Nonlinear Mathematical Modeling in Pneumatic Servo Position Applications

This paper addresses a new methodology for servo pneumatic actuators mathematical modeling and selection from the dynamic behavior study in engineering applications. The pneumatic actuator is very common in industrial application because it has the following advantages: its maintenance is easy and simple, with relatively low cost, self-cooling properties, good power density power/dimension rate , fast acting with high accelerations, and installation flexibility. The proposed fifth-order nonlinear mathematical model represents the main characteristics of this nonlinear dynamic system, as servo valve dead zone, air flow-pressure relationship through valve orifice, air compressibility, and friction effects between contact surfaces in actuator seals. Simulation results show the dynamic performance for different pneumatic cylinders in order to see which features contribute to a better behavior of the system. The knowledge of this behavior allows an appropriate choice of pneumatic actuator, mainly contributing to the success of their precise control in several applications.


INTRODUCTION
This work presents a new methodology to identify the main nonlinear characteristics in pneumatic actuators and its mathematical modeling in engineering applications.The pneumatic actuator is very common in industrial application [1] because it has the following advantages: it maintenance is easy and simple, relatively low cost, self cooling properties, good power density (power/dimension rate), fast acting with high accelerations [2] and installation flexibility.Also, compressed air is available in almost all industrial plants [3].
However, there are difficulties of control due to various nonlinear characteristics of the system [4,5].The nonlinearities present in pneumatic actuators are motivated by it's very low stiffness (caused by air compressibility), inherently nonlinear behavior, parameter variations and low damping of the actuator systems, which make it difficult to achieve precise motion control.The main nonlinearities in pneumatic servo systems are the servovalve dead zone [6], air flow-pressure relationship through valve orifice [1,7], the air compressibility and friction effects between contact surfaces in actuator seals [8,9].
Several recent authors present a study on the characteristics of nonlinear pneumatic actuators [1,5,7,9,10,11].Valdiero et al. [6] presents a mathematical model to dead zone in pneumatic servovalves, followed by the method used to compensation that is made with the addition of an inverse dead zone function in control system.Rao and Bone [1] presents a modeling approach where they use the bipolynomial functions to model the valve flow rates, but is used a poor classical friction model.Perondi [10] developed a nonlinear accurate model of a pneumatic servo drive with friction, where the nonlinear airflow relationship between the pneumatic valve's driving voltage and the upstream/downstream pressures.Endler [7] used the methodology of optimal feedback control for nonlinear systems proposed by Rafikov et al. [12] in servo pneumatic system and simulation results show that a full nonlinear mathematical model is important in pneumatic robot applications.
The main paper contribution is to systematize its nonlinear mathematical model with some innovations such as a new equation for valve flow rate and to show how it is important to the success in control applications.The paper is organized as follows.Section 2 brings a description of servo pneumatic positioning system with its main components, the used test rig and a schematic drawing with the nonlinearities presents in the actuator.In section 3 is shown the systematic methodology of the pneumatic actuator nonlinear mathematical modeling.Results are presented in section 4. Conclusions are outlined in section 5.

