Barrier Lyapunov Based Adaptive Control for Hydraulic Servo Systems with Parametrical Nonlinearities and Modeling Uncertainties

, uncertainties, Barrier-Lyapunov function.


Introduction
Hydraulic servo systems are widely used in industrial fields, such as earthquake simulator, vehicle active suspension, heavy mining equipment in virtue of their high power-to ratios, fast response, large force/torque output and good robustness [1][2][3][4].However, the parametrical nonlinearities including input saturation, dynamic friction and valve overlap hinder the high precision position tracking of hydraulic equipment [5][6][7].In addition, the hydraulic system uncertainties (e.g., bulk modulus, load variation, oil leakage, external disturbance) also complicate the controller design [8].Hence, the traditional linear models cannot meet the requirement of high-performance dynamic response and small steady-state precision for modern industrial process control [9].More importantly, these nonlinearities and uncertainties may degrade performance and even result in control system unstable [10].Nevertheless, current hydraulic servo systems demand both transient response performance and tracking precision.To meet this requirement, it is very necessary to enhance the control response performance in hydraulic system.
To retain satisfying tracking performance, numerous control schemes have been developed, such as proportional-integral-derivative (PID) control [11], adaptive control (AC) [12][13][14], robust control [15][16][17], backstepping control [18][19][20], fuzzy control [21] and neural network control [22,23].Classical PID control may not achieve satisfy tracking performance when there are model uncertainties, parameter fluctuations and external disturbance.To cope with the drawbacks of traditional PID control, intelligent optimization algorithm is integrated with PID to on-line adjust controller's parameters [24].Yao et.al developed an adaptive controller to deal with parametric uncertainties along with nonlinear friction compensation of hydraulic systems, where adaptive control law and robust control law is designed in backstepping controller to achieve excellent tracking performance [25].Deng et al utilized robust controller to addresses the high precision position control of hydraulic system with parametric uncertainties and unmodeled disturbances by introducing a nonlinear robust term [26].Guo et.al proposed backstepping controller to handle unknown load disturbance and uncertain nonlinearity by using extended-state-observer [18].Shen et.al proposed fuzzy controller to solve the parametric coupling and intrinsic nonlinearity of hydraulic cylinder, where fuzzy system is used to approximate the minimum sliding mode gain [21].Guo et al presented a neural network-based adaptive composite dynamic surface controller, where neural network is employed to estimate the system state and unknown nonlinearity of dynamic friction [27].
Despite the aforementioned control methods have good tracking performance, they did not take state constraints into consideration.In real world, the practical systems abide the effect of the constraints, such as pressure of hydraulic cylinder and physical limitation of system [28].Song et al. investigated an adaptive controller f to solve the full-state and input constraints of motors, where log-type barrier Lyapunov function (BLF) is used to deal with output performance constraint, input constraints, and unknown external disturbances [29].Liu et al used adaptive learning controller to cope with the state constraints of nonlinear systems, where the BLF were designed to guarantee that the state constraints are bounded [30].The abovementioned control methods can solve the constant constraint problem well to some extent.In this paper, in order to relax the conditions of the system constant constraint, the time-varying asymmetric barrier Lyapunov function (TABLF) is selected the intermediate function in each step of backstepping design.In addition, the adaptive law can also be used to update the disturbance upper bounds online, which improves the operability of the controller in practical applications.
This paper is organized as follows.Problem formulation and dynamic model are described in section 2. In section 3, the controller design is presented and the system stability is analysed.Section 4 proposes comparative simulation results.Some conclusions are provided in section 5.

Problem formulation and dynamic model
In this section, the nonlinear dynamic model of the hydraulic servo system is given.Fig. 1 shows the considered hydraulic servo system, which mainly contains pump, actuator, servo valve and other components.The hydraulic actuator drives the load to move the desired position.The servo valve determines the direction of the motion and the speed of hydraulic actuator by controlling the spool position.The accumulator installed next to the pump acts as energy replenishment device.The relief valve plays an important role in protecting equipment operation by controlling the maximum of work pressure.The control goal hydraulic servo system is to make the load track the desired signal as closely as possible by adjusting the output pressure of hydraulic cylinder.
The dynamics equation of load is described as follows ( ) where m is the displacement of load, y is the mass of load; PL=P1-P2 is the load pressure; A is the efficient ram area; B is the viscous coefficient; f is the unmodeled disturbance.

