WEBMATHEMATICA BASED TOOLS FOR NONLINEAR
CONTROL SYSTEMS
Heli Rennik, Maris T
˜
onso and
¨
Ulle Kotta
Institute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, Tallinn, 12818, Estonia
Keywords:
Web-based education, webmathematica, nonlinear control, linear algebraic framework.
Abstract:
Algebraic approach of differential one-forms provides simple theoretical framework for several typical prob-
lems of nonlinear control theory that makes it useful for educational purposes. Additional assistance is pro-
vided by Mathematica functions, developed by us and made available by creating a web-based application
using web-Mathematica. These symbolic computation tools provide solutions for several modelling and anly-
sis problems like accessibility, identifiability, realizability and realization and require no other software except
for an internet browser.
1 INTRODUCTION
Over the many years we have developed a software
package NLControl in Mathematica for nonlinear
control systems (Kotta and T
˜
onso, 1999; Kotta and
T
˜
onso, 2003), based on algebraic approach of differ-
ential one-forms (Aranda-Bricaire et al., 1996; Kotta
et al., 2001; Conte et al., 1999). This package pro-
vides basic tools for modelling, analysis and synthesis
both of discrete- and continuous-time systems.
The Mathematica functions developed by us and
described partly in (Kotta and T
˜
onso, 1999; Kotta and
T
˜
onso, 2003) cannot be used outside of Mathematica
environment. Our task was to make our tools avail-
able via the world-wide-web, in such a way that no
other software except for an internet browser needs
to be installed in a computer to use these tools. Re-
cently, we developed a first set of web-based tools.
The reason for this was twofold. First, the computer
has become an integral part of the educational pro-
cess (Moog et al., 2003a). Second, we want to make
the tools, developed by us, available to a larger con-
trol community. The other web-based tool for nonlin-
ear control system is described in (Ondera and Huba,
2005). Since the intention was to create a web-based
application that would re-use the original program
code and whose outputs would mimic the original
tools as much as possible, we used webMathematica.
The paper is organized as follows. Section 2 de-
scribes webMathematica technology both the fron-
tend, which is available to everyone over the web,
and web server technology, which includes descrip-
tion of kernels, kernel pools and configuration op-
tions. Section 3 provides a description of nonlinear
control systems, both discrete- and continuous-time,
together with the necessary assumptions and our ba-
sic tool to handle the considered problems - the se-
quence of subspace H
k
of one forms, associated to the
nonlinear control system. Next, a brief description of
modeling and analysis problems, implemented in our
webMathematica website is given together with some
simple examples and program codes. Finally, the last
section concludes the paper by discussing our contri-
bution and providing some future perspectives.
2 WEB-BASED
IMPLEMENTATION
This section describes a webMathematica website
for nonlinear control systems, both discrete- and
continuous-time. We have implemented several
Mathematica functions programmed by us to web-
Mathematica for public use. WebMathematica is an
option to make our functions available for students
178
Rennik H., Tõnso M. and Kotta Ü. (2007).
WEBMATHEMATICA BASED TOOLS FOR NONLINEAR CONTROL SYSTEMS.
In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 178-183
DOI: 10.5220/0001624501780183
Copyright
c
 SciTePress

and science community without revealing our pro-
gramming code. Our webMathematica website is
simple, including at moment five different function
pages for discrete- and continuous-time systems de-
scribed either by input-output equations or by state
space equations.
2.1 About Webmathematica Frontend
WebMathematica frontend is a web based user inter-
face for Mathematica. WebMathematica adds inter-
active calculations and visualizations to a website by
integrating Mathematica with web server technology.
WebMathematica makes all Mathematica functional-
ity available over the web and is easy to use even for
non-professional programmer. For using webMathe-
matica one has to know HTML basics and use a web
browser.
Websites (or in other words user interfaces) can
use standard web graphical user interface elements,
such as text fields, check boxes, and drop-down lists.
The major browsers, like Internet Explorer, Mozilla
and Netscape Navigator support webMathematica.
WebMathematica allows a site to deliver HTML
pages that are enhanced by the addition of Mathe-
matica commands and uses the request/response stan-
dard followed by web servers. The process how
webMahtematica works is following:
1. Browser sends a request to webMathematica
server.
2. WebMathematica server acquires Mathematica
kernel from the pool.
3. Mathematica kernel is initialized with input pa-
rameters, it carries out calculations, and returns
result to server.
4. WebMathematica server returns Mathematica ker-
nel to the pool.
