Full-state time-varying asymmetric constraint control for non-strict feedback nonlinear systems based on dynamic surface method

We investigate the tracking control problem for a non-strict feedback nonlinear system with external disturbance and time-varying asymmetric full state constraints. Firstly, the unknown nonlinear term with external disturbance in the system are estimated by fuzzy logic system. The backstepping method is applied to the design of adaptive fuzzy controller. However, to prevent that the constraints are overstepped by introducing an improved log-type time-varying asymmetric barrier Lyapunov function (TABLF) in each step of backstepping design. Secondly, the dynamic surface control (DSC) is introduced in the designed algorithm to solve the computational explosion problem of controller caused by the derivative of control law. The proposed control scheme can speed up the tracking speed of the system. Compared with the previous work, it is verified that the combination of DSC and TABLF can obtain good performance within the constraint range, and can ensure fast and stable tracking convergence under external disturbance. Finally, two simulation examples verify the performance of the adaptive controller.

at any time during the different production progress, and the constraint interval is asymmetric. In this case, the time-varying asymmetric barrier Lyapunov function(TABLF) is required to constrain the system state with time-varying asymmetry. TABLF provides more flexibility in dealing with state constraints. Constraint control methods based on TABLF have received wide attention in recent years. The reference 11 designs a robust adaptive controller for nonlinear systems with dynamic characteristics based on the TABLF, which limits the system output to the specified range. The reference 12 applies TABLF to impose a time-varying asymmetric constraint on the full state of the input unmodeled dynamics system. The reference 13 applies tan-type BLF working for both constrained and unconstrained scenarios to constrain all states of the nonlinear system with time-varying asymmetry. In addition to the common logarithmic BLF, there are integral BLF and tan-type BLF. Different BLF have their own characteristics and scope of application. Different types of BLF can be selected according to the control conditions. The TABLF has also made many achievements in practical application. The reference 14 is combined with the finite-time stability theory, the log-type BLF is constructed to constrain state variables such as angular speed and stator current of permanent magnet synchronous motor in a predefined compact set. The reference 15 uses TABLF to improve the control accuracy of aircraft. The reference 16 uses asymmetric integral barrier Lyapunov functions are adopted to handle the fact that the operating regions of flight state variables are asymmetric in practice, while ensuring the validity of fuzzy-logic approximators. The reference 17 applies log-type TABLF are utilized to confine flight states within some predefined compact sets all the time provided. System state constraint is a problem that must be carefully considered in the actual system. The constraint control for nonlinear systems is worth further studying.
Inspired by previous work, in comparison with the strict feedback systems and pure feedback systems, the non-strict feedback systems have more applicability in practical application. However, the traditional backstepping method can not be directly applied in the non-strict feedback systems. For this problem, the reference 18 uses the method of variables separation to design the controller and provided a solution to the adaptive control problem of the non-strict feedback nonlinear systems. Compared with the variable separation method, the control method proposed in this paper removes the limitation of the unknown functions f i (x) ≤ �(|x|) in references 18,19 , making the new method more widely applicable. However, the repeated differentiation in backstepping will result in the requirement of high-order differentiability and the complexity of controllers in the multiple-state high-order systems. This study introduces dynamic surface control (DSC) to deal with these problems. The controller constructed by backstepping DSC method is much simpler and has been well studied to solve the asymptotical tracking problem of non-strict feedback nonlinear systems. In recent years, many experts and scholars have applied the DSC method 14,[20][21][22][23][24] to solve the problem of computational complexity. The reference 14 proposes an adaptive fuzzy finite-time DSC method for PMSM with full-state constraints. The reference 22 introduces DSC to handle constraints for a class of nonlinear systems. The introduction of DSC technology further optimizes the design process of the adaptive backstepping control method, making it easier to design an adaptive controller for a nonlinear system. Therefore, this paper presents a class of full state time-varying asymmetric constraints for non-strict feedback nonlinear system. It is different from strict feedback system and pure feedback system [25][26][27][28] . Firstly, an adaptive fuzzy controller for non-strict feedback systems is designed by using the adaptive backstepping method. TABLF is introduced in the design process to set the lower and upper bounds of the system state, thus, the full state time-varying asymmetric constraint of the system is realized. Secondly, by introducing DSC technology in the adaptive backstepping design process. The first-order filter is used to process the virtual control function, which solves the problem of repeated differential technology and reduces the computational complexity.
According to the above control methods, the main contributions and advantages of this paper are summarized as follows: (1) Different from the references 9,25-29 that only focuses on the state constraints of strict feedback systems, this paper proposes a adaptive fuzzy control scheme considering full state constraints is investigated for nonstrict feedback nonlinear systems and removes the limitation of the unknown functions f i (x) ≤ �(|x|) in references 18,19 . (2) Compared with time-invariant symmetric constraint in references [30][31][32] , an improved TABLF method is used to solve time-varying asymmetric constraint control for non-strict feedback systems. And the DSC is introduced in the design process, which is used to reduce the order of TABLF, thus simplifying the design process of the controller.

