Parameter Identification and Model Validation of a Servopneumatic Actuator

Copyright © IFAC System Identification, Copenhagen, Denmark, 1994

PARAMETER IDENTIFICATION AND MODEL VALIDATION OF A SERVOPNEUMATIC ACTUATOR H. HAHN, A. PIEPENBRINK UnivtTsity of Kasul. Department of Mechanical Engineering (FB 15J. Control Engineering and System Theory Group. M6nchebtTgstr. 7. 34109 Kasul. Germany. Phone 05611804 3260. Fax 05611804 2330. e·Mail: [email protected]·lcJJsul.de

Abstract. This paper presents an experimental parameter identification and a model validation concept applied to a servopneumatic actuator. The mathematical actuator model includes the nonlinear differential equations of essential thermodynamical, fluid mechanical and mechanical processes of the actuator components. Special sensing elements for experimental identification of the model parameters have been developed and successfully applied. The identification algorithms used are different types of LS and RLS methods. Excellent agreement between measured and predicted time histories of laboratory experiments have been achieved based on the identified model parameters. Key Words. Identification; parameter estimation; fluidic devices; servopneumatic actuator; special purpose sensors

I. INfRODUCnON

identification schemes used in (Isermann, 1987; Ljung, 1987; Unbehauen and Rao, 1987; Hahn, 1993). It roughly includes the following identification steps:

Servopneumatic actuators are widely used in various fields of industrial practice. Systematic theoretical investigations as well as the development of control concepts of pneumatic actuators and of pneumatic shock absorbers are based on non linear model equations (Backe, 1979; Hahn, 1988; Lin, Reid, 1993). This paper provides a systematic block oriented model identification scheme (Section 2) which includes as parts the model building (Section 3), a rough specification of the identification tests (Section 4), the identification procedures (Section 5), the numerical results obtained in the parameter identification experiments and in the model validation steps (Section 6). The results obtained show excellent agreement between theoretical calculations and associated experimental measurements. These investigations show, that it is possible to obtain accurate estimates of fluid mechanical and thermodynamical parameters from simple identification experiments. The results of these investigations will be used for identification based active servopneumatic shock absorber design and control.

mathematical modelling of the servopneumatic actuator from the basic laws of physics (cf. I, Block A, Section 3), specification of the identification scenario (cf. I, Block B, Section 4) including choice of sensing elements and selection of identification test signals, run of identification experiments (cf. I, Block C) including preliminary experiments for model refinements and main experiments providing the input data for parameter estimation algorithms, choice of an identification strategy and algorithm (cf. I, Block D, Section 5), estimation of model parameters (cf. I, Block F, compare Section 5),

2. MODEL IDENTIFICAnON SCHEME The general model identification scheme used in this paper is drawn in Fig. I. It is similar to

comparison of measured and predicted output variables based on the identified model parameters (cf. 1, Block H, compare Section 6.2).

This work has been supported by the German Science Foundation "DFG" in the framework of the Graduiertenkolleg "Identifikation von System- und Materialeigenschaften".

233

Fig. 2a.

Photo of the laboratory identification experiment

Fig. 2b.

Computer animated drawing of the pneumatic actuator

system disturbances n l and n 2 including electronic measurement noise, system inherent noise, model error noise and various disturbances from the laboratory environment. Fig. I.

Identification scheme used The input signal u controls the motion of the actuator piston for identification and for data recording purposes. The analog multisensor controller provides the input voltage of the servovalve amplifier.

3. PNEUMATIC ACTUATOR MODEL EQUATIONS The pneumatic actuator considered (cL 2a, 2b) includes the following components (cf. 3,4):

m,. ,....-----,

"-'

electronic multisensor controller, plant components torque motor (I ), servovalve (1), actuator including

..,

"

.-.................-P,.,.......

----

Fig. 3.

.

~tar.....,Jn-_

o

. . . .

ca...Mrte~T,.

. "--'1" • •

r.

thefh&icl_ppJ:r

::::=::i/~t..~.~., .~tar"""~ ......, .... tiIo• .lr..~.i.

Block diagram of the controlled servopneumatic actuator

pressure evolution «2.1), (2.2», mass flow «3.1), (3.2», mechanical load «4.1), (4.2», sensing elements for data recording (compare Fig. 4) input signal u of the servovalve amplifier, servovalve piston displacement xv' actuator chamber pressures PI' P", pressures Ps, PR of the fluid supply, actuator chamber temperatures T" T", temperatures Ts ' TR of the fluid supply, actuator chamber mass flow mZ/, mZ/l , actuator piston displacement, velocity and acceleration x t ' xt ' it.

