Open Journal of Applied Sciences, 2013, 3, 51-55
doi:10.4236/ojapps.2013.32B010 Published Online June 2013 (http://www.scirp.org/journal/ojapps)
Specification and Analysis of Discrete Behavior
of Hybrid Systems in the Workbench ISMA
Yury V. Shornikov, Dmitry N. Dostovalov, Maria S. Myssak, Artem N. Komarichev,
Andrey M. Tolokonnikov
Department of Automated Control Systems, Novosibirsk State Technical University, Novosibirsk, Russia
[e-mail], [e-mail]
Received 2013
ABSTRACT
Hybrid systems are important in applications in CAD, real-time software, robotics and automation, mechatronics, aero-
nautics, air and ground transportation systems, process control, and have recently been at the center of intense research
activity in the control theory, computer-aided verification, and artificial intelligence communities. In the past several
years, methodologies have been developed to model hybrid systems, to analyze their behavior, and to synthesize con-
trollers that guarantee closed-loop safety and performance specifications. These advances have been complemented by
computational tools for the automatic verification and simulation of hybrid systems. Modern technologies of computer
simulation tools include preparing, debugging, analysis and calculation of effective program models, meaningful inter-
pretation of research results.
Keywords: Hybrid System; Statechart; Event Detection; Differential-algebraic Equations; Simulation Tools
1. Introduction
There are many systems (mechanical, electrical, chemi-
cal, biological, etc.), the behavior of which can be con-
veniently described as a sequential change of continuous
modes. These systems are referred to as hybrid (HS) or
event-continuous. Each mode is given by a set of differ-
ential-algebraic equations with the following constraints:
 


 
0000
0
y
x
yy
x
y
x
y
x
k
N
N
NN
N
N
NNx
N
NS
yfx,y,t,x x,y,t,
pr:gx,y, t,
tt,t,xt x,yty
xR ,yR,tR,
0
,
f
:RRRR ,
:RRRR ,
g
:RRRR .










)
(1)
The vector-function (
g
x,y,t
0
is referred to as event
function or guard. A predicate determines the con-
ditions of existence in the corresponding mode or state.
Inequality means that the phase trajectory
in the current mode should not cross the border
. Events occurring in violation of this con-
dition and leading to transition into another mode with-
out crossing the border are referred to as one-sided.
Many practical problems are characterized by stiff modes,
and the surface of boundary has sharp
angles or solution has several roots at the boundary [1].
Numerical analysis of such models by traditional meth-
ods is difficult or impossible, as it gives incorrect results.
Therefore it is necessary to use special methods to detect
events accurately.
pr
()gx ,y,t
)0(gx ,y,t
()gx ,y,t0
Computer analysis of these systems is typically per-
formed in simulation tools, best of which are Charon
(USA), AnyLogic (Russia), Scicos (France), MVS (Rus-
sia), Hybrid Toolbox and HyVisual (USA), DYMOLA
(Sweden) and etc.
The specification of the discrete behavior of HS re-
flects the instantaneous discrete transitions from one state
to another one. State diagrams allow to easily describe
HS models of any formalism and to represent semantics
of HS modes and mechanism of discrete transitions in
intuitive manner.
This paper describes the features of design and HS
models analysis in the ISMA instrumental environment
[2].
2. Discreet Behavior of Hybrid Systems and
Its Specification
Statecharts have been widely used since a variant has
become a part of the Unified Modeling Language (UML).
This way of describing hybrid systems commonly used

Y. V. SHORNIKOV ET AL.
52
in information technology and related fields. This ap-
proach was proposed by D. Harrel [3] and is a descrip-
tion of complex dynamic systems. Statecharts represent
directed graph whose nodes correspond to states of con-
tinuous object, arcs – to discrete transitions (changing of
states). System can be represented by statecharts if it is
characterized by a finite number of continuous states.
The method allows to describe the operating logic of the
object in visual form in order to represent its behavior at
the continuous parts of the phase trajectory and to specify
the continuous state changes conditions. For example,
Figure 1 shows a Harel statechart, where A,B,C corre-
sponds to the continuous states; α, β, γ – to the predicates
(conditions).
Graphics Editor
Graphics editor was designed by our team for a clearer
representation of statecharts (Figure 2).
The editor is developed on the object-oriented pro-
gramming language Java. The main reason for choosing
this language is that programs written in Java are cross-
platform.
α
β
γ
A B
C
Figure 1. Example of statetchart.
Figure 2. Graphics editor.
The right panel is for editing of the base statechart. On
the left panel there are elements that can be dragged to
the right panel for further work with them (drag-and-
drop). In addition, there was developed menu that is
standard for applications running on the Windows.
The editor commands are controlled by hot keys.
These keys are responsible for copy, paste, delete, and
edit the selected state or arc. Thus the editing of states
and transitions is a convenient process.
Designed application allows you to convert a graphical
representation of simulation model. The transformation
algorithm includes the steps of the analysis and the selec-
tion of all the states and transition, relationships defini-
tion and generation in computational model taking into
account the defined relations.
3. Event Detection in Hybrid Systems
The correct analysis of hybrid models is significantly
depends on the accuracy of detection of the change of the
local states of the HS. Therefore, the numerical analysis
is necessary to control not only the accuracy and stability
of the calculation, but also the dynamics of the event-
function. The degree of approximation by the time the
event occurred is defined by the behavior of event driven
function.
Consider the mode of one-sided HS as a Cauchy prob-
lem with constraints (1).
Any non-linear guard (g)
x
,y,t can be reduced to
linear form by adding the phase variable ()zg
x
,y,t.
As a result, problem (1) can be rewritten as follows (for
simplicity we omit the algebraic equations and proceed
to the autonomous Cauchy problem:
0
gg
yf(y),zf(y) z
yt
,

 .