5. WebMathematica server returns result to Browser.
Requests are sent to the server with webMath-
ematica web pages that are based on two standard
Java technologies: Java Servlet and JavaServer Pages
(JSP) technologies. JavaServer Pages (JSPs) use a
special library of tags that work with Mathematica.
These tags have the form hmsp : tagi. The hmsp :
allocatei tag causes a Mathematica kernel to be al-
located to use for computations. The contents of
the hmsp : evaluatei tags are sent to Mathematica
for computation with the result inserted into the fi-
nal page. The h/msp : allocatei tag frees the Mathe-
matica kernel to be used for another computation.The
library of tags is called the MSP Taglib. In our site
we use also JavaScript and Java. We use JavaScript
for opening and closing windows and communicating
between windows and Java for calculating random in-
puts.
2.2 Webmathematica Web Server
Technology
There are many different combinations of hardware
and operating systems that support webMathematica
components, for example Windows, Linux, Solaris,
Mac OS X. Before one starts to install webMathe-
matica, one has to install Java and a servlet container.
WebMathematica is tested with Apache Tomcat and
JRun. We are using Linux operating system and Tom-
cat as a web container.
WebMathematica server can be configured as nec-
essary. The number of pools of Mathematica kernels
can be specified. Also one can set the number of ker-
nels that are launched when the system starts. The
parameters, as the number of times that each kernel
can be taken from the kernel pool before being shut
down, can be specified in order to clean it up from
unnecessary processes. It is a good idea to shut down
each kernel at a regular time-period. Another param-
eter specifies the maximum number of milliseconds
for processing a page. When this time is exceeded,
the kernel is shut down and restarted. This is useful in
case one is computing large operations.
We are using only one kernel and one kernel pool
and we have got problems with functions that do not
work properly. They intend to let Tomcat ending
up crashing. We have specified the time 30 seconds
in which all responses must be got from the server.
When the web site is made available for the students,
we will increase the number of kernels.
3 IMPLEMENTED FUNCTIONS
Several different functions programmed by us are
gathered into one Mathematica package called
NLControl for solving different control prob-
lems. At moment we have implemented five
functions from this package into webMathemat-
ica website. These functions are
Submersivity,
SequenceHk, Realization, Accessibility
and
Identifiability . We have chosen functions that
are based on subspaces H
k
and cover different model-
ing and analysis problems.
Consider the continuous-time
˙x = f(x,u) (1)
or the discrete-time nonlinear control system
x
+
= f(x,u), (2)
WEBMATHEMATICA BASED TOOLS FOR NONLINEAR CONTROL SYSTEMS
179

respectively, where x ∈ IR
n
, u ∈ IR
m
, f is an analytic
function and x
+
denotes the forward shift of x. In
order to make computations with these equations in
Mathematica, we have created two special objects for
NLControl package: CStateSpace[ f,x,u,t ] and
DStateSpace[ f,x,u,t ] , that represent systems
(1) and (2), respectively. Objects
CIO[ f,u,y,t ]
and DIO[ f,u,y,t ] are used for representing
continuous- and discrete-time input/output equations,
respectively. DStateSpace, CStateSpace, DIO
and CIO are used by all the implemented functions
Submersivity, SequenceHk, Realization,
Accessibility
and Indentifiability .
3.1 Submersivity
The function f in (2) is said to be a submersion, if
generically (i.e. everywhere, except on a set of mea-
sure zero)
rank
∂f
∂(x,u)
= n. (3)
Submersivity assumption has to be satisfied for
discrete-time control systems for the other func-
tions to be applicable.
Submersivity gives True if
discrete-time state equations are submersive. If the
result is
False then the other Mathematica functions
cannot be used since they may yield wrong results.
For checking
Submersivity assumption the sys-
tem has to be given in the state space form with the
list of state and input variables.
The Mathematica block sent to the server looks as
follows:
<msp:evaluate>
MSPBlock[ {$$f, $$Xt, $$Ut},
MSPFormat[ Submersivity[
DStateSpace[$$f, $$Xt, $$Ut, t]],
OutputForm
]
]
</msp:evaluate>
Example: Consider the bioreactor model, where cells
are being grown through the consumption of a sub-
strate (Kazantzis and Kravaris, 2001):
x
1
(t + 1) = x
1
(t) + [
x
1
(t)x
2
(t)
x
1
(t)+x
2
(t)
− 0.08x
1
(t)]T
x
2
(t + 1) = x
2
(t) + [
−x
1
(t)x
2
(t)
x
1
(t)+x
2
(t)
− 0.08x
2
(t)
+0.008]T
(4)
The result is
True .