Problem formulation
System description. Consider the following SISO non-strict feedback nonlinear system, an adaptive fuzzy controned to realize the full state time-varying asymmetric constraints of the system.
where x i = [x 1 , x 2 , · · · , x i ] T ∈ R i represents state vector, f i (x n ), i = 1, 2, · · · , n denotes unknown smooth nonlinear function. y ∈ R and u ∈ R are the output and input of the system, respectively. ε i (x n , t) is the external disturbance, and ε i (x n , t) satisfies |ε i (x n , t)| ≤ε i , ε i is a positive constant.

Assumption 1
Ref. 24 It is assumed that the controlled system (1) is controllable and observable. (1)

Remark 1
The system (1) is a class of non-strict feedback nonlinear systems with external disturbances. The non-strict feedback system in (1) is usually applied to the study of adaptive control, such as in references [33][34][35] . The one-link manipulator [36][37][38][39] can be expressed in the form of the system.
The control objectives of this paper: (1) All signals in the closed-loop systems are bounded.
(2) The system state does not violate the constraint conditions.
(3) The tracking error of the system can remain within a prescribed constraint interval.

Assumption 2
For the lower and upper bounds k ci (t) and k ci (t) of the time-varying asymmetric constraint intervals, There exist the constants K ci , K ci , D cij , D cij , i, j = 1, 2, · · · n such that k ci (t) ≤K ci , k ci (t) ≥ K ci and k j ci (t) ≤D cij and k j ci (t) ≤ D cij , where k j ci (t) and k j ci (t) denote j − th time derivative of K ci and K ci .

Assumption 3
For reference signal y r (t) and its derivatives y , and there also exist some positive parameters

Remark 2
In order to meet the system control request, the above assumptions need to be made. Assumption 2 and 3 ensure that the lower and upper bounds of the constraint, the reference signal and its derivatives are all bounded, so that the functions involved in the derivation are bounded. The above assumptions are often used in the research of constrained control of nonlinear systems. For example, there are similar assumptions in reference 40 .

Lemma 1
Ref. 41 On account of the unknown function, we draw into the unknown function of FLS to approximate it. The form of function can be described as follows: The log-type TABLF construction. In the controller design process in this paper, all states of the nonlinear system are constrained to a specified interval by the BLF. The log-type TABLF construction can make the selection of the constraining interval of the system more flexible and can satisfy the constraining requirements of actual systems. Definition 1 For the nonlinear system ẋ = f (x) , the smooth positive definite function V(x) is defined on the intervalU containing the origin. Within interval U, V(x) has a first-order continuous partial derivative. If X approaches the boundary of interval U, Then it is the BLF. The essence of the log-type TABLF is still BLF.

