Original Article Proc IMechE Part I: J Systems and Control Engineering 227(2) 230–242 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0959651812459303 pii.sagepub.com Accurate pressure control of a pneumatic actuator with a novel pulse width modulation–sliding mode controller using a fast switching On/Off valve Babak Hejrati 1 and Farid Najafi 2 Abstract Pressure control of a pneumatic actuator using fast switching On/Off valves continues to remain a major challenge for researchers. This article proposes a novel pulse width modulation–sliding mode controller that improves pressure track- ing of pneumatic actuators. First, a comprehensive mathematical model of the pneumatic system was developed that consists of several submodels. The model comprises pressure and temperature equations describing the thermodynamic process inside the pneumatic chamber, an orifice flow model for the On/Off valve, and a model for the dynamic response of the On/Off valve to the control signal. Second, computer simulations were carried out using the model, and then, experimental tests were performed to verify the simulation results. The comparison between simulation and experimental results demonstrates good accuracy of the presented model. Since the governing equations of pneumatic systems are highly nonlinear in terms of parametric (including discharge coefficients of the valve) and structural uncer- tainties (the lack of knowledge about the exact type of the thermodynamic process), a robust controller was designed for such a system. In this study, a novel pulse width modulation–sliding mode controller is proposed that demonstrates a significant improvement in pressure control of pneumatic actuators compared to other proposed controllers from the literature. Keywords Pressure control, sliding mode control, pulse width modulation algorithm, mathematical modeling, On/Off valve Date received: 19 February 2012; accepted: 2 August 2012 Introduction Recently, growing demand for micromachining with minimal capital investment has inspired further studies for investigating the capabilities of different methods and actuators. 1,2 Since pneumatic systems have many distinguishable features such as energy saving, cleanli- ness, ease of maintenance, accessibility of air, simple structure and operation, and low cost and high power- to-weight ratio, they have been extensively used for many years in industrial automation (Moor 3 ). Since pneumatic actuators are soft and compliant, their appli- cation in rehabilitative robots has been studied by a growing number of researchers. Due to air compressibil- ity, delay of the valves, and friction, pneumatic systems are known as highly nonlinear systems which, in turn, makes servo-control of these systems very difficult and challenging. In the past, modeling of pneumatically driven systems has been studied by many researchers including Shearer 4,5 With the advent of powerful com- puters, nonlinear models, which are more suitable for pneumatic systems, were proposed by Wang and Kim 6 and in the recent decade by Richer et al. 7 and Najafi et al. 8 Moreover, the type of valves that are used for imple- menting the control strategy is of high importance both in adaptation of control strategy and in the 1 Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA 2 Mechanical Engineering Faculty, K. N. Toosi University of Technology, Tehran, Iran Corresponding author: Babak Hejrati, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA. Email: [email protected]
Transcript
Original Article
Proc IMechE Part I:J Systems and Control Engineering227(2) 230–242� IMechE 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0959651812459303pii.sagepub.com
Accurate pressure control of apneumatic actuator with a novelpulse width modulation–slidingmode controller using a fast switchingOn/Off valve
Babak Hejrati1 and Farid Najafi2
AbstractPressure control of a pneumatic actuator using fast switching On/Off valves continues to remain a major challenge forresearchers. This article proposes a novel pulse width modulation–sliding mode controller that improves pressure track-ing of pneumatic actuators. First, a comprehensive mathematical model of the pneumatic system was developed thatconsists of several submodels. The model comprises pressure and temperature equations describing the thermodynamicprocess inside the pneumatic chamber, an orifice flow model for the On/Off valve, and a model for the dynamicresponse of the On/Off valve to the control signal. Second, computer simulations were carried out using the model, andthen, experimental tests were performed to verify the simulation results. The comparison between simulation andexperimental results demonstrates good accuracy of the presented model. Since the governing equations of pneumaticsystems are highly nonlinear in terms of parametric (including discharge coefficients of the valve) and structural uncer-tainties (the lack of knowledge about the exact type of the thermodynamic process), a robust controller was designedfor such a system. In this study, a novel pulse width modulation–sliding mode controller is proposed that demonstrates asignificant improvement in pressure control of pneumatic actuators compared to other proposed controllers from theliterature.
Date received: 19 February 2012; accepted: 2 August 2012
Introduction
Recently, growing demand for micromachining withminimal capital investment has inspired further studiesfor investigating the capabilities of different methodsand actuators.1,2 Since pneumatic systems have manydistinguishable features such as energy saving, cleanli-ness, ease of maintenance, accessibility of air, simplestructure and operation, and low cost and high power-to-weight ratio, they have been extensively used formany years in industrial automation (Moor3). Sincepneumatic actuators are soft and compliant, their appli-cation in rehabilitative robots has been studied by agrowing number of researchers. Due to air compressibil-ity, delay of the valves, and friction, pneumatic systemsare known as highly nonlinear systems which, in turn,makes servo-control of these systems very difficult andchallenging. In the past, modeling of pneumatically
driven systems has been studied by many researchersincluding Shearer4,5 With the advent of powerful com-puters, nonlinear models, which are more suitablefor pneumatic systems, were proposed by Wang andKim6 and in the recent decade by Richer et al.7 andNajafi et al.8
Moreover, the type of valves that are used for imple-menting the control strategy is of high importance bothin adaptation of control strategy and in the
1Department of Mechanical Engineering, University of Utah, Salt Lake
City, UT, USA2Mechanical Engineering Faculty, K. N. Toosi University of Technology,
Tehran, Iran
Corresponding author:
Babak Hejrati, Department of Mechanical Engineering, University of Utah,
performance of the pneumatic systems. Most of theresearchers have used proportional valves because oftheir good accuracy and continuous behavior.However, bulkiness and high cost of these valves haveencouraged many researchers to study On/Off valves asgood alternatives for proportional valves. AlthoughOn/Off valves are inexpensive and less bulky, becauseof their discrete nature, a suitable method should beused to convert their discrete behavior to a quasi-continuous behavior. Pulse width modulation (PWM),which has been applied to direct current (DC) electricalmotors, was first used by Noritsugu9 for position con-trol of an electropneumatic servomechanism. VanVarseveld and Bone10 proposed a new scheme forPWM algorithm to eliminate dead band, delay and alsoto achieve a linear correlation between control inputand velocity output. Furthermore, Najafi et al.11 usedPWM along a sliding mode controller (SMC) for accu-rate positioning of a servo-pneumatic system.
