n − 1) known nonnegative smooth functions. To remove this limitation, an intelligent tracking control method was firstly developed in [17] without involving any restrictive growth condition. The same strategy, together with the common Lyapunov function method, was extended to deal with switched high-order nonlinear systems in [18]. Most recently, a new error compensation method was further derived in [19] to improve the tracking accuracy. Despite those efforts above, these results rarely focus on the rate of convergence. To be specific, only the exponential convergence of tracking error is guaranteed in the aforementioned works, which reveals the convergence time tends to be infinite.
From a practical perspective, the rate of convergence is of great significance to the transient tracking performance [20,21]. Recently, the finite-time tracking control for lower-/high-order nonlinear systems is investigated in [22][23][24], which makes the tracking error converge into the compact set within a finite time. Nevertheless, the convergence time achieved in [22][23][24] depends on the initial states of the system. This inevitably brings up a problem, that is, the convergence time cannot be accurately settled when the initial states of the system are unknown. To solve such problem, the fixed-time control [25][26][27][28] is proposed skillfully, by which the tracking error can converge into the compact set within a fixed time and its upper bound of convergence time is independent of system initial conditions. The authors in [29] established an adaptive fuzzy feedback fixed-time control mechanism for uncertain highorder nonlinear systems. However, the Lyapunov function in [25][26][27][28][29] will be augmented by additional estimation error variables when uncertainties exist in lower-/high-order model, leading to the result that the state tracking errors only can converge to a bounded compact set whose size cannot be set explicitly in advance (spec-ified tracking accuracy) within fixed time. On the other hand, the above results neglect crucial aspects, such as state constraints, whose importance is explained hereafter.
Because state constraints generally exist in actual systems, ignoring these state constraints may deteriorate system performance and even endangers system safety [30,31]. In order to prevent violations of state constraints, barrier Lyapunov function (BLF) is exploited to guarantee the non-violation of state constraints, while ensuring closed-loop stability [32]. Generally, the commonly seen forms include log-type BLF [33], tan-type BLF [34] and integral-type BLF [35]. Compared with common log-type BLF and integraltype BLF, the tan-type BLF can integrate constraint analysis into a general method, which can handle constrained/unconstrained situations [36][37][38]. Thus, the tan-type BLF has become a popular research topic. However, most of the existing results of tan-type BLF are concentrated on low-order nonlinear systems subject to full-state constraints and cannot be directly applied to high-order cases. On the other hand, the existing tan-type BLF is symmetric time-invariant, which cannot be used to handle asymmetric timevarying cases [34,37,38]. To our best knowledge, such issue has not been considered in the literature studies. In addition, the actuator faults are ubiquitous in engineering applications, which may cause the system instability and performance degradation [39][40][41][42][43]. Motivated by the above observations, the main innovative points of this work are listed as follows: (1) Compared with existing fixed-time control designs [25][26][27][28][29], our proposed method can not only ensure fixed-time convergence, but also can make the tracking errors converge to a compact set whose size can be specified/predefined a priori. (2) As opposed to the existing symmetric time-invariant tan-type BLFs [34,37,38], a new type of high-order tan-type BLF is proposed to guarantee that state variables are confined within some asymmetric time-varying sets all the time, and the initial conditions are inside of corresponding sets. (3) A new BLF-based tuning function and a projection operator are delicately embedded into the control design to make the size of the convergence regions of state tracking errors adjustable in the framework of fixed-time stability. A variable-separable lemma is utilized to extract the virtual control signals and actuator faults in a "linear-like" manner.

