Keywords

1 Introduction

Control of the hydraulic servo drive has already been an object of examinations in different centers of education and research for many years [1, 2]. Nonlinear dynamic characteristics of the hydraulic actuator as well as servo valve are caused by large inertia of the movement, friction forces, deformations and springiness of mechanical elements, compressibility of working fluid, and characteristic flows [3, 4]. Changes in dynamic parameters of hydraulic servo drive are associated with load and velocity of the movement, maladjustment of the control structure, and influence of many other factors associated with characteristics of working fluid and exploitation parameters. They all have a significant influence on reducing the resistance of control system. Frequent maladjustment of the control structure results from large forces or load moments of the servo drive hydraulic system. Moreover, requirements concerning high accuracy of control positional and velocity in wide scope cannot be fulfilled, because these conditions are often changeable in time depending on the external load. Turbulent character of the fluid flows in valves and appearance of structural nonlinearities, such as saturation pressure or rate of fluid flow, zone of overlap caused by positive windows overlap of the control slider as well as hysteresis caused by magnetizing the armature of control slider. They all cause that the process of physical phenomena in unknown states in all hydraulic systems can be described only in the nonlinear way. Model of the hydraulic servo drive should include dynamic characteristics of the hydraulic actuator or hydraulic engine, flow characteristics of the servo valve or proportional valve, characteristics of electromechanical converters, compressibility (capacity) hydraulic in distinguished servo drive areas, friction forces appearing in elements of the system, efficiency of hydraulic elements, and other factors like the accuracy of carrying slide steam, accuracy of the filtration as well as characteristics of the working fluid [5]. We should pay attention to nonlinear static characteristics of the control valve. Proportional control valves can have positive or negative overlap [6]. In amplifiers, valve sliders are applicable with the value of overlap to 5 % nominal jump, which takes about 0.5 mm up to 1.0 mm. Edges of the spool valve and cornet cooperating with it are carried out with the lower tolerance from ±2.5 μm [6, 7]. It allows for keeping the nonlinear scope in the vicinity of zero for about ±3 % of jump. In this range, the movement of slider may change the rate of strengthening the valve to 200 % of its value appearing at normal opening the valve. Such large changes of control parameters may lead to the unstable work of servo drive, e.g., during positioning of the hydraulic actuator or hydraulic engine. Slight leakages that appear in the valve are caused by inaccuracies of making the spool and valve body, which corresponds to the negative overlap. Such a situation also appears, when the control system is unable to hold a slider in the position corresponding to turning off the hydraulic actuator from the power supply. Also, the work instability can be caused by pollutants, which block the flow of the valve. It causes the delays of valve action as at positive overlap. Disadvantageous feature is also appearing of losses caused by leakages, and they cause the movement of the piston at the zero control signal. The value of control signal must then change even in the steady state, at the lack of spool valve movement. Also, undervalued cause of the nonlinearity of hydraulic servo valves is friction between the spool valve and the sleeve valve. The threshold of insensitivity causes that to the coil of valve must be given the minimal intensity of current in order to trigger corresponding slider movement and flow of working fluid. The friction force as well as action of the well-proportioned electromagnet introduces the hysteresis into static characteristics of the control valve and electromechanical converter [8–10].

2 Dynamic Model of the Hydraulic Servo System

The mathematical model of occurring physical phenomena in the studied drive system was created after assuming the following [7, 8]: volume module of the oil compressibility is fixed in the entire scope of change pressures and temperatures in the hydraulic system, the ps pressure in the crowded wire of pump is permanent during the system work, hydraulic control valve has zero overlap, output pt pressure from the control valve into the container is negligibly small toward pressures in the cylinder chambers, leaks of pressures between the pump and control valve are being omitted, temperature and viscosity of the oil are established during the system work, and delay time of the control valve is equal to zero. Calculation diagram of the hydraulic servo drive model, consisting of the double-acting cylinder with one-sided piston rod and directional control valve, is presented in Fig. 1.

Fig. 1
figure 1

Scheme of the hydraulic servo model

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Marked parameters presented in the mathematical model of analyzed hydraulic servo drive were compared in Table 1.