In the literature, pressure control has been per-formed for two purposes: to improve the performanceof pneumatic position servosystems and to carry outforce control of pneumatic actuators. Lai and Meng12
used an inner loop pressure controller together with anouter position feedback to improve the positioning of apneumatic actuator controlled by PWM method.Belforte et al.13 used four 2/2 On/Off valves andpresented a modified PWM method to control thepneumatic chamber’s pressure. In that study, aproportional–derivative (PD) controller was designedto track sinusoidal pressure inputs, in which by increas-ing the frequency, the tracking accuracy was decreased.A linear quadratic Gaussian (LQG) self-tuning pres-sure regulator was proposed by Wang et al.14 Liuet al.15 utilized a hybrid fuzzy controller for high preci-sion pressure control of a pneumatic chamber. Theircontroller comprised a proportional–integral–derivative(PID) controller, for accurately controlling the steady-state pressure, and a fuzzy controller, for providing afast rise time as well as low overshoot of the pressure intheir system. In the study by Liu et al.,15 the controllerwas tested for multistep inputs, and similar to Wang etal.’s study,14 a servo valve was used for the control pur-pose. Ying et al.16 developed a hybrid fuzzy controlmethod using a set of On/Off valves for the preciseforce-feedback control of the pneumatic actuators ofan arm exoskeleton. More intelligent controllers basedon adaptive, fuzzy, and neural network methods havebeen applied to other nonlinear systems with uncertain-ties that are essentially similar to pneumatic systemsin terms of nonlinearity and uncertainty. Chen CWet al.17–19 have proposed genetic based adaptive neuralnetwork controller, and genetic-based fuzzy controllerfor nonlinear systems with uncertainties, while ElOuafiet al.20 have enhanced the accuracy of multiaxis com-puter numerical control (CNC) machines throughonline neurocompensation. In addition, the use of anadaptive fuzzy SMC for nonlinear systems has beeninvestigated by Chen PC et al.21,22 who have observed
acceptable performance of such a SMC in the numeri-cal simulations. The stability analysis of such proposedcontrollers for nonlinear systems with uncertainties wasthe objective of some studies.18,21,23
Despite the variety of controllers used for accuratepressure control, there is still a need for more improve-ment. One of the main issues with other controllers isthat due to high nonlinearity and uncertainties of pneu-matic systems, high bandwidth control is a big chal-lenge such that all the mentioned control methods aresuitable for step inputs or sinusoidal inputs with lowfrequencies (the highest reported frequency is 0.6Hz inBelforte et al.13). The novelty of the proposed controlleris that it comprises a SMC and PWM method for pres-sure control of the pneumatic system using an On/Offvalve. As presented in this article, accurate modeling ofpneumatic system is challenging, therefore, model-based controllers (such as state space, linear, LQG, andadaptive fuzzy controllers) are not suitable. However,the SMC is capable of controlling nonlinear systemswhere an accurate model of the system is not required.In addition, sliding mode control approach is an effi-cient method for designing robust controllers for com-plex nonlinear systems in the presence of parametricand structural uncertainties. It is worth mentioning thata SMC was utilized by Richard and Hirmuzlu24 in aproportional valve to carry out force control of a pneu-matic actuator. One of the main objectives of this studyis to reduce the cost of the control system by replacingexpensive proportional valves by inexpensive On/Offones. Therefore, PWM method is used to facilitate theimplementation of sliding mode control by means of anOn/Off valve. As a result, the proposed PWM-SMC iscapable of more accurate pressure control at higher fre-quencies (up to 2Hz) when an inexpensive valve is used.The advantages of the proposed controller are itsrobustness to the uncertainties of the pneumatic system,which previous control methods including PID andfuzzy controllers had some difficulties in dealing with.
The main goal of this research is to enhancepressure-tracking performance in comparison with pre-viously proposed methods. First, a comprehensive non-linear model of the pneumatic air cylinder, containing athermodynamic process and the On/Off valve’s flowand delay models, was presented and validated by per-forming some experiments. The purpose of the com-puter model was that the proposed controller could befirst tested in simulations to provide a better insightinto the controller design. Simulations were performedto evaluate the performance and capabilities of thedesigned controller before its physical realization.Finally, the performance of the proposed PWM-SMCin tracking step, multistep, and sinusoidal pressureinputs was illustrated by the experimental tests. Theexperimental results demonstrated the benefits andpotentials of the proposed controller in improving thetracking accuracy.
