Control Engineering Practice 12 (2004) 409–416
Recursive spline interpolation method for real time engine
control applications
Alexander Stotsky
a,
*, Attila Forgo
b
a
Volvo Car Corporation, Engine Design and Development, Real-Time Control, Volvo Cars, Dept. 97542, HA1N, SE-405 31 Gothenburg, Sweden
b
Volvo Car Corporation, Engine Design and Development, Dept. 97568, HA1S, SE-405 31 Gothenburg, Sweden
Received 6 May 2002; accepted 1 May 2003
Abstract
In this paper, new computationally efficient algorithms for spline interpolation method are explained. Theoretical comparative
analysis of the spline interpolation method with combined high-gain observer and spline interpolation method is presented. New
spline interpolation algorithms are implemented for estimation of the engine angular acceleration from the crankshaft angle
measurements.
r 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Real-time system; Spline interpolation method; Observers; Engine crankshaft acceleration; Combustion quality monitoring; Spark
ignition engines
1. Introduction
Numerical calculation of the derivatives of a signal is
an old problem in numerical analysis and digital signal
processing. The backward difference method gives one
of the simplest numerical differentiators. Despite the
fact that it is quite common in engineering applications
the behavior of the derivative is very often accompanied
with peaking phenomena. Spline interpolation method
proposed by Diop, Grizzle, Moraal, and Stefanopoulou
(1994) is based on on-line least-squares polynomial
fitting over the moving in time window of a size w: The
advantage of this method over the backward difference
method is its good transient behavior. The idea for the
spline interpolation method is to fit a polynomial of a
certain order as a function of time in least squares sense
and take the derivatives analytically. Properties of
this method are described by Diop et al. (1994)
and Dabroom and Khalil (1999). However, several
practical problems remain. Relatively large window size
w requires more on-line computations and makes
practical implementation of the method difficult. This
necessitates the development of computationally effi-
cient recursive algorithms. Moreover, in papers by Diop
et al. (1994) and Dabroom and Khalil (1999) only
constant intersampling time is considered, which in
many practical applications is not constant. For
example, crankshaft angle measurements in auto-
motive engines are based on the measurements of
the crankwheel tooth number and therefore the elapsed
time between two teeth passing a fixed point varies.
This necessitates the development of recursive compu-
tationally efficient algorithms for variable discretization
step.
The contributions of this paper are the following:
*
new computationally efficient recursive spline inter-
polation algorithms for variable discretization step;
*
theoretical comparative analysis of the spline inter-
polation method and combined spline method with
high-gain observer; and
*
real-time implementation of the spline interpolation
method for the crankshaft acceleration estimation.
The paper is organized as follows. In the next section
recursive computationally efficient algorithms for spline
interpolation method are presented. Section 3 is devoted
to the second-order spline example where spline inter-
polation method is described in the first part of this
section. In the second part, comparative analysis of
ARTICLE IN PRESS
*Corresponding author. Tel.: +46-31-325-59-72; fax: +46-31-59-
18-08.
E-mail addresses: astotsky@volvocars.com (A. Stotsky),
aforgo@volvocars.com (A. Forgo).
0967-0661/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0967-0661(03)00114-X