A novel method of prescribed constraint control without initial condition of nonlinear systems

—A novel constraint control strategy without initial condition of constrained variables is investigated based on backstepping technique for nonlinear systems. In this paper, the novel constraint control strategy is presented for a class of strict-feedback nonlinear systems with actuator saturation and external disturbances by using a nonlinear mapping and a novel performance constraint function. In this control strategy, there are two prescribed constraint functions, the design of these functions is not related to the initial conditions of the constrained variables. Unlike the existing constraint control method without initial condition, the proposed method gives a new solution. It can guarantee that the constraint variable gets into a prescribed constraint region from any initial value no later than a setting time. And the setting time is a design parameter, it can be set arbitrarily. A prescribed performance constraint tracking controller is designed in this paper. It can make that the tracking error of the nonlinear system is constrained to a given region no later than the given setting time, and the transient and steady state performance of the system are ensured. Finally, the proposed method is compared with the existing method, the e ﬀ ective and superiority of the proposed method are demonstrated by two practical examples.

A novel method of prescribed constraint control without initial condition of nonlinear systems

Hui Liu, Xiaohua Li, Xiaoping Liu
Abstract-A novel constraint control strategy without initial condition of constrained variables is investigated based on backstepping technique for nonlinear systems. In this paper, the novel constraint control strategy is presented for a class of strict-feedback nonlinear systems with actuator saturation and external disturbances by using a nonlinear mapping and a novel performance constraint function. In this control strategy, there are two prescribed constraint functions, the design of these functions is not related to the initial conditions of the constrained variables. Unlike the existing constraint control method without initial condition, the proposed method gives a new solution. It can guarantee that the constraint variable gets into a prescribed constraint region from any initial value no later than a setting time. And the setting time is a design parameter, it can be set arbitrarily. A prescribed performance constraint tracking controller is designed in this paper. It can make that the tracking error of the nonlinear system is constrained to a given region no later than the given setting time, and the transient and steady state performance of the system are ensured. Finally, the proposed method is compared with the existing method, the effective and superiority of the proposed method are demonstrated by two practical examples.
Index Terms-Nonlinear systems; prescribed constraint control; nonlinear mapping; performance constraint function; without initial condition.

I. Introduction
B Ecause the prescribed performance control can guarantee the transient performance of systems, including overshoot, convergence rate and convergence accuracy, it has attracted a great deal of attentions in control field. In fact, the prescribed performance control can be regarded as a kind of constraint control. At present, the existing constraint control methods include prescribed performance control, funnel control, output constraint control, and so on. And the constraint control is usually designed by barrier Lyapunov function (BLF) method. The types of constraint boundary can be summarized as the constant value constraint (CVC) and the time-varying function constraint (TVFC).
In CVC control design approaches, there are mainly four categories at present. The first is the logarithm-type BLF method [1]- [7]. The logarithm-type BLF was originally investigated for a class of systems with the Brunovsky normal form in [1]. Inspired by [1], [2]- [7] discussed the control problem on tracking error constraints or state constraints based on the logarithm-type BLFs. The second is the tan-type BLF method [8]- [11]. The third is the integral-type BLF method, see [12]- [15]. They were presented to constrain the tracking errors of systems or the full states of systems. The fourth is the nonlinear mapping method, see [16]- [19]. In this method, a one-to-one nonlinear mapping is introduced in order to achieve the constraint performance of systems.
TVFC control is usually called the prescribed performance control. In this control method, the system tracking error is constrained by two time-varying boundaries. The control idea was firstly proposed in [20]. Generally, in this kind of control strategy, an error transformation function is adopted firstly to transform the original inequality constraint for the tracking error into an equality constraint, then the prescribed performance controller is designed according to the transformed error so that the prescribed performance of the system is guaranteed, see [21] and [22]. At present, the prescribed performance control strategy has been widely used in many researches, such as [21]- [24]. Specially, it should be noted that the funnel control method is actually a prescribed performance control method without the error transformation before control design, see [25], [26].
Although the aforementioned constraint control schemes have satisfactory control performance, they have still serious flaws, that is, the choice of prescribed performance function depends on the initial condition of the constrained variable. In other words, the initial condition of the constrained variable must be known before the system controller is designed. But it is difficult to obtain the initial condition in practical systems. Therefore, the developed prescribed performance control design methods will be difficult to be applied in practice. How to removes the effect of initial condition is an extremely challenging and meaningful topic in constraint control design. Therefore, in [28]- [31], a class of improved prescribed performance control design schemes, which are independent of the initial condition, were proposed. The control schemes in [28]- [31] circumvented the initial condition of the constrained variables by the similar error transformations. The transformations employed a shifting function or a tuning function to make the transformed error being zero at initial time. Compared with the existed methods, the proposed method in this paper presents a new transformation idea. A nonlinear mapping and a new prescribed performance function are adopted, and the constrained variable is transformed to a prescribed scope at initial time rather than to zero. At present, the similar work is not still found. However, actuator saturation is not considered in the existing works [28]- [31].
Actuator saturation is a widespread phenomenon in control systems [32]. It should be mentioned that the actuator saturation in practical systems may limit or degrade the system control performance, even destabilize the system, see [33] and [34]. Therefore, the control research for the systems with actuator saturation is meaningful. In this paper, the actuator saturation problem is considered simultaneously while the novel prescribed constraint control scheme is proposed.
Motivated by the above discussions, this paper proposes a low-complexity and novel constraint control design scheme for nonlinear systems with actuator saturation. The main contributions of this paper are summarized as follows: 1) Compared with the traditional constraint control works [5]- [7], [9]- [11] and [22], a novel prescribed constraint control scheme is presented for a class of nonlinear systems with actuator saturation, which circumvents completely the problem that the traditional performance constraint control schemes depend on the initial conditions of the constrained variables. 2) Unlike the existing works [28]- [31], this paper presents a new constraint idea. A nonlinear mapping and a new prescribed performance function are adopted, and the constrained variable is transformed to a defined scope rather than to zero at initial time.
3) The setting time, at which the constrained variable of the system gets into the prescribed constraint region, is a design parameter and can be set arbitrarily.
The rest of this paper is organized as follows. Section II presents the problem formulation. Section III formulates a novel constraint control idea. The design process for adaptive neural network prescribed performance constraint controller is shown in Section IV. The simulation examples are given in Section V. Section VI gives the conclusion of the paper.