This article is organized as follows: in ‘‘System math-ematical model’’, the mathematical model of the system
Hejrati and Najafi 231
is derived while ‘‘Open-loop system’’ will present thePWM-SMC. Simulation and experimental results ofthe open-loop system are presented and compared in‘‘Sliding mode controller’’. Moreover, simulation andexperimental results of the proposed controller arecompared in section ‘‘Simulation and experimentalresults’’ to show the high performance of the PWM-SMC in tracking of various desired inputs. Finally,conclusions are drawn.
System mathematical model
The test rig used in this study includes a pneumaticcylinder, a pressure transducer, a 3=2 On/Off valve, anelectronic board (an interface), and a computer and airsupply unit. However, the complete test rig includesmore components, as shown in Figure 1. In thisresearch, all the components of the test rig were notused, therefore, just the components used in thisresearch are described.
A sealed cylinder (type: FESTO DSNU-25-140-P-A)was used as the pneumatic chamber. A high precisionpressure transducer (type: BD 26.600-1002-1-100-300-500) was employed to detect the pressure in the cham-ber. The sensor output is 4–20 mA; therefore, anelectronic board was used to convert this output to a0–5 V signal readable by a data acquisition card (type:ADVANTECH multifunction card PCI 1710HG).Consequently, the pressure signal was transferred tothe computer for comparing with the desired pressure.The control input generated in the computer was sentto the 3=2 On/Off valve (FESTO MHE2-MS1H-3/2G-M7) via the electronic board. During the experimentaltests, the pressure of air supply is 580kPa and its tem-perature is 295K. The comprehensive model of the
pneumatic system that consists of several submodels isderived in the following subsections.
Air cylinder chamber model
A general model of a gas control volume comprisesthree equations: the equation of state (ideal gas law),the mass conservation law (continuity equation), andthe energy conservation law. The following assump-tions have been made while deriving the mathematicalmodel of the air cylinder: (1) the air is an ideal gas,(2) the pressure and temperature distribution within thecylinder chambers are homogeneous, (3) kinetic andpotential energies are negligible, (4) due to short lengthof air tubes, the delay caused by tubes is negligible, and(5) there is no leakage neither in the tubes nor in thechamber. The energy equation can be written in theform of equation (1) based on the book by Van Wylenet al.25
where _qin and _qout are in and out heat transfer rates; kis the specific heat ratio; Cv is the specific heat at con-stant volume; TS is the temperature of the incominggas flow that is equal to the ambient temperature; _min
and _mout are the mass flows entering and leaving the aircylinder chamber, respectively; Tout is the temperatureof the outlet gas; _W is the rate of change of control vol-ume work ( _WCV =P _V); and _U is the change of internalenergy of the system. The total change in internalenergy is in equation (2) based on the book by VanWylen et al.25
_U=1
k� 1_PV� P _V� �
ð2Þ
2
1 3
PressureTransducer
ElectronicBoard
Computer I/O Card
3/2 On/Off Solenoid
Valve
P, V, APneumaticCylinder
On/Off Valve
Orifice Load Cell
Pneumatic Actuator
Potentiometer
Pressure Transducer
(a) (b)
Figure 1. (a) Schematic diagram of the test rig used in the present research and (b) complete test rig.I/O: input/output.
232 Proc IMechE Part I: J Systems and Control Engineering 227(2)
Using equations (1) and (2), we have
_qin � _qout �k
k� 1P _V� �
=k
k� 1
P
rT_moutTout � _minTSð Þ
+1
k� 1V _P� �
ð3Þ
The thermodynamic process within the chamber wassupposed to be adiabatic ( _qin � _qout=0); therefore,equation (3) can be written as follows
_P= kR
V_minTS � _moutToutð Þ � k
P _V
Vð4Þ
Therefore, the rate of change of temperature can beexpressed by
_T= kTS � Toutð ÞR _min
PVTout � k� 1ð Þ
R _mout
PVT2out � k� 1ð Þ
_V
VTout ð5Þ
Equations (4) and (5) express the thermodynamic pro-cess of the system assuming an adiabatic thermody-namic process. On the other hand, the chamber volumeV was constant; therefore, the time derivative of thechamber volume was zero ( _V=0); therefore, equations(4) and (5) could be written as equation (6)
_P= kR
V_minTS � _moutToutð Þ
_T= kTS � Toutð ÞR _min
PVTout � k� 1ð ÞR _mout
PVT2out
8>><>>:
ð6Þ
where _P is pressure rate; k is the specific heat ratio; R isideal gas constant; V is the chamber’s volume; TS is thetemperature of the incoming gas flow that is equal tothe ambient temperature; _min and _mout are the massflows entering and leaving the air cylinder chamber,respectively; Tout is the temperature of the outlet gas;and P is the chamber’s pressure. Equation (6) has beenused in simulations and experimental implementation.
