Abstract—The paper presents a method for energy efficiency analysis of servo pneumatic actuator systems. Simulation study indicated that different quantities of compressed air will be consumed for one cycle of piston movement when the same controller is adopted but with different profiles for servo pneumatic actuators. This motivated the authors to investigate the profile which leads the system to use the least amount of compressed air, that is, the most energy efficient profile. To avoid solving a set of complicated nonlinear differential equations, the nonlinear system is linearized through input/output state feedback and energy efficient optimal control theory is then applied to the linearized system. An optimal control strategy is developed with respect to the transformed states of the linear system model. The solution of the state trajectory results in an energy efficient profile. Through an inverse transformation, the system is converted back to the original states and control variables. The generalized controller has been proved to be a sub-optimal control with respect to the original nonlinear system model. Simulation study has conducted and the results obtained from linear and nonlinear system models are analyzed and compared. Key words: pneumatic actuators, optimal control, nonlinear systems, energy efficiency.

I. INTRODUCTION

NEUMATIC cylinders are widely used in industry since pneumatic actuator systems have distinct

advantages: clean for environment, rapid point-to-point positioning, high load-carrying capacity-to-size ratio, mechanical simplicity, low cost, and ease in maintenance. In the UK, a massive energy consumer, over 10% of the National Grid output is used to generate compressed air ([1], [2]). It is estimated that of all electricity used at a typical production plant, at least 20% is consumed by compressors [3]. However, the energy efficiency of pneumatic actuator systems is low, only 23%-30% energy efficiency achieved in working systems. Some efforts have been made to improve energy efficiency of pneumatic actuators. It is reported that an additional air tank was connected into the downstream side of a pneumatic system in order to form a closed-loop circuit of compressed air [4]. A new type of air compressor was recently manufactured with improved energy efficiency [1]. Energy efficiency of pneumatic actuators has been analyzed using air “exergy”

_________________________________________________________ *J Wang, the Author for correspondence, Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UK, Fax: +44 151 794 4540, E-mail: jhwang@liverpool.ac.uk

by Kagawa, et al 2000 [5]. Norgren, one of the leading American companies in manufacturing pneumatic components, has taken the initiative in helping compressed air users to increase energy efficiency [6]. Although much effort has been made, it is considered that no substantial improvement in energy efficiency has been made and there is therefore considerable scope for study in this subject.

In authors' previous work, theoretic analysis of energy efficiency of servo-pneumatic cylinders has been conducted, which is based on optimal control theory [7]. The resulted eight first order nonlinear differential equations with partially known boundary conditions are almost impossible to be solved analytically. Numerical solution may be the only way to find the energy efficient control and state trajectories. To avoid the problems of solving the eight complicated nonlinear differential equations, an alternative way has been investigated in 2004 ([8]), which employed the input/output state feedback linearization. The method reported in [8] can be broken down into three stages: 1) The pneumatic cylinder model is linearized through an input/output linearization with state feedback; 2) Applying the optimal control theory with the linear system model, a generalized energy efficient control strategy is developed and the optimal trajectories are obtained with respect to the state variables after the transformation; 3) The generalized energy efficient feedback control is then substituted back to the original system control input variable. In this way, an analytic solution of energy efficient control is obtained.

The work described in this paper is a continuation of the work reported in [8]. The goal of the paper is to investigate if the generalized optimal control with respect to the nonlinear system model is still a most energy efficient control for the original pneumatic actuator system. Simulation study reveals that it is an optimal control but the initial chamber pressure conditions are slightly away from the results obtained from that obtained using the linear system model. The conditions to guide the optimal control design are obtained for the original pneumatic cylinder systems. The simulation study has shown that the new profiles will lead the cylinders to use the less quantity of compressed air comparing with the traditional trapezoidal profile.