Fig. 4.

-_

.........

Scheme of the pneumatic actuator with full sensor equipment

The model equations are collected in (1) to (4) (compare Backt~, 1979; Pachnicke, 1986; Hahn, 1988). The model parameters and the model variables used are collected in the appendix.

234

The servovalve is modeled by a linear differential Equation (I) based on the following assumptions: torque motor dynamics omitted, valve dynamics independent of the load, servovalve piston friction neglected. (I)

4. SPECIFICAnON OF THE IDENTlFICAnON SCENARIO

The pressure evolution PI' PI/ in the actuator chambers is described by the two nonlinear differential Equations (2.1) and (2.2), assuming ideal isentropic processes in the actuator chambers.

This step includes among others the following activities (cf. 1. Block B): choice and development of sensing elements and selection of suitable identification test signals.

The temperatures Ts ' TR of the fluid supply and the system pressures Ps, PR of the fluid supply are assumed to be constant.

/(' (R· ~ .mn

(2.1 )

(Vo, +AI·X k )

/('(-R'~"mnl +AI/'x k , PI/)

P"

4.1. Choice and Development of Special Purpose Sensor Elements

- AI' x k 'PI)

The results and the quality of experimental parameter identification severely depends on the quality of the recorded data. The accuracy of the measurements of the actuator chamber pressures and of the actuator mass flows severely affects the identification results of the fluid mechanical parameters in Equations (2). (3). Special purpose sensors for recording these variables have been developed and will be roughly described subsequently by discussing

(2.2)

(VO" -A" 'Xk)

The two nonlinear Equations (3.1) and (3.2), describing the mass flow m n • m nl into the actuator chambers. include the following simplifications: zero control egde overlapping. no bypass mass flow across the actuator. (3.1 )

mn

= 1C·a J ·d·x '(J(-x v' O).III(PR).p I,' Y' I P,

+

.~ R.T, 2

their principle functioning and their hardware realization.

,

~

1C·a·d·x '(J(x v' O)-1I1(-)·p . PI 2 2" Y' S R.T, Ps s

Hi&h response mass flow sensor. The identification of the actuator model equations requires measurements of the dynamics of the mass flow into the actuator chambers. The "Hot Wire Anemometer" measurement device developed, provides an electric signal directly from the mass flow without knowledge of the density of the fluid. A heated platinum wire is cooled by the mass flow, where an electrical circuit either controls the main sensor current to be constant or the sensor temperature to be constant. In our application control of the sensor temperature has been realized, which provides dynamic mass flow measurements with an upper frequency of 1 kHz. The wire as sensing element can be even used inside standard plastic pipes. Fig. 5 shows a photo of the electronic control circuit and of the sensing element. Actuator piston with inte&rated pressure sensor and temperature sensor. The identification of the friction force in Equation (4) requires the measurement of both chamber pressures in close neighbourhood to the actuator piston to exclude a pressure phase lag due to the pipe dynamics. Two pressure sensors of DMS bridge principle and their electronic control circuits (Fig. 6) have been

(3.2)

.~

m nl -- 1C·a3 ·d3 'x v .(J(-x v' O)-1I1(P,,).p Y' s R .2T, Ps s

+

~

1C.a.d.x '(J(x v' O).III(.!2..).p . R.2T, 4 4 v Y' /I PI/ /I

where

The actuator piston mechanics is modeled by the Equations (4.1) and (4.2) including a sophisticated friction model (compare Pachnicke. 1986). mk

,xi + PI·AI- pl/·AI/-m

(4.1) k

'g=

sign(xk)·(dk +ck '~kl)

235

developed. The pressure sensors have integrated temperature sensors, which provide both, temperature compensation in the pressure measurement device and temperature measurements in both actuator chambers. The cables of the electronic control circuits are lead through the hollow body of the piston pole (cf. 6). The elastomer density element of the piston has been replaced by PTFE material, so that the modeled friction in Equation (4) describes the real friction forces at the actuator piston sufficiently accurate.

on the type of the individual equations. on the system disturbances and on the location of the disturbances within the system.

-......... - -.........-

J~ I

- 1')ope I

TypeD

T)'peDJ

Type IV

1J

J~ I

If Fig. 7.

Fig. 5.

Photo of the mass flow sensor

Fig. 6.

Photo of the actuator piston

Classification of model equations from the parameter identification point of view

Table 1

Classification of the actuator model

Type I equations:

Equation (1), Equation (4.1)

The control signals of the parameter identification experiments are time discrete signals. The types and shapes of the signals used are restricted

TypeD equations:

Equation (2.1) Equation (2.2), Equation (4.2).

by the maximum displacement of the servovalve piston, by the measurement range of the mass flow sensors limited to 4gs- 1 and by the maximum displacement of the actuator piston.