Pneumatic valve model
For controlling the air flow through the system, pneu-matic On/Off valves are used in the pneumatic system.On/Off valves can be considered as a standard orifice;therefore, the mass flow rate through an orifice of areaAv is given by equation (7)25
where _m is the mass flow through the valve orifice, Cd isa nondimensional discharge coefficient, Pu is theupstream pressure, Pd is the downstream pressure, Tu is
the upstream temperature, and Pcr is a critical value forthe pressure ratio of Pd=Pu, and its value is accompaniedby C1 and C2, which are expressed as in equation (8)25
For a given flow, C1, C2, and Pcr in equation (8) wereconstant values, and for air, values of k=1:4 andPcr =0:528 were used in equation (8). The upstreamand downstream pressures in equation (7) were differ-ent for charging and discharging processes within thechamber. For charging, the pressure in the supply tankshould be considered as the upstream pressure, whereasthe pressure in the chamber is considered as the down-stream pressure, and for the charging process, thereverse is true.
Av is the cross-sectional area of the valve, which foran On/Off valve could be either completely open orcompletely closed, depending on the control signal.Therefore, Av varied from zero to its maximum value;however, the fall and rise times and delays do not letthe valve to be open or closed immediately after receiv-ing the input signal. In equation (9), trise and tfall arethe rise and fall times, respectively, which were given inthe catalog of the valve, and t was the time measuredfrom the moment that the valve starts responding tothe signal until it is either completely open or closed.Equation (9) represents the valve’s area during openingand closing period after the valve started responding tothe signal
Av =Avmax
triset
Av =Avmax
tfallt
8>>><>>>:
ð9Þ
0 0.005 0.01 0.015 0.02 0.025 0.03
0.5
1
1.5
2
2.5
3 x 10-6
Time(s)
Val
ve's
Are
a (m
2 )
Valve's Actual AreaValve's Desired Area
SonT riset SoffT fallt Time(s)0
Figure 2. Implementation of PWM method for using the On/Off valve.
Hejrati and Najafi 233
Figure 2 depicts the desired and actual valve’s areasregarding the response times and the delays where TSoff
is the switching ‘‘Off’’ delay of the valve and TSon is theswitching ‘‘On’’ delay of the valve. The applied modelwas proposed by Najafi et al.8 to account for changesin cross-sectional area Av of the valve. The model hasbeen validated in the mentioned study and seems to besufficient for modeling of the valve.
PWM method
PWMmethod offers the ability to provide servo-controlof pneumatic systems at a significantly lower cost by uti-lizing On/Off valves. In a PWM-controlled system, theair delivered to the chamber is metered discretely by deli-vering packets or quanta of fluid mass via an On/Offvalve that is either completely On (Av =13Av) or com-pletely Off (Av =0). If the delivery of air packets occurson a time scale that is significantly faster than the systemdynamics (thermodynamic process), then the systemresponds to the average mass flow rate into and out ofthe cylinder in a continuous manner.12 Figure 3 schemati-cally demonstrates how a control signal turns into pulsewaves appropriate for an On/Off valve.
In the context of PWM method, the ratio of timeduring which the valve is On to the PWM time periodis called duty cycle and expressed by equation (10)
d=TOn
TPWMð10Þ
where TOn is ‘‘On’’ time of the valve, TPWM is the PWMperiod, and d is duty cycle of the valve. TPWM shouldbe chosen small enough, so that continuous behavior isachieved. On the other hand, the value of TPWM waslimited by the delay of the valve. Therefore, if TPWM
was chosen to be smaller than the valve’s delays, thenthe valve might not have enough time to open or close,and the desired duty cycle would not be achieved. Inthis research, TPWM was chosen as 20ms, which wasobtained by performing experiments on the existingvalve. Equation (11) presents minimum and maximumduty cycles of the valve based on the switching On/Offdelays of the valve
TPWM=TSon+TSoff + trise + tfall
dmax=TPWM � TSoff
TPWM
dmin =TSon
TPWM
8>>>>>><>>>>>>:
ð11Þ
Since most control systems are generally symmetric inthe control effort,10 using a linear transformation, thecontroller output u was mapped to the duty cycle of thevalve (d 2 ½dmin, dmax�). Equation (12) shows the correla-tion between the duty cycle d of the valve and the con-trol effort u
d= dmaxu5 umax
d=0:5u+0:5umin \ u\ umax
d= dminu4umin
8>><>>: ð12Þ
Table 1 shows the values for PWM method where umax
is the control input by which dmax was obtained(dmax=0:5umax+0:5), and similarly, umin is defined asthe control input by which dmin was obtained(dmin=0:5umin+0:5).
Open-loop system
So far, a comprehensive model for all the componentsof the system has been introduced based on thermody-namic relations and dynamics of opening and closingof the valve. In order to evaluate the used model andthe numerical simulations and see how accurately theydescribed the real system’s behavior, open-loop testswere performed. It was expected that if the proposedmodel presented the real system accurately, the open-loop results of tests verified the open-loop results fromsimulations. In both numerical simulations and experi-ments, the system was subjected to three input pulseswith different duty cycles because the authors couldinvestigate the model for different levels of the cham-ber’s pressure. First, a pulse with a duty cycle of 25%was applied to the system that is shown in Figure 4.Figure 4 also illustrates the comparison between simu-lation and experimental results for the mentioned inputsignal and the error between simulations and experi-ments. Two other signals with duty cycles of 50% and75% were applied to the system for charging processes.
The comparison between simulations and experi-ments demonstrates high accuracy of the proposedmodel. The discharge process was studied as well tocomplete the investigation of the accuracy of the model.In this case, the chamber was first connected to the airsupply and then it was disconnected from the air sup-ply, and after a while, the pressure inside the chamber
Table 1. Parameters and their values usedin PWM method.