3.2 SequenceHk
Let
K be the field of meromorphic functions in a
finite number of system variables {x,u
(k)
,k ≥ 0} or
{x(0),u(k),k ≥ 0} for the continuous- and discrete-
time system, respectively. Over the field
K one
can define a vector space
E := span
K
{dϕ | ϕ ∈ K }.
The elements of
E are called one-forms. The rel-
ative degree r of one-form ω in
X = span
K
{dx} is
given by r = min{k ∈ IN | span
K
{ω,..., ω
(k)
} 6⊂ X }
or r = min{k ∈ IN | span
K
{ω(0),... , ω(k)} 6⊂
X }.
Let us define a decreasing sequence of subspaces
H
0
⊃
H
1
⊃
H
2
⊃ . . . such that each
H
k
, for k > 0, is
the set of all one-forms with relative degree at least
k(Aranda-Bricaire et al., 1996; Conte et al., 1999):
H
0
= span
K
{dx,du}
H
k
= {ω ∈
H
k−1
|
˙
ω ∈
H
k−1
}
or
H
k
= {ω ∈
H
k−1
| ω
+
∈
H
k−1
}.
There exists an integer k
∗
> 0 such that
H
k
⊃
H
k+1
,
for k ≤ k
∗
, and
H
k
∗
+1
=
H
k
∗
+2
= ...
H
∞
,
H
k
∗
6⊇
H
∞
.
The existence of k
∗
comes from the fact that each
H
k
is a finite dimensional vector space, so that at each
step either the dimension decreases by at least one or
H
k+1
=
H
k
.
H
∞
contains the one-forms with infinite relative
degree so that these one-forms will never be influ-
enced by the control.
An one-form ω ∈
E is exact if dω = 0, and closed
if dω
ω = 0 where by is denoted the wedge prod-
uct. We say that the subspace is completely integrable
if it admits the basis which consists only of closed
one-forms.
The function SequenceHk computes first N ele-
ments in the sequence of subspaces Hk associated to
state equations, where N is an positive integer speci-
fied by us. For running
SequenceHk , the system has
to be given in the state space form with the list of state
and input variables.
The Mathematica block sent to the server looks as
follows:
<msp:evaluate>
MSPBlock[ {$$f, $$Xt, $$Ut},
MSPFormat[ SequenceHk[
DStateSpace[$$f, $$Xt, $$Ut, t], All],
OutputForm, TimeArgument-> True
]
]
</msp:evaluate>
Example 1: Consider another bioreactor model,
where the microorganisms grow by consuming the
substrate (Benamor et al., 1997):
x
1
(t + 1) = x
1
(t) + T
a
1
x
1
(t)x
2
(t)
a
2
x
1
(t)+x
2
(t)
− Tu(t)x
1
(t)
x
2
(t + 1) = x
2
(t) − T
a
3
a
1
x
1
(t)x
2
(t)
a
2
x
1
(t)+x
2
(t)
− Tu(t)x
2
(t)
+Ta
4
u(t)
y(t) = x
1
(t)
(5)
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180

The result is the following:
H
1
= Span[dx
1
(t),dx
2
(t)]
H
2
= Span[dx
1
(t)]
H
3
= {0}
(6)
Example 2: The dynamic model of a current-driven
induction motor expressing the rotor flux and the
stator currents in a reference frame rotating at syn-
chronous speed is given by the continuous-time state
equations(Bazanella and Reginatto, 2000):
˙x
1
(t) = −c
1
x
2
(t) − u
1
(t)x
2
(t) + c
2
u
3
(t)
˙x
2
(t) = −c
1
x
2
(t) + u
1
(t)x
1
(t) + c
2
u
2
(t)
˙x
3
(t) = −c
3
x
3
(t) + c
4
(c
5
(x
2
(t)u
3
(t) − x
1
(t)u
2
(t))
−Tm)
(7)
The result is the following:
H
1
= Span[dx
1
(t),dx
2
(t),dx
3
(t)]
H
2
= {0}
(8)
3.3 Realization
Consider a higher order i/o differential
y
(n)
= Φ(y,. . .,y
(n−1)
,u,... , u
(s)
) (9)
or difference equation
y(t + n) = Φ(y(t), . . . ,y(t + n− 1),
u(t),...,u(t + s)),
(10)
where n and s are nonnegative integers, s < n and Φ
is a real analytic function. The realization problem is
to construct the state equations of order n,
˙x = f(x,u) x
+
= f(x, u)
or
y = h(x) y = h(x)
for the i/o equation (9) and (10), respectively. It is im-
portant to stress that the state-space realization does
not exist for every i/o equation. In order to find state
equations, one has to compute the sequence
H
k
for
the i/o equations (9) or (10). The detailed procedures
can be found in (Kotta et al., 2001) and (Moog et al.,
2003b), respectively.