II. Problem Formulation
Consider the tracking control problem of a class of strictfeedback systems with actuator saturation and unknown nonlinear functions, and external disturbances. The mathematical model of the system is described by the following differential equation.
where x = [x 1 , x 2 , . . . , x n ] T ∈ R n is the state vector of the system,x i = [x 1 , x 2 , . . . , x i ] T ∈ R i . And y ∈ R is the output of the system. v is the designed control input, u(v) denotes the plant input with saturation type nonlinearity. g i (·) and f i (·), (i = 1, 2, . . . , n) are unknown, continuous and smooth nonlinear functions, d i (t), (i = 1, 2, . . . , n) are the bounded external disturbances. The tracking error of the system (1) is defined as e 1 = x 1 − y r , where y r is a reference signal. Specially, the system (1) satisfies the following assumptions: Assumption 1 [32] The reference signal y r (t) and its derivatives y (i) r (t), i = 1, 2, . . . , n are known and bounded.
Assumption 2 [35]: The sign of g i (x i ) is known and unchanged. Without loss of generality, it is further assumed as g i (x i ) > 0 and there exists an unknown constant b m such that The saturation model in (1) is asymmetric, it is described as [36] where u R = (u + + u − ) + (u + − u − )sign(v)/2; erf(·) denotes a Gaussian error function (GEF) which can be defined as To facilitate the control design later, we define where λ is an unknown positive constant, see [36], [37]. Hence, the asymmetric smooth model (2) can be written as Assumption 3 [37]: There exist positive constants ∆ max and ∆ min such that In this paper, the following RBFNN is used to approximate a continuous functionf nn (Z) : R q → R.
where Z ∈ Ω Z ⊂ R q is the input vector, W = [w 1 , w 2 , . . . , w l ] T ∈ R l is the weight vector, l > 1 is the node number of the neural network, S (Z) = [s 1 (Z), s 2 (Z), . . . , s l (Z)] T ∈ R l means the basis function vector with s i (Z) being chosen commonly as Gaussian function defined by the following form.
where µ i = [µ i1 , µ i2 , . . . , µ iq ] T and η stand for the center of the receptive field and the width of the Gaussian function, respectively. Just as [34], [38], the RBFNN (7) can be used to approximate a smooth nonlinear functionf (Z) over the compact set Ω Z to arbitrary accuracy as where W * = [w 1 , w 2 , . . . , w l ] T ∈ R l is the ideal constant weight vector. W * is defined as the value of W that minimizes δ(Z) for all Z, i.e., is the approximation error, ε is an unknown bound constant.
Lemma 1 [40]: Let function V(t) 0 be a continuous function defined ∀t ∈ R + and V(0) bounded, and ρ(t) ∈ L ∞ be real-valued function. If the following inequality holdṡ where a 0 > 0, b 0 are constants, then V(t) is bounded.
The control objective of this paper is summarized as follows.
Design an adaptive neural prescribed constraint tracking controller by using a nonlinear mapping and a novel performance constraint function which make the prescribed performance constraint function be independent of the initial condition of the constrained variable. The designed controller can guarantee that all the signals in the closed-loop system (1) are bounded, and the tracking error e 1 (t) is constrained by a prescribed performance constraint function and can converge to a given compact set no later than the given setting time. And the setting time can be set arbitrarily. Specially, the pre-specified constraint performance of the system can't be affected by external disturbances.

