Development of energy efficient optimal control for servo pneumatic cylinders

Control 2004, University of Bath, UK, September 2004

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DEVELOPMENT OF ENERGY EFFICIENT OPTIMAL CONTROL FOR SERVO PNEUMATIC CYLINDERS Jia Ke, Kary Thanapalan, Jihong Wang†, Henry Wu Department of Electrical Engineering and Electronics, University of Liverpool Brownlow Hill, Liverpool L69 3GJ, UK, E-mail: [email protected] Nonlinear system, optimal control, actuators, feedback linearisation, energy

solution is the only way to find the energy efficient control and state trajectories.

The paper conducts energy efficiency analysis of servo pneumatic actuator systems and presents a method of energy efficient optimal control. The system is linearised through input/output state feedback and then energy efficient optimal control theory is applied to the linearised system. An optimal control strategy is developed with respect to the transformed states of the linear system model. An energy efficient velocity profile is derived which depends on the final state and initial state. Through the inverse transformation, the new states are converted back to the original system state and control variable. In this way, a nonlinear energy efficient controller is derived for the servo pneumatic actuator system.

This paper is to investigate if there is an alternative way to develop an energy efficient control strategy and to avoid the problems of solving the eight complicated nonlinear differential equations. The method described in the paper can be broken down into three stages: (a) The pneumatic cylinder model is linearised through an input/output linearisation with state feedback. With the linearised model, the well developed linear control theory can be applied. (b) 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. (c) Then the generalized energy efficient feedback control is substituted back to the original system control input variable.

Keywords: pneumatic efficiency.

Abstract

1. Introduction A wide range of industries now rely on pneumatics since pneumatic actuators have distinct advantages. In the UK, a massive energy consumer, over 10% of the National Grid output is used to generate compressed air ([4], [5]). However, the energy efficiency of pneumatic actuator systems is low. 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 ([9]). A new type of air compressor was recently manufactured with improved energy efficiency ([4]). Energy efficiency of pneumatic actuators has been analysed using air “exergy” by Kagawa, et al 2000 [6]. Norgren, a leading American company in manufacturing pneumatic components, has taken the initiative in helping compressed air users to increase energy efficiency ([8]). 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 ([12]). Since the pneumatic actuators are modelled by four first order nonlinear differential equations, the above theoretical analysis will result in a group of eight first order non-linear differential equations with partially known boundary conditions. It is almost impossible to find the analytic solutions for the group of resulted equations. Numerical † The author for correspondence.

In this way, an analytic solution of energy efficient control is obtained. The controller has a very complicated structure and it is impossible to be implemented into the practical pneumatic systems. So further simplification is necessary. The final goal of the paper is to achieve energy efficient control for servo-pneumatic systems. The key issue for servo systems is the profile which the state variable is required to follow. Therefore, after the controller is developed, a set of energy efficient system state trajectories can be obtained. These trajectories can be used as energy efficient profiles for servo-controlled pneumatic cylinders. The trajectories derived are still too complicated for implementation so an approximation is carried out to simplify the profiles. The simplified profiles are adopted by the servo-pneumatic systems although it may only give out a sub-optimal control results. With the newly approximated profiles, simulation studies have been conducted. 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.

2. Pneumatic system model and linearisation An analysis of the dynamic behavior of a pneumatic system usually requires individual mathematical descriptions of dynamics of the three component parts of the system (see, [1], [2], and [3]): (i) the valve, (ii) the actuator, and (iii) the load. Such an analysis has been reported in [10] and [11]. Therefore, the detailed modelling procedure is not included in this paper. The co-ordinate system illustrated in Figure 1 is

Control 2004, University of Bath, UK, September 2004 adopted in the paper for system modelling. Pneumatic actuators can be modelled as a fourth order nonlinear system ([10] and [11]), which is affine in the control inputs. The system equations are as follows:

ID-119 k

J / Kg  2  k −1 , Pe = 1 × 10 N 2 , C r =   = 0.528 , R = 287 m K  k +1 5

k +1

and Ck =

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

Chamber B

Chamber A

Pneumatic Cylinder Load

0

x

x

l

Figure 1 Co-ordinate system of a pneumatic cylinder (1a)

x&1 = x 2

1 [ − K f x 2 − K S − c S ( x 2 , x 3 , x 4 ) + Aa x 3 − Ab x 4 ] M

x& 2 =

− k [ x3 x 2 − x& 3 =

RTs C d C0 wa fˆ ( x 3 , Ps , Pe )u1 ] Aa l / 2 + x1 + ∆

RT k [ x 4 x 2 + s C d C 0 wb fˆ ( x 4 , Ps , Pe )u2 ] Ab x& 4 = l / 2 − x1 + ∆

2  k + 1  k −1   = 3.864 . k −1  2 

(1b)

 1, ~  f ( pr ) =  1 C [ p 2 / k − p ( k +1) / k ]2 , r  k r

Patm < pr ≤ Cr Pu ,

(2)

C r < p r < 1.

fˆ ( Pa , Ps , Pe )  ~ Pa  Ps f ( P ) / Ts , Chamber A is a drive chamber s  =  ~ Pe  Pa f ( P ) / Ta , Chamber B is a drive chamber a 

(3a)

and

(1c)

(1d)

fˆ ( Pb , Ps , Pe )  ~ Pe  Pb f ( P ) / Tb , Chamber A is a drive chamber b  =  ~ Pb  Ps f ( P ) / Ts , Chamber B is a drive chamber s 

(3b)

where x1 = x , x 2 = x& , x3 = Pa , x4 = Pb , u1 = X a and

The term, − K f x 2 − K S − c S ( x 2 , x 3 , x 4 ) , in Equation (1b)

u 2 = X b . The symbols are list below:

represents the summing effects of static and dynamic friction forces of the system, where

a, b A Cd ∆ Kf

Subscripts for inlet and outlet chambers respectively Ram area (m2) Discharge coefficient The generalized residual chamber volume Viscous frictional coefficient

k l m M Pd

Specific heat constant Stroke length (m) and x ∈ ( −l / 2, l / 2) Mass flow rate (Kg/s) Payload (Kg) Down stream pressure ( N / m 2 )

Pe

Exhaust pressure ( N / m 2 )

Ps

Supply pressure ( N / m 2 )

Pu

Up stream pressure ( N / m 2 )

R

Universal gas constant (

Ts

Supply temperature (K) Port width (m) Load position (m) Spool displacement of Valve 1 or Valve 2 (m)

w x X 1, 2

J / Kg ) K

The following constants appear in the system model:

k = 1.4 , Ps = 6 × 10 5 N

m2

,

T

s

= 293 K

, C d = 0 .8 ,

K S − c ( x ) S ( x& , Pa , Pb ) :=  ( Aa Pa − Ab Pb ),    K ( x ) sign ( x& ),  c

x& = 0

and

Aa Pa − Ab Pb ≤ K S ( x )

x& ≠ 0

or

Aa Pa − Ab Pb > K S ( x )

,

which describes the static frictions. The detailed analysis for the influences of friction forces can be found in [2]. Pneumatic system model validation was reported in [10]. 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 linearisation method described in Wang and Kotta 2003 ([13]), a set of new state variables, z , are chosen to linearize the above nonlinear system. The state variables are: z1 = x1 z 2 = x2 z3 = −

Kf M

x2 +

A A x3 − x4 M M

z 4 = x4 Then, the pneumatic system is transformed into:

Control 2004, University of Bath, UK, September 2004 where αz 32 (T )

z&1 = z 2 z& 2 = z3

    Kf k ( z 2 z 3 + Ab z 2 z 4 / M + K f z 22 / M )  z&3 = − z3 − M l / 2 + z1 + ∆   ˆ kRTs C d C0 wa f ( z , Ps , Pe )  + u1  M (l / 2 + z1 + ∆ )  Ab kz 4 z 2 kRTs C d C0 wb fˆ ( z 4 , Ps , Pe )  − − u2  M (l / 2 − z1 + ∆ ) M (l / 2 − z1 + ∆ )   ˆ kz 4 z 2 kRTs C d C0 wb f ( z 4 , Ps , Pe )  + z& 4 = u2  l / 2 − z1 + ∆ Ab (l / 2 − z1 + ∆ ) Let M (l / 2 + z1 + ∆ ) u1 = kRT C C w fˆ ( z , P , P ) s

d

a

0

s

T

d

0

(4)

and define V1 = V3 −

s

movement), z 2 (0) = 0 , z 2 (T ) = 0 , z3 (0) = z 30 , and z 4 (0) = z 40 .