Theorem 1. The i/o equation has a state-space re-
alization iff for 1 ≤ k ≤ s + 2 the subspaces
H
k
are
completely integrable. The state coordinates can be
found by integrating the basis functions of
H
s+2
.
Function
Realization determines whether the
nonlinear higher order input-output difference equa-
tion can be realized in the classical state-space form
and the if the i/o equation is realizable, finds the state
equations. For running
Realization the system has
to be given by the input-output equations and the list
of input and output variables.
The Mathematica block sent to the server looks as
follows:
<msp:evaluate>
MSPBlock[ {$$eqs, $$Ut, $$Yt},
MSPFormat[ Realization[
DIO[$$eqs, $$Ut, $$Yt, t], [#][t]&],
OutputForm, TimeArgument-> True
]
]
</msp:evaluate>
Example 1: The fed-batch bakers’ yeast fermentation
process 1 is described by the following i/o equations
(Keulers et al., 1993).
y(t + 1) = 0.9106y(t) + 2.072u(t + 1) − 1.903u(t)
+120.7y(t)u(t +1) − 107.3y(t)u(t) + 299.6u(t + 1)
2
−232.8u(t + 1)u(t) − 84.17u(t)
2
(11)
The result is that the classical state space form does
not exist for system (11).
Example 2: Consider the continuous-time i/o
equation:
...
y
(t) = ¨y(t)u(t) + ˙u(t)y(t) + u(t) (12)
The classical state equations for (12) are:
˙x
1
(t) = x
2
(t)
˙x
2
(t) = u(t)x
1
(t) + x
3
(t)
˙x
3
(t) = u(t)(1+ u(t)x
1
(t) − x
2
(t) + x
3
(t))
y(t) = x
1
(t)
(13)
3.4 Accessibility
A function ϕ
r
in
K is said to be an autonomous vari-
able for system (1) or (2), if there exist an integer
µ ≥ 1 and a non-zero meromorphic function F so that
F(ϕ
r
,sϕ
r
,..., s
µ
ϕ
r
) = 0 or F(ϕ
r
,δϕ
r
,..., δ
µ
ϕ
r
) = 0
for continuous- and discrete-time system, respec-
tively. The system (1) or (2) is said to be accessible
if there does not exist any non-zero autonomous vari-
able in
K .
Theorem 2. (Aranda-Bricaire et al., 1996; Conte
et al., 1999) The nonlinear system is strongly accessi-
ble iff
H
∞
= {0}. (14)
Note that the Theorem 1 and 2 holds both in
continuous- and discrete-time cases though the rules
to calculate the subspaces
H
k
are different.
Accessibility returns True if the nonlinear sys-
tem is strongly accessible and
False otherwise.
For computing Accessibility the system has to
be given in the state space form with the list of state
and input variables.
The Mathematica block sent to the server looks as
follows:
WEBMATHEMATICA BASED TOOLS FOR NONLINEAR CONTROL SYSTEMS
181

<msp:evaluate>
MSPBlock[ {$$f, $$Xt, $$Ut},
MSPFormat[ Accessibility[
DStateSpace [$$f, $$Xt, $$Ut, t]],
OutputForm
]
]
</msp:evaluate>
Example 1: Consider a grain drying process (Kotta
and Nurges, 1985).
x
1
(t + 1) = −0.0081u(t)x
1
(t) + x
2
(t)
x
2
(t + 1) = −0.01772892u(t)x
1
(t)
+ 1.6332x
2
(t) + x
3
(t)
x
3
(t + 1) = (−0.1751− 0.00360073u(t))
x
1
(t) − 0.4567x
2
(t)
(15)
The result is
True , that is system (15) is accessible.
Example 2: Consider a simple academic example:
x
1
(t) = x
1
(t)(x
2
3
(t) + 1)
2
x
2
(t) = x
2
(t)(x
2
3
(t) + 1)
3
x
3
(t) = x
3
(t) + u(t)
(16)
The result is
False . H
∞
= span{3x
2
dx
1
− 2x
1
dx
2
}.
