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Linear Control System study material very important course

1. Chapter 5
Instructor: Muhammad Imran

3. In this chapter we will learn
▪How to reduce block diagrams of multiple sub-
systems to a single block representing equivalent
transfer function from i/p to o/p
▪How to analyze and design transient response for
a system consisting of multiple sub-systems
▪How to represent in state space a system
consisting of multiple sub-systems
▪How to convert between alternate representation
of a systems in state space

4. So far we have worked with individual sub-
system with its i/p and o/p
More complicated systems are represented
by interconnection of many sub-systems
Response of single transfer function cab be
calculated if we represent multiple
subsystems as a single transfer function
Apply analytical techniques to obtain
transient response information about entire
system

6. Multiple sub-systems are represented in two
ways
▪Block Diagram
▪(generallyused for Frequency domain analysis)
▪Signal Flow Graph
▪(generally used for Time domain analysis)
Block diagrams we use blocks to represent
transfer function
Signal flow graphs represent transfer
function as lines and signals as circular nodes

8. We use different techniques to reduce each
representation to a single transfer function
Block diagram algebra will be used to reduce
block diagrams
Mason’s Rule will be used to reduce signal
flow graphs

9. Sub-system is represented as a block by an
i/p, o/p and a transfer function
For multiple sub-systems are interconnected
a few more schematics elements must be
added to a block diagram which includes
▪Summing Junctions
▪Pick Off points

10. In Summing Junction O/p is an algebraic sum of i/p signals
A pick off point distributes i/p signal R(s) undiminished to several
o/p points

11. Now we will examine some common
topologies used for interconnecting sub-
systems and derive single transfer function
representation of each topology
Three Common topologies are
▪CASCADE FORM
▪PARALLEL FORM
▪FEEDBACK FORM

12. Intermediate signal values are shown at o/p
of each element
Each signal is derived from product of i/p
times the transfer function

13. In cascaded system it is assumed that
▪inter connected sub systems do not load adjacent
subsystems
▪System o/p remains same whether the subsystem
is connected or not
If there is change in o/p, then subsequent
subsystem loadsthe previous subsystem
▪Consequence: equivalent transfer function does
not remain the product of individual transfer
function (cascaded system)

15. If network is placed in cascade from then
transfer function becomes

16. But according to definition of cascaded
subsystem
Thus loadinghas changed the two transfer
functions
Solution is to use amplifier between two
networks to compensate loading effect

17. Amplifier has high i/p impedance, so it does
not load previous network
With amplifier included , equivalent transfer
function is the product of transfer function
and gain Kof amplifier

18. Parallel sub-system have a common i/p and o/p is formed
by an algebraic sum of o/p from all other subsystems

19. It is more frequently encountered topology in control systems
Equivalent transfer function is obtained by simplified feedback
form

21. So far we have seen three different
configurations of multiple subsystems
For each configuration we have derive transfer
function expression
Since these forms are combined in to a
complex arrangement in physical systems,
recognizing these topologies is a pre-requisite
to obtain equivalent transfer function of
complex systems

22. Next we will reduce complex systems
composed of multiple subsystems to obtain
equivalent single transfer function

23. Since subsystems are not always present in
familiar forms (i.e. Cascade, Parallel or Feed
back from)
So we need to transform given subsystem
into familiar form in order to reduce block
diagram for single transfer function
Here we discuss basic block moves that can
be made to achieve familiar form

24. We learn how to move blocks left and right
past summing junction and pick off points
By block moves equivalence can be verified
by tracing the signal at the i/p through to the
o/p and recognizing that the o/p signals are
identical

25. Signal R(s) and X(S) are multiplied before reaching o/p
Both block diagrams are equivalent with

26. Signal R(s) is multiplied with G(s) before reaching o/p but
x(s) is not
Both block diagrams are equivalent with

28. Example 5.1: Reduce given block diagram to
single transfer function

31. Example 5.2: Reduce given block diagram to
single transfer function

36. Alternate approach to block diagram
Block diagram consists of blocks, signal,
summing junction pick off points etc
Signal flow graph consists of
▪Branches(to represent system)
▪Nodes(to represent signals)
A system is represented by a line and an
arrow showing direction of signal flow
through system

37. Adjacent to line we write transfer function
A signal is a node with signal name written
adjacent to node

38. Fig shows interconnection of systems and
signals
Each signal is sum of signals flowing into it

39. Summing negative signals we associate
negative sign with the system not with
summing junction
In order to convert block diagram to signal
flow graph we need to
▪First convert signals into nodes
▪Interconnect the nodes with system branches

40. Example 5.5: Convert the cascaded, parallel and
feedback form of following block diagrams?

43. Example 5.6: Convert the following block
diagram into signal flow graph?

46. It is a technique used to reduce signal flow
graph to single transfer function that relate
the o/p of system to its i/p
Since block diagram reduction we require
successive application of fundamental
relationship to arrive at system G(s)
On the other hand, Mason’s rule requires
application of one formula for reduction
Following key definitions must be kept in
mind while solving Mason’s Formula

47. Loop gain: It is the product of branch gains
found by traversing the path that starts and
ends at the same node, following the
direction of signal flow , w/o passing through
any other node more than once

49. Forward Path gain: The product of gains
found by traversing a path from the i/p node
to the o/p node of signal flow graph in the
direction of signal flow

