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consists in finding the parameters that minimize the
prediction error weighting the past data in order to
reduce its influence on the present model parameters
(Young, 2011). This technique can be applied to con-
tinuous systems which have been discretized, and the
models outputs accurately represent the outputs sam-
pled from the real system.
An alternative method that allows the recursive es-
timation of the continuous system parameters for a
slowly time variant single-input single-output (SISO)
system is the Recursive Least Squares State-Variable
Filter (RLSSVF) method, introduced in (Padilla et al.,
2016). In that paper, filtering and estimation ap-
proaches are proposed and applied in scalar form to
SISO systems. In this paper, we propose an arrange-
ment to perform the filtering and estimation in state
space form, which allows the direct application of the
method to multivariable systems.
When dealing with the identification of conti-
nuous time systems, the first difficulty encountered is
the need to know a priori the temporal derivatives of
the input and output plant signals. Several methods
have been devised to circumvent this difficulty and
reconstruct the signals temporal derivatives. In (Gar-
nier et al., 2003) comparisons were made between
the offline identification methods from experiments to
study the sensitivity of each approach to design pa-
rameters, sample period, signal-to-noise ratio, noise
power spectral density and the input signal type. The
article (Young and Garnier, 2006) provides an intro-
duction to the key aspects of existing time domain
methods for identifying continuous time linear mod-
els from discrete time sampled data. Each method is
characterized by specific advantages such as: math-
ematical convenience, simplicity in numerical im-
plementation and computation, treatment of initial
conditions, physical vision, precision, among others
(Young, 2011). From those approaches, the State-
Variable Filter (SVF) (Garnier et al., 2008) presents
some advantages such as smoothing of the variables
and, mainly, it deals satisfactorily with the system in-
puts and outputs derivatives. The method is originally
developed for monovariable systems described by dif-
ferential equations.
Once the inputs and outputs of a dynamical system
are known, it is possible to estimate with the Recur-
sive Least Squares Method a set of parameters, which
minimizes the error between the model and the sys-
tem outputs. However, for the state space form, the
system inputs, outputs and states must be known to es-
timate the model parameters. This makes their physi-
cal implementation more complex, since generally the
states of the real system are not measurable. Thus, to
circumvent such a difficulty, the Kalman Filter (KF)
algorithm uses the system inputs and outputs to esti-
mate the states given that a model is known (Rayyam
et al., 2015). Therefore, it allows the implementation
of algorithms for state space system identification.
This algorithm is based on linearizing the model
around the current state. With this method it is
also possible to estimate linear system parameters by
rewriting their state space model, so that the parame-
ters to be estimated are computed as part of the state
vector, making the model non-linear. Examples of
that can be found on the references (Rayyam et al.,
2015).
In this paper a hybrid algorithm composed of three
stages that perform the sampling, the filtering and
the estimation of a continuous time variant dynami-
cal state space model, is shown. In the first stage of
the algorithm, sampling is performed together with
the filtering of the system input and output signals by
the SVF method. In the second stage, the Extended
Kalman Filter (EKF) is applied based on the current
estimated model to estimate the states from the fil-
tered signals. And in the third stage, from the fil-
tered input and output signals and the estimated states,
the RLSSVF method is executed for recursive esti-
mation of the system parameters in state space form.
The motivation for the hybrid algorithm development
is the possibility to estimate recursively continuous
time system parameters in the state space form only
from the system sampled inputs and outputs. This has
strong appeal to real applications where the multiva-
riable continuous systems parameters are required.
The article is organized as follows: in the section
2 the fundamentals of sampling, filtering and identifi-
cation of the continuous time systems are presented.
In the section 3 the methodology and the case study
are illustrated. In the section 4 a case study is pre-
sented, in the section 5, the numerical results of the
method application are presented and in the section 6
the conclusions found in this work are reported.
2 FILTERING AND DIRECT
IDENTIFICATION OF
CONTINUOUS TIME MODEL
In this section, the main algorithms that are used in
the study proposed in this paper are presented. In
the subsection 2.1, the formulation and arrangements
for sampling and filtering the system input and out-
put signals are presented. In the subsection 2.2, the
fundamentals of state variable filtering are reported.
In the subsection 2.3 the filtering and identification
method for linear continuous time slowly time variant
Recursive Identification of Continuous Time Variant Dynamical Systems with the Extended Kalman Filter and the Recursive Least Squares
State-Variable Filter
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