Laboratoire d'Automatique Industrielle, BBt 303-20, Avenue Einstein, 69621. Villeurbanne Cedex. France. ABSTRACT: The singular perturbation method ...
Order Reduction of Multi-time Scale Systems Using Bond Graphs, the Reciprocal System and the Singular Perturbation Method by G.DAUPHIN-TANGUY
and
P.BORNE
Laboratoire d’Automatique et d’lnformatique Industrielle, Institut Industriel du Nord, B.P. 48,59651 Villeneuve d’Ascq Cedex, France and M.LEBRUN Laboratoire d’Automatique Villeurbanne Cedex. France
Industrielle,
BBt 303-20,
Avenue
Einstein,
69621
The singular perturbation method applied to multi-time scale processes enables ABSTRACT: the reduction of dimensionality by considering only one part of the system-the slow or the fast part-depending on thefrequency domain of interest. Thefast and slow dynamics of bond graph models can be estimated by determination of causal loop-gains. We define here the notion of a reciprocal system which, with singular perturbation techniques, can obtain more accuracy on the fast time scale behaviour of the system.
I. Introduction When a system has the multi-time scale property, it is often justified and necessary to separate dynamics, to make an order reduction or to eliminate calculation errors due to very different order of magnitude terms. This is important particularly in real time problems. Many various application examples could be presented such as robotic systems with a slow mechanical part and a fast electronic part, chemical processes with different reaction speeds, systems with sensors and actuators or controlled systems with high gain (1). The representation tool used here is the bond graph modeling method and its associated state equations. Two cases are considered : (1) If the important part of the system is the slow part, the singular perturbation method can be applied and a comparison between these results and those obtained by simplification of the bond graph model is presented. (2) If the study concerns the fast part of the system (fast transients or highfrequency behaviour), this decoupling method cannot be applied directly. The reciprocal system is then defined for all the representation forms previously described. The dynamics are inverted and the fast part becomes the slow reciprocal part, which can be separated by singular perturbation technique. An example concerning a mechanical system is shown.
The Franklin Institute 001&0032/8533
oO+O.OO
157
G. Dauphin-Tanguy, P. Borne and M. Lebrun
Dissipative elements
Storage elements
FIG. 1. Symbolic
II. Representation
form of a bond graph.
of a Continuous Process
A bond graph is composed ofbasic elements, associated to ports, I and C (storage), R (dissipation), S, and S, (sources). Elements 0, 1, TF (transformer), GY (gyrator) compose the junction structure which exchanges energy between the different parts of the dynamic system, and which enforces constraints (2). The symbolic form of a bond graph in integral causality is presented in Fig. 1 which gives, by considering effort vectors and flow vectors, the block-diagram form (Fig. 2) (3). The state variables (energy variables : impulse p, charge q), associated to storage elements in integral causality satisfy a state matrix equation, defined in the linear case by : ~=AX+BU
with
X=
[I i
(1)
and obtained by standard techniques. A can be decomposed as : A = A,S = (A”, + A”,)S
(2)
where S is a symmetrical matrix deduced of storage multiports, A”, characterizes the energy conservation property of the junction structure and is symmetrical. The antisymmetrical matrix A”, concerns the energy dissipation of the dissipative multiport.
~$&z--p-j Disslpatlon
Storage
FIG. 2. Block-diagram
158
form of Fig. 1. Journal of the Franklin Institute Pergamon Press Ltd.
Reduction
of Multi-time Scale Systems
III. Example
Let us consider the mechanical system shown in Fig. 3 corresponding to the suspension of a car. The bond graph representation is shown in Fig. 4. The input vector is composed of a flow source and two effort sources associated with gravity. The inertial, capacitive and resistive elements associated with masses, spring stiffnesses, damping coefficients are linear but can be generalized to non linear laws (4). The bond graph vectors are
_g=
f4-
q4
f3
q3
x.=
,
e6
P6
: e,
Pl.
1 1[ e4
)
z=
e3
fs
fs
,
u=
ell
(3)
e12
fi
Din =fi
DO,, = e2
where q3 and q4 correspond to springs deformations and p1 and p6 are the momenta of the masses. The energy variable state space representation is fs ell
.1
(4)
e12
A basis change (5) transforming the energy state vector in phase variables gives the
x t --
M
33 f
T
9
K
1-2
m
I
k
I I
8
FIG. 3. Physical model. Vol. 319. No. l/Z, pp. 157-171, January/February Printed m Great Britain
1985
159
G. Dauphin-Tanguy, P. Borne and M. Lebrun
TAl r IO
0
CR 3
c
f I/K
7
-mg:
s, +
I
I
-.&pm
0
$&C.I/k
x
I9
1:
5 S:z. f dt
FIG.4. Bond graph model.
following
formulation,
i.e.