Fig. 1 Control mechanism of hydraulic servo system
The fluid dynamic equation of the actuator is described as where Vt is the total volume of the actuator; βe is the effective bulk modulus, Ct is the coefficient of the internal leakage, QL=(Q1+Q2)/2 is load flowrate, in which Q1 and Q2 are the supplied flowrate and return flowrate.The load flowrate QL can be described by () where Cd is the discharge coefficient, w is the spool valve area gradient, xv is the spool valve displacement, Ps is the supply pressure, ρ is the oil density.The sign function is given by ( ) The relationship between spool position xv and servo valve control input voltage u can be described as xv=kiu, where ki is a positive electrical constant.Therefore, Eq. ( 3) can be rewritten as TT L x x x x y y P ==, combining Eqs. ( 1)-( 5), the system state space can be described as Remark 1.The hydraulic parameters Cd, ρ, K, B, βe, Ct are always uncertain positive constants.Remark 2. The unmodeled disturbance f is bounded by |f|≤fmax.,where fmax is a positive constant.
Based on remarks 1and 2, the dynamical model of hydraulic servo system ( 6) is written as where four model function are designed as

Controller design
The nonlinearities and modeling uncertainties of hydraulic servo system is given by ( ) ( ) where () i-1 is a virtual law which is designed in backstepping controller.For the sake of narrative, the definition is first introduced.
Then one can see that Substituting ( 20) into (19), it yields where 1  is an adaptive parameter.
Step 2: The time derivative of z2=x2-α1 is The time derivation of V2 is given by The virtual control law α2 is chosen as where  is a design parameter, Let z3=x3-α2, and substituting ( 26) into (23), it yields Substituting ( 30) into (28), it yields where ) .ˆp According to (23), the following inequality holds  17) and ( 24) into ( 23), the derivative Step 3: The controller is designed as The adaptation law is designed as The Lyapunov function is chosen as The time derivative of V is obtained by Then, one can see that Hence, if the state initial values satisfy (0) (

Comparative simulation studies
To verify the effectiveness of the proposed control scheme, PID and AC are compared in simulation.The simulation parameters of hydraulic servo system are listed in Table 1.For evaluating the tracking performance of the compared controllers, the maximum Me, average μ, and standard deviation σ of the tracking errors are used to quantitatively assess the quality of different controllers.The performance indices are defined as follows. ) The controller parameters of three different controllers are selected by multiple debugging.More importantly, all the control strategies operate under their optimal parameters to guarantee that the results are convincing.

AC:
The control law is given by:   To further validate the performance of the proposed controller, a multi-frequency desired reference signal xd=50sin(5πt)+40sin(10πt)+20sin(25πt) is applied.The comparative position tracking and corresponding tracking error of three controllers are shown in Figs.7 and 8.The simulation results indicate that the proposed BLBAC controller obtain the best tracking performance among three controllers.It demonstrates the superiority of the proposed control scheme in dealing with t parametrical nonlinearities and model uncertainties encountered by the hydraulic servo system.
And the performance indices of multi-frequency sinusoidal signal are collected in Table 3.It is obvious that the proposed BLBAC controller obtains the tracking performance in terms of three performance indices, i.

Conclusion
In this paper, an adaptive controller based on Barrier-Lyapunov has been proposed for trajectory tracking control of hydraulic servo systems in the presence of parametric uncertainties, unmodeled dynamics and external disturbances.To improve the position tracking performance, the dynamic model is first derived.To ensure the tracking trajectory and pressure within the boundaries of expectation, the time varying Barrier-Lyapunov is employed, which guarantees the tracking error is asymptotic stable.Furthermore, a novel adaptive law is adopted to deal with the hydraulic parametric nonlinearities and modeling uncertainties.Comparative simulation results have proved the superiority of the proposed controller PLBAC over PID and AC.

Step 1 :
The time derivative of z1=x1-yd is

Fig
Fig. 2 Trajectories of

Fig. 3 Fig. 4
Fig. 3 Trajectory of adaptation law to prevent the constraints overstepped.Contrastive control inputs of PID, AC, BLBAC are shown in Fig.6, which are continuous and bounded.It is worth noting that the control input of BLBAC is smaller than that of PID and AC, which is owing to that BLBAC has better tracking performance than PID and AC control methods.

Fig. 8
Fig. 6 Control input of sinusoidal signal

Fig. 9
Fig. 9 Control input of multi-frequency sinusoidal signal

Table 2
Performance indices of sinusoidal signal

Table 3
Performance indices of multi-frequency sinusoidal signal