Figure 3. Implementation of PWM method for using the On/Off valve.PWM: pulse width modulation.
234 Proc IMechE Part I: J Systems and Control Engineering 227(2)
dropped to the atmospheric pressure. Figure 5 demon-strates the discharge behavior of the chamber.
In order to have a quantitative measure for evalua-tion of the accuracy of the numerical simulations, rootmean square (RMS) of error between simulations andexperiments is used. In Table 2, RMS of errors andfinal values of the chamber’s pressure for all chargingand discharging processes are presented.
As illustrated in Table 2, by increasing the duty cycle,the final pressure and RMS of error increased. This phe-nomenon can be explained by the fact that increasingduty cycle caused a larger volume of air to enter thechamber, so that the final pressure increased. In addi-tion, the values of RMS of errors indicate that the model
accurately describes the real system’s behavior in bothcharging and discharging processes.
0 1 2 3 4 596
150
200
250
Time (s)
Pre
ssur
e (K
Pa)
ExperimentalSimulation
0 1 2 3 4 5−0.1
0
0.1
Time (s)
Err
or (
Bar
)0
20 40 60 80 100 120 140 160 180 200 220 240
0.5
1
1.5
Time(ms)
Sig
nal V
alue
(a) (b)
(c)
Figure 4. Simulation and experimental results for the duty cycle of 25%: (a) input pulse, (b) experimental and simulation responses,and (c) modeling error.
0 1 2 3 4 50.961.5
22.5
33.5
44.5
55.5
6
Time (s)
Pre
ssur
e (K
Pa)
ExperimentalSimulation
0 1 2 3 4 5−1
0
Time (s)
Err
or (
Bar
)
(a) (b)
Figure 5. Simulation and experimental results for discharge process: (a) experimental and simulation responses and (b) modelingerror.
Table 2. RMS error between simulation and experimentalresults.
Pneumatic actuation systems are highly nonlinear, andcontrol in such systems is more complicated thanhydraulic or electrical systems. Complex flow propertiesof the valve including discharge coefficient Cd, lack ofthe exact knowledge about the thermodynamic process(adiabatic or isothermal), and parametric and structuraluncertainties within the system make control in suchsystems difficult. In previous studies, different types ofcontrollers were utilized that included a PD controlleras in the study by Belforte et al.,13 a combined PID andfuzzy controller as in the research of Liu et al.,15 LQGself-tuning regulator as in the study by Wang et al.,14
and a hybrid fuzzy controller for pressure and forcecontrol of a pneumatic arm exoskeleton.16
In the aforementioned researches, different controllerswere applied for the purpose of accurate pressure con-trol. However, most of the desired trajectories were stepinputs and the obtained accuracies implied that the con-trollers had difficulties to achieve better accuracy sincepneumatic systems were highly nonlinear and there wereparametric and structural uncertainties in such systemsthat affected the performance of the previously proposedcontrollers. On the other hand, SMC enables accuratecontrol of highly nonlinear systems in the face of alltypes of uncertainties as stated in the study of Slotine.26
Therefore, combining SMC with the PWM algorithmled to a PWM-SMC controller for pressure tracking,which is the main contribution of this article. Figure 6presents the schematics of the closed-loop control systemthat was implemented in simulations and experiments.
Sinusoidal desired inputs as well as multistep inputswere used to evaluate the performance of the proposedcontroller that was expected to achieve acceptably highaccuracy. Standard form of a dynamic system for asingle-input system is as follows26
x(n) = f X!� �
+ b X!� �
u ð13Þ
According to equation (6) and in order to have a simpli-fied equation for designing the controller, the commonprocess between charging and discharging processescan be expressed by equation (14)
_P=RT
Vain _min � aout _moutð Þ ð14Þ
where ain and aout took values between 1 and k depend-ing on the actual heat transfer during the process. Inequation (14), it was not necessary to know the exact heat
transfer characteristics but solely estimate the coefficientsain and aout. From a control perspective, it was importantto know that the uncertainties of the estimations werebounded by k� 1 and k. For the charging process, whichwas considered to be close to an adiabatic process, thevalue of ain should be chosen close to k, while for the dis-charge process, which is close to an isothermal process,the value of aout was recommended to be chosen as 1.7
The control input was embedded in the mass flowterms of _min and _mout that makes equation (14) a nonaf-fine equation in the form of equation (13). As a result, amethod had to be applied to convert a nonaffine equa-tion stated by equation (14) to an affine equation suit-able for presenting in the form of equation (13). Themass flow _m at each sampling time can be written as
_m=Cf Pu,Pdð ÞAv pulseð Þ ð15Þ
In equation (15), f(Pu,Pd) represents all the terms inequation (7) that contains upstream and downstreampressures. On the other hand, C stands for all constantcoefficients in equation (7) while Av(pulse) is the pas-sage area of the valve that was a function of the excit-ing PWM pulse. The control input u was embedded inAv(pulse) since the exciting pulse stems from duty cycleand consequently from the control input u. To extractu from equation (15), the average mass flow over aPWM period should be written as equation (16)
_m=Cf Pu,Pdð ÞAvd ð16Þ
Over a TPWM, Av(pulse) became the multiple of two con-stant values of the nominal cross-sectional area of thevalve Av and duty cycle of the valve d over a PWMperiod. Substituting equation (12) in equation (16) andwriting equation (16) for inlet and outlet mass flows, theaverage mass flows could be described by equation (17)
Figure 6. Block diagram of SMC and the system.PWM: pulse width modulation; SMC: sliding mode controller.