Energy Efficiency Analysis and Optimal Control of Servo Pneumatic Cylinders

J Ke, J Wang*, N Jia, L Yang, Q H Wu Department of Electrical Engineering and Electronics, University of Liverpool, Liverpool L69 3GJ, UK

P

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MC5.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 541

II.PNEUMATIC SYSTEM MODEL AND LINEARIZATION The following symbols are used for system modeling in the paper: a b, Subscripts for inlet and outlet chambers respectively A Ram area (m2)

dC Discharge coefficient The generalized residual chamber volume fK Viscous frictional coefficient

k Specific heat constant l Stroke length (m) and )2/,2/( llx m Mass flow rate (Kg/s)M Payload (Kg) Pd Down stream pressure ( 2/ mN )

eP Exhaust pressure ( 2/ mN )

sP Supply pressure ( 2/ mN )

uP Up stream pressure ( 2/ mN )R Universal gas constant ( )/(KgKJ )

sT Supply temperature (K) w Port width (m) x Load position (m)

2,1X Spool displacement of Valve 1 or Valve 2 (m)The following constants appear in the system model:

4.1k , 25106 m

NPs , KT s 293 , 8.0dC ,

25101 mNPe , 528.0)1/(2 1k

k

r kC ,

KKgJ

R/287 , and 864.3

21

12 1

1kk

k

kk

C .

Pneumatic actuators can be modelled as a fourth order nonlinear system [9], [10], which is affine in the control inputs. If a co-ordinate system illustrated in Fig. 1 is adopted, the system equations can be modeled by:

21 xx (1a)

]),,([14343222 xAxAxxxSKxK

Mx bacSf (1b)

1

13023

3 2/

]),,(ˆ[

xl

uPPxfwCCA

RTxxk

xesad

a

s

(1c)

1

24024

4 2/

]),,(ˆ[

xl

uPPxfwCCA

RTxxk

xesbd

b

s

(1d)

where xx1 , xx2 , aPx3 , bPx4 , aXu1 and

bXu2 .The functions in Equations (1a)-(1d) are defined as

.1,

,1)(

~

21

/)1(/2rr

kkr

krk

rr

s

atm

r

pCppC

CpP

P

pf , (2)

x

AChamber BChamber CylinderPneumatic

0

l

Load

Fig. 1. Co-ordinate system of a pneumatic cylinder.

chamberdriveaisBChamberTPP

fP

chamberdriveaisAChamberTPP

fP

PPPf

aa

ea

ss

as

esa

,/)(~

,/)(~

),,(ˆ

(3a)

chamberdriveaisBChamberTPP

fP

chamberdriveaisAChamberTPP

fP

PPPf

ss

bs

bb

eb

esb

,/)(~

,/)(~

),,(ˆ

(3b)

The term, ),,( 4322 xxxSKxK cSf , in (1b) represents the summing effects of friction forces of the system, where

)(0),()(

)(0),(

:),,()(

xKPAPAorxxsignxK

xKPAPAandxPAPA

PPxSxK

Sbbaac

Sbbaabbaa

bacS

which describes the static frictions. The detailed analysis for the influences of friction forces can be found in [9] [11]. For the convenience of analysis, the friction forces are ignored initially and they will be treated as uncertainties at the stage of designing a feedback control for the pneumatic system.

For the case of using two independent three-port valves, adopting the similar linearization method described in [12], a set of new state variables, z , are chosen to linearize the above nonlinear system. The state variables are:

11 xz , 22 xz , 4323 xMA

xMA

xM

Kz f , 44 xz .

Then, the pneumatic system is transformed into:

21

40

1

244

21

40

1

24

11

0

1

224232

33

32

21

)2/(),,(ˆ

2/

)2/(),,(ˆ

)2/(

)2/(),,(ˆ

2/)//(

uzlA

PPzfwCCkRTzlzkz

z

uzlM

PPzfwCCkRTzlMzkzA

uzlM

PPzfwCCkRT

zl

MzKMzzAzzkz

M

Kz

zz

zz

b

esbds

esbdsb

esads

fbf

(4)

Let

542

331

224232

0

11

2/)//(

),,(ˆ)2/(

VzM

K

zl

MzKMzzAzzk

PPzfwCCkRT

zlMu

ffb

esads

21

42

40

12 2/),,(ˆ

)2/(V

zlzkz

PPzfwCCkRT

zlAu

esbds

b

and define 231 V

MA

VV b . Substitute 1u and 2u back into

(4), we have

24

13

32

21

Vz

Vz

zz

zz

(5)

So the system (5) is completely a linear system with two independent inputs 1V and 2V .