Table 1 Markings presented in the model
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Considering such nonlinearities as evolution characteristics of the flow, friction, and stiffness of working fluid in cylinder chambers from the position, such equations were determined in the following form:

  • equation of the movement of piston

    $$\left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}x_{p} \left( t \right)}}{{{\text{d}}t}} = v_{p} \left( t \right)} \hfill \\ {\frac{{{\text{d}}v_{p} (t)}}{{{\text{d}}t}} = \frac{1}{{m_{p} }}\left[ {A_{1} p_{1} \left( t \right) - A_{2} p_{2} \left( t \right) - F_{tc} - F} \right]} \hfill \\ \end{array} } \right.$$
    (1)

  • equation of flow through control inter space of the valve

    for \(x_{s} \left( t \right) > 0\)

    $$\left\{ {\begin{array}{*{20}l} {Q_{R1} \left( t \right)_{ + } = K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{s} - p_{1} \left( t \right)} \right|} } \hfill \\ {Q_{R2} \left( t \right)_{ + } = - K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{2} \left( t \right) - p_{t} } \right|} } \hfill \\ \end{array} } \right.$$
    (2)

    for \(x_{s} \left( t \right) < 0\)

    $$\left\{ {\begin{array}{*{20}l} {Q_{R1} \left( t \right)_{ - } = K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{1} \left( t \right) - p_{t} } \right|} } \hfill \\ {Q_{R2} \left( t \right)_{ - } = - K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{s} - p_{2} \left( t \right)} \right|} } \hfill \\ \end{array} } \right.$$
    (3)

  • equation of pressures in the cylinder chambers

    for \(x_{s} \left( t \right) > 0\)

    $$\left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}p_{1} \left( t \right)}}{{{\text{d}}t}} = \frac{{E_{ol} }}{{V_{1} + A_{1} x_{p} \left( t \right)}}\left( {K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{s} - p_{1} \left( t \right)} \right|} - A_{1} v_{p} \left( t \right) - Q_{w1} \left( t \right)} \right)} \hfill \\ {\frac{{{\text{d}}p_{2} \left( t \right)}}{{{\text{d}}t}} = \frac{{E_{ol} }}{{V_{2} - A_{2} x_{p} \left( t \right)}}\left( { - K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{2} \left( t \right) - p_{t} } \right|} + A_{2} v_{p} \left( t \right) + Q_{w1} \left( t \right) - Q_{w2} \left( t \right)} \right)} \hfill \\ \end{array} } \right.$$
    (4)

    for \(x_{s} \left( t \right) < 0\)

    $$\left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}p_{1} \left( t \right)}}{{{\text{d}}t}} = \frac{{E_{ol} }}{{V_{1} - A_{1} x_{p} \left( t \right)}}\left( {K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{1} (t) - p_{t} } \right|} - A_{1} v_{p} \left( t \right) + Q_{w1} \left( t \right)} \right)} \hfill \\ {\frac{{{\text{d}}p_{2} \left( t \right)}}{{{\text{d}}t}} = \frac{{E_{ol} }}{{V_{2} + A_{2} x_{p} \left( t \right)}}\left( { - K_{q} x_{s} \left( t \right)\sqrt {\left| {p_{s} - p_{2} (t)} \right|} + A_{2} v_{p} \left( t \right) - Q_{w1} \left( t \right) - Q_{w2} \left( t \right)} \right)} \hfill \\ \end{array} } \right.$$
    (5)

Flow coefficient \(K_{q}\) through the valve is determined as

$$K_{q} = C_{q} \pi d\sqrt {\frac{2}{\rho }}$$
(6)

where d is the diameter of the spool valve, C q is the coefficient of resistance flow through the valve, and \(\rho\) is the oil density.

Leakages appearing in the cylinder are proportions to the difference of pressures in the cylinder chambers:

for \(x_{s} \left( t \right) > 0\)

$$\left\{ {\begin{array}{*{20}l} {Q_{w1} \left( t \right)_{ + } = k_{w1} \left( {p_{1} \left( t \right) - p_{2} \left( t \right)} \right)} \hfill \\ {Q_{w2} \left( t \right)_{ + } = k_{w2} p_{2} \left( t \right)} \hfill \\ \end{array} } \right.$$
(7)

for \(x_{s} \left( t \right) < 0\)

$$\left\{ {\begin{array}{*{20}l} {Q_{w1} \left( t \right)_{ - } = k_{w1} \left( {p_{2} \left( t \right) - p_{1} \left( t \right)} \right)} \hfill \\ {Q_{w2} \left( t \right)_{ - } = k_{w2} p_{2} \left( t \right)} \hfill \\ \end{array} } \right.$$
(8)

