Adaptive Barrier Control for Nonlinear Servomechanisms with Friction Compensation

This paper proposes an adaptive barrier controller for servomechanisms with friction compensation. A modified LuGre model is used to capture friction dynamics of servomechanisms. This model is incorporated into an augmented neural network (NN) to account for the unknown nonlinearities. Moreover, a barrier Lyapunov function (BLF) is utilized to each step in a backstepping design procedure. Then, a novel adaptive control method is well suggested to ensure that the full-state constraints are within the given boundary. The stability of the closed-loop control system is proved using Lyapunov stability theory. Comparative experiments on a turntable servomechanism confirm the effectiveness of the devised control method.

Moreover, the unknown disturbances can also deteriorate the control precision of servomechanisms.To overcome this issue, an effective method is to design a disturbance observer (DOB) to estimate the unknown disturbances [30][31][32].In [31], Ohnishi proposed a DOB-based control method.Then, the disturbance observer technique-based control methods were widely utilized in practice.In [32], a neural network disturbance observer (NNDOB) was developed for servosystem to estimate the unknown dynamics.In [33], a nonlinear disturbance observer (NDOB) was devised for robotic manipulators to compensate for the unknown friction.Recently, Han proposed a new disturbance estimation technique named extended state observer (ESO) [34].The main feature is that the ESO can not only estimate the unmeasured system states but also observe the unknown disturbances.The ESO has been widely applied in many control fields [35][36][37][38][39].In [40], an ESO was employed to estimate mismatched disturbances of power converter systems.In [41], an ESO was used to observe state vectors and system uncertainties and an adaptive controller was designed using the feedback linearization method for a robotic system.In [42], to estimate unknown disturbances and system uncertainties, an event-triggered active disturbance rejection control (ADRC) was proposed for physical servosystems.In [1], an ESO based on an adaptive funnel control scheme was developed to achieve position tracking control of servomechanisms, where the ESO was used to estimate the unknown disturbance and system states.Although the aforementioned control methods can improve the control performance of the servosystem, it is noted that the transient performance can not be guaranteed.
In recent years, a new constraint function called prescribed performance function (PPC) was proposed by Rovithakis et al. in [43,44].The main feature of this technique is to provide the prescribed performance function (PPF) so that the tracking error of an original nonlinear system can be transformed into a new error of a transformed system.Then, the tracking error can be guaranteed within a given prescribed boundary.The PPC method has been widely applied in practice such as vehicle active suspension systems [45,46], unmanned surface vehicle systems [47], nonlinear systems [48,49], fault-tolerant control [50], and robot manipulators [51].On the other hand, the barrier Lyapunov function (BLF) is used as a constraint control which is utilized to transform the error into a new error without constraint [52].In [53], a boundary controller with BLF was proposed for flexible marine riser to suppress the riser vibration.In [54], a BLF is used to constraint the error and integrated into the adaptive backstepping control design to guarantee the states within the barrier.A BLF-based learning control was presented for hypersonic flight vehicle with AOA constraint and actuator faults [55].In [56], an adaptive BLF controller was devised for PMSM with full-state constraints.An adaptive neural control strategy was designed for multiple input multiple output (MIMO) nonlinear systems with various constraints [57].
This paper proposes an adaptive barrier control method for servomechanisms with friction compensation.A modified continuous friction model is developed to capture the friction dynamics.The NN is employed to estimate unknown dynamics (e.g., nonlinear friction, external disturbances, and system uncertainties) and is incorporated into an adaptive controller to reduce the effect of unknown dynamics.Moreover, a BLF is introduced to each step in a backstepping design procedure to improve the control performance.Then, a new adaptive controller is devised using a recurrent feedback form to ensure the states within the given constraints.The semiglobal boundedness of all closed-loop signals is ensured, and the tracking error converges to a neighborhood of zero.Finally, the effectiveness of the proposed control method is validated via experimental results.
The main contributions are summarized: (1) A new friction model is presented by using the modified LuGre friction model to describe the nonlinear friction.The NN is used to approximate friction dynamics and unknown disturbances of servomechanisms.
(2) To improve the control performance, a BLF is used to transform the error into a new error without constraint, a novel adaptive neural backstepping design procedure is designed, and the new-type adaptation law is developed.We prove that all signals of the closed-loop system are bounded and the tracking error can converge to a small region.
(3) Extensive comparative experiment results provide evidence of the advantage of suggestion approach in comparison to adaptive neural dynamic surface controller (ANDSC) and PID.
The remainder of this brief is organized as follows.System model is given in Section 2. Controller design is provided in Section 3, and Section 4 presents system stability analysis.Experiment results are shown in Section 5 and Section 6 describes the conclusions.