III. A Novel Constraint Control Idea
In the traditional prescribed performance control method, the tracking errors of systems need to satisfy the following constraint: where In control design, the achievement of the prescribed performance control requires the initial condition of the controlled variable e 1 (t) to satisfy −µ(0) < e 1 (0) < µ(0). However, the initial conditions of practical systems maybe difficult to be obtained, therefore, for this case, the traditional prescribed performance control method is limited in application. In order to avoid the defect, a novel prescribed constraint control method is given in this paper. The method is independent of the initial condition of the constrained variable. The novel idea of the method is described as follows.
According to the control objective, the tracking error e 1 (t) needs to satisfy −ρ 1 (t) < e 1 (t) < ρ 1 (t) when t ≥ T > 0 without the limit of the tracking error e 1 (0). Here, T is a designable time parameter and it is called the setting time. The constraint function ρ 1 (t) is defined as where ρ 1 (t) is a direct constraint function, ρ 0 > ρ ∞ > 0, λ 1 > 0 and T > 0 are the design parameters. The statement for the novel constraint control idea is clearly illustrated by the graphical representation in Fig. 1.
In order to make the control effect of the tracking error be independent of the initial condition, a new nonlinear mapping of e 1 (t) is proposed. The error e 1 (t) is mapped to z 1 by the following hyperbolic tangent mapping.
Remark 1: From (12), it can be observed that z 1 (t) and e 1 (t) is approximately linear around the origin. Based on (12), an indirect constraint function ρ 2 (t) of e 1 (t) is proposed to constrain z 1 into the following prescribed performance range: where and the constants M ≥ 1,τ ≥ 0,v 0 ≥ 1,v 1 > 0 are the design parameters. Remark 2:v(t) is a smooth function, which has been proven in [41]. In addition, it is easy to prove that ρ 2 (t) is a smooth function.
Proof: From (12) and (13), it follows that If t ≥ T , according to (14), the following mathematical relationships can be obtained.
The control objective of the tracking error e 1 (t) can be reduced to constrain z 1 (t) according to (13). Next, the constraint control design method for z 1 (t) will be presented.