236 Proc IMechE Part I: J Systems and Control Engineering 227(2)
If equation (17) is substituted in equation (14), an affineequation is obtained. However, in equation (17), theaverage mass flows for the first and third conditions(if u. umax, if u\ umin) led to constant values that werenot used for designing the SMC because they did notcontain the control input u. Equation (14) should berewritten by the use of equation (17)
_P=RT
VainCf Ps,Pð ÞAv 0:5u+0:5ð Þð
�aoutCf P,PSð ÞAv 0:5� 0:5uð ÞÞ ð18Þ
Then, equation (18) could be written in the form ofequation (13) as equation (19)
f Xð Þ= 0:5CRT
Vain f PS,Pð ÞAv � aout f P,PSð ÞAvð Þ
b Xð Þ= 0:5CRT
Vain f PS,Pð ÞAv +aout f P,PSð ÞAvð Þ
8>><>>:
ð19Þ
A time-varying surface was defined by equation (20) asfollows
S X!, t
� �=
d
dt+ l
� �n�1 ððt0
~xdt
0@
1A ð20Þ
In equation (20), ~x is the pressure error (~x=P� Pd),and l is the control bandwidth that was strictly posi-tive. Zhu and Barth27 experimentally demonstrated thatchoosing
ÐÐ t0
~xdt as the variable of interest for definingthe sliding surface led to a superior performance com-pared to the conventional choices of ~x or
Ð t0
~xdt. Whena different variable rather than ~x was chosen, then n inequation (20) would be defined as the relative degree ofthe model to the variable of interest as expressed bySlotine.26 Therefore, the relative degree of the model tothe variable of interest
ÐÐ t0
~xdt is represented by n=3.On the other hand, once on the sliding surface, the
dynamics of the system could be expressed by _S=0. Itimplies that in the face of all uncertainties, the best esti-mate of the control law u was obtained such that itmaintains _S=0 (Filippov’s construction of the equiva-lent dynamics). If the best estimate of f and b in equa-tion (19) is denoted by f and b, then the best estimateof the control law, u, is obtained by substituting _~x with_P� _Pd in equation (20) that leads to equation (21)
u= �~f+Pd � 2l~x� l2
ðt0
~xdt ð21Þ
The equivalent control effort ueq was obtained by com-bining our best estimate of the control effort u fromequation (21) with a term discontinuous across the sur-face S=0. As a result, the sliding condition,1=2(d=dt)S24�hjSj, was satisfied. Since chatteringhad to be eliminated for the proper performance of thecontroller, the control discontinuity was smoothed outin a thin boundary layer neighboring the sliding surface.
In order to minimize chattering and smoothen the con-trol input, Sign function was replaced by Sat functionfor the discontinuous part. Equation (22) demonstratesthe control effort that comprises a discontinuous termas well as the best estimate of the control effort u. Forsimplicity, we dropped eq for u and we call the controleffort u in short
u= b�1
u� KSatS
f
� �� �ð22Þ
In equation (22), b was the geometric mean of thecontrol gain bounds bmin and bmax (b=(bminbmax)
(1=2)).For ensuring the stability of the closed-loop systemin the Lyapunov sense, the sliding condition1=2(d=dt)S24�hjSj had to be satisfied that requiresS(t) _S(t)\ 0. In equation (22), u was replaced by theexpression in equation (21). If b was the gain margin ofthe design, F was the bound on the estimation error onf and h was a strictly positive constant value that guar-antees that given any initial condition, the systemwould reach the sliding surface in a finite time smallerthan treach4(jS(t=0)j)=h.
Therefore, after simplification for evaluating thevalue of K, if K5b(F+h)+ (b� 1)juj, then1=2(d=dt)S24�hjSj, and consequently, sliding condi-tion was satisfied which in turn led to the proof of sta-bility. The boundary layer thickness of the designedcontroller was represented by f while e was the bound-ary layer width. These two boundary layer dimensionswere related by f= eln�1. Furthermore, e was inter-preted as the desired pressure-tracking precision (ratherthan perfect tracking). In Table 3, the values of theSMC parameters are presented.
In Table 3, e is the boundary layer width, l is controlbandwidth, h was a strictly positive constant used insliding condition, and it guaranteed that given any ini-tial condition, the sliding surface would be reached in afinite time smaller than (jS(t=0)j)=l, and f was theboundary layer thickness. The values for these para-meters were chosen by the procedure stated in Slotine,26
and then the evaluated values were tuned by observingthe performance of the controller.
It should be stated that the control input must beconstant in a PWM period (TPWM=20ms). The cham-ber’s pressure was read by the sampling rate of theReal-Time Toolbox of MATLAB, which wasTSampling=1ms, and all the parameters were calculatedwith this rate. Because the control effort was constantover a PWM period, the mass flows calculated at each
Table 3. PWM-SMC controller parameters.
Parameter Value
e 0.01l 25h 0.1f 4
Hejrati and Najafi 237
sampling time were averaged over each PWM period(TPWM =20ms), and the averaged values were used forconstructing the control effort. By averaging all theparameters over a PWM period and then feeding themto the controller at the end of each TPWM, it was guar-anteed that the control input got updated at eachTPWM, so that it remained constant within this period.