Lemma 2
Ref. 42 For any positive constant k bi , when e i satisfies |e i | < k bi , there are the following inequality: 43 Considering the nonlinear system f(x), for smooth positive definite function V(x), if there exist scalars > 0 and µ > 0 , such that Then the solution of the nonlinear system is uniformly bounded.

Lemma 4
Ref. 44 Let k a (t) and k b (t) be arbitrary functions, Z = {e ∈ R : −k a < e < k b } ⊂ R and N = R l × Z ∪ R n+1 are open sets. For the system (1), it is assumed that there are continuously differentiable positive definite functions V : Z → R + and U : R l → R + such that where ζ 1 and ζ 2 are k ∞ type functions.
In order to impose time-varying asymmetric constraints on all states, the TABLF in references 44 is introduced at each step of the controller design process where It can be seen from (6) that the TABLF is a piecewise, continuous differentiable, positive definite function. The asymmetric BLF can design the lower and upper intervals of the constraint interval respectively. Compared with the symmetric BLF, it has more flexibility and wider application range, but the design process of the controller is also more difficult. Symmetric constant BLF can be regarded as a special case of (6), that is, the constraint interval is constant and symmetric up and down.

Controller design
In order to design the controller, define the error variables as follows: The backstepping design process of the adaptive controller is as follows Step 1: According to the system (1) and the defined error (8), we obtain Then the introduction of first-order filter with a time constant τ 1 has been used for virtual function.
Thus, we could obtain the first-order filter error Further we can get that According to (8), we can get Substituting (11) and (13) into (9), it can be written as Then, we choose the TABLF candidate combined with quadratic Lyapunov function as where where ζ 1 is a positive design parameter, θ 1 denotes the estimation of θ * 1 , θ 1 = θ * 1 − θ 1 stands for the estimation error.
The time-varying constraints k a1 (t) and k b1 (t) on output tracking error e 1 in (15) corresponding to output constraints k c1 (t) , k c1 are given by By Assumptions 2 and 3, there exist positive constants K a1 (t) , The derivative of V 1 is given by , n the following inequalities can be obtained where θ * 1 = �� 1 � 2 κ 1 , ω 1 , κ 1 and η 1 are positive design parameters. By substituting (18)-(22) into (17), the following inequality can be obtained: Select the virtual control function α 1 and adaptive law θ 1 as where σ 1 > 0 and γ 1 > 0 are design parameters, and the time-varying gain is given v 1 (t)by According to (24), (25) and (27), (23) can be written as where then (28) can be further expressed as Therefore, the selection range of constant gain and σ 1 time constant τ 1 should be limited to 1 and in order to guarantee the closed-loop stability.
Step i (i = 2, 3, · · · , n − 1) : According to the system (1) and the defined error (8), we obtain Then the introduction of first-order filter with a time constant has τ i been used for virtual function α i .
Thus, we could obtain the first-order filter error We can further obtain that According to (8), we can get that According to (36) and (38), (34) can be written as Then, we choose the TABLF candidate combined with quadratic Lyapunov Function as where ζ i is a positive design parameter, θ i denotes the estimation of θ * i , θ i = θ * i − θ i stands for the estimation error.
The time-varying constraints k ai (t) and k bi (t) on output tracking error e i in (15) corresponding to output constraints k ci (t) , k ci are given by By Assumptions 2 and 3, there exist positive constants k ai (t) , k ai , k bi (t) , k bi such that K ai ≤ k ai (t) ≤K ai , The derivative of V i , we can obtain that where According to Lemma 1, we can have: where i (x n ) ≤¯ i and ¯ i > 0 are constants. By applying Young's inequality, the following inequality can be obtained where θ * i = �� 1 � 2 k i , ω 1 , k i and η i are positive design parameters. According to the derivation process in the previous step, we can get that Based on (39)- (44), (38) can be expressed as q(e i ) = 1, e i > 0 0, e i < 0 2 . Therefore, the selection range of constant gain σ i and time constant τ i should be limited to σ 1 > 1 and in order to guarantee the closed-loop stability.
Step n: According to the system (1) and the defined error (8), we obtain the derivative of e n Then, we choose the TABLF candidate combined with quadratic Lyapunov Function as (45) e n =ẋ n −α n−1 = f n (x n ) + u + ε n (x n , t) −α n−1 where ζ n is a positive design parameter, θ n denotes the estimation of θ * n , θ n = θ * i − θ n stands for the estimation error.
The time-varying constraints k an (t) and k bi (t) on output tracking error e n in (15) corresponding to output constraints k cn (t) , k ci are given by By Assumptions 2 and 3, there exist positive constants K an (t) , K ai , K bn (t),K bi such that K ai ≤ k ai (t) ≤K ai , K an ≤ k bi (t) ≤K bi , ∀ ≥ 0.
According to (52) and (53), we can get that where From step n-1 of the derivation process, we can get that According to Lemma 1, we can have : By applying Young's inequality, the following inequality can be obtained where θ * n = �� n � 2 k n , ω 1 , κ n and η n are positive design parameters. Substituting the (57)-(60) into (55), so that (53) In order to apply backstepping method to the design of controller for non-strict feedback nonlinear system, the control method proposed removes the limitation of the unknown functions f i (x) ≤ �(|x|) in references 18,19 , which makes the proposed control scheme more widely used.