51. Non-touching loops: Loops that does not
have any node in common

52. Non-touching loop gains: the product of loop
gains from non-touching loops taken two,
three , four or more at a time

54. The transfer function C(s) / R(s) of a system
represented by signal flow graph is
G(s) = C(s) / R(s) = Σ
kT
k∆
k/ ∆

55. The transfer function C(s) / R(s) of a system
represented by signal flow graph is
G(s) = C(s) / R(s) = Σ
kT
k∆
k/ ∆

56. Example 5.7: Find the transfer function,
C(s)/R(s) for the signal flow graph?

61. By replacing all values in Mason’s formula we get transfer
function
Home Assignment : Solve skill assessment Exercise 5.4

62. Now we draw signal flow graph from state
equations
First identify three nodes for three state
variables x1,x2 & x3

63. Identify three nodes , placed at left of each
respective state variable , to be the
derivatives of state variables
Also identify one node for each i/p and o/p
Interconnect state variables and their
derivatives with the defining integration (1/S)

64. By using state equation feed each node with
the indicated signal

68. Signal flow graph helps us to visualize the
process of determining alternate
representation in state space
A system having same i/p & o/p terminal can
have different state space representation
In chapter 3 , we discuss only phase variable
form
System modeling in state space can take on
any form other than phase variable form

69. In this section we will see how state space
representation can be written from signal
flow graph
Five Common forms are
▪Phase Variable Form
▪Cascade Form
▪Parallel Form
▪Controller Canonical Form
▪Observer Canonical Form

70. System are expressed in state space
representation with state variables chosen to
be phase variables i.e. variables are
successive derivative of each other

71. Here choice of state variables is different
from phase variables
Consider transfer function of Example 3.4
In block diagram of cascaded form it can be
shown as

72. Here o/p of each first order system has been
labeled as state variables
Now we show how signal flow graphs are
used to obtain state space representation
Signal flow graph of first order system can be
formed by transforming each block into an
equivalent differential equation

75. B matrix is I /pmatrix that contains the term that
couples i/p r(t) to the system
C matrix is o/p matrix that contains the term that
couples the state variable x1 to o/p c(t)
A matrix is system matrix it contains the terms
relative to internal system itself
In this form, the system matrix Aactually
contains the system polesalong diagonal

76. Another form to represent system in parallel
form
This form leads to System matrix A that is
purely diagonal, provided that no system
pole is repeated root
It is obtained by partial fraction expansion of
transfer function
To arrive at signal flow graph we first solve for
o/p C(S)

77. Considering same example by partial fraction
Since C(S) is sum of three terms & each term
is first order subsystem with R(S) as i/p

78. We use signal flow graph to write state & o/p
equations
State variables are o/p of each integrator

80. Parallel form yield diagonal system matrix
Advantage
▪Each equation is first order differential equation in
one variable only
▪We can solve three equations independently
▪Equations are said to be decoupled
If denumerator of transfer function has
repeated real roots , parallel form can still be
obtained with A will not be diagonal

81. Jordan Canonical Form

82. Another representation that uses phase
variable form
Name is given because this form is used in
design of controller (Chap 12)
It is obtained from phase variable form simply
by ordering the phase variables in reverse
order
Consider Example 3.5

84. Phase variable form was derived as
Renumbering the phase variables in reverse
order

85. Finally by rearranging in ascending numerical
order yields the Controller Canonical form

86. Controller Canonical form is obtained by
renumbering phase variable in opposite
direction
Phase variables form contains coefficients of
characteristic equation in bottom row
Controller Canonical form contains
coefficients of characteristic equation in
upper row
Signal flow graph is shown as

88. Name is given because of its use in observer
design (Chap 12)
Beginning with same example and dividing
numerator and denumerator with highest
power of s i.e. s
3

89. By cross multiplication
Combining power of like powers of
integration gives

90. Above expression can be used to draw signal
flow graph
Starting with three integrators

93. Controller Form
Observer Form

94. This form is similar to phase variable from
except that coefficients of denumerator are in
first column
Coefficients of numerator are in i/p matrix B
System matrix Ais transposeof Controllerform
Bis transposeof controller form matrix C
Cis transposeof controller form matrix B
Hence two forms are dual
If system is represented by A, B & C its dual are

98. Example 5.8: Represent the given feedback
control system in state space. Model the
transfer function in cascade form?

99. By modeling in cascaded form gain of 100,
the pole at -2& -3 in cascaded form are shown
as
Now by adding the feedback and i/p paths

102. Parallel form of signal flow graph yields a
diagonal matrix form
Advantage was decouples state equations at
the expanse of partial fraction expansion
To achieve same thing we now apply matrix
transformationto decouple a system without
partial fraction
if we can find correct matrix Pthen
transformed system matrix P
-1
APyields
diagonal matrix

103. Eigenvector: The eigenvector of matrix A are all
vectors Xi ≠ 0 which under the transformation
becomes multiples of themselves i.e.
where λiare constant

104. Eigenvalue: The eigenvalues of matrix A are
values of λithat satisfy
To find eigenvectors

105. Example 5.10: Find the eigenvectors of the
matrix

106. Transformation matrix P consists of
eigenvectors of A i.e.
P=[x1 x2 x3 ……….. Xn]
Diagonal matrix D is obtained by applying
transformation P
-1
AP

107. Example 5.11: For a given system find diagonal
system?
Since same system of example 5.10 we have
already calculated
Eigenvalues: -2 & -4
Eigenvectors : X1 = [1 1]’ & X2 = [1 -1]’

108. Form matrix P from eigenvectors
P=[x1 x2 x3 ……….. Xn]
Apply transformation P
-1
AP

109. In matrix form