Lecture A2 : Mathematical models (1)
Jan Swevers
July 2006
Aim of this lecture :
• Teach you how to derive a mathematical model starting from a physical
model
• Explain you the analogies/similarities between different kinds of systems
• Introduce system graphs: an instrument to derive the system equations
in a systematic way
0-0
Lecture A2 : Mathematical models (1)
1
Contents of this lecture
• Introduction
• Dynamic system elements
• System graphs
• Generalized continuity and compatibility
• Formulation of the system equations
• Examples
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
2
Introduction
The behavior of a physical system is determined by the flow, storage and
exchange with its environment of different types of energy.
The analysis of the behavior of a system requires physical models :
• consist of ideal system elements
• these elements are realizations of relations between physical variables (for
example: force, flow rate, voltage, etc)
• these relations are called equations of motion or system equations :
mathematical models.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
3
Formulation of the system equations: three steps
• Step 1: Selection of the physical variables :
– Through variables measure the flow of a quantity though the system
element: current through a resistor, force through a spring, fluid flow
through a pipe. A through variable has the same value at the two
terminals or ends of the element, and in order to measure it at a point,
we must sever the system at that point and insert the measurement
device.
– Across variables measure the difference between a quantity at both ends
of an element: voltage across a resistor, pressure drop across a pipe,
difference between the velocity at both ends of a damper. An across
variable is specified in terms of a relative value or difference between the
terminals. In order to measure it, the measurement device is placed in
parallel to the considered elements, and its terminals are connected to
the terminals of the element.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
4
Formulation of the system equations: three steps (2)
• Step 2: Formulate the elemental equation between the through
and across variable
There exists for each system element, an elemental equation between the
across and through variable: physical laws, element equations, etc ...
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
5
Formulation of the system equations: three steps (3)
• Step 3: Generalized continuity and compatibility : describe how the
different system elements interact, exchange energy.
– Continuity requirements : vertex or continuity requirements between the
through variables. (Newton’s law, Krichhoff’s current law, law of
conservation of matter and energy)
– Compatibility requirements : path or compatibility requirements
between across variables. (geometric constraints, Kirchhoff’s voltage
law, and in fluid and thermal systems, compatibility is a statement of
the fact that pressure and temperature are scalar quantities and must
add as one moves from point to point)
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
6
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
7
Dynamic system elements
• Two-terminal or single-port elements
– generalized inductance
– generalized capacitance
– generalized resistance
– pure energy sources
• Four-terminal or two-port elements
– pure transformers
– pure gyrators
– pure transducers
– dependent sources
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
8
The dynamic behavior of a system element is determined by the relationship
between its through and across variables:
• the through variable f has the same value at both ends of the element
• the across variable v is specified in terms of a relative value or difference
between the terminals
• we introduce here the integrated through variable h and the integrated
across variable x :
f
=
v
=
dh
dt
dx
dt
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
9
• The power or energy that enters a system element equals:
P
E
= fv
Z t
Z t
P.dt =
f vdt
=
0
0
• Exception in thermal systems : the power IS the through variable.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
10
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
11
Two-terminal or single-port elements
• generalized inductance
• generalized capacitance
• generalized resistance
• pure energy sources
• system graph representation
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
12
Generalized inductance
Defined by a single-valued relationship between their through variables f and
their integrated across variables x:
• Examples: pure translational and rotary springs, pure inductance, and
pure fluid inertance.
• x21 = f1 (f ) with x21 = 0 if f = 0 and f1 is a single-valued function.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
13
• The inductance is ideal (linear) if :
x21 = L . f of v21 = L . df /dt
with L constant : called the generalized inductance.
– for an ideal spring, L is called the compliance (= (stif f ness)−1 ).
– for an ideal inductance, L is called the inductance.
– for ideal fluid elements, L is called the inertance.
– there is no inductive thermal element.
• The energy E(t) supplied to a pure inductance equals:
Z f (t)
Z t
E(t) =
f.v21 .dτ =
f Ldf = 1/2 L f (t)2
0
0
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
14
Generalized capacitance
Defined by a single-valued relationship between their across variables v21 and
their integrated through variables h:
• Examples: pure translational and rotary masses, pure electrical, fluid, and
thermal capacitances.
• h = f2 (v21 ) with h = 0 if v21 = 0 and f2 is a single-valued function.