where \(k_{w1} ,k_{w2}\) are the coefficients of leakages in the cylinder equation of the spool valve movement:

$$\left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}x_{s} \left( t \right)}}{{{\text{d}}t}} = v_{s} \left( t \right)} \hfill \\ {\frac{{{\text{d}}v_{s} \left( t \right)}}{{{\text{d}}t}} = \frac{1}{{m_{s} }}\left( { - F_{ts} \left( t \right) - c_{s} x_{s} \left( t \right) + F_{e} \left( t \right)} \right)} \hfill \\ \end{array} } \right.$$
(9)

where

force of the sticky friction is

$$F_{ts} \left( t \right) = f_{s} v_{s} \left( t \right)$$
(10)

and the force of the electromagnet spool valve is [5]

$$F_{e} \left( t \right) = K_{e} \left( {0.3u\left( t \right) + 6.5} \right)$$
(11)

In the simulated system a friction force appearing in the hydraulic cylinder was taken into account:

$$F_{tc} (t) = F_{tl} (t) + F_{tz} (t)$$
(12)

Friction in servo-hydraulic drives is a phenomenon causing standbys of the movement and reducing the efficiency of system as well as at the same time influences on attenuation mechanical oscillation [8]. The fundamental element of Eq. (12) is the sticky friction force, which has a primary importance in the modeling of dynamics drive system:

$$F_{tl} = f_{l} v_{p} \left( t \right)$$
(13)

The coefficient of sticky friction is determined by the relation \(f_{l} = \frac{\mu A}{h}\) where A is the total area of the joint, h is the layer of the oil film (value of float), and \(\mu\) is the dynamic coefficient of the oil viscosity.

Dry friction force \(F_{tz}\) and factor of the sticky friction \(f_{l}\) were appointed stimulating the system of constant tension \(u\left( t \right)\) and measuring the pressure \(p_{1} \left( t \right)\) as well as \(p_{2} \left( t \right)\) in the cylinder chambers of in the equilibrium of velocity v p what allows for calculating the total force of friction. During the movement of the piston rod with the total constant velocity, friction force \(F_{tc}\) is expressed as

$$F_{tc} = f_{l} \cdot v_{p} + F_{tz}$$
(14)

Value of the friction force \(F_{tz}\) appearing in the system is determined by Eq. (15) according to the Stribeck model [8]:

$$F_{tz} \left( t \right) = \left( {F_{tk} + F_{tp} } \right)\text{sgn} \left( {v_{p} \left( t \right)} \right) + F_{tu}$$
(15)

where \(F_{tk} = f_{tk} F_{N}\) is the kinetic friction force, \(f_{tk}\) is the coefficient of the kinetic friction, and \(F_{N}\) is the normal force.

Friction appearing in sealing is determined in following relation (16) [8]:

$$F_{tu} = \frac{{f_{u} A_{u} }}{2}\left( {p_{1} (t) + p_{2} (t)} \right)$$
(16)

where \(f_{u}\) is the coefficient of the friction, \(A_{u} = \pi dl\) is the surface of the sealing, and d and l are the diameter of the piston and length of the sealing.

3 Designing the Control System—Linearization with Feedback

In examining and designing hydraulic servo drive, computer simulation of the model played an important role. Natural way to adapt algorithms of control the nonlinear object is to use the classic method of linearization model process around the point of work and use the linearization of model algorithm. Applying the linearization toward the nonlinear system, the algebraic transformation is explored which eliminates nonlinearities of the object. Therefore, it is necessary to apply the so-called input–output feedback linearization [9]. Nonlinearities are being eliminated (entirely or partly) from the object so that after closing the system was linear.