Preliminaries and Problem Statement
2.1.System Model.Considering the motion tracking control of a class of nonlinear servomechanisms [39] (see Figure 1), the dynamic mathematical model of such system can be described as where i d and i q are the d-axis and q-axis stator voltages, respectively; u d and u q are the d-axis and q-axis stator currents, respectively; n p is the number of pole pairs; R and L are the stator resistance and stator inductance, respectively; T l is the load torque; K T is the torque constant; T d is the disturbance torque; ψ f is the rotor flux linkage; q is the angular position; q is the angular speed; T f is the friction torque; J is the motor inertia; and f q, q stands for the unknown resonances and uncertainties.
In practice, the parameter L/R is smaller than the mechanical time constant.Thus, the parameter Ldi q /dt decays very rapidly to zero.In addition, to eliminate the couplings between the angular speed and current, the d-axis reference current i * d is set to zero.In this case, (1) can be simplified as where Choose the state vector x = x 1 , x 2 T = q, q T ; then, the (2) can be simplified as where 2.2.Modified LuGre Friction Model.Over the past years, various friction models have been proposed to describe the friction dynamics.Among these models, the LuGre model is widely used to capture the friction behaviors.To describe the friction dynamics of servomechanisms, the LuGre model is described by where σ 0 , σ 1 , and σ 2 are the friction coefficients; ω denotes the angular velocity; and z is the unmeasurable internal friction state which can be represented by where g w is the nonlinear function.Usually, the nonlinear function is given as where f c denotes the coulomb friction and f s is the stiction force; ω s is the Stribeck velocity.
It is worth to note that the Stribeck function g ω is a discontinuous function due to signum function in g ω .The conventional LuGre model can not be used in the backstepping controller design.To overcome this issue, we will employ a modified Stribeck function g ω in this paper.Hence, the modified nonlinear function g ω [58] is given as where c 1 , c 2 , and c 3 are the positive constants.
Defining a new function N ω = ω/g ω , the modified model can be written as When z = 0, the stead-steady friction f ss can be obtained 2.3.NN Approximation.NNs have been widely used in modeling and control of nonlinear functions using their approximations and learn abilities [59][60][61].In this paper, the NN is employed to approximate a continuous function f x .f x : R m → R over a compact domain Ω ∈ R m is defined as where ε * denotes the approximation error of NN and ε * ≤ ε m ; W * 0 is the ideal value of NN weights that minimizes the approximation error ε * .Therefore, Because the ideal NN weight W * 0 is unknown, we can only use the estimation value Ŵ * 0 of W * 0 in the control design, which can be updated online via an adaptive law.

Controller Design
3.1.Barrier Function.Barrier Lyapunov function (BLF) is used as constraint control, which has been widely used in some fields [55][56][57].In this paper, we will adopt the following BLF candidate [52]: where log • represents the natural logarithm of • and k b1 is the constraint on z 1 , that is, z 1 < k b1 (see Figure 2).Then, we have the following Lemma 1.
Lemma 1 [52].For any positive constant k b1 , the following inequality holds for all z 1 in the interval Assumption 1.The desired trajectory x d and its firstand second-time derivatives exist and satisfy where k c1 , A 0 , and A 2 are positive constants.

Adaptive Controller Design.
In this section, an adaptive barrier controller will be designed using the backstepping technique for servomechanisms with error constraints to achieve position tracking.The control structure is shown in Figure 3.The design steps are given as follows.
Step 1.The tracking error is defined as z 1 = x 1 − x d , where x d represents the reference signal.Thus, the derivative of z 1 is Select a BLF as where k b1 is a design parameter and k b1 = k c1 − A 0 .Defining the second error surface The virtual controller α 1 is designed as where λ 1 is the design parameter.
Then, (15) can be written as Step 2. The time derivative of z 2 is Substituting the second equation of system (3) into ( 19), we have where is the unknown function which cannot be measured.Thus, we employed the NN to approximate F x which can be written as where ε is the approximation error.
Choose the barrier Lyapunov function as where W = W − Ŵ denotes the estimation error and Γ is the design parameter.Then, differentiating V 2 yields Using (21), ( 22) becomes Using Young's inequality, we obtain 4 Complexity where a is a design parameter.The control signal u can be designed as

26
where λ 2 is a positive constant.The adaptation law Ŵ is designed as where Γ and σ are the design parameters.