PNEUMATIC SERVO POSITION SYSTEM
The servo pneumatic positioning system considered in this paper is formed by a proportional servo valve (component 4 in Fig. 1) and a double action rodless cylinder (component 2 in Fig. 1).This actuator permits to position one load in desired position of the actuator curse or follow a desired trajectory.The Fig. 1 shows the schematic drawing of used experimental setup with main components for the purpose of investing the nonlinear mathematic model.The acquisition and control system used is a dSPACE DS 1102 board.It is composed by 4 analog inputs (ADCs) and 4 analog outputs (DACs) as shown in Dspace [13].Sensors permit measure air system inlet pressure (1), the actuator position (3) and actuator chamber pressures (p a and p b ), ( 5) and (6).
Figure 2 shows the schematic drawing of a servo pneumatic actuator for better understanding of system behavior.During the operation, the control signal u energizes valve's solenoid so that a resulting magnetic force is applied in the valve's spool, producing the spool displacement.The spool displacement opens control orifices so that one port is connected to the supply's pressure line and the other is connected to the atmosphere.Consequently, there is the pressure difference between cylinder chambers, resulting in a force that moves the mass M in a positive or negative displacement y, depending on the control signal input.
Figure 3 shows block diagram of the main dynamics in the nonlinear mathematical model of the pneumatic actuator.The main nonlinear characteristics of this dynamic system are servovalve dead zone, air flow-pressure relationship through valve orifice, air compressibility and friction effects between contact surfaces in actuator seals.
Dead zone is common in pneumatic valves because the spool blocks valve orifices with some overlap, so that for a range of spool positions there is no air flow [6].It is located at the dynamic system as a block diagram shown in Fig. 2, and is characterized in section 3.1.The air flow-pressure relationship through valve orifice is a nonlinear function that depends on pressure difference across the valve orifice and valve opening [7].In this paper, we present a new mass flow rate equation in section 3.2.
The pressures dynamic model is obtained from continuity equation and results in nonlinear first order differential equation.This dynamic behavior depends on pneumatic cylinder size.Small cylinder bore size produces significant effects such as it results in a faster pressure response [1], the bore size is reduced the ratio of friction force to maximum pneumatic force increases, and the chamber pressures are more sensitive to small variations in the mass flow rate.Therefore the precise tracking control is more difficult with smaller bore sizes.This detail nonlinear dynamics is presented in section 3.3.
The nonlinear friction is the more important factor that affects the motion equation.Friction is a nonlinear phenomenon difficult to describe analytically [8].The friction often changes with time and may depend in an unknown way on environmental factors, such as temperature and lubricant condition.Even so is important the modeling of their main characteristics.In this paper, we consider the actuator friction dynamics described by the LuGre model, proposed in Canudas et al. [14] and improved by Dupont et al. [15] in order to include stiction effects.This model is presented in section 3.4.

NONLINEAR MATHEMATICAL MODELING
The systematic methodology of the pneumatic actuator nonlinear mathematical modeling is presented from experimental data and recent literature information.The full system constitutes a fifth order nonlinear dynamic model of the pneumatic positioning system and considers the nonlinearity of the dead zone, the mass flow rate, the pressure dynamic and the motion equation, that includes the friction dynamics.

Dead Zone NonLinearity
This section presents the mathematical model for dead zone nonlinearity and its graphical representation.Dead zone is a static input-output relationship which for a range of input values gives no output.Figure 4 shows a sectional view sketch of typical spool valve with main mechanical elements.
The mathematical model for dead zone in pneumatic servovalves presents in this section was obtained from Tao and Kokotovic [16].The dead zone analytical expression is given by Eq. (1).
Figure 5 shows a graphical representation of dead zone.In general, neither the break-points zmd and zme ) nor the slopes ( md and me ) are equal.
In current fluid power literature, dead zone in valves is expressed as a percentual of spool displacement.Valdiero et al. [6] presents a new methodology for dead zone nonlinearity identification in proportional directional pneumatic valves.It is based on observing the dynamic behavior of the pressure in the valve gaps.
where u is the input value, zm u is the output value, zmd is the right limit of dead zone, zme is the left limit of dead zone, md is the right slope of output and me is the left slope of output.The dead zone nonlinearity is a key factor that limits both static and dynamic performance in feedback control of fluid power systems.The usual method to cancel the harmful effects of dead zone is to add its fixed inverse function into the controller.This inverse is modeled by a set  of parameters that need to be identified.The classic dead zone parameter identification uses expensive flow transducers and special test rig, while our proposed methodology needs only pressure transducers shown in Fig. 1.Experimental results are presented in Valdiero et al. [6] and illustrate the efficacy of this methodology that is cheaper and faster.

Mass Flow Rate
According to Rao and Bone [1], the mass flow rate model of the proportional valve is a key part of the system model.In this paper, we use an innovator model to mass flow rate equation ma q and mb q developments by Endler [7], given by equations: where 1 g and 2 g are signal functions given by equations ( 4) e (5).Equations ( 2) and ( 3) are a fitting of a surface obtained experimentally [5,7] in test rig of Fig. 1, considering that the piston is stopped, in that way the volume is constant and the speed of the piston is null.The mass flow rates at different pressures and valve input voltages were first estimated from the pressure versus time responses obtained for step inputs in valve voltage and a fixed piston position.
The fitted mass flow rate in valve orifice, ma q , is plotted versus input voltage and pressure difference in Fig. 6.Rao and Bone [1] used a 2nd order bipolynomial equation to fit this function.In a similar way, Perondi [10] used a third order polynomial one.Bobrow and McDonell [17] use a curve fit for the change in internal energy as a function of cylinder pressure which is quadratic in u.One of the greatest problems in these equations found in the literature is the difficulty in isolate the signal u, necessary when is used a control methodology that considers the nonlinear characteristics of the system.Equations to mass flow rate proposed by Ritter et al. [5] are innovations that possess advantages as easiness of computational implementation and differentiation.