In solving these problems by using explicit methods
[4], we obtain 11nnn
yyh
n



, . Then
the event dynamic is described as
012n, ,,...
11nnnnnn1
g
g( yh,th)




.
Decomposing the 1n
g

1n
g
 in a Taylor series and
taking into account the linearity of

g
y,t, we obtain
the dependence of 1n
g

of the projected step 1n h

:
11
n
nnn n
n
g
g
ggh yt


 
 



(2)
Theorem. The choice of the step according to the
formula
11n
nnn
n
g
h()
g
g
yt






(3)
where 01y(,)

, provides the event-dynamics behavior

Y. V. SHORNIKOV ET AL. 53
as a stable linear system, the solution of which is asymp-
totically approaching to the surface .
0g(y,t) 
Proof. Substituting (3) in (2), we have 1nn
g
g



,
. Converting recurrently this expression we
get 0
012n,,,
1n1n
g
g


. Given that 1

, then
takes place when . In addition, condition
0
n
g
0
n


implies that function n
g
does not change sign. There-
fore, when 0, will be valid for all n. Then
the guard condition will potentially never cross poten-
tially the dangerous area
0g0
n
g


0
nn
t
n
t
gy, , which com-
pletes the proof.
Let us formulate integration algorithm, taking into ac-
count the forecast of step by an event function. Let the
solution n in the point n is calculated with step n.
Then the approximate solution at the point
y th
1

is cal-
culated as follows.
Step 1. Calculate the function nn
f
f(y )
n
.
Step 2. Calculate the functionsn
g
g(n y ,t),
nnn
g
/y g(y,t)/y, nnn
g
/t g(y,t
1
)/y .
Step 3. Calculate the step
p
n
h by (3), where
nn
f

.
Step 4. A new step 1n is calculated by the formula
1
1, where 1 is step by selecting
the numerical method of integration.
h

1
pn
nn
n
hminh,h


n
n
h
Step 5. Go to next step of integration.
In the practical implementation of the algorithm it is
necessary to consider the following. Near the boundary
regime denominator (3) will be positive, and away from
the boundary becomes negative. Then, de-
fining the direction of change event-function, we cannot
perform step 4 of the algorithm and do not impose any
further restrictions on the integration step if the event-
function is removed from the boundary of state.
0g(y,t ) 
Test of the Event-detection Algorithm
To illustrate the event-detection algorithm we consider a
hybrid system of two oscillating masses on springs [5],
shown in Figure 3.
The system can be in one of two local states: when
masses move separately or together. Mathematical model
is not presented here because of the proximity to descrip-
tion of the computer model. A computer model of system
in the ISMA shown in Figure 4.
Qualitative simulation results are obtained with en-
abled event-detection algorithm (Figure 5). Traditional
analysis of the system without using the event-detection
algorithm does not allow to obtain valid results as shown
in (Figure 6).
Figure 3. The system of two oscillating masses.
Figure 4. A computer model of the system in ISMA instru-
mental environment.
Figure 5. Calculation results (using the event-detection al-
gorithm).
Figure 6. Calculation results (excluding the event-function
dynamics).
4. Dry Friction Dynamics Simulation
Simulation object shown in the Figure 7 represents a
system described in [6].
C opyright © 2013 SciRes. OJAppS

Y. V. SHORNIKOV ET AL.
54
Figure 7. Simulation object.
The motion of body has an oscillation nature, where
the motion stage and the resting stage are changing peri-
odically. The motion stage is divided on the motion in
range of Shtribeck effect at low speeds and the motion in
range of Amonton-Coloumb law.
Figure 8. The hybrid model of dry friction.
Obtained simulation results are shown in the Figure 9
and Figure 10. Time charts of tension of spring x, body
velocity V, friction force F, displacement of the body
relative to the origin l and local state z are presented in
Figure 9. Figure 10 illustrates the phase diagram.
The Hybrid model in the formalism of hierarchical
behavioral maps designed in the ISMA is shown in Fig-
ure 8.
Figure 8(a) corresponds to the top-level behavioral
map, Figure 8(b) illustrates the motion stage.
Figure 9. Simulation results. Time characteristics.
Figure 10. Simulation results. Phase diagram.
5. Acknowledgements
This work was supported by the Russian Foundation for
Basic Research under grant 11-01-00106-a.
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