Simulation and experimental results
In order to observe the ability of the controller, its per-formance in response to multistep and sinusoidal inputswas studied. Figure 6 is a block diagram of the system.Moreover, Figure 7 is the response of the closed-loopsystem to a multistep input under PWM-SMC in bothsimulations and experiments.
In addition, Figure 7 compares the control efforts inboth simulations and experiments for a multistep input.Similarity of the results as well as good performance ofthe controller in both cases is significant as shown inFigure 7.
Control signals were almost limited within ½�1, 1�range, and if they exceeded this range, PWM schemeresulted in either maximum or minimum duty cycleaccording to equation (12). Figure 7 confirms that inboth cases, during steady state, control signals werewithin the desired range.
To assess PWM-SMC controller performance thor-oughly, sinusoidal desired inputs with frequencies rang-ing from 0.1 to 5Hz, and different amplitudes were
tested. Some of the simulation and experimental resultsare presented in Figures 8 and 9, in which input fre-quencies of 0.5 and 2Hz are depicted. It can be notedthat by increasing the input frequency, the trackingerror got worse. This phenomenon can be explained bythe fact that the dynamic system was not capable tocatch up with very high-frequency inputs. In otherwords, the bandwidth of the pneumatic system couldnot be exceeded.
The results also show that there was a reasonablecorrelation between control signals and the input fre-quencies. Three observations were made by studyingthe control signals: first, as it was mentioned before,the control signals were mostly within ½�1, 1� range inall the cases. If the control signals exceeded the range,it is for a very short period of time (about 1ms) anddid not cause the saturation of the system at all.
Second, as the input frequency increased, the controlsignals approached the boundaries of ½�1, 1� range. Inother words, if the input frequencies exceeded the band-width of the system, control signals would be mostlyoutside the ½�1, 1� range, which led to the saturation ofthe system and instability.
Third, although experimental results were consistentwith simulation ones, there were differences betweenthese two that were due to some unmodeled dynamicsubsystems. Since the dynamics of electronic subsys-tems was faster than other mechanical subsystems, theywere not included in the modeling. However, in prac-tice, all of these subsystems influenced the performanceof the system. Comparison between simulation control
0 5 10 15 20 25 302
3
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6
Time (s)0 5 10 15 20 25 30
Time (s)
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ssur
e (K
Pa)
2
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ssur
e (K
Pa)
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0 5 10 15 20 25 30−1.5
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0
0.5
1
1.5
Time (s)
Con
trol
inpu
t
SimulationExperimental
Desired PressureActual Pressure
(a) (b)
(c)
Figure 7. Multistep pressure tracking controlled by PWM-SMC (a) in simulation, (b) in experiment, and (c) comparison betweencontrol signals.
238 Proc IMechE Part I: J Systems and Control Engineering 227(2)
signal and the experimental one indicated that pressuretracking in experiments was slightly more demanding
than simulations, which was in agreement with whatwe expected. Although the accuracy of both open-loop
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51
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6
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Desired PressureSimulation Pressure
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51
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ssur
e (K
Pa)
Desired PressureActual Pressure
0 1 2 3 4 5 6 7 8 9 10−1.5
−1
−0.5
0
0.5
1
1.5
Time (s)
Con
trol
inpu
t
SimulationExperimental
(a) (b)
(c)
Figure 9. The system’s response controlled by PWM-SMC and error to a sinusoidal input with the frequency of 2 Hz (a) insimulation, (b) in experiment, and (c) comparison between control signals.
0 1 2 3 4 5 6 7 8 9 101
2
3
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ssur
e (K
Pa)
Desired PressureSimulation Pressure
0 1 2 3 4 5 6 7 8 9 101
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ssur
e (K
Pa)
Desired PressureActual Pressure
0 1 2 3 4 5 6 7 8 9 10−1.5
−1
−0.5
0
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1
1.5
Time (s)
Con
trol
inpu
t
(a) (b)
(c)
SimulationExperiment
Figure 8. The system’s response controlled by PWM-SMC to a sinusoidal input with frequency of 0.5 Hz (a) in simulation, (b) inexperiment, and (c) comparison between control signals.
Hejrati and Najafi 239
and closed-loop simulations were outstanding, themain goal of the simulations was to provide a goodinsight into the control system before carrying out anyexperiments. Therefore, as long as the overall behaviorobserved in simulations could be found in the experi-ments, the main goal of the research was attained.
Furthermore, transient response existed in the sys-tem because the initial pressure in the cylinder, whichwas equal to the atmosphere pressure, was differentfrom the desired input pressure with a mean value of350 kPa. However, the transient response period of thesystem in both simulations and experiments was muchshorter than the steady-state period (0.4 s in the worstcase compared to an operating period of 10 s).
There were some differences between simulationsand experiments in terms of the transient responses.These differences stemmed from subtle inaccuracies inthe modeling of the real system. It can be inferred thatsmall inaccuracies in modeling can lead to significantdifferences in transient responses. Moreover, becausethe transient response was dependent on the initial con-dition of the system, even a small deviation in the initialcondition of the experimental setup and simulationmodel could lead to a noticeably different transientresponse. In other words, before starting the experi-ment, the pressure inside the chamber was not exactlythe atmospheric pressure, and the residual air in thecylinder could influence the initial condition. Since thesystem did not become unstable during the transientperiod and settling time of the system was very short,these differences did not affect the performance of thesystem. Moreover, similarities between the results interms of the RMS of error and the bandwidth of thesystem imply that the simulation results were reliablefor predicting the behavior of the real system.
