Transcript Mathematical Models of Systems

‫بسم ا‪ ...‬الرحمن الرحيم‬
‫سیستمهای کنترل خطی‬
‫پاییز ‪1389‬‬
‫دکتر حسین بلندي‪ -‬دکتر سید مجید اسما عیل زاده‬
Recap.
• State Space Equation:
– Canonical Forms,
– Transfer Function,
• Block Diagram
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Signal-Flow Graph Models
Outline
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Terms and concepts
Mason’s signal-flow gain formula
Numerical examples
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A signal-flow graph
• A diagram consisting of nodes that are
connected by several directed branches.
• A graphical representation of a set of linear
relations.
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The basic elements of a signalflow graph
• branch - a unidirectional path segment,
which relates the dependency of an input
and an output variable.
• nodes - the input and output points or
junctions.
• path - a branch or continuous sequence of
branches that can be traversed from one
node to another node.
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• All branches leaving a node will pass the
nodal signal to the output node of each
branch ( uniderectionally ).
• The summation of all signals entering a
node is equal to the node variable.
• The relation between each variable is
written next to the directional arrow.
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• A loop - a closed path that originates and
terminates on the same note, and along
the path no node is met twice.
• Two loops are said to be nontouching if
they do not have a common node.
• Two touching loops share one or more
common nodes.
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Block and branch of DC motor
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Two-input, two-output system
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Y1(s) = G11(s) R1(s) + G12(s) R2(s)
Y2(s) = G21(s) R1(s) + G22(s) R2(s)
Gik - transfer function relating the i-th output to
the k-th input
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Interconnected system
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• The flow graph is a pictorial method of
writing a system of algebraic equations so
as to indicate the interdependencies of the
variables.
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A set of simultaneous equations
• 1. Write the system equations in the form
X1 = A11 X1 + A12 X2 + …+ A1n Xn
X2 = A21 X1 + A22 X2 + …+ A2n Xn
….……………………………………
Xm= Am1 X1 + Am2 X2 + …+ Amn Xn
Note: An equation for X1 is not required if X1 is an input
node.
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• 2. Arrange the m or n (whichever is larger) nodes
from left to right.
• 3. Connect the nodes by the appropriate
branches A11, A12, etc.
• 4. If the desired output node has outgoing
branches, add a dummy note and unity branch.
• 5. Rearrange the nodes and/or loops in the graph
to achieve pictorial clarity.
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Set of simultaneous algebraic equations
a11 x1 + a12 x2 + r1 = x1
a21 x1 + a22 x2 + r2 = x2
r1, r2 - input variables
x1, x2 - output variables
x1 (1 - a11) + x2
(- a12) = r1
x1 ( - a21) + x2 (1 - a22) = r2
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Mason’s signal-flow gain formula
Tij(s) = ∑kPijk ∆ijk /∆
Pijk = k-th path from variable xi to variable xj
∆ = determinant of the graph
∆ijk = cofactor of the path Pijk
∑k = all possible k path from xi to xj
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Step by step construction
y2 = a12 y1 + a32 y3
y3 = a23 y2 + a43 y4
y4 = a24 y2 + a34 y3 + a44 y4
y5 = a25 y2 + a45 y4
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• The nodes representing the variables y1,y2,y3,y4
and y5 are located in order from left to right.
• The first equation states that y2 depends upon
two signals a11 y1 and a32 y3.
• The second equation states that y3 depends upon
the sum of a23 y2 and a43 y4,therefore a branch of
gain a23 is drawn from node y2 to y3 and a branch
of gain a43 is drawn from y4 to y3, with directions
of the branches indicated by arrows.
• Similarly, with consideration of the third and
fourth equation.
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Mason’s signal-flow gain formula
∆
= 1 - (sum of all different loop gains)
+ ( sum of the gain products of all
combinations of two nontouching loops)
- ( sum of the gain products of all
combinations of three nontouching loops)
+ …,
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∆ijk = cofactor of the path Pijk is the the
determinant with the loops touching the
k-th path removed.
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T(s) = Y(s)/R(s)
T(s) = Y(s)/R(s) = ∑kPk ∆k /∆
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• The path gain or transmittance Pk is defined
as continuous succession of branches that
are traversed in the direction of the arrows
and with no node encountered more than
once.
• A loop is defined as a closed path in which
no node is encountered more than ones per
traversal.
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Ex. 2.8 Interacting system
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The paths connecting the input
R(s) and output Y(s)
path 1
P 1= G 1 G 2 G 3 G 4
path 2
P 2= G 5 G 6 G 7 G 8
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Four self-loops
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•
L1 = G2 H2
L2 = H3 G3
L3 = G6 H6
L4 = G7 H8
• Loops L1 and L2 do not touch L3 and
L4
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The determinant :
∆ = 1 - (L1 + L2 + L3 + L4) + ( L1L3 +L1L4 +
L2L3 + L2L4)
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The cofactor along path 1 is evaluated
by removing the loops that touch path 1
from ∆.
L1 = L2 = 0
∆1 = 1 - (L3 +L4)
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The cofactor along path 2 is evaluated
by removing the loops that touch path 2
from ∆.
• L3 = L 4 = 0
∆2 = 1 - (L1 +L2)
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The transfer function of the
system
• T(s) = Y(s)/R(s) = (P1 ∆1 + P2 ∆2 )/∆
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Ex. 2.7 Block diagram reduction
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Ex. 2.7 Mason’s signal-flow gain
P1 = G1 G2 G3 G4
L1 = - G2 G3 H2
L2 = G3 G4 H1
L3 = - G1 G2 G3 G4 H3
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• All the loops have common nodes and therefore
are all touching.