Stability Analysis
In this section, the stability of the closed-loop system is proven by the Lyapunov stability theory.Based on the design procedure in Section 3, our main result can be summarized in the following theorem.
Theorem 1.Consider servomechanism (1) with Assumption 1.By designing an adaptive controller ( 26) and virtual controller (17) and choosing the adaptation law (27), it can be guaranteed that all signals of the closed-loop system are semiglobally bounded and the tracking error z 1 can be kept within a compact set.
Proof 1. Choose a Lyapunov function: Deriving V and substituting ( 18) and ( 23) into ( 28), we have Using Young's inequality, one has Then, ( 29) can be written as From ( 32), ( 31) can be represented as Multiplying both sides by e μt , (33) can be written as d V t e µt /dt ≤ ϱe µt , and integrating it over 0, t , one has From ( 28) and (34), one can see that log k 2 bi / k 2 bi − z 2 i and W, i = 1, 2 are bounded.Since Because W and W are bounded, the boundness of Ŵ = W + W can be obtained.From z 2 ≤ k b2 and x 1

Output
Servo system with friction NN estimator

Adaptive barrier controller
From ( 35), we have it can be concluded that given any △>k b1 1 − e −2ϱ/µ 1/2 , there exists T such that for any t > T, one has z 1 ≤ △.As t → ∞, z 1 ≤ k b1 1 − e −2ϱ/µ 1/2 .One can see that z 1 can be made arbitrarily small by choosing the appropriate design parameter.
The proof is completed.In the following, the tuning guideline for controller parameters is given.The controller parameters include three parts: barrier variables k b1 and k b2 should fulfill the initial conditions; the controller gains λ 1 and λ 2 ; and the adaptive parameters Γ, and σ should be chosen based on the estimation error.Here, the tuning steps are listed as follows: (1) Select the adaptive parameters Γ and σ to make the estimation values to precisely achieve true values.
(2) The barrier parameters k b1 and k b2 can be selected to fulfill with z i 0 < k bi , i = 1, 2.
(3) The feedback term λ i i = 1, 2 should be depended on a trial-and-error way to maintain tracking performance and smoothness of the tracking signals.Large gains λ i will lead to the vibration.

Experiment Setup.
To validate the effectiveness of the developed adaptive barrier control scheme, a turntable servomechanism is used as the experimental platform.The servosystem constituted of servomotor as a controlled plant, a DSP (TMS3202812) connected a personal computer via an A/D converter.The controller is implemented by C++ program on a PC.The sampling time is 0.01 s.In this experiment, the motor position signal is measured by an encoder.The schematic of the servomotion control system is illustrated in Figure 4, and the system parameters are given in Table 1.

Controller Design.
To test the effectiveness of the suggested method.An adaptive neural dynamic surface controller (ANDSC) [62] and PID are employed as a comparison.
( (3) PID: the control law is defined as u , and control gains are chosen as k p = 30, k i = 0 05, and k d = 25.

Experimental Results
Case 1.To test the effectiveness of the proposed adaptive barrier controller, a sinusoidal signal x d = 0 5sin 2πt/4 with amplitude 0.5 and period 4 is adopted in this experiment.The experimental results are shown in Figures 5 and 6. Figure 5 describes the position tracking result for the different controllers (AC, ANDSC, and PID), and Figure 6 gives the tracking error of three controllers.From these figures, we can see that the tracking effectiveness of the proposed adaptive control scheme is better than the other two controllers (e.g., ANDSC and PID).This is mainly because the proposed adaptive controller scheme contains the state constraints and friction compensation.Moreover, ANDSC method produces the smaller tracking error than PID scheme because ANDSC employed NN to compensate for the 6 Complexity friction dynamics of the servomechanism.Among the three controllers, PID provides the worst tracking performance.
Case 2. To further test the effectiveness of the proposed adaptive controller, another sinusoidal signal x d = 0 8sin 2πt/5 5 with large amplitude 0.8 and large period 5.5 is employed in this experiment.The experimental results are depicted in Figures 7 and 8. From these figures, one can find that the proposed adaptive control method provides the best control performance in three controllers.The tracking error is smaller than the ANDSC and PID control schemes.Nevertheless, we may find that the position tracking signal produces fluctuations in comparison to Case 1.This is reasonable because the proposed adaptive controller is able to capture the triggered high-frequency time-varying dynamics and thus to compensate for their effects by calling for corresponding control actions.

Discussion
All the aforementioned simulation and experiment results show that the control performance of the suggested adaptive control scheme is better than the ANDSC and PID control schemes in terms of the friction dynamics for different trajectories.The reason is that the NN is employed to estimate the friction dynamics and incorporated into the controller design to compensate for the friction.Thus, the developed adaptive control method is more useful for the position tracking of the servosystem.

Conclusions
This paper proposed an adaptive barrier controller for servomechanisms with LuGre friction compensation.A modified LuGre friction model is used to capture unknown friction dynamics.Then, the NN is employed to approximate unknown dynamics (i.e., friction, disturbance, and unknown nonlinearities).Moreover, a barrier Lyapunov function (BLF) is introduced to each step in a backstepping design procedure.Then, a novel adaptive control scheme is well suggested to ensure that the full-state constraints are not

Figure 1 :
Figure 1: Schematic diagram of position control for PMSM.