Problem statement
Consider the uncertain high-order nonlinear systems with the following form: where y ∈ R is the system output; u ∈ R is the control input (to be designed);ξ i = [ξ 1 , ξ 2 , . . . , ξ i ] T ∈ R i and ξ = [ξ 1 , ξ 2 , . . . , ξ n ] T ∈ R n are the states. For i = 1, . . . , n, Ψ i (·) and Γ i (·) are unknown continuous functions, and p i are positive odd integers. In this paper, we consider the system (1) to be the time-varying fullstate constraints, i.e., ξ i (t) is required to remain in the set Remark 1 When p i = 1, dynamics (1) reduces to the models of works in [6][7][8][9]21,22,26,27,32,44,45] which means (1) includes most of the existing nonlinear systems as special cases. Besides, the nonlinear functions Ψ i (·), here, are completely unknown, and they do not need to satisfy the growth constraints in [14][15][16], i.e., The actuator fault is taken into account, which contains no fault, partial fault, bias fault and stuck fault. In line with [38] and [43], suppose the actuator output when fault occurs is expressed as where 0 ≤ H ≤ 1 denotes the efficiency factor of the actuator and η u represents the unknown time-varying bias fault of the actuator. The actuator fault can represent the following fault patterns: (1) H = 1 and η u = 0. In this case, the actuator is working normally, and free of failures and faults.
(3) H = 0. The case is said to be the failure with total loss of effectiveness (TLOE). This means that no matter the control u c takes any values, u is stuck at an unknown value. Assumption 1 It is assumed that the efficiency factor H satisfies 0 < H ≤ H < 1 for the PLOE faults, where H is a known constant. In addition, the bias fault is assumed to be bounded by η u ≤η withη being the known positive constant.
Remark 2 Please note that the actuator fault (2) in our paper has been extensively used in the literature [10,[39][40][41][42][43]. However, the common point in aforementioned works is to require the systems to be in low-order form, making the above methods fail due to the existing high powers, i.e., (Hu c + η u ) p n . A new design must be sought beyond the available schemes.
Without loss of generality, the following standard assumptions from the literature are made.

Assumption 2
For a continuously differentiable desired trajectory y r (t), there exists a known positive func-

Assumption 3 The signs of Γ i (·) are known and there exist known positive constants
Remark 3 Assumption 2 is standard and it is essential for the state constrained systems to select suitable bounded reference signal y r (t) to be tracked. Thus, Assumption 2 is reasonable, and we can always provide a robust estimate about the upper bound of y r (t). Assumption 3 is given to ensure the controllability of dynamics (1).
The control objective is to design a fixed-time singularity-free fault-tolerant controller such that: (1) The asymmetric and time-varying state constraints under fault-tolerant control are not transgressed; (2) The tracking errors can converge into the specified compact sets within fixed time.
The following lemmas are useful for deriving the main results.

Radial basis function neural network
In theory, RBFNN can approximate any nonlinear function with arbitrary precision by choosing appropriate parameters [44], which is formulated as where δ (χ ) is the minimum approximation error of the RBFNN, the weight vector is expressed as where n is the number of the RBFNN nodes, ι i = ι i1 , ι i2 , . . . , ι iq T and ℵ i are the center and width of the Gaussian function, respectively. It is worth noting that, if the ith node is viewed to be activated, θ i (χ ) has to be in the compact set Ω θ = {θ i (χ ) |1 > θ i (χ ) ≥ ε min > 0} with ε min denoting the user-defined activation threshold. If the value of x is in the active region, i.e., the node i is called to be activated. Therefore, we can conclude that the smaller the value of ε min is, the larger the size of active region is. However, the amount of computation will increase dramatically if the value of ε min is chosen very small, which reveals that the importance of the selection of ε min .