Type IV equations:

Equation (3.1), Equation (3.2)

4.2. Choice of Test Signals

The following identification approach is done in several steps (cf. 17): Identification of the parameters XUrit in preliminary experiments.

The test signals used in the experimental parameter identification step are

Pkri,

and

Inserting Pkrit into (3.1) and into (3.2) and inserting XUrit into (4.1) and into (4.2) provides model equations, which are linear in the remainder unknown parameters.

rectangular control signals to identify (1) and (4.1), sinusoidal control signals to identify (2.1), (2.2), (3.1), (3.2) and (4.1), additional forces of rectangular time histories acting on the actuator load, used to identify the model parameter /( in (2.1) and in (2.2) and system inherent noise in the various components of the actuator.

The remainder unknown parameters are identified from these equations using sophisticated LS and RLS methods. The parameters ~y , wy , k y of Type I Equation (1) are obtained using a least squares procedure (Eykhoff, 1974; Isermann, 1987; Ljung, 1987; Sooerstrom, 1989), taking into account the recorded variables u , X By filtering the step function input signal u and the step response output signal x. by means of a second order lowpass filter (Young. 1981; Unbehauen and Rao. 1987). approximations of the time derivatives Xv' y are calculated as functions of the filter input signals u and xv' respectively. Fig. 8 shows the filter outputs F(u) , F(x.). F(xy) and F(Xy). The identified parameters ~y • w y and ky are shown in Fig. 9. y •

5. PARAMETER IDENTIFICATION AND MODEL VALIDATION CONCEPT

x

The model Equations (1) to (4) of Section 3 are of Types I, n and N of Fig. 7 (compare Table 1). The choice of the identification algorithms (compare Fig. I, Block F, G) depends

236

"ffi"··)IV'

"'.)1_1

~.

.., ._._.-. .

.j

.

••

I.'

~I-J

Fig. 8.

...

1I'.lh.,rl

m. .

.

: . . . . . . . . . . • . . . . . . __

:

: . ·········,·········H··························

_._._.

.

IIi"H.. .1

compressibility flow terms PIJI .(Vow ± AJ./I . x k ) of (2.1) and (2.2). This is achieved by introducing a suitable dynamic external actuator load in addition to the control input signal u. The identified thermodynamic parameters /C. R of Equations (2.1). (2.2) are shown in Fig. 11.

4

I :

.

••

~l-I

...'

••

~I~J

-El : , _: :

.

,

.•.,..

......

~I_I

n

Time histories of the filter output signals F(u) , F(x.). F(x.) and F(i.)

c.

a

1.1)

:

,...

....... :

Ej_".'o'ill

rn "'IO"'J

:

: . . . . . ...•. .. . ... ....

..

I

.

':

~t.

•. . . . . . . . . ... . . . . .. . ..."

~'

.

._-......

:..

~10Cl0

:

.

Fig. 11.

5O.0.cno

Time lIftS]

The parameters /C. R of Type II Equations (2.1), (2.2) are obtained using a recursive least squares method with recorded variables PI' PII' x k • i k • mZJ' mZJI' ~ and ~I (cf. 11) and with known model constants VOI' VOII ' AI and All'

-:: . .

::

.

..

..

....

;...:

.

~ •• ' ,

50.

.,

o

.

::..

- -

..

..

:

-.'

_:

: :

I.

0.'"''

;" .....

.-

1

,.

.

eo·····

1.

_.

..

""'1;'1

.......•. ;

.

. . ~ . ._.

:

1.

..

.

0.

:

.1.

.- .i .._.

.

: " ......•

. . ; . ._.

.,

IJ

-4,

Fig.

-

•

/C

and R

:....

JO

T1lIlC1'1

10. Time histories of recorded signals PI' PII' Xk • i k • mZJ' mZJI' ~ and ~I

•

..

_.:. ..

Jo· 1

I.J

~I'J

..

.

G·-

1-'

:

4'"

:

J.O ·1.

~lsJ

._. .

U

JO

~I~

'EE"'·""'" 'fa"'·"'-' ~.

t

&.

.......... :

1.

I

lO ..

~(sJ

Fig. 13.

The signals shown in Fig. 10 have been recorded inside the control loop (cf. 3) using a sinusoidal input signal u of the control system (cf. 3). The input data of the identification algorithm shown in Fig. 10 have been chosen in agreement with the following measurement restrictions:

j .._.