III. DEVELOPMENT OF ENERGY EFFICIENT CONTROL USING THE LINEARIZED MODEL

The energy efficient control for this particular pneumatic cylinder actuator system implies to move a piston from one position to another within a pre-specified time period limit and the motion consumes the least compressed air. Choosing 1zy as the system output, the linear system can be rewritten in a matrix form as follows:

1zy

BVAzz (6)

where Tzzzzz 4321 , TVVV 21 ,

0000000001000010

A , and

10010000

B .

The aim of energy efficient control is to derive a feedback control )(zV , for system (6) to minimize the following performance index:

TTVdtVTzJ

0

23 )( (7)

where )(23 Tz means the squared final acceleration and

TTVdtV

0 represents the integration of the control effort or

the energy consumption since V is directly proportional to the valve displacement. It is desired that less quantity of compressed air is used to move a load from one position to another. Smaller valve displacement represents smaller valve opening and, in turn, smaller quantity of compressed air is used. As the piston will stop at the desired position, it is certainly expected that the final acceleration )(3 Tz as small as possible. If the piston is assumed to move from one end to another end of the cylinder, the boundary conditions are 2/)0(1 lz or 2/)0(1 lz , 2/)(1 lTz or

2/)(1 lTz , which depends on the directions of the

piston movement, 0)0(2z , 0)(2 Tz , 033 )0( zz , and

044 )0( zz . )(3 Tz and )(4 Tz are free boundary

conditions. This can be considered as a class of continuous-time optimal control problem with a function of final state partially fixed. To obtain a optimal control solution, the first step is to construct a Hamiltonian function - ),,,( tVzH with an associated multiplier 4 shown below [13]:

)(),,,( BVAzVVtVzH TT (8) Then, we have

BVAzH

z (9)

TAzH (10)

TBVVH 20 (11)

From (11), the optimal control will be obtained as

31 21

V (12)

42 21

V (13)

Expending (10), the following equations can be written

0

0

4

23

12

1

(14)

The solutions for (14) are

44

322

13

212

11

21

tt

t (15)

where 1 ~ 4 are the constants to be determined. By substituting (15) into (12) and (13) and then, substituting (12) and (13) back into (6), the analytic solution of energy efficient control is obtained below:

544 21

tz (16)

632

23

13 21

61

21

tttz (17)

762

33

24

12 21

61

241

21

ttttz (18)

872

6

33

42

511

21

41

121

481

2401

tt

tttz (19)

where 5 ~ 8 are the constants to be determined. Suppose that the piston moves to the positive direction and substitute the boundary and initial conditions into the solutions (16)-(19), part of the unknown constants can be

543

determined to have the following values: )0(45 z ,

)()0(2444 Tzz

T, 06 , 07 , 2/8 l , and

212

1481

2401

2)( 3

34

25

11

lTTT

lTz

23

32

412 2

161

241

210)( TTTTz

TTTTz 32

23

13 21

61

21)(

Then

)(12014403351 Tz

TTl

)(487203242 Tz

TTl

)(6120333 Tz

TTl .

IV. SIMULATION STUDIES OF THE LINEARISED SYSTEM

From the analytic solution of energy efficient control, the simulation studies have been conducted. The conditions specified for the simulations are as follows: rodless cylinder with the bore size m032.0 and length ml 1 ,compressed air supply pressure 25 /106 mNPs (6 bar), initial position mx 5.0 , initial velocity smx /0 ,initial chamber pressures 25 /105.3 mNPP ba

(3.5 bar),

the static frictions are neglected, 0SK , 0CK and the viscous frictional coefficient mNsK f /15 , simulation time T=2s. The simulation results and the analysis are presented below.