Fixed-time adaptive controller design
A novel fixed-time singularity-free fault-tolerant control scheme is proposed for high-order nonlinear sys- is utilized to handle the approximation of the unknown functionf i (Z i ) that will be specified latter. We use δ i (Z i ) to denote the approximation error which is bounded byδ, i.e., δ i (Z i ) ≤δ. By lumping w i and θ i (Z i ) into the following vectors: and we further havē To quantify the control objective, we define a constant q > 0 as q = max 1≤i≤n { p i } and consider the following change of coordinates: where ξ i,c represents the virtual control law which will be specified later. Before performing the controller design, define the following switch functions: where i (s i (t)) is the asymmetric time-varying tracking error constraints, i ∈ {1, 2, . . . , n}.
From the definitions of (10) and (11), one has and where m represents positive integer.
Step 1: Using (1) and (8), differentiating s 1 with respect to time yieldṡ Consider the high-order tan-type BLF as where ζ 1 = |s 1 | − β 1 . From (9), we know that b1 (t) and a1 (t) represents the lower and upper bound of tracking error, respectively, with b1 (t) > β 1 > 0, Remark 4 If 1 (t) approaches to infinity, the term will tend to ζ q− p 1 +2 1 q− p 1 +2 , and thus the proposed high-order tan-type BLF can deal with both the constrained case and the unconstrained case. The conventional BLF in [30,31,33] can only handle the constrained case due to the fact Remark 5 It is well known that there have been many studies on the tan-type BLF constraint control problem [34,37,38], but they only consider the symmetric timeinvariant scenarios. However, due to the influence of faults, asymmetric time-varying constraints are more common in actual systems, so the fault-tolerant control of asymmetric time-varying constraints is more practical. That is the reason why we construct asymmetric time-varying BLF in our paper. (15), the switch function ψ 1 is applied such that the bounds of steady-state tracking errors can be preset as β 1 . By utilizing a smooth switch sg 1 (s 1 (t)), the chattering phenomenon is effectively eliminated, and thus the transient performance of system is improved.

Remark 6 In
In view of (14) and (15), the time derivative of V 1 can be expressed aṡ and Z 1 = [ξ 1 ,ẏ r ] T . Noting (7), it can be known that (16) can be rewritten aṡ where Φ = max w T i w i +δ 2 , i = 1, 2, . . . , n, ρ 1 = sg 1 (s 1 ) ϑ T 1 (Z 1 ) ϑ 1 (Z 1 ) + λ 0 with λ 0 > 0 being the parameter to be designed. We are now in the position to handle the term ξ Substitute (18) into (17), and the derivative of V 1 iṡ According to Lemma 1, one has Substitute (20) into (19) yieldṡ Design the singularity-free switching controller ξ 2,c as where c 1 , a 1 , κ > 0 are the parameters to be designed. The time-varying gain termh 1 is chosen as in which o 1 > 0 is the parameter to be designed. The switched tuning function γ 1 is chosen as Remark 7 In (22), when ζ 1 → 0, it has sin q πζ q− p 1 +2 According to the L'Hospital's rule, one has that is to say, there is no singularity problem if 1 (t) = 0 in our proposed fixed-time controller. Furthermore, we note that the inequation tan > 0 holds since ζ 1 < 1 ; thus, it is feasible to design the tuning function (24).
In view of (23), it can be derived that Substituting (22) and (25) into (21) reacheṡ Step i (i ∈ {2, . . . , n − 1}): The design process for step i follows recursively from step 1. From (1) and (8), the time derivative of variable s i can be written aṡ  (9), we know that bi (t) and ai (t) represent the lower and upper bound of tracking error, respectively, with bi (t) > β i > 0, where

being the parameter to be designed. Along similar lines as (18)-(20), one haṡ
Design the singularity-free switching controller ξ i+1,c as here the time-varying gain termh i is chosen as (32) in which o i > 0 is the parameter to be designed. The switched tuning function γ i is chosen as Substituting (31) to (30) and using the fact that we can further havė where It should be worth noting that ℘ i < 0 is always satisfied.
Step n: Invoking (1) and (8), the dynamics ofṡ n are given bẏ ∂ y rẏ r + Γ n (ξ )υ n η p n u and Z n = [ξ 1 , ξ 2 , . . . , ξ n ,ẏ r ] T . Take the high-order tan-type BLF as where ζ n = |s n |−β n . κ is the parameter to be designed. Φ is the estimate of Φ withΦ = Φ −Φ denoting the estimation error. Similarly to step i, define , the time derivative of V n can be given bẏ and therefore, the inequation n ≤ 0 always holds. Then, we can derive thaṫ

Stability analysis
At this point, the main result of this paper is given in Theorem 1. (22) and (31), the state-constrained fault-tolerant controller (39), and the adaption laws (40). By choosing appropriate design parameters, there exists a settling time independent of initial states such that tracking errors converge into the user-defined intervals and all signals of the closed-loop system are fixed-time stable and the time-varying state constraints will never be violated.