IV. Adaptive Neural Network Prescribed Performance Constraint Controller
Based on the proposed constraint control idea in the above section, an adaptive neural constraint controller will be designed by adopting backstepping technique. The following coordinate transformation is chosen firstly. where ] T ,β denotes the estimated value of the unknown constant β, which is defined as β = max{ W * T i 2 ; i = 1, 2, . . . , n}, and the estimation errorβ = β −β.
Step 1: Choose a Lyapunov functional candidate as where k 1 is a positive design parameter. The derivative of V 1 is obtained aṡ where Q = π 2 tan . The following inequalities can be obtained by invoking Young's inequality.
V ≤b m η 1 ( 4 (e e 1 + e −e 1 ) 2 g 1 α 1 +f 1 (Z 1 ) where η 1 = Q/ρ 2 and are the auxiliary functions to compensate the term ∂α i−1 ∂ββ in (38), they will be given in Appendix I. Here, each unknown functionf i (Z i ) is approximated by a RBFNN in this control design, namely, With the aid of Young's inequality, it follows that where b i > 0 (1 ≤ i ≤ n) are design parameters. Based on (38)∼(40), it can be obtained thaṫ According to (41), we can design the virtual control laws and the adaptive laẇ where c i (i = 1, 2, . . . , n) and d 0 are positive constants. Based on the above deduction, we give the following theorem.
Theorem 2: Consider the nonlinear system (1) satisfying Assumptions 1-3, if the direct constraint function ρ 1 and indirect constraint function ρ 2 are chosen respectively in (11) and (14), the virtual control laws, actual control law and adaptive laws of the system are designed according to (42)∼(45), then the tracking error of the controlled system (1) can be driven into the prescribed scope given by ρ 1 and −ρ 1 when t ≥ T . And all the signals in the closed loop system (1) are bounded.
Proof : 1) On stability for the closed-loop system (1) By substituting (42)∼(45) into (41), the following inequalities can be obtaineḋ Remark 4: According to (45),β(t) ≥ 0 always holds for all t ≥ 0 ifβ(0) ≥ 0.β(t) ≥ 0 has been used in the deduction of (46). In (46), it can be seen that V is bounded if Λ ≤ 0 holds. In fact, Λ ≤ 0 can be proven. In order to clarify the derivation, the proof of Λ ≤ 0 is placed in Appendix I. Therefore, (46) can be rewritten aṡ For the term d 0 k 1 b 2 mββ in (47), the following inequality is verified easily by using Young's inequality.
Then, substituting (48) into (47) yieldṡ From the formula (49) and Lemma 1, we can know that V is bounded, which means that ζ 1 , η i (i = 2, . . ., n) andβ are bounded. From (43) and (44), we also know α i (i = 2, . . . , n−1) and v are bounded, so x 3 , x 4 , . . . , x n are bounded due to the coordinate transformation (19). However, the boundedness of x 1 and x 2 can not be obtained from the above analysis. The proof of the boundedness of them will be given in the part 3).
3) On constraint performance and boundedness of the tracking error e 1 (t) From (50) and Theorem 1, it can be known that the control object has been achieved, that is, −ρ 1 (t) < e 1 (t) < ρ 1 (t) when t ≥ T . With the help of the continuity of the tracking error e 1 (t), it is easy to be known that e 1 (t) is bounded in whole time domain by mathematical logic. Thus, x 1 is bounded. Furthermore, α 1 can be proved to be bounded easily, so x 2 is also bounded. Therefore, the control objective of this paper is achieved.
The proof of Theorem 2 has been completed. Remark 5: There are the design parameters ρ 0 , ρ ∞ , λ 1 and T in the performance function ρ 1 (t). From (11), it can be found that increasing the design parameter λ 1 and decreasing the design parameters ρ 0 , ρ ∞ , T can bring better tracking performance. The smaller the parameter ρ ∞ , the smaller the steady state error. Increasing λ 1 can accelerate the convergence rate of the tracking error. However, the shorter setting time T , the smaller ρ ∞ and the larger λ 1 can induce the larger control input. Therefore, the design parameter T and the function ρ 1 (t) should be chosen depending on the physical limitations and performance requirements of the output in practical applications. For the design parametersτ, M,v 0 and v 1 in ρ 2 (t), we find that increasing the value ofτ can achieve a better tracking performance, but the design parameters M,v 0 andv 1 are only need to satisfy the design condition given in this paper. The larger the parameter c i , the larger the control input. The smaller the parameter b i , the larger the control input. The design parameters k 1 and d 0 slightly affect the control performance.

V. Simulation Studies
In order to illustrate the effectiveness and superiority of the proposed control method, the developed adaptive neural prescribed constraint controller is applied to two practical examples.
Example 1: Consider a rigid robot manipulator described in [42], and input saturation and external disturbance are added to the system, then the dynamic model of the system is where g 2 = 1 J 0 , f 2 = − 1 J 0 m r g v l r cos (x 1 ). x 1 , x 2 , m r , g v and l r are the angular position of manipulator, the relative angular velocity, the load mass, the gravity and the length of manipulator, respectively. And J 0 = 4m r l 2 r /3 is the inertia coefficient, m r = 5mg, g v = 9.8m/s 2 and l r = 0.25m.
The simulations is conducted with the system initial con- In order to compare with the method in [30], according to the constraint control idea in [30], an adaptive constraint tracking controller is designed for the rigid robot manipulator system. The designed controller (70)∼(72) can be founded in Appendix II. For the purpose of comparison, the same constraint function is chosen as the same as the prescribed performance function in this paper for the tracking error e 1 , that is, F 1 = F 2 = 0.07e −0.4(t−3) + 0.01. The other design parameters and neural networks of the method in [30] are as the same as the proposed method in this paper. The designed controller is applied to the rigid robot manipulator, and the simulation results are also given simultaneously in Figs. 2-7. Fig. 2 shows the tracking effect of the output y. Fig. 3 gives the control effect of the tracking error e 1 . Fig. 4 illustrates the actual controller input v and the asymmetric saturation output u. The boundedness of the system state x 2 is shown in Fig.  5. Fig. 6 shows the boundedness of the adaptive parameterβ. Fig. 7 gives the external disturbance d 2 . From Figs. 2-3, we can observe that the tracking performance of the system output y is satisfactory, e 1 is constrained by the constraint function ρ 1 no later than the given setting time T = 3s when e 1 (0) isn't between ρ 1 (0) and −ρ 1 (0) in the two design methods. Figs 2-7 demonstrate clearly that the proposed control method is effective. From Fig. 3, it can be observed that the tracking performance of our method is better than the method in [30].