Pressure Dynamics
The cylinder used in this modeling is symmetric and without spindle.In mathematical modeling the pressure changes in the chambers are obtained using energy conservation laws.Figure 7 shows a schematic drawing of cylinder used.The relationship between the air mass flow rate and the pressure changes in the chambers is obtained using energy conservation laws.According to Perondi [10], the energy balance yields Eq. ( 6). ) ( 1 where T is the air supply temperature, ma q is the air mass flow rate into chamber A, is the volumetric flow rate.Assuming that the mass flow rates are nonlinear functions of the servovalve control voltage (u) and of the cylinder pressures, that is, ) , ( u p q q a ma ma = and ) , ( u p q q b mb mb = .The total volume of chamber A is given by where A is the cylinder cross-sectional area, y is the piston position and 0 a V is the initial volume of air in the line and at the chamber A extremity, include the pipeline.The change rate for this volume is , where y  is the piston velocity. In this manner, calculating the derivative term in the right hand side of Eq. ( 6), and using

Friction Dynamics in motion equation
Applying Newton's second law to the piston-load assembly results in where M is the mass of the piston-load assembly, y  is the cylinder acceleration, atr F is the friction force, p F is the pneumatic force related to the pressure difference between the two sides of the piston, that is given by ) ( b a p p A − .In this section the dynamic model to friction is based in the microscopic deformation of asperities in surface contact.It is possible to perceive an evolution in friction models that are based in the asperity microscopic deformations and depicted in recent papers.
The Dahl model describes friction in the presliding movement phase, in similar way with the rigid spring with damping behavior, but has not included the Stribeck friction effect.The LuGre model, proposed by Canudas-De-Wit et al. [14], is an improved model that includes the Stribeck Friction and describes many complex friction behaviors, but is limited in the presliding movement phase, according to simulations results presented by Dupont et al. [15] and experimental tests carried out by Swevers et al. [18].These authors propose also improvements in LuGre model through the inclusion of a model to hysteresis with non local memory and sliding-force transition curves in presliding movement phase.This improved model is named Leuven model and used in friction modeling to a pneumatic servo positioning system by Nouri et al. [19].Dupont et al. [15] also propose improvements in LuGre model through its interpretation as an elasto-plastic friction model that are used in this paper.
Figure 8 represents the contact between surfaces through a lumped elastic asperity, considering a rigid body where the displacement y is decomposed into its elastic and plastic (inelastic) components z and w.The friction force is described according to the LuGre friction model proposed by Canudas-de-Wit et al. [14].In this model the friction force is given by where z is a friction internal state that describes the average elastic deflection of the contact surfaces during the stiction phases, 0 σ is the stiffness coefficient of the microscopic deformations z during the presliding displacement, 1 σ is a damping coefficient, 2 σ represents the viscous friction, y  is the velocity The dynamics z  of the internal state z is modeled by the equation where ) ( y g ss  is a positive function that describes the steady-state characteristics of the model for constant velocity motions and is given by  al. [15] and is used to represent the stiction.This function is defined by where ba z is a breakaway displacement, such that to ba z z ≤ , all movements in friction interface consists in elastic displacements only and max z is the maximum value of microscopic deformations and is velocity dependent. Is possible to note that, with z represented by Eq. ( 14), when sliding movement is in steady state, y  is constant, Substituting the equation ( 16) into equation ( 14) is obtained the friction force at steady state: This dynamic properties of friction model presented are shown by Dupont et al. [15] and follow similar analysis carried out by Lyapunov method, as presented by Canudas-De-Wit et al. [14] and Canudas-De-Wit [20].Among model main properties, is cited that z state variable is limited, the model is dissipative, satisfies the stick and slip conditions and represents adequately the pre-sliding movement phase.
The applied force of Fig. 10(a) was chosen to challenge the stiction capability of the model, the force ramps up to cause break-away, and then returns to a level below that of Coulomb friction.Additionally, an oscillation is present such as could be introduced by sensor noise or vibration.The response of friction model is seen in Fig. 10(b).The friction dynamic model renders both presliding displacement and stiction.