3.5 Identifiability
Identifiability property characterizes the possibility to
find the unknown parameters of the system from the
measured input-output data. Consider the continuous-
time nonlinear system
˙x = f(x,u,θ)
y = h(x,u,θ),
(17)
where x ∈ IR
n
, u ∈ IR
m
and the parameter vector θ ∈
IR
q
. There are two concepts for identifiability - with
and without the knowledge of initial conditions. At
the moment only identifiability without knowledge of
the initial state is implemented on our website. Our
website offers two alternative methods to check iden-
tifiability of the system. The first is algebraic method,
that enables to find the parameters by solving certain
algebraic equations depending only on input-output
information, see (Xia and Moog, 2003). The sec-
ond method checks if the parameters can be found
by the least squares method (LSM). To check iden-
tifiability by the LSM one has to find the i/o repre-
sentation E(y, ˙y, . . .,y
(n)
,u, ˙u,...,u
(s)
,θ) = 0 for sys-
tem (17). If the above i/o equation admits a separa-
ble form
∑
n
i
i=1
g
i
(θ)E
i
(y, ˙y,...,y
(n)
,u, ˙u,...,u
(s)
) and
the condition g(θ) = g(θ
∗
) ⇔ θ = θ
∗
is satisfied, the
least squares method described in (Pearson, 1989), is
applicable for parameter identification.
For computing
Identifiability the system has
to be given in the state space form with the list of state,
input and output variables.
The Mathematica block sent to the server looks as
follows:
<msp:evaluate>
method = Switch[ MSPToExpression[ $$method],
algebraic, Algebraic, pearson, Pearson];
MSPBlock[
{$$f, $$Xt, $$Ut, $$h, $$Yt, $$theta},
MSPFormat[ Identifiability[
CStateSpace[f, Xt, Ut, t, h, Yt],
{theta},Method->method],
OutputForm
]
]
</msp:evaluate>
Example: Consider the simple academic example
˙x
1
(t) = θ
1
x
1
(t)
2
+ θ
2
x
1
(t)x
2
(t) + u(t)
˙x
2
(t) = θ
3
x
1
(t)
2
+ θ
4
x
1
(t)x
2
(t)
y(t) = x
1
(18)
The system is identifiable neither with LSM nor with
algebraic method.
3.6 Comparative Evaluation
In this section our web-based tools are compared by
web-based tools for nonlinear control systems in (On-
dera and Huba, 2005). Besides the fact the problems
handled are different ((Ondera and Huba, 2005) is
dedicated solely to the exact static state feedback lin-
earization problem) there are other points to be men-
tioned.
Our web-site enables user to specify the system
equations and calculate the state transformation and
linearizing control law (work in progress, not de-
scribed in this paper). The web-tools in (Ondera and
Huba, 2005) allow the user additionally to submit a
desired closed-loop poles of a pole-placement con-
troller and to perform a simulation.
There is also a difference in chosen technology.
The web-tools in (Ondera and Huba, 2005) are based
on MATLAB and its Symbolic Math Toolbox. This
toolbox is a Maple 8 symbolic kernel that was bought
from Maplesoft and implemented into MATLAB by
The MathWorks. The different platform also implies
a different internet implementation. In (Ondera and
Huba, 2005) tools are web-accessible via MATLAB
Web Server that is based on CGI technology, whereas
webMathematica is java and javascript-based.
Both web-tools relieve users from installing Math-
ematica or MATLAB on their computers and help to
make programs available to everyone without seeing
program code.
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182

4 CONCLUSION
WebMathematica is a web version of Mathemat-
ica that uses a web server technology, HTML and
Java Servlet Pages. Calculations entered via web
pages are sent to kernel, where the result is cal-
culated and sent back to web pages. Several dif-
ferent functions programmed by us are gathered
into one Mathematica package called NLControl
for solving different modeling, analysis and synthe-
sis problems. At moment we have implemented
five functions from this package into webMathemat-
ica website. These functions are
Submersivity,
SequenceHk, Realization, Accessibility
and
Identifiability . In the future we are expand-
ing our website with functions
Linearization and
PrimeForm . The function Linearization checks if
the state equations can be linearized via the static state
feedback and coordinate transformation, and finds
necessary transformations. The function
PrimeForm
transforms the system into the prime form, when-
ever possible, using the static state feedback, and
the coordinate transformations in the state and output
spaces. Besides continuous- and discrete-time sys-
tems we are also programming functions for systems,
described on homogeneous time scales (Bartosiewicz
et al., 2007).
ACKNOWLEDGEMENTS
This work was partially supported by the Estonia Sci-
ence Foundation Grant No 6922.
The authors thank M. Ondera for discussions on com-
parative evaluation of web-based symbolic tools pre-
sented in this paper and in (Ondera and Huba, 2005).
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