12. Time histories of recorded signals mZJ' mZJI and Xv

...........

Fig.

'0

~ls)

: : :~:: .,.... :Ej'.•. .

I

TIlDe "I

~lsl

~...

).0....

'fEJm'.d"_1

I.'

U

J.O ..

Identified model parameters

~I'I

")0

U

'~ I~'I :~-a.: I. ;'1 ttl"«~~) ,.., t;J~«."'~ ,'

,

'111'

,

8

.

__

....•...........

~u

'0 0

~I'I

1Ml : :B~l~AtJ :gg~'~' ...

,

:......

'JIt;I(I,. 01 (12)

The signals shown in Fig. 12 have been recorded inside the control loop (cf. 3) using a sinusoidal input signal u of the control system. Applying LS methods. the identified parameters a, A ..... a.· d. of Equations (3.1). (3.2) are shown in Fig. 13.

:E&J~~I~I . :B~':h' tB····~"~_1 ~~II~=I _.. .. .... . ...

1.

a

The fluid mechanical parameters a I ·dI' ••• , a 4 ·d4 of Type IV Equations (3.1), (3.2) are obtained by inserting the identified parameters /C. R into (3.1) and (3.2). Inserting the parameter Pk"' • obtained in preliminary experiments. into (3.1) and into (3.2) modifies Type IV Equations (3.1) and (3.2) to Type II equations.

Time Ims)

Identified model parameters kv • wand}: '" -"v

Fig. 9.

IJ

J.O ..

~Is)

.

~lCno

Tune Imsl

.:.

:

.

.

.

.

r u C 0(2.2).

(1.1)

•.....

_

IJ

. _. .. . . . . .. " ,. . .

I :

'''''''''I 01

Ej ...... , : ......

.. .

lO.

:...

···

..

7·· · ..··

~ ~(sJ

1.

J.O ..

_.;...

........•

;

U

10 11 ..

m,·d,I_1 .

1.

).0

..

~lsl

Identified parameters a.·d I'

..

.

.

IJ

)0

~I~

••• ,

a 4 ·d4

The parameters ek • d k of Type I Equation (4.1) are obtained. by inserting the parameter value of xk~. (d etermmed in preliminary measurements (cf. 1. Block C» into (4.1) and applying least squares methods using the

limited range of 4gs- 1 of the mass flow sensors. limited signal to noise ratio characteristics of the mass flow sensors.

recorded variables x t • i k ,PI and PII and the known model constants AI' All and mk • The signals shown in Fig. 14 have been recorded inside the closed loop (cf. 3) using a sinusoidal input signal u of the control system. The identified parameters c k and d k are shown in Fig. 15.

The identification of the thermodynamic parameter /C requires a significant contribution of the

237

6.

_E;l~I~. I "El~I' .• "1 -

...•.•....

)lID.

~IM. I. :il··~'M~. . .

.

,;ta

I.J

D

Ij

.-

;

JO ·1.

-

I.'

·16

1.0'.

6.1. Final Parameter Identification Results

.

.

I'

All identified parameters are collected in Table 2. They have been obtained by using the procedures of "State Variable Filters" as well as different types of "Least Squares" methods.

J.O

TDC'"

T.chJ

Fig. 14.

IDENTIFICAnON

m ... '

...

Time histories of recorded signals P" P". t and t

x

x

=0 .

Table 2

~P''''I :rgJ':NI..

••........ '4'

-

Ila·

...

ni,

,

Identified model parameters

identified parameter

.

name

value

unit

.":'"

J0

~.

0

Tamcl'!

).0

torque motor coefficient k,.

TDeI'J

Identified model parameters ct and d t

Fig. 15.

RESULTSOFEXPE~NTAL

l.e-4

servovalve coefficients

Inserting the identified parameters ct and d t into (4.2), the unknown parameter f;,Qft in Type 11 Equation (4.2) is obtained by applying least squares method using the

0.35 1000 flow coefficients and servovalve diameters

x

recorded variables xt ' t ,P, and p" and the known model constants A" A" and mt . Fig. 16 shows the identified parameter Fltaft using a rectangular input signal u of the control system.

a,·d, a 2 ·d2 a 3 ·d3 a 4 ·d4

6 6 6 6

R

290 1.4

mm mm mm mm

fluid constants /(

friction coefficients

Fltaft Identified parameter Fltaft

Fig. 16.

X k .krit

ct dt

i

~.T•.••.••• ...

,,~.'::':"~:..-rt.W;;~ ::":!..