• In all cases except the electrical capacitance, the elements must have one
terminal attached to a constant through-variable reference.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
15
• The capacitance is ideal (linear) if :
h = C v21 or f = C . dv21 /dt
with C constant : called the generalized capacitance.
– For ideal masses, C is called the mass or moment of inertia.
– For ideal electical, fluid and thermal elements, C is called the electical,
fluid, and thermal capacitance, respectively.
• The energy supplied to a generalized capacitance equals :
Z t
Z v21 (t)
1 2
E=
v21 f dτ =
Cv21 dv21 = Cv21
2
0
0
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
16
Generalized resistance (pure energy dissipator)
Defined by a single-valued relation between their across variables v21 and their
through variables f :
• Examples: pure translational and rotary dampers, electrical, fluid, and
thermal resistances.
• f = f3 (v21 ), with f = 0 if v21 = 0 and f3 is an single-valued function such
that the signs of f and v21 are always equal.
• The resistance is ideal (linear) IF :
f = R1 v21 with R constant : called the generalized resistance.
• The power supplied to the generalized resistance (always positive, that is,
the power flows into the resistance) equals:
P = f v21 = v21
2
v
. f (v21 ) = f 2 R = 21
R
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
17
Pure energy sources
A device that is capable of delivering energy continuously to a system.
Two types of pure sources:
• v-source :
– The source variable is the across variable.
– Ideal if v is independent of the f -variable supplied by the source to the
system.
– battery (voltage source), melting ice (temperature source), ...
• f -source :
– The source variable is the through variable.
– Ideal if f is independent of the v-variable across the source terminals.
– hydraulic pump (flow source), torque motor (torque source), ...
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
18
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
19
System graph representation of two-terminal or single-port elements
• The ends or terminals of the graph indicate the across variable for the
element.
• The line between the terminals represents the continuity of the through
variable in the element.
• The sign convention for the through and across variables can be
incorporated into the graph by adding an arrow on the graph, thus forming
an oriented graph. The arrow implies that f is positive in the direction of
the arrow and that v21 is positive when v2 > v1 .
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
20
Oriented graph for a pure source :
• v-source
– If the arrow is oriented from v2 to v1 : v2 − v1 > 0
– The positive direction of the f -variable is also according to the
orientation of the arrow, indicating that the f -variable through the
v-source is negative.
• f -source
– The arrow indicates the positive direction of the f -variable
– The positive direction of the v-variable across this source is also
according to the orientation of the arrow, indicating that the v-variable
of the f -source is negative.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
21
Four-terminal or two-port elements
• pure transformers
• pure gyrators
• pure transducers
• dependent sources
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
22
Pure transformers
Pure transformers transforms an across (through) variable into an across
(through) variable within the same physical medium, such that the total power
flow into the device is identically zero. Defined by a single-valued relation
between the integrated across variables (or integrated through variables) at
both ports.
• xb = f (xa ) of hb = f (ha )
• for which the total energy flux in the device is zero : p = fa va + fb vb = 0
• an ideal (lineair) transformer : xb = nxa so vb = nva and fb = − n1 fa , with
n constant, called the transformation ratio
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
23
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
24
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
25
Here, the transformation ration n = −1 because vb is negative if va is positive.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
26
Pure gyrators
Pure gyrator transforms an across (through) variable into an through (across)
variable within the same physical medium, such that the total power flow into
the device is identically zero. Defined by a single-valued relation between the
integrated across variable (or integrated through variable) at one port and the
integrated through variable (or integrated across variable) at the other port.
• an ideal (lineair) gyrator is determined by : vb = r . fa and fb = − 1r va ,
with r constant and called the gyrator ratio
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
27
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
28
Pure transducers
Transducers changes energy from one physical medium to another without
energy loss or storage. Two types of transducers exist: the transforming
transducer and the gyrating transducer.
• Transforming transducers:
– electric motors and generators transduce current into torque, or
conversely,
– thermo-couples are thermal-to-electrical transforming transducers
• Gyrating transducers:
– piezoelectric crystals are used in accelerometers to produce a voltage
depending on applied force: an electro-mechanical gyrating transducer.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
29
Dependent sources
A pure dependent source is a pure source in which the source variable (f or v)
is a function of a second, independent, variable.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
30
System graphs
Procedure:
• Identify the different lumped system elements
• Establish a reference point within the system. This point serves as a
reference for all across variables. A reference vertex in the system graph
will correspond to this reference point. If a mechanical or a fluid-flow
system is being considered, the reference must be an inertial reference.
