Hydraulic servo system plays an important role in industrial fields due to the advantages of high response, small size-to-power ratio and large driving force. However, inherent nonlinear behaviors and modeling uncertainties are the main obstacles for hydraulic servo system to achieve high tracking performance. To deal with these difficulties, this paper presents a backstepping sliding mode controller to improve the dynamic tracking performance and anti-interference ability. For this purpose, the nonlinear dynamic model is firstly established, where the nonlinear behaviors and modeling uncertainties are lumped as one term. Then, the extended state observer is introduced to estimate the lumped disturbance. The system stability is proved by using the Lyapunov stability theorem. Finally, comparative simulation and experimental are conducted on a hydraulic servo system platform to verify the efficiency of the proposed control scheme.

Hydraulic servo system has become increasingly popular in modern industrial automation and has been used in many fields [

To obtain satisfactory control performance, many control methods have been developed, such as PID control [

Compared to other control methods, backstepping control has attracted great interests due to its robustness to external disturbances and low sensitivity to parameter variations. As backstepping control implies, the high-order nonlinear system is decomposed into multiple first-order systems. The variables at the next step are taken as virtual input and acted on the current subsystem, and the Lyapunov function is established according to the system at the upper step to achieve system stability. Until the last step, the true control expression and the actual update law are obtained. To avoid the differential explosion of traditional backstepping method, Guo et al. used dynamic surface control to ease the computation burden and system singularity [

The rest of this paper is organized as follows. Section 2 depicts the hydraulic servo system and presents the dynamical mathematical model. Section 3 provides the design of backstepping sliding mode controller based on extended state observer. Meanwhile, the stability analysis and convergence proof of the proposed controller are given respectively. Simulation and experiment results for the desired trajectory tracking control are given in Sections 4 and 5, respectively. Finally, some conclusions are drawn in Section 6.

The hydraulic servo systems are mainly composed of the electrical part, the hydraulic part and the mechanical part. The coupling between these parts deteriorates the nonlinear behaviors and modeling uncertainties [

Due to the fact that the dynamics of servo valve are much faster than the rest parts of system such that the dynamics can be neglected without loss of significance control performance. Thus, the dynamic equation of servo valve can be simplified as [

where _{v} is the displacement of servo valve, _{sv} is the gain constant of servo valve,

The flow rate of servo valve is controlled by the valve orifice. Regardless of the leakage, the load flow equation of servo valve is given as

where _{L} = (_{1} + _{2}) is the real-time flow rates, where _{1} is the supply flow and _{2} is the return flow; _{d} is the flow discharge coefficient; _{s} is the supply pressure; _{L} = _{1} − _{2} is the differential pressure of actuator, where _{1} and _{2} are the pressures in each side of the cylinder. The sign function represents the change in the direction of flow fluid through the servo valve, which is the main nonlinear factors.

According to the flow conservation law, the flow-pressure equation of hydraulic cylinder is described as follows

where _{t} is the total leakage coefficient of the hydraulic actuator; _{t} is the total volume of the actuator; _{e} is the effective bulk modulus of the hydraulic fluid.

The piston of hydraulic servo system can be modeled as the classical mass-spring-viscous system. According to Newton’s second law, the force balance equation of hydraulic actuator can be described by

where

Define the system state variables

For hydraulic servo systems, the parameters _{d}, _{e} and _{t} are unknown positive constants. Considering nonlinear behaviors and modeling uncertainties, the state space model

where

The control objective is to design a controller to let the _{1} track the desired trajectory _{d} as closely as possible in the presence of nonlinear behaviors and modeling uncertainties.

Extend _{2} as an additional state variable by defining _{3} as an additional state variable by defining

The goal of the observer design is to estimate the lumped disturbances

where

Combined with

where

Define

where

Note that _{1} and _{2} are Hurwitz, so there exists positive definite matrixes _{11} and _{12} satisfying the following equations

Define a positive definite function _{o} as

The time derivative of _{o} can be obtained as

The upper bound of

Define

where

Integrating

According to

Define sliding surface for each state variable

where

The sliding mode surface for the first subsystem is defined as

Choose the virtual control item

where _{1} is positive definite. Then the derivative of _{1} can be derived as

The sliding mode surface for the second subsystem is defined as _{2} can be written as

In order to make the second subsystem to reach the sliding mode surface, the virtual control item is defined as

where _{2} is positive definite,

The Lyapunov function for the second subsystem is given by

Thus, the time derivative of _{2} can be derived as

Substituting

To ensure the stability of the second subsystem,

Similarly, the sliding mode surface for the third subsystem is defined as _{3} can be written as

Now, choosing the real control signal as

where _{3} is positive definite. The candidate Lyapunov function for the third subsystem is given by

The first-order differential of the candidate Lyapunov function is given as

where

According to Lyapunov theory, the dynamic system is globally asymptotically stable, which implies that the error will converge to zero even in the face of nonlinear behaviors and modeling uncertainties.

The comparative simulations are conducted in MATLAB/Simulink software. The parameters of hydraulic system are selected as: _{t} = 9 × 10^{−5}m^{3}, ^{3}, ^{−4}m^{2}, _{t} = 4 × 10^{−3}, _{sv} = 5.9 × 10^{−7} m/V, _{d} = 0.62, _{e} = 6.9 × 10^{8} Pa. In order to verify the effectiveness of the proposed controller, PID and backstepping controllers are tested for comparison. (1) PID: the control gains are set as _{p} = 200, _{i} = 80, _{d} = 0.01, which is adjusted by a heuristic tunning method for small steady-state error and good transient response. (2) Backstepping: the constants are set as _{1} = _{2} = _{3} = 2, _{2} = 1.5, _{2} = 0.6. (3) Presented controller: the bandwidths are set as _{1} = _{2} = _{3} = 5, _{2} = 1.5, _{2} = 0.6. In particular, the parameters of these three controllers are well tuned with many times trial-and-error procedure to obtain satisfactory performance.

To simulate the nonlinear characteristics and external disturbances, the measurement noise of the displacement sensor and the input noise are 0.2% and 0.5% of the maximum ranges, respectively. The comparative tracking performance for sine motion with amplitude 10 mm and frequency 1 Hz is shown in

Both the estimations _{e1} and _{e2} are shown in

To quantitatively evaluate the tracking performance of different controllers, three error indices mean value of the absolute error, standard deviation of absolute error and integral of time multiplied by absolute error are defined as follows

The performance indices of the different controllers are given in

Indices | |||
---|---|---|---|

PID | 0.6879 | 0.3295 | 0.6847 |

Backstepping | 0.4888 | 0.2373 | 0.3952 |

Presented | 0.2156 | 0.1056 | 0.1767 |

The position comparative tracking performance for sine signal is given in

The comparative tracking performance of the above controllers for square signal is shown in

The comparisons of position tracking and position tracking error for multi-frequency sine signal are shown in

The concerned performance indices of different controllers for sine signal, square signal and multi-frequency sine signal are summarized in

In this paper, a backstepping sliding mode control based on extended state observer is developed for the high precision tracking control of hydraulic servo system. The dynamic mathematical model of state space representation considering nonlinear characteristics and external disturbances is first constructed. Then, the extended state observation is introduced to estimate the lumped uncertainties. The controller is capable of forcing the position of the hydraulic servo system to track the desired trajectory in finite time and has robust behavior. The system stability is verified by Lyapunov criteria. Simulation and experimental comparative results validate the effectiveness of the proposed controller as compared to the PID controller and backstepping control.

The authors thank their families and colleagues for their continued support.

The work is supported by the

The authors declare that they have no conflicts of interest to report regarding the present study.