e

Ab V2 . Substitute u1 and u2 back into M

(4), we have

z&1 = z 2 z&2 = z3

(5)

z&3 = V1 z&4 = V2

energy consumption. As the piston will stop at the desired position, it is certainly expected that the final acceleration z3 (T ) as small as possible. If the piston is assumed to move from one end to another end of the cylinder, the boundary conditions are z1 (0) = − l / 2 or z1 (0) = l / 2 , z1 (T ) = l / 2 or

z1 (T ) = −l / 2 (which depends on the directions of the piston

e

4

b

T

0

 − kz2 z 4  Ab (l / 2 − z1 + ∆) + V2   ˆ kRT C C w f ( z , P , P )  l / 2 − z1 + ∆  s

means the squared final acceleration and

∫ V Vdt represents the integration of the control effort or the

 k ( z 2 z 3 + Ab z 2 z 4 / M + K f z 22 / M ) K f  z 3 + V3  +  l z M / 2 + + ∆ 1  

u2 =

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So the system (5) is completely a linear system with two independent inputs V1 and V2 .

3. Development of energy efficient control suing the linearised model

z3 (T ) and z 4 (T ) are free boundary conditions. This optimal control problem can be considered as a class of continuoustime optimal control with a function of final state partially fixed. To obtain the optimal control solution, the first step is to construct a Hamiltonian function - H ( z ,V , t , λ ) with an associated multiplier λ ∈ R 4 below ([7]): H ( z ,V , t , λ ) = V TV + λT ( Az + BV ) Then, we have ∂H z& = = Az + BV ∂λ ∂H = AT λ − λ& = ∂z ∂H = 2V + B T λ 0= ∂V From (11), the optimal control will be obtained as 1 V1 = − λ3 2 1 V2 = − λ 4 2 Expend (10), the following equations can be derived λ&1 = 0 λ& = −λ 2

1

λ&3 = −λ2 λ& = 0

(8) (9) (10) (11)

(12) (13)

(14)

4

The energy efficient control for this particular pneumatic cylinder actuator system is 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 y = z1 as the system output, the linear system can be rewritten in a matrix form as follows: z& = Az + BV (6) y = z1 where z = [z1

z2

0 0 A= 0  0

z3

z 4 ] , V = [V1 V2 ] , T

1 0 0 0 0  0 1 0  , and B =  1 0 0 0   0 0 0 0

T

0 0 . 0  1

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

∫

T

J = αz32 (T ) + V TVdt 0

(7)

The solutions for (14) are λ1 = µ1

λ2 = − µ1t + µ 2 λ3 =

1 2 µ1t − µ 2 t + µ 3 2

(15)

λ4 = µ 4 where µ1 ~ µ 4 are the constants to be determined. Substitute (15) into (12) and (13), then substitute (12) and (13) back to (6), we have: 1 1 z&4 = V2 = − λ4 and z 4 = − µ 4 t + µ5 (16) 2 2 1 z&3 = V1 = − λ3 2 11 3 1  and z3 = −  µ1t − µ 2 t 2 + µ3t + µ6  (17) 26 2  z&2 = z 3

and

1 1 1 1  z 2 = −  µ1t 4 − µ 2 t 3 + µ3t 2 + µ6 t + µ 7  (18) 2  24 6 2  z&1 = z 2

Control 2004, University of Bath, UK, September 2004

1 1 1 µ1t 5 + µ 2 t 4 − µ3t 3 240 48 12 and (19) 1 1 2 − µ 6 t − µ 7 t + µ8 4 2 where µ5 ~ µ8 are the constants to be determined. Suppose 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 determined to have the 2 following values: µ5 = z 4 (0) , µ 4 = [z 4 (0) − z 4 (T )] , µ6 = 0 , T µ7 = 0 , µ8 = −l / 2 , and 1 1 1 l l z1 (T ) = = − µ1T 5 + µ2T 4 − µ 3T 3 − 2 240 48 12 2 1 1 1 1  z 2 (T ) = 0 = −  µ1T 4 − µ 2T 3 + µ 3T 2  2  24 6 2  11 1  3 2 z3 (T ) = −  µ1T − µ 2T + µ 3T  26 2 

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z1 = −

Then

1440l 120 − 3 z 3 (T ) T5 T −720l 48 − 2 z 3 (T ) µ2 = T4 T −120l 6 − µ3 = z 3 (T ) T3 T

µ1 = −

Figure 3. Chamber A pressure and Chamber B pressure for different chosen terminal chamber pressures Figure 3 illustrates the pressures for chambers A and B. From the figure, it can be seen that the both chambers have same initial and terminal pressures for the energy optimal control under the assumption that the static friction is neglected.