In effect of the linearization carried out according to the scheme [7], obtained function of coupling the valve for displacement of the spool valve \(x_{s} \left( t \right)\) is as follows:

$$x_{s} \left( t \right) = f\left( {p_{1} ,p_{2} ,x_{p} ,v_{p} ,v} \right)$$
(17)

for \(x_{s} \left( t \right) > 0\)

$$\begin{aligned} x_{s} \left( t \right) = & \left[ {a_{0} x_{p} (t) + a_{1} v_{p} (t) + a_{2} \frac{{\left( {p_{1} (t)A_{1} - p_{2} (t)A_{2} } \right)}}{m}} \right. \\ & \quad \left. { + v(t)_{p} \left( {\frac{{A_{1} A_{2} E_{{ol}} \left( {V_{2} (t) + V_{1} (t)} \right)}}{{m_{p} \left( {V_{1} + x_{p} (t)A_{1} } \right) \cdot \left( { - V_{2} + x_{p} (t)A_{2} } \right)}}} \right) - \nu (t)} \right] \cdot \alpha _{ + } (t) \\ \alpha _{ + } \left( t \right) = & \frac{{m_{p} \left( {V_{1} (t) + x_{p} (t)A_{1} } \right) \cdot \left( { - V_{2} (t) + x_{p} (t)A_{2} } \right)}}{{E_{{ol}} K_{q} (A_{1} \sqrt {\left| {p_{s} - p_{1} (t)} \right|} \left( {V_{2} (t) - x_{p} (t)A_{2} } \right) + A_{2} \sqrt {\left| {p_{2} (t) - p_{t} } \right|} \left( {V(t)_{1} + x_{p} (t)A_{1} } \right))}} \\ \end{aligned}$$
(18)

for \(x_{s} \left( t \right) < 0\)

$$\begin{aligned} x_{s} \left( t \right) = & \left[ {a_{0} x_{p} (t) + a_{1} v_{p} (t) + a_{2} \frac{{\left( {p_{1} (t)A_{1} - p_{2} (t)A_{2} } \right)}}{m}} \right. \\ & \quad \left. { + v_{p} (t)\left( {\frac{{A_{1} A_{2} E_{{ol}} \left( {V_{2} (t) + V_{1} (t)} \right)}}{{m_{p} \left( {V_{1} (t) + x_{p} (t)A_{1} } \right) \cdot \left( { - V_{2} (t) + x{}_{p}(t)A_{2} } \right)}}} \right) - v(t)} \right] \cdot \alpha _{ - } (t) \\ \alpha _{ - } \left( t \right) = & \frac{{m_{p} \left( {V_{1} (t) + x_{p} (t)A_{1} } \right) \cdot \left( { - V_{2} (t) + x_{p} (t)A_{2} } \right)}}{{E_{{ol}} K_{q} (A_{1} \sqrt {\left| {p_{1} (t) - p_{t} } \right|} \left( {V_{2} (t) - x_{p} A_{2} } \right) + A_{2} \sqrt {\left| {p_{s} - p_{2} (t)} \right|} \left( {V_{1} (t) + x_{p} (t)A_{1} } \right)}} \\ \end{aligned}$$
(19)

where \(v = k_{s} \left( {x_{p}^{\text{ref}} - x_{p}^{{}} } \right)\), \(x_{p}^{\text{ref}}\) is the set signal, and \(a_{0} ,a_{1} ,a_{2}\) are the parameters of the model [8].

For the simulation purposes, constant values of parameters were implemented, see Table 2.

Table 2 Simulation parameters
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In Fig. 2, the chosen dynamic characteristics such as displacement \(x_{p} \left( t \right)\), velocity \(v_{p} \left( t \right)\), acceleration \(a_{p} \left( t \right)\), pressures in individual chambers \(p_{1} \left( t \right)\) and \(p_{2} \left( t \right)\) as well as displacement the spool valve \(x_{s} \left( t \right)\), and velocity \(v_{s} \left( t \right)\) at mass load 100 kg received as a result of the simulation model are presented.

Fig. 2
figure 2

Dynamic characteristics of the model

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4 Conclusions

The considered solution regards algorithms of the control hydraulic servo drive for which characteristics do not change in time. The accepted theoretical description does not change during the system work. In such assumption, we may accept the sufficient theoretical description only once and select parameters of the adjuster in the course of enforcing the movement drive. Unfortunately, the disadvantage of presented solution is the large sensitivity to mistakes appearing in the description of the controlled object. A lack of the system resistance to interferences appears in case of the parametric model uncertainty [2]. Additionally, all variables of the state must be available for analysis. Since the values of many parameters are not possible to be appointed by direct measurements, they must be appointed as a result of the parametric identification of created mathematical model of the examined object. It is a difficult and laborious process, often loaded by a large dose of the uncertainty. As a result, to assure the right regulation during the work, we must adopt parameters of control process.