Remark 4
Note that ζ is a positive constant and can guarantee v 1 (t) > 0 when k al and k b1 are both zero.

Remark 5
Note that ζ is a positive constant and can guarantee v i (t) > 0 when k ai and k bi are both zero. (61) (σ k − 1)e 2 k K ek + e n−1 e n K en−1 − ξ k + e n K en κ n e n K en θ * n 2ω 2 n ϕ T n (x n )ϕ n (x n ) + κ n e n K en 2η 2 n + e n K en 2 + χ n−1 τ n−1 + u k an (t) e n + q(e n ) k bn (t) k bn (t) e n + ω 2 Proof In order to facilitate the calculation process, the following parameters are defined.
Then (66) can be expressed as follows Because of x 1 (t) = e 1 (t) + y r (t) , z i (t) ∈ Z i = {−k ai (t) < z i < k bi (t)} , i = 1, 2, · · · , n and according to Assumptions 2 and 3, we can obtain In the derivation process, it has been proved that α i , i = 1, 2, · · · , n is bounded, so it can be obtained that all states in the system (1) are satisfied Remark 6 It can be seen from (73) that the selection of upper and lower boundaries k ai and k bi of time-varying asymmetric constraint intervals will affect the tracking error of the system. According to Lemma 4 and (62) and the simulation results, when the constraint interval increases, the system tracking error increases and the system control effect becomes worse. When the constraint interval is reduced, the tracking effect of the system becomes better, but the peak and fluctuation of the system input u will become larger. Therefore, we should choose the appropriate constraint interval to balance the system.