Example 2:
To further illustrate the effectiveness of the proposed constraint control scheme, consider a inverted pendulum system taken from [4], and actuator saturation and external disturbance are added to the inverted pendulum system. The where g = 9.8m/s 2 , M 0 = 1kg, m = 0.1kg, l 0 = 0.5m are the acceleration of gravity, mass of cart, the mass of pole and the half length of pole, respectively. After the controller is designed according to Theorem 2, the simulation is conducted with the initial condition  Fig. 8 shows the tracking effect of the output y. Fig. 9 exhibits the control effect of the tracking error e 1 , and e 1 (t) converges to the prescribed constraint region no later than the given setting time T = 1s when e 1 (0) isn't between ρ 1 (0) and −ρ 1 (0). Fig. 10 gives the actual controller input v and the asymmetric saturation output u. The boundedness of the system state x 2 and the adaptive parameterβ is depicted in Fig. 11 and 12, respectively.

VI. Conclusion
In this paper, a novel adaptive neural network prescribed constraint control design scheme is presented based on backstepping technique, which is independent of the initial condition of the constrained variable. Be different from the existing constraint control method, in this paper, the purpose of constraint control is achieved by using a nonlinear mapping and a novel performance constraint function. In this method, the setting time T , at which the constrained variable of the system gets into the prescribed constraint region, is a design parameter. The designed controller not only guarantees the system stability but also achieves the desired constraint performance no later than the setting time for any initial condition even if the system is affected by external disturbance. The effectiveness of the proposed control method has been demonstrated via the simulations of two practical systems. Therefore, the proposed constraint control strategy is more practical in application. Our future work will focus on the application research of the proposed method to stochastic systems, switched systems, large-scale systems and multiagent systems.

Appendix I
Proof of Λ ≤ 0 Proof: As we know, it always holds that To clarify the deducing procedure of the proof, the following two inequalities are given firstly by using the formula (45) and Lemma 1 in [39].
The first inequality is where The second inequality is Here, s is the an upper bound of ∥S k (Z k )∥. Substituting (53) and (54) into (52), one has If the auxiliary functions ξ i (Z i ), i = 2, . . . , n are chosen as So far, the proof has been completed.

Appendix II
The controller design based on the method in [30] According to the design idea in [30], the design process is presented as follows: According to (4), the rigid robot manipulator in (51) can be rewritten as Define the coordinate transformation where e 1 is the tracking error, y r is the desired signal, α 1 is the virtual control.
Adopting the idea of [30], the shifting function is chosen as then Step 1: Choose a barrier Lyapunov function candidate as where b m , k 1 andβ are as the same as the definition in this paper, F 1 and F 2 are the prescribed constraint function of ζ 1 . Then, the time derivative of V 1 iṡ The following inequality can be derived by using Young's inequality.
Substituting (59) into (58) generateṡ where Step 2: Consider a Lyapunov function candidate as where Applying Young's inequality, the following inequalities can be obtained.
which means that F 1 and F 2 will become the prescribed performance function of the tracking error e 1 . So far, the design process is completed.

Declarations
Funding This work is supported by the Natural Science Conflict of interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Availability of data and material Not applicable. Code availability Not applicable. Authors contributions 1) Compared with the traditional constraint control works [5]- [7], [9]- [11] and [22], a novel prescribed constraint control scheme is presented for a class of nonlinear systems with actuator saturation, which circumvents completely the problem that the traditional performance constraint control schemes depend on the initial conditions of the constrained variables. 2) Unlike the existing works [28]- [31], this paper presents a new constraint idea. A nonlinear mapping and a new prescribed performance function are adopted, and the constrained variable is transformed to a defined scope rather than to zero at initial time.
3) The setting time, at which the constrained variable of the system gets into the prescribed constraint region, is a design parameter and can be set arbitrarily.