This paper deals with the position control of a hydraulic servo system rod. Our approach considers the surface design as a case of virtual controller design using the backstepping method. We first prove that a linear surface does not yield to a robust controller with respect to the unmatched uncertainty and perturbation. Next, to remedy this deficiency, a sliding controller based on the second-order sliding mode is proposed which outperforms the first controller in terms of chattering attenuation and robustness with respect to parameter uncertainty only. Next, based on backstepping a nested variable structure design method is proposed which ensures the robustness with respect to both unmatched uncertainty and perturbation. Finally, a robust sliding mode observer is appended to the closed loop control system to achieve output feedback control. The stability and convergence to reference position with zero steady state error are proven when the controller is constructed using the estimated states. To illustrate the efficiency of the proposed methods, numerical simulation results are shown.

Actually, the hydraulic servo systems are very popular in several industrial applications such as robotics, aerospace flight-control actuators, heavy machinery, aircrafts, automotive industry, and a variety of automated manufacturing systems. This is mainly due to their ability to produce high power and accurate and fast responses. However, these systems have a high nonlinear behavior due to the pressure flow characteristics [

Owing to their simplicity, linear controllers of PID type [

To improve the controller performances, several strategies have been adopted such as using the self-tuned PID controller [

In the present paper, we are interested in controlling the position of the rod in a hydraulic servo system that consists of a four-way spool valve supplying a double effect linear cylinder with a double-rodded piston. The piston is driving a load modeled by a mass, a spring, and a sliding viscous friction. Our work aims to design a controller that may achieve the reference position in presence of mismatched parameter uncertainty and perturbation in addition to actuator saturation. To realize this objective, we start in the second section by formulating the problem and presenting the effects of using first- and second-order SMC with a linear surface. In Section

The electrohydraulic system that we will deal with in this paper is depicted in Figure ^{2}, ^{3}, ^{3} s^{−1} A^{−1} Pa^{−1/2}, ^{3} s ^{−1} Pa^{−1}, and

Hydraulic servo system controlled using a servovalve.

The most difficult aspect in this model is the existence of a mismatched perturbation as well as a mismatched parameter uncertainty. In addition, the leakage model includes nonlinearity with respect to the control signal

The SMC design consists of two phases. In the first phase the sliding surface is designed such that the system is asymptotically stable when it is confined to it and in the second phase a switching controller is designed to ensure the existence of the sliding mode. Our idea consists in viewing the sliding surface design as a special case of backstepping design. Therefore, at sliding mode,

Thus, should we choose a linear virtual controller,

Using the pole placement method and imposing a stable multiple pole at

Despite the perturbation

Figure

The behavior of the system under sliding mode controller defined by (

Since the system is third-order single input, then we may think of designing the higher-order SMC up to second order. Let

The solution should be understood in the Filippov sense [

By defining

Given the sliding variable

Consider the nonempty real second-order sliding set

The variable structure control law

We can determine the control parameters

From Figure

Behavior of system under second-order sliding mode controller (

The boundedness of

To overcome the problem of mismatched perturbation and uncertainty, we suggest in this section designing a sliding surface

In Section

When attempting to achieve the attractivity of

From Figure

Behavior of system under sliding mode controller using nonlinear surface defined by (

Behavior of system under sliding mode controller using nonlinear surface defined by (

As we can notice, the controller conceived in the foregoing section uses all three state variables. However, measuring the differential pressure

Let us consider the observer model given in (

To prove the efficiency of the observer and the fact that the estimated states based controller can also achieve accurate positioning in presence of perturbation and uncertainty, we will proceed by a step-by-step proof.

Let

Let

Finally, the differential pressure error dynamics can be roughly expressed as

Figure

Behavior of the observer and the hydraulic servo system controlled with sliding mode controller with states feedback.

Behavior of the observer and the hydraulic servo system controlled with sliding mode controller with estimated states feedback.

The controllers designed in this paper used the sliding mode theory which is the most used approach to deal with systems running under uncertainty conditions. However, we have seen in the second section that the first-order sliding mode controller with a linear sliding surface is not robust with respect to perturbation and mismatched uncertainty. In fact, by using the fact that on the sliding surface the system behaves in a similar way to a linear second-order system, it can be easily shown that as far as the sliding motion is preserved, the system is asymptotically stable if the closed loop eigenvalues are chosen as in (

Therefore, the steady state error gets smaller as the closed loop eigenvalues have a larger amplitude. This is illustrated by the simulation results delineated in Figure

To circumvent the problem, we have suggested a second-order sliding mode controller. Applying this controller, we can notice that it achieves a better performance than the first SMC. In fact, the second-order SMC outperforms the first-order SMC in sense of robustness with respect to mismatched perturbation but it cannot guarantee the robustness with respect to the constant perturbation. The steady state error due to the perturbation is equal to

To overcome the problem of mismatched perturbation and uncertainty, we have suggested a sliding mode controller with nonlinear surface. The idea is based on the backstepping method and using a robust variable structure virtual controller. The obtained results achieved robustness with respect to parameter uncertainty and perturbation. The zoomed curve on Figure

Finally, in Section

Compared to other methods such as that presented in [

In this paper, we developed several controllers based on the sliding mode theory. Our aim was to control the position of a hydraulic servo system piston in presence of mismatched uncertainty and perturbation. We have shown that a first-order sliding mode controller did not achieve any robustness. Next we have developed a second-order sliding mode controller that has shown robustness with respect to parametric uncertainty but was not robust to the perturbation. Finally, we have suggested a sliding mode controller based on a nonlinear sliding surface. The design is based on the backstepping method where on each step a variable structure virtual controller design leads to the design of the sliding surface. This controller emphasized the chattering phenomenon due to the nested sliding modes. As a remedy, we suggested substituting the discontinuous function with a smooth saturation function. Eventually, we have designed a robust sliding observer in order to substitute the unmeasured states with their estimates. A step-by-step proof has shown that the controller issued from the estimated states achieved the position tracking.

The authors declare that there is no conflict of interests regarding the publication of this paper.