Pmr

6.2. Model Validation Results

' •• T•••_ ....... _ •. ,., T.....

-r-='':-:--'-::-1

_.'tI....

·11.

A final overall judgement of the identified model (Isermann, 1979; Ljung, 1987; Unbehauen, 1987) has been obtained by comparing the measured and the model predicted variables (compare Fig. 1, Block H) and/or by comparing the time histories between simulated and measured output signals (compare Fig. I, Block I). Subsequently predicted and measured signals of only a few significant thermodynamic variables are compared in Figures 18 and 19. Fig. 18 shows time histories of the measured and the predicted pressure evolution of P, and p".

u.· ....s.w--.....,. ~

. .e;---.1C; .......

-...-V

(1'

•• u...t.

t C ; ~ IMt

...".,....·

twl;~ . . . . . . HI7)

~

T.. , . - . .:_.,'_ ~

0.25

_"11.- ......

<••

SVA

N -I m·s N·s·m- ' N

flow function coefficient

---

--.......

85 0.1 120 55

'-F~,..~

,

~........,...........

Fig. 17. Identification scheme of the actuator model parameters

238

Soderstrom, T. and Stoica, P. (1989). System identification. Prentice Hall: New York, London. Unbehauen. H. and G.P. Rao (1987). Identification of continuous time systems. North Holland Publishing Company. Amsterdam. Young, P.c. (1981). Parameter estimation for continuous-time models - a survey. Automatica, 17,23-39.

II~-'I • ·1:~PWlI .

ij ..

_.

............ . . -. -.-.

-

...

I.J

T.c It.

U

..

.._

_. .

-

_., I.J

. J.e

r .. ltl

Fig. 18. Time histories of predicted variables P'p and PUp and of measured variables P'm and PUm

8. APPENDIX

The measured and the predicted signals of the nonlinear mass flows mZJ and mZJl in (3.1), (3.2) are shown in Fig. 19.

Fig. 19.

Time histories of predicted variables and m ZJlp and of measured variables and mZJlm

Model variables: u = input voltage of servovalve amplifier, x. = servovalve piston displacement, P, = pressure of actuator chamber I, P" = pressure of actuator chamber 2, mZJ = mass flow into actuator chamber I, mZJl = mass flow into actuator chamber 2, T, = temperature in actuator chamber I, T" = temperature in actuator chamber 2, x. = actuator piston displacement, x. = actuator piston velocity and x. = actuator piston acceleration.

mZJp

m

ZJm

Model parameters: These results show, that the pneumatic processes have been sucessfully estimated.

~.

= servovalve damping factor, = servovalve frequency constant. k. = gain factor of the torque motor, /( = adiabatic exponent, R = gas constant, a l ,a2 ,a3 ,a4 = flow coefficients of the control

w.

7. REFERENCES Backe, W. (1979). Grundlagen der Pneumatik. Vorlesungsumdruck RWTH Aachen. Eykhoff, P. (1974). System identification. 1. Wiley, London. Hahn, H. (1988). Mathematisch/physikalisches ModeII eines ungeregelten elektro servopneumatischen Antriebs. IMAT-Bericht RT-2, Kassel. Hahn, H. (1993). Identification of Linear Systems. IMAT-Bericht RT-12, Kassel. Isermann, R. (1987). Digitale Regelsysteme. Springer, Vol. I and 2, 2nd Edition. Iserrnann, R. (1987). Practical Aspects of Process Identification. 5th IFAC Symposium of System identification. Darmstadt. 1979 Lin, K. C.• Reid. K. N. (1993). An Approach to modeling and identification of systems with signal-dependant parameters. IFAC World Congress. Sydney Ljung, L. (1987). System identification: theory for the user. Englewood Cliffs: Prentice Hall. Pachnicke, E. (1986) Entwicklung von Methoden zur Verbesserung des Positionierverhaltens servopneumatischer Linearantriebe durch Mikroprozessoreinsatz. Dissertation TH Aachen

edges 1,2,3,4 of the servovalve, dl'd 2 ,d3 ,d4 = diameters of the pistons over the

control edges of the servovalve, d. c.

= friction coefficient of the actuator,

X Urir

= parameter of the friction model and

F'"aft

=

viscous damping coefficient.

=

static friction coefficient. Model constants:

= = = =

actuator piston mass, actuator piston area of chamber I, actuator piston area of chamber 2. VOI volume of actuator chamber I for x. =0. Vou = volume of actuator chamber 2 for x. =0, Ps, PR= pressures of fluid supplies and Ts' TR = temperatures of fluid supplies.

m.

A, Au

239