Simulation analysis
In this section, two simulation examples are given to demonstrate the effectiveness of the adaptive fuzzy controller proposed in this paper. Two control methods are adopted for each simulation example, and the two control methods are compared in the simulation results.
Case 1: The full state time-varying asymmetric constraint control scheme for non-strict feedback nonlinear systems based on the DSC proposed in this paper is applied.
Case 2: The traditional time-varying asymmetric constraint control scheme is used to the control of non-strict feedback nonlinear systems. Example 1: Numerical example. Consider the following non-strict feedback nonlinear state constrained system with external disturbances  Figure 1 shows the trajectories of the system output y, the reference y r and constraint intervals. Figures 2 and 3 are the trajectories of x 2 and x 3 and constraint intervals. Figure 4 shows the trajectories of adaptive law θ 1 , θ 2 and θ 3 . Figure 5 shows the trajectory of the system input u. Figure 6 shows the trajectory of tracking error e 1 .
It can be seen from Figs. 1, 2, 3, 4, 5 and 6 that the controller designed in this paper can realize the effective tracking control of the non-strict feedback nonlinear system (76) with external disturbance. The system output can achieve the desired tracking effect, and the output tracking error do not violate constraint conditions. All variables of the system are bounded. Compared with the traditional time-varying asymmetric constraint control scheme, the time-varying asymmetric constraint control scheme based on DSC method can full states and the tracking error do not violate constraint conditions, and all variables of the system are bounded. The above numerical simulation shows that the adaptive fuzzy controller designed in this paper can satisfy the control requirements. Example 2: Application example. In the face of more and more complex production processes, the control requirements of industrial manipulators are also increasing. How to effectively control industrial manipulator has always been a hot research direction, and many research results have been obtained in recent years. In some work tasks that need to interact with people or high-precision, in order to ensure production safety and control accuracy, the motion space, motion speed and tracking error of the manipulator need to be limited. Therefore, it is of great practical significance to study the constraint control of manipulator.
Therefore, in the simulation design of this section, the system model of one-link manipulator 37-39 is adopted, the adaptive fuzzy controller designed in this paper is applied to the control of one-link manipulator, and the time-varying asymmetric constraint interval is designed to restrict the rotation angle, rotation speed and torque of one-link manipulator.
The system model of one-link manipulator can be expressed as the following , j = 1, 2, · · · , 7, i = 1, 2, 3   www.nature.com/scientificreports/ where x 1 = q is the angular position of the one-link manipulator, x 1 =q is the angular velocity, x 3 is the torque, and the reference signal is y r = 0.5 sin(t).
The simulation results are shown in Figs. 7, 8, 9, 10, 11 and 12. Figure 7 shows the trajectories of the system output y, the reference y r and constraint interval.The adaptive fuzzy controller designed can ensure the one-link , j = 1, 2, · · · , 5, i = 1, 2, 3    www.nature.com/scientificreports/ manipulator full state and the tracking error do not violate constraint conditions, and the system output y r can remain within a prescribed constraint interval. Figures 8 and 9 show are the trajectories of x 2 and x 3 and constraint intervals, system states x 2 and x 3 are constrained within intervals. Figures 10 and 11 shows the trajectories of adaptive law θ 1 , θ 2 and θ 3 and input u. It can be seen that all variables in the system are bounded. Figure 12 shows the trajectory of tracking error e 1 , which satisfies the constraints. From the above simulation results, it can be seen that the time-varying asymmetric constraint control scheme based on the DSC method designed in this paper can effectively control the one-link manipulator, time-varying asymmetric constraints on the rotation angle, rotation speed and torque of the manipulator, and reduce the stabilization time of the one-link manipulator.

Conclusion
In this paper, based on the DSC method, time-varying asymmetric constraints are applied to a class of non-strict feedback nonlinear systems. In the design process, the fuzzy logic system is used to estimate the unknown nonlinear function in the system. In each step of the controller design process, the time-varying asymmetric BLF is introduced to design the lower and upper time-varying constraint boundaries of the system state respectively, in order to time-varying asymmetric constraints on all states of the system. Based on the DSC method, a firstorder filter is introduced to process the virtual control function in the design process, which solves the problem that the traditional adaptive backstepping design method needs to perform repeated differential calculations on the virtual control function, reduces the order of TABLF, reduces the computational burden and speeds up the tracking speed of the system. Finally, through numerical simulation and one-link manipulator system simulation, it is proved that the adaptive fuzzy controller designed in this paper can meet the predetermined control  www.nature.com/scientificreports/ requirements. The simulation results show that all signals of the system are bounded, and all states of the system do not violate the time-varying asymmetric constraints during operation. The adaptive tracking control for a class of switch nonlinear systems or stochastic nonlinear system with full state constraints will be our future works.