• The path P1 touches all the loops, so ∆1 = 1
T(s) = Y(s)/R(s) = P1 ∆1 /(1 - L1 - L2 - L3)
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• P1 = G1 G2 G3 G4
• L1 = - G2 G3 H2
• L2 = G3 G4 H1
• L3 = - G1 G2 G3 G4 H3
T(s) = Y(s)/R(s) = P1 ∆1 /(1 - L1 - L2 - L3)
G1 G2 G3 G4/(1+ G2 G3 H2 - G3 G4 H1G1 G2 G3 G4 H3)
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T(s) = G1 G2 G3 G4/(1 - G3 G4 H1 +
G2 G3 H2 + G1 G2 G3 G4 H3)
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Ex. 2.11
A complex system
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The forward paths:
• P1 = G1 G2 G3 G4 G5 G6
• P2 = G1 G2 G7 G6
• P3 = G1 G2 G3 G4 G8
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Ex. 2.11
A complex system
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The feedback loops:
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•
•
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L1 = - G2 G3 G4 G5 H2
L2 = - G5 G6 H1
L3 = - G8 H1
L4 = - G7 H2 G2
L5 = - G4 H4
L6 = - G1 G2 G3 G4 G5 G6 H3
L7 = - G1 G2 G7 G6 H3
L8 = - G1 G2 G3 G4 G8 H3
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The determinant and cofactors:
• ∆ = 1 - (L1 + L2 + L3 + L4 + L5 + L6 + L7 +
L8) + ( L5L7 + L5L4 + L3L4)
• ∆1 = ∆3 = 1
∆2 = 1 - L5 = 1 + G4H4
Loop L5 does not touch loop L7or loop L4,
and loop L3 does not touch loop L4;
but all other loops touch.
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The transfer function:
• T(s) = Y(s)/R(s) = (P1 ∆1 + P2 ∆2 + P3 ∆3 )/∆
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Important properties of SF-G
• A SF-G applies only to linear systems.
• The equations for which a SF-G is drawn must
be algebraic equations in the form of effects
as function of causes.
• Nodes are used to represent variables.
Normally, the nodes are arranged from left to
right, following a succession of causes and
effects through the system.
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• Signal travel along branches only in the direction
described by the arrows of the branches.
• The branch directing from node yk to yj represents
the dependence of variable yj upon yk, but not the
reverse.
• A signal yk traveling along a branch between
nodes yk and yj is multiplied by the gain of the
branch, akj, so that a signal akj yk is delivered at
note yj.
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In the case when the system is represented by
a set of integrodifferencial equations, we must
first transform these into Laplace transform
equations and then rearrange the latter into the
form of
Yj(s) = ∑Gkj(s) Yk(s) for k, j =1,2,…,N
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Summary
An alternative use of T(s) models in S-FG was
investigated. Mason’s SF-G formula was found to be
useful for obtaining the relationship between system
variables in complex feedback system.
Mason’s SF-G formula provides the relationship between
system variables without any reduction or manipulation
of the flow graph.
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• Time Domain Performance
Specification
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Test Input Signal
• Since the actual input signal of the system
is usually unknown, a standard test input
signal is normally chosen. Commonly
used test signals include step input, ramp
input, and the parabolic input.
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General form of the standard test
signals
r(t) = t
n
n+1
R(s) = n!/s
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Test signals r(t) = A tn
n=0
r(t) = A
R(s) = A/s
n=1
r(t) = At 2
R(s) = A/s
n=2 2
r(t) = At 3
R(s) = 2A/s
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Table 5.1 Test Signal Inputs
Test Signal
Step
position
Ramp
velocity
Parabolic
acceleration
r(t)
R(s)
r(t) = A, t > 0
R(s) = A/s
= 0, t < 0
r(t) = At, t > 0 R(s) = A/s2
= 0, t < 0
r(t) = At2, t > 0 R(s) = 2A/s3
= 0, t < 0
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Test inputs vary with target type
step
ramp
parabola
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Steady-state error
• Is a difference between input and the
output for a prescribed test input as
t 
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Application to stable systems
• Unstable systems represent loss of
control in the steady state and are
not acceptable for use at all.
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Steady-state error:
a) step input, b) ramp input
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Time response of systems
c(t) = ct(t) + css(t)
The time response of a control system is divided
into two parts:
• ct(t) - transient response
• css(t) - steady state response
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Transient response
• All real control systems exhibit transient
phenomena to some extend before steady
state is reached.
lim ct(t) = 0
for t 
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Steady-state response
• The response that exists for a long time
following any input signal initiation.
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Poles and zeros of a first order system
Css(t)
Ct(t)
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Poles and zeros
1.
A pole of the input function generates the form of the
forced response ( that is the pole at the origin generated
a step function at the output).
2. A pole of the transfer function generate the form of the
exponential response
3. The zeros and poles generate the amplitudes for both the
transit and steady state responses
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Effect of a real-axis pole upon transient
response
A pole on the real axis generate an exponential response
of the form Exp[-t] where - is the pole location on real axis.
The farther to the left a pole is on the negative real axis,
the faster the exponential transit response will decay to zero.
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Evaluating response using poles
K1 K2
K3
K4
C(s)  


s s2 s 4 s 5
Css(t
)
Ct(t)
2t
c(t)  K1  K 2e
4 t
 K 3e
5t
 K4e
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