Theorem 1 Consider the high-order nonlinear systems (1) subject to actuator faults (2), under Assumptions 1-3, the virtual controllers
Proof Consider the total high-order BLFs as Invoking (18), (19) and (21), it yields thaṫ Substituting (45) into (48), we get where ω i m stands for the upper bound of estimation errorω i . Then, the time derivative ofV i becomeṡ where . With the results obtained in (46), it leads to that the inequations V n ≤ 0 hold; therefore, we can derive that ζ i andΦ are bounded. Due to that ζ i = |s i | − β i and the value of β i is user-defined, s i is bounded. SinceΦ = Φ −Φ and Φ is a constant, we haveΦ is bounded. Since ξ i ; V ∈ L ∞ . Therefore, all the signals of closed-loop system are bounded. In view of Lemma 2 and previous analysis, we can conclude that the tracking errors s 1 will asymptotically converge to β within fixed time . The proof is thus completed.

Remark 8
Although some existing works such as [44] and [45] have dealt with the predefined accuracy issue, these works require the systems to be in low-order form and cannot preserve the fixed-time stability. Compared with the existing FTC strategies [39][40][41][42][43], our proposed method can not only achieve fixed-time convergence, but also guarantee the asymmetric time-varying state constraints. In other words, our proposed method includes existing approaches as special cases.

Remark 9
In Theorem 1, we can prove that the tracking error s 1 asymptotically converges to a predefined interval β 1 and similar analysis can be applied to prove that other state errors s i also converge to prescribed interval β i . With this result, along with the adaptation law designed in (40), we have lim t→∞Φ = 0 andΦ ultimately converges to a constant. Such result is also demonstrated by the evolutions in Fig. 4b.

Simulation example
To verify the effectiveness of the proposed scheme in practical application, a poppet valve system which is one of the most commonly used components in hydraulic systems is considered as shown in Fig. 1 [1 A poppet valve is typically utilized to control the timing and quantity of gas or vapor flow into an engine, and its behavior can be modeled by the annular leakage equation. The input force F drives the poppet to move for regulating the volumetric flow rate Q vol = λc 3 of oil from the high-pressure to the low-pressure chamber, where λ = πr 6μL ΔP is a lumped coefficient, c = αy is the effective clearance of the annular passage with α a constant and y the displacement of poppet, and where r , μ and L are constants independent of the axial motion of poppet, and ΔP is the pressure drop between two chambers. The dynamics of oil volume V in upper chamber is given bẏ where R is the lumped reduction rate of oil attributed to consumption and other leakages. Likewise, the equation of motion of the poppet is where m is the mass of the poppet, k is the viscous friction coefficient, T is the lumped elastic force, and F represents the input force. At this point, let us introduce the following notation substitutions: Then, the dynamic of systems (52) and (53) comes down tȯ where Γ 1 = λα 3 Table 1.
It can be perceived from Figs. 2a, 3a and Table 1 that the system output y tracks the desired trajectory y r closely and the tracking error converges to a specified interval β 1 = 0.03ft/s within fixed time, and the proposed control scheme possesses a better performance than the method in [17]. Figures 2b, c and 3b, c show the profiles of ξ 2 , ξ 3 , s 2 and s 3 . The asymmetric and timevarying constraints are guaranteed via the proposed control scheme, while the system state constraint is violated with the control method in [17]. Besides, compared with the existing results, the fluctuation of tracking error is smaller by utilizing the proposed method. The boundness of the controller and adaptive parameters is illustrated in Fig. 4. As shown in Fig. 4, the proposed controller owns good robustness to the actuator faults.

Conclusions
An asymmetric time-varying barrier Lyapunov functions-based nonsingular fixed-time adaptive FTC method is constructed for high-order nonlinear systems in the presence of asymmetric time-varying state constraints and actuator faults. By constructing a new tuning function and a projection operator, the size of the convergence regions of state tracking errors in our case can be adjustable. The singularity and chatting problems are avoided by utilizing the switch function and smooth sign function. An interesting problem to be investigated in the future is how to solve the adaptive control problem for the high-order nonlinear systems in the presence of time-varying actuator fault. Besides, how to reduce the complexity of controller is still a very challenging problem [47][48][49], and this may be regarded as a possible future research topic.