From the above solutions, the simulation studies have been conducted. The conditions specified for the simulations are: rodless cylinder, cylinder bore size φ = 0.032m , cylinder length l =1m,compressed air supply pressure Ps = 6 × 105 N / m 2 , initial position x = −0.5m , initial velocity x& = 0m / s , initial chamber pressures Pa = Pb = 3.5 ×105 N / m 2 , K S = 0 , K C = 0 and K f = 15Ns / m . The results and the analysis are presented below.

Figure 4. The relationship between the terminal value of chamber pressures with the performance index The above figure clearly indicates that the minimum value of J is achieved at the condition that the initial chamber pressures are equal to the terminal chamber pressures. It should be noted that the minimum value is not zero here. This result will be very useful in practical controller design as it indicates that the controller should aim at driving the chamber to reach the same chamber initial and terminal pressures. Certainly, the simulation is obtained with the assumption of zero static frictions.

Figure 2. Piston position z1/x1 and velocity z2/x2 Figure 2 shows the dynamic responses with the derived optimal control for 20 different terminal chamber pressures, which are from 2.0bar to 6.0bar. The interesting finding is that the cylinder has the same position and velocity curves for all different chamber pressure patterns. The velocity diagram looks like a “SINE” wave, which may be used as the desired profile for servo control.

The authors’ previous study ([12]) 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 trapezoidal shape profile. This result is obtained through the simulation study using a PID controller. Based on the simulation results in [12], it was predicted that there exists a most energy efficient profiles for servo control pneumatic systems. Comparing the previous results and the result presented in Figure 2, Figure 5 is illustrated below. Obviously, the “sine” wave shape profile is very close to the energy efficient profile obtained in the paper, especially, they have same variation trend. The results are very encouraging as it has verified the previous findings at a certain level ([12]).

Control 2004, University of Bath, UK, September 2004

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velocity profile, which verified the results obtained in the authors’ previous work. From the simulation results, it reveals that the minimum value of the performance index J happens at the situation of the system achieving the same initial and terminal values for its two chamber pressures. The optimal velocity is very useful for practical applications, which suggested that the industry may need to reform their traditional trapezoidal velocity profile to the newly developed velocity profile. One system may not save a huge amount of energy consumption but with the big population of the users, the amount of energy saving may be considerable. Figure 5. Optimal profile and “sine” wave profile The true control variables u1 and u2 are expressed as: M (l / 2 + z1 + ∆ ) u1 = kRT s C d C 0 wa fˆ ( z , Ps , Pe )  k ( z 2 z 3 + Ab z 2 z 4 / M + K f z 22 / M ) K f z3 +  l / 2 + z1 + ∆ M  −

11  2  µ1 t − µ 2 t + µ 3 + µ 4   22 

 − kz 2 z 4  l / 2 − z1 + ∆ + µ4   kRTs Cd C0 wb fˆ ( z 4 , Ps , Pe )  l / 2 − z1 + ∆  The generalized energy efficient feedback control is then substituted back to the original system control input variable. Further simulation work has been conducted. The piston position, velocity and both chamber pressures profiles are exactly the same as those of the linearised system.

u2 =

References [1]. [2].

[3]. [4]. [5]. [6].

[7]. [8].

[9]. Figure 6. The relationship between the terminal chamber pressure values with the performance index J’ It can be seen that the profiles of Figure 4 and Figure 6 are consistent. The performance index of the original system T J ' = ( u1 + u 2 )dt also can be minimized. But the minimum

∫

0

value of J’ is different with that of J in the linearised system. The on-going work is trying to find out more suitable problem formulation for the original system.

[10].

[11].

[12].

4. Conclusion The paper described an energy efficient control strategy for servo pneumatic actuator systems. The paper starts from linearsing the system via input/output nonlinear state feedback. Then an energy optimal control strategy is proposed based on the linearised model. The solution of the energy optimal control problem results an energy efficient

[13].

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