Fig. 2 shows the dynamic responses with the derived optimal control for 20 different terminal chamber pressures from 2.0 to 6.0 bars. From the figure, the interesting finding is that the optimized trajectories for velocity and position are unique. So the trajectory can be chosen as the energy efficient profiles in practice.

0 0.5 1 1.5 2-0.5

0

0.5

Pis

ton

Pos

ition

(m)

Time(s)

0 0.5 1 1.5 20

0.5

1

Time(s)

Pis

ton

Vel

ocity

(m/s

)

Fig. 2. Piston position and velocity

Fig. 3 illustrates the pressures of chambers A and B with the same initial pressure 3.5 bar but 20 different terminal pressures from 2.0 to 6.0 bars. The figure illustrated 20 different terminal chamber pressures. Fig. 4 is to show the both chamber pressures. It has the same initial and terminal

pressure value since a rodless cylinder is used for this study.

0 0.5 1 1.5 22

3

4

5

6

Time (s)

Cha

mbe

r B p

ress

ure

(bar

)

0 0.5 1 1.5 22

3

4

5

6

Time (s)

Cha

mbe

r A p

ress

ure

(bar

)

Fig. 3. Chamber A and Chamber B pressures for different chosen terminal chamber pressures.

0 0.5 1 1.5 22.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

Time (s)

Cha

mbe

r pre

ssur

e (b

ar)

Chamber AChamber B

Fig. 4. Chamber A and Chamber B pressure with a terminal pressure of 2.5 bar.

The authors’ previous study [7] has discovered that the servo pneumatic system uses less compressed air when a “sine” wave shape profile is adopted comparing with the situation of using parabola and trapezoidal shape profiles. Based on the simulation results in [7], it was predicted that there exists a most energy efficient profile for servo controlled pneumatic systems. From the simulation, it is found that the energy efficient velocity response obtained here looks alike the sine wave. Fig. 5 compares the “sine” wave shaped profile and the optimal velocity trajectory obtained in the paper. Obviously, the “sine” wave shape profile is very close to the energy efficient profile, especially, they have same variation trend. The results are very encouraging as it has verified the previous findings at a certain level [7]. The curve also tells that it is require a lower maximum velocity to move a piston from one position to another within the same length of time period comparing with the sine wave.

To investigate the condition under which the minimum value of performance index J occurs, the initial chamber pressure is fixed firstly, whereas the terminal chamber pressure is free. Through the simulation study, it is found that the minimum performance index will be obtained at the condition that the initial chamber pressures are equal to the terminal chamber pressures. Fig. 6 clearly indicates this situation with the initial pressures equal to 2.5 bar, 3.5 bar, and 4.5 bar, respectively. For each case, the minimum J hasthe same value. This result will be very useful in practical controller design as it implies that the controller should aim

544

at driving the chamber to reach the same initial and terminal chamber pressures. Certainly, the simulation is obtained with the assumption of zero static frictions. Some modification may need to be addressed when the static frictions are considered.

Fig. 5. Optimal velocity trajectory comparing with a “sine” wave profile.

2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5x 10

10

Terminal chamber pressures (bar)

Per

form

ance

inde

x J

P0=2.5barP0=3.5barP0=4.5bar

Fig. 6. The relationship between the performance index J and terminal chamber pressures

The above results were obtained under the same compressed air supply pressure. To investigate if the supply pressure affects the results illustrated in Fig. 6. Simulation study has been conducted under different air supply pressures. The results are shown in Fig. 7. From the figure, it can be seen that the minimum J happens at the point while the terminal chamber pressures are same as the initials no matter how high the supply pressure is.

