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Robust identification of pneumatic servo actuators in the real situations

Robuste Identifikation von pneumatischen Servo-Aktuatoren in der realen Situationen

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Abstract

Intensive research in the field of mathematical modelling of the pneumatic cylinder has shown that its mathematical model is nonlinear and that a lot of important details cannot be included in the model. Selection of the model and the identification method have been conditioned by the following facts:

  1. (a)

    The nonlinear model of the system can be approximated by a linear model with time-variant parameters.

  2. (b)

    There is the influence of the combination of heat coefficient, unknown discharge coefficient and change of temperature on the pneumatic cylinder model. Therefore it is assumed that the parameters of the pneumatic cylinder are random (stochastic parameters).

  3. (c)

    In practical conditions, observations have a non-Gaussian distribution.

Due to the abovementioned reasons, it is assumed that the pneumatic cylinder model is a linear stochastic model with variable parameters. The Masreliez-Martin filter (robust Kalman filter) was used for identification of parameters of the model. For the purpose of increasing the practical value of the filter, the following two heuristic modifications were performed:

  1. (1)

    It was adopted that T(k)=1 holds for the scalar transformation of residuals.

  2. (2)

    Fisher information was approximated by a derivative of the Huber’s function.

The proposed modifications were confirmed through intensive simulations. In order to provide persistent excitation, the autocovariance function “1/f” of the signal was used. The behaviour of the new approach to identification of the pneumatic cylinder is illustrated by simulations.

Zusammenfassung

Intensive Forschung auf dem Gebiet der mathematischen Modellierung des pneumatischen Zylinders hat gezeigt, dass sein mathematisches Modell nichtlinear ist und dass viele wichtige Details nicht in das Modell einbezogen werden können. Die Auswahl des Modells und die Art der Identifikation werden durch folgende Tatsachen bedingt:

  1. (a)

    Das nichtlineare Modell des Systems kann durch ein lineares Modell mit zeitvarianten Parametern angenähert werden.

  2. (b)

    Es besteht ein Einfluss der Kombination von Wärmedurchgangs-Koeffizient, unbekanntem Durchflusskoeffizienten und Änderungen der Temperatur auf das pneumatischen Zylinder-Modell. Es wird daher angenommen, dass die Parameter des pneumatischen Zylinders zufälligen Charakters sind.

  3. (c)

    Unter praktischen Bedingungen haben die Beobachtungsergebnisse eine nicht-Gaußsche Verteilung.

Aufgrund der vorgenannten Gründe wird davon ausgegangen, dass das Pneumatikzylinder Modell ein lineares, stochastisches Modell mit variablen Parametern sein muss. Der Masreliez-Martin-Filter (robust Kalman-Filter) wurde für die Identifizierung von Parametern des Modells verwendet. Zur Erhöhung des praktischen Werts des Filters, wurden die beiden folgenden heuristischen Modifikationen durchgeführt:

  1. (1)

    Es wird angenommen, dass T(k)=1 für das skalare Transformation der Residuen hält.

  2. (2)

    Die Fisher-Information wird durch ein Derivat des Hubers Funktion approximiert.

Die vorgeschlagenen Änderungen werden durch intensive Simulationen bestätigt. Um für eine anhaltende Erregung zu sorgen, wird die Autokovarianzfunktion “1/f” des Signals verwendet. Das Verhalten des neuen Ansatzes zur Identifikation des pneumatischen Zylinders wird durch Simulationen aufgezeigt.

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Abbreviations

P S :

supply pressure

P i :

pressure in the chamber i=a,b

P o :

outer absolute pressure

m :

total mass of the piston and the load

β e :

nonlinear viscous friction coefficient

k e :

load spring gradient

F ext :

load force disturbance on the piston

F f :

friction forces

A i :

effective area of the piston

T S :

supply temperature

R :

the universal gas constant

\(\dot{m}_{l}\) :

leakage mass flow rate between the cylinder chambers

\(\dot{m}_{i}\) :

mass flow rates through the orifice i=a,b

C d (t):

valve discharge coefficient

α(t):

heat coefficient

β(t):

uncertain bound parameter

τ(t):

variation of the temperature

W :

port width

y :

displacement of the piston

y(k):

discrete scalar observations of the piston displacement

x(k):

discrete time n×1 state vector

u(k):

input signal

F(k):

n×n state transition matrix

H(k):

1×n observation matrix

w(k):

process noise

v(k):

measurement noise

T(k):

a linear transformation

W(k):

covariance for process noise w(k)

P(k|k−1):

a priori covariance matrix

P(k|k):

a posteriori covariance matrix

\(E_{P_{\varepsilon}} \{ \cdot\}\) :

expectation with respect to the least favourable pdf

I(p):

Fisher information for the least favourable pdf

ε :

degree of contamination

ψ p [⋅]:

vector influence function

φ(k):

1×n regression vector

θ(k):

n×1true parameter vector

\(\bar{\theta}\) :

mean value of true parameter vector

Σ θ :

covariance matrix of true parameter vector

\(\hat{\theta} (k)\) :

estimation of true parameter vector

C :

a priori known non-singular matrix

\(\hat{y}(k)\) :

adjustable predictor

a i ,b j :

system parameters (i=1,…,n;j=1,…,m)

\(r_{k}^{d}, \hat{r}_{k}\) :

desired and non central autocovariances of excitation signal

s t :

excitation signal

ϕ 1/f :

spectrum of bandlimited noise

\(\omega, \underline{\omega} ,\bar{\omega}\) :

frequency and its lower and upper limits

a :

head side of the piston

b :

rod side of the piston

ARX:

autoregressive with exogenous input

ARMA:

autoregressive moving average

ARMAX:

autoregressive moving average with exogenous input

RLS:

recursive least square

OE:

output error

MSE:

mean square error

SISO:

single input/single output

pdf:

probability density function

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Authors and Affiliations

  1. Faculty of Mechanical Engineering Kraljevo, Department of Energetics and Automatic Control, University of Kragujevac, Dositejeva 19, 36000, Kraljevo, Serbia

    V. Filipovic, N. Nedic & V. Stojanovic

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  1. V. Filipovic
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  2. N. Nedic
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  3. V. Stojanovic
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Correspondence to V. Stojanovic.

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Filipovic, V., Nedic, N. & Stojanovic, V. Robust identification of pneumatic servo actuators in the real situations. Forsch Ingenieurwes 75, 183–196 (2011). https://doi.org/10.1007/s10010-011-0144-5

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