As presented in Figures 7–9, the designed SMCalong with the On/Off valve and the PWM method per-formed very well in tracking various desired inputsincluding sinusoidal ones. Despite the complication ofequations, SMC was successfully implemented bymeans of an On/Off valve, and good pressure trackingof the closed-loop system confirms the high perfor-mance of the proposed controller. Tables 4 and 5demonstrate maximum absolute error (maxAE) andRMS of error for simulations and experiments,respectively.
Comparing the obtained results with ones presentedin Belforte et al.,13 Wang et al.,14 Liu et al.,15 and Yinget al.16 suggests that PWM-SMC was more robust intracking of high-frequency inputs than conventionalcontrollers like PID. In order to prove the merit of theproposed controller, the authors carried out someexperiments using a PID controller to compare theresults with the proposed controller. The PID controllerwas designed based on the second method of Ziegler–Nichols since sustained oscillations were observed inthe step response of the system28
Gc =Kp 1+1
TiS+TdS
� �ð23Þ
where Kp is the proportional gain, Ti is the integraltime constant, and Td is the derivative gain. First, whilekeeping Ti =0 and Td =0, Kp was increased to a criti-cal value Kcr at which the output first exhibited sus-tained oscillations. Thus, the critical gain and thecorresponding period Pcr were experimentally deter-mined, and then Kp =0:6Kcr, Ti =0:5Pcr, andKp =0:125Pcr were set. Finally, the obtained gainswere tuned experimentally by observing the trackingerror. Figure 10 demonstrates tracking performance ofthe PID controller for a sinusoidal input of 2Hz.Higher frequency inputs were tested on the systemusing a PID that led to poor performance of the con-troller for frequencies higher than 2Hz. To provide abetter quantitative comparison, maxAE and RMS oferror are presented in Table 6.
Table 4. Results of sinusoidal pressure tracking in thesimulations using PWM-SMC.
Table 6. Results of sinusoidal pressure tracking in thesimulations using a PID controller.
Inputamplitude (bar)
Input frequency(Hz)
maxAE(bar)
RMS oferror (bar)
1.5 0.1 0.13 0.061.5 0.5 0.6 0.181 2 1.25 0.6
RMS: root mean square.
240 Proc IMechE Part I: J Systems and Control Engineering 227(2)
It can be seen in the figures and the table that forlower frequency inputs, both controllers performedsimilar to each other, which means that the PID con-troller was capable of handling low-frequency inputs.However, the big advantage of the proposed controllerwas associated with its capability in tracking higher fre-quency inputs where the PID controller appeared to beproblematic. Therefore, considerable error wasobserved in the PID performance when the input fre-quency was raised to 2Hz and further increase in theinput frequency led to instability, whereas PWM-SMChad an acceptable performance even at frequencies asbig as 5Hz.
Conclusion
In this article, first, a comprehensive mathematicalmodel including thermodynamic process of charge anddischarge of the pneumatic chamber and also behaviorof the 3=2 fast switching On/Off valve was developedfor pressure tracking of a pneumatic cylinder. It wasobserved that by increasing the duty cycle, the errordecreased from 0.1 to 0.06 bar, and for the dischargeprocess, the error was 0.4 bar. The ratio of the magni-tude of error over the pressure amplitude varied from0:06% in discharge to 0:01% in the charging processfor the duty cycle of 75%. The comparison betweensimulation results and experiments indicates acceptableaccuracy of the proposed model for the pneumatic
system to be used in designing the controller beforeimplementing it experimentally.
To accomplish pressure tracking, a PWM-SMC wasdesigned and applied to the mathematical model insimulations and experimental tests. The performance ofPWM-SMC controller was evaluated by studying thepressure-tracking error of the system for multistep andsinusoidal inputs (inputs with different frequenciesand amplitudes). The tracking errors (RMS of errorsand maxAE) for different inputs demonstrate good per-formance of the control strategy.
The results demonstrate significant improvements inthe pressure-tracking performance of the proposed con-troller compared to the previous studies utilizing PIDcontroller such as the studies by Belforte et al.13 andWang et al.14 and hybrid fuzzy controller such as thestudy by Liu et al.15 In this article, high-frequency sinu-soidal inputs were tracked with lower tracking errors.To prove that the proposed PWM-SMC was morerobust than a conventional controller, a PID controllerwas tested on the system. A comparison of the RMS oftracking errors at different input frequencies showedthat PWM-SMC controller performed better, especiallyat high-frequency inputs. The proposed PWM-SMC ismore advantageous than other types of controllersfrom the literature since it was designed for dealingwith a highly nonlinear system. The proposed control-ler and experimental setup were shown to be efficientand cost-beneficial for pressure and force control appli-cations of pneumatic actuators.
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Err
or (
KP
a)
(a) (b)
(c)
Figure 10. System’s response controlled by a PID controller to a sinusoidal input with the frequency of 2 Hz: (a) trackingperformance, (b) control input, and (c) tracking error.
Hejrati and Najafi 241
Funding
This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.
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