RESULTS
The most common and simple industrial application is a positioning task.By a positioning task is meant the objective of bringing the load position to a specified target in the actuator´s curse.The proposed nonlinear mathematical model of a pneumatic servo position system was used in computer simulations of three cases of different cylinder size where desired target position is 0.045 m.
The pneumatic servo position system model dynamics is given by equations ( 1), ( 2), ( 3), ( 8), ( 9), ( 10), (11) and (12).This model was implemented on the MatLab/Simulink software of which block diagram is shown in Fig. 11 and using parameters presented in Tab. 1 and Tab. 2. The classic Proportional controller (P) was chosen because it is easy to implement and has only one parameter to adjust.Also, the results are easier to see with P controller.The choice curse length for case (c) in Tab. 2 is determined such that it results in same chamber volume of the case (a).It is good idea because chamber volume has a great influence in pressure dynamics given by equations ( 8) and ( 9).The results presented in case (a) outline the faster response with oscillating and overshoot in actuator position.Also there are hunting problems that are oscillations caused by limit cycles around desired position.In many applications as robotics and aerospace engineering, the faster response is one of the requirements for the positioning task and we can design pneumatic positioning systems with smaller cylinder diameter and increase the supply pressure obtaining necessary actuator force.To solve this overshoot problem we can used an optimal control design for nonlinear systems as Rafikov et al. [12].Also, the friction compensation is especially important so that there are no hunting problems [4] and the actuator has an accurate response.
Despite being slow, the results in case (b) are very good for some engineering applications as automatic welding, machining processes, surface finishing and agricultural machinery, where application requirements don't permit overshoot and task velocity is smaller.In this case, we can design pneumatic positioning systems with larger cylinder diameter that it results in the damping increase turns the system more slowly.Besides, we can design a classical feedback control system depending on necessary accuracy in application.The case (c) presented dynamic behavior similar to case (b) and it shows that the chamber volume doesn't have significant influence in this positioning task.

CONCLUSION
In this paper was presented full nonlinear mathematical model for pneumatic servo position system that can be used in numerical simulations to mechanical design and control system design of industrial applications.There was bibliographical revision in recent international literature.However, these works don't address completely all important nonlinearities in mathematical model.So, the main paper contribution was present its nonlinearities and their completed mathematical modeling with some innovation and application results.The proposed systematic methodology is important to help researches in the nonlinear modeling and precision control success.Future research will include an optimal nonlinear control strategy to overcome problems of the servo pneumatic system in agricultural machinery applications with high performance.

Figure 1 .
Figure 1.Experimental setup with main components

Figure 2 . 1 Figure 3 .
Figure 2. Schematic drawing of a pneumatic servo system

Figure 4 .
Figure 4. Sectional view sketch of typical spool valve whit main mechanical elements of the proportional valve with input dead zone

Figure 5 .
Figure 5. Graphical representation of the dead zone.

4
pressure, atm p is the atmospheric pressure and ench β and esv β are the constant coefficients.

Figure 6 .
Figure 6.Fitted model of mass flow rate.
a p is the absolute pressure in Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP -ISSN 2178-3667 202 5 chamber A, p C is the specific heat of the air at constant pressure, v C is the specific heat of the air at constant volume v p C C = γ is the ratio between the specific heat values of the air, R is the universal gas constant,

Figure 8 .
Figure 8. Model of body subject to friction force showing elastic (z) and inelastic (w) displacement components.
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP -ISSN 2178-3667 203 NonLinear Mathematical Modeling in Pneumatic Servo Position Applications Antonio Carlos Valdiero, Carla Silvane Ritter, Cláudio Fernando Rios, Marat Rafikov 6 where c F is the Coulomb friction force, s F is the static friction force and s y  is the Stribeck velocity.Figure 7 illustrate the behavior of the friction force as a function of velocity in steady-state [8].

Figure 9 .
Figure 9. Friction force characteristics combined in steady-state.

Figure 10 .
Figure 10.Applied force in pneumatic actuator and position response in both preslinding displacement and stiction

Figure 11 .Table 1 .
Figure 11.Proportional feedback control structure block diagram with nonlinear mathematical model of pneumatic servo positioning system.

Table 2 .Figure 12 .
Figure 12.Simulation results to positioning task: target position yd=0.045m, case (a) cylinder with small diameter (pink color), case (b) cylinder with large diameter (blue color), and case (c) cylinder with large diameter and small curse length (red color).
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP -ISSN 2178-3667 201 NonLinear Mathematical Modeling in Pneumatic Servo Position Applications Antonio Carlos Valdiero, Carla Silvane Ritter, Cláudio Fernando Rios, Marat Rafikov (