Computers and Chemical Engineering 28 (2004) 487–499
On-line adaptive parameter estimator tuning
Michael Shagalov, Hector Budman
∗
Department of Chemical Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1
Received 14 January 2003; received in revised form 15 July 2003; accepted 19 August 2003
Abstract
This paper discusses a systematic technique for tuning a parameter estimator. The estimator used in this work is a discrete version of a
projection algorithm estimator. Although the development is explained for linear models, the method is equally applicable to any type of
linear or non-linear model that is linear with respect to the estimated parameters.
The estimation scheme includes a first-order filter in the error dynamics equation and individual parameter adaptation equations. A simple
analytical formula was derived for on-line tuning of the filter gain based on the maximisation of the decay of the norm of the output and
parameter errors. The tuning of the parameter adaptation gains is also done based on the maximisation of the decay of this norm. The
measurement noise problem was addressed using a specific dead zone approach. The algorithm performance is compared to the conventional
RLS algorithm.
© 2003 Elsevier Ltd. All rights reserved.
Keywords: Adaptive estimation; Tuning; Lyapunov stability
1. Introduction
Chemical systems are generally non-linear due to nonlin-
earity of reaction rate expressions or non-linear dependency
of physical parameters with respect to temperature, pressure
or flow conditions. Often, these non-linear models cannot
be easily generated using first principle equations and con-
sequently, empirical non-linear models are used for estima-
tion, model based filtering and control. New developments
in neural networks (NNs) models based on wavelets or ra-
dial basis functions have generated renew interest in the area
of adaptive estimation and control of non-linear systems
based on these novel empirical non-linear models (Sanner &
Slotine, 1992a, 1992b; Kavchak & Budman, 1999).
For example, a model based on non-linear basis functions
is formulated as follows:
y
k
=
n
i=1
a
i
f
1
(y
k-i
) +
m
j=1
b
j
f
2
(u
k-j
) (1.1)
In order to represent the non-linear behaviour of the
system different types of basis functions may be used for
∗
Corresponding author. Tel.: +1-519-888-4601;
fax: +1-519-746-4979.
E-mail address: hbudman@uwaterloo.ca (H. Budman).
f
1
(y
k-i
) andf
2
(u
k-i
) such as Gaussians or Wavelet func-
tions. In Eq. (1.1), y
k -i
and u
k-j
denote past output and
input measurements, respectively.
This superposition structure is generally referred to as a
NN (Sanner & Slotine, 1992a, 1992b). The coefficients a
i
and b
j
are identified from experimental data. These coeffi-
cients are often learned on-line because processes are either
time-varying or non-linear so they will require a large num-
ber of experiments at different operating points for complete
off-line identification. When linear dynamics is considered,
a discrete dynamic equation can be expressed as follows:
y
k
=
n
i=1
a
i
y
k-i
+
m
j=1
b
j
u
k-j
(1.2)
Thus, it is evident from Eqs. (1.1) and (1.2) that the pa-
rameter estimation problem is similar for both the linear
and non-linear models given by Eqs. (1.1) and (1.2), since
both methods are linear with respect to the parameter set
[a
1
,a
2
,...,a
n
,b
1
,b
2
,...,b
m
] to be learned. Eq. (1.2) may
be seen as a special case of Eq. (1.1) when f(x) = x. This
fact is important for the present study since the methodol-
ogy in this work has been illustrated for the simpler linear
case but is equally applicable to the non-linear models given
by Eq. (1.1).
0098-1354/$ – see front matter © 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compchemeng.2003.08.003