• For each element of the system, a branch is entered in the system graph.
• Orient the system graph by placing arrows on each branch.
• If the system is mechanical, a positive reference for velocity should be
chosen at this time.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
31
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
32
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
33
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
34
Continuity and compatibility requirements
Continuity
Continuity is an expression of the following physical laws:
• in electric systems : Kirchhoff’s current law
• in mechanical systems: Newton’s law
• in fluid systems: law of conservation of matter
• in thermal systems: law of conservation of energy
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
35
The vertex law is a statement of the continuity condition.
It states that, for an oriented graph of a system, the algebraic sum of
all through-variables entering any vertex must be zero:
X
fik = 0
i
for each vertex k.
Given a system graph of two-terminal elements with n vertices, only
n − 1 of the vertex equations are linearly independent.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
36
Compatibility
Compatibility is an expression of the following physical laws:
• in electrical systems: Kirchhoff’s voltage law
• in mechanical systems: geometric constraints
• in fluid systems: the sum of pressure drops in a closed path is equal to zero
• in thermal systems: the sum of temperature drops in a closed path is equal
to zero
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
37
The path law is an expression of the compatibility condition.
It states that, for an oriented graph, the algebraic sum of the across
variables around any closed path is zero ;
X
vqp = 0
q
for every closed path p.
Given a system graph of two-terminal elements with n vertices and b
branches, only b − (n − 1) of the path equations are linearly
independent.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
38
System equations : a mathematical model
The total number of equations that can be formulated is:
• (n − 1) independent vertex equations
• b − (n − 1) independent path equations
• b − s element equations (s is the number of sources)
• In total : 2b − s independent equations
Number of unknown variables : 2b − s unknowns (two for each branch except
for the branches that contain a source, where we have only on unknown).
If not all variables of all branches have to be known, there exist two systematic
approaches to reduces the number of equations: the node method and the
loop method.
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
39
The node method
Procedure:
• The across variables from each vertex to some reference vertex are chosen
as the unknowns in terms of which the final set of equations will be
formulated: the node variables. These variables automatically satisfy the
path laws since the across variable between nodes is simply the difference
between the appropriate node variables.
• The vertex equation is then written at each node, and the through
variables are then expressed directly in terms of the node variables as
related by the elemental equations.
• We then have eliminated all variables except the node variables, and have
(n − 1) equations and (n − 1) node variable (unknowns).
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
40
• Choose the node variables to be from vertices a, b, and c to e (the reference
vertex).
vae
= v
vce
= v4
vbe
= v2
vde
= v6
(known source)
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
41
• Apply the vertex law at b, c and d :
b
: −f1 + f2 + f3 = 0
c : −f3 + f4 + f5 = 0
d
: −f5 + f6 = 0
• Substitute the through variables f in terms of the node variables by using
the element equations:
b
c
d
v − v2
v2
v2 − v4
: −
+
+
=0
R1
R2
R3
v4
v4 − v6
v2 − v4
+
+
=0
: −
R3
R4
R5
v4 − v6
v6
: −
+
=0
R5
R6
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
42
The loop method
Procedure :
• Introduce circulating through variables in each of the meshes: the loop
variables. The branch through variables will be the difference between the
loop variables on each side of the branch. In this way, the loop variables
automatically satisfy the vertex law.
• The path law for each mesh is then written and substitutions are made for
the across variables in terms of the loop variables using the element
equations.
• In this way the system is quickly reduced to a number of equations equal to
the number of meshes (b − n + 1).
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
43
• Choose the meshes shown as A, B en C, and define the loop variables.
• Apply the path law to these meshes:
A
: −v + v1 + v2 = 0
B
: −v2 + v3 + v4 = 0
C
: −v4 + v5 + v6 = 0
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
44
• Introduce the element equations:
A : −v + R1 fA + R2 (fA − fB ) = 0
B
: −R2 (fA − fB ) + R3 fB + R4 (fB − fC ) = 0
C
: −R4 (fB − fC ) + R5 fC + R6 fC = 0
Control Theory [H04X3a]
Lecture A2 : Mathematical models (1)
45
Aim of this lecture :
• You have to know how to derive a mathematical model starting from a
physical model
• You have to understand the analogies/similarities between different kinds
of systems
• A system graph is an interesting instrument to derive the system equations
in a systematic way (use is not mandatory)
Control Theory [H04X3a]