Fig. 7. Performance indices as functions of terminal chamber pressure with different air supply pressures

V.SIMULATION STUDIES USING NONLINEAR FEEDBACK CONTROL

When the controls are substituted back to the input state feedback transform, we have the generalized nonlinear control as follows:

4322

1

31

224232

0

11

21

21

2/)//(

),,(ˆ)2/(

tt

zM

K

zl

MzKMzzAzzk

PPzfwCCkRT

zlMu

ffb

esads

41

42

40

12 2/),,(ˆ

2/zl

zkz

PPzfwCCkRT

zlu

esbds

A typical control variables are illustrated in Fig. 8, which represent the true valve displacements.

0 0.5 1 1.5 2-8

-6

-4

-2

0

2

4

6x 10

-4

Time (s)V

alve

dis

plac

emen

t (m

)

u1u2

Fig. 8. Typical control variables for the nonlinear control

Further simulation study has been conducted to find out if the minimum value of the performance index still happens under the same conditions derived from the linearised systems. The performance index is set as:

TdtuuJ

0 211 )( (20)

J1 can be explained as that the total spool displacement of the two valves on the pneumatic cylinder is minimized, that is, the integration of the control effort or the energy consumption is minimized. Similar to the simulation conducted for the linearised system model, by fixing the initial chamber pressures but varying the terminal pressures, the relationship between the values of J1 with the terminal pressures are shown in Fig. 9. Three sets of results are presented in the figure with three different supply air pressures – 6, 9, and 12 bars. It is noticed that the minimum value of J1 does not happen at the point of the terminal pressures same as the initial pressures. The minimum J1 appears when the terminal pressure is slightly lower or higher than the initial values. The supply pressure variations affect the trends of the curves in Fig. 9 obviously. If the supply pressures increase, the terminal pressures increase to achieve the point of minimum J1.Therefore, in principle, the conclusion reached in Section IV still applies for the original nonlinear system. It is realistic to set up the initial and terminal chamber pressures to have same value. But, in this case, only sub-optimal control can be achieved as the minimum J1 happens at the point slightly away from the optimal terminal pressures. In practice, this will guide the profile design. Once the initial

2 3 4 5 6 7 8 9 10 11 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

11

Terminal chamber pressures (bar)

Per

form

ance

inde

x J

Ps=4barPs=5barPs=6barPs=9barPs=12bar

2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6 x 109

545

pressure is determined the terminal pressure will settle at the same level of its initial pressure.

2 3 4 5 6 7 8 9 10 11 12

1

1.5

2

2.5

x 10-3

Terminal chamber pressure (bar)

Per

form

ance

inde

x J1

P0=2.5barP0=3.5barP0=4.5bar

Ps=12bar

Ps=9bar Ps=6bar

Fig. 9. The performance index J1 as functions of terminal chamber pressures with different supply pressures

How does the initial chamber pressure affect the performance index? The simulation study with different initial chamber pressures was conducted and the results are shown in Fig. 10. The results indicated that the initial chamber pressures should be chosen as around 10% ~ 20% lower than the 50% of the supply pressure.

2 4 6 8 10 122

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Initial chamber pressure (bar)

Term

inal

cha

mbe

r pre

ssur

e (b

ar)

Ps=6barPs=8barPs=12bar

Fig. 10. Terminal chamber pressure when the minimum J1

value is achieved as a function of the initial chamber pressure with different air supply

pressures

VI. CONCLUSION

This paper describs an energy efficient control strategy for servo pneumatic actuator systems. It starts from linearzing the system model via input/output nonlinear state feedback. Then an energy optimal control strategy is proposed based on the linearized model. The solution of the energy optimal control problem results in an energy efficient velocity profile, which verified the results obtained in the authors’ previous work. The new findings can be summarized below:

1) The optimal control velocity profile is obtained which has a shape close to “sine” function;

2) If the initial chamber pressure is same as the terminal chamber pressure, the sub-optimal control can be achieved;

3) The initial chamber pressures should be set at the level 10%-20% lower than the 50% of the supply pressures;

4) For servo controlled pneumatic system, the velocity profiles can be designed to have the shape derived in the paper. The control strategy will be required to drive the real piston velocity to follow the profiles.

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