1. Chapter 5. Reduction of Multiple Subsystems
Pusan National University
Intelligent Robot Laboratory

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3. Table of Contents Introduction Block Diagrams
Analysis and Design of Feedback Systems
Signal-Flow Graphs
Mason’s Rule
Signal-Flow Graphs of State Equations
Alternative Representations in State Space
Similarity Transformations

4. Introduction Multiple subsystems
Represented by the interconnection of many subsystems
Block diagrams and signal-flow graphs representation
Figure a. Transfer function ;
b. Equivalent block diagram showing phase variables;
Note y(t) = c(t);

5. Block Diagrams Example: Configuration of multiple subsystems
Subsystem is represented as a block with an input, an output, and a transfer function. Many systems are composed of multiple subsystems, as in Figure 5.2.
Figure The space shuttle consists of multiple subsystems.
Can you identify those that are control systems or parts of control systems?

6. Block Diagrams Components of a block diagram
Figure 5.3 Components of a block diagram for a linear, time-invariant system

7. Three basics forms: cascade, parallel, and feedback forms.
Cascaded Form
Intermediate signal values are shown at the output of each subsystem.
Each signal is derived from the product of the input times the transfer function.
Figure a. Cascaded subsystems ;
b. Equivalent transfer function

8. Parallel Form
Parallel subsystems have a common input and an output formed by the algebraic sum of the outputs from all of the subsystems.
(a)
Figure a. Parallel subsystems ;
b. Equivalent transfer function

9. Feedback Form Basis for our study of control systems engineering
Figure a. Feedback control system ;
b. Simplified model ;
c. Equivalent transfer function

10. Directing our attention to the simplified model,
The typical feedback system, described in detail in Chapter 1, is shown in Figure 5.6(a); a simplified model is shown in Figure 5.6(b).
Directing our attention to the simplified model,
But since ,
Substituting Eq. (5.2) and solving for the transfer function, , we obtain the equivalent, or closed-loop, transfer function shown in Figure 5.6(c),
The product, , in Eq. (5.3) is called the open-loop transfer function, or loop gain.
(5.1)
(5.2)
(5.3)

11. Moving blocks Moving Block to create familiar Forms
These equivalences, along with the forms studied earlier in this section, can be used to reduce a block diagram to a single transfer function.
Figure 5.7 Block diagram algebra for summing junctions-equivalent forms for moving a block
a. To the left past a summing junction ;
b. To right past a summing junction

12. Moving blocks
Figure 5.8 Block diagram algebra for pickoff points-equivalent forms for moving a block
a. To the left past a pickoff point ;
b. To the right past a pickoff point

13. Moving blocks Ex 5.1 Block diagram reduction via familiar forms
Reduce the block diagram shown in Figure 5.9 to a single transfer function.
Figure 5.9 Block diagram for Example 5.1

14. Moving blocks
Sol ) We solve the problem by following the steps in Figure First, the three summing junctions can be collapsed into a single summing junction, as shown in Figure 5.10(a).
Figure Steps in solving Example 5.1:
a. Collapse summing junctions

15. Moving blocks
Second, recognize that the three feedback functions, , are connected in parallel. They are fed from a common signal source, and their outputs are summed. The equivalent function is Also recognize that are connected in cascade. Thus, the equivalent transfer function is the product, The results of these steps are shown in Figure 5.10(b).
Figure Steps in solving Example 5.1:
b. Form equivalent cascaded system in the forward path and
equivalent parallel system in the feedback path

16. Moving blocks
Finally, the feedback system is reduced and multiplied by to yield the equivalent transfer function shown in Figure 5.10(c).
Figure Steps in solving Example 5.1:
c. Form equivalent feedback system and multiply by cascaded

17. Reduction of blocks Ex 5.2 Block diagram reduction by moving blocks
Reduce the system shown in figure 5.11 to a single transfer function.
Figure Block diagram for Example 5.2

18. Reduction of blocks
Sol ) In this example we make use of the equivalent forms shown in Figures 5.7 and 5.8 First, move to the left past the pickoff point to create parallel subsystems, and reduce the feedback system consisting of and This result is shown in Figure 5.12(a).
Figure 5.12 Steps in the block diagram reduction for Example 5.2

19. Reduction of blocks
Second, reduce the parallel pair consisting of and unity, and push to the right past the summing junction, creating parallel subsystems in the feedback. These results are shown in Figure 5.12(b)
Figure 5.12 Steps in the block diagram reduction for Example 5.2

20. Reduction of blocks
Third, collapse the summing junctions, add the two feedback elements together, and combine the last two cascaded blocks. Figure 5.12(c) shows these results.
Figure 5.12 Steps in the block diagram reduction for Example 5.2

21. Reduction of blocks
Fourth, use the feedback formula to obtain Figure 5.12(d).
Finally, multiply the two cascaded blocks and obtain the final result, shown in Figure 5.12(e).
Figure 5.12 Steps in the block diagram reduction for Example 5.2

22. Analysis and Design of Feedback Systems
Secondary feedback system analysis
Percent overshoot, settling time, peak time, and rise time can then be found from the equivalent transfer function.
Figure Second-order feedback control system
(5.4)

23. Analysis and Design of Feedback Systems
Real pole : over damped
The real pole overlap : critically damped
Complex number pole : under damped
K value increases
Increase in the imaginary part of pole
Fixed settling time
Peak time reduction
Percent overshoot increases
(5.5)
(5.6)
(5.7)

24. Analysis and Design of Feedback Systems
Ex 5.3 Finding transient response
For the system shown in Figure 5.14, find the peak time, percent overshoot, and settling time.
Sol ) The close-loop transfer function found from Eq. (5.3) is
From Eq. (4.18),
Figure Feedback system for Example 5.3
(5.8)
(5.9)

25. Analysis and Design of Feedback Systems
From Eq.(4.21),
Substituting Eq. (5.9) into (5.10) and solving for yields
Using the values for and along with Eqs. (4.34), (4.38), and (4.42), we find, respectively,
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)

26. Analysis and Design of Feedback Systems
Ex 5.4 Gain design for transient response
Design the value of gain, K, for the feedback control system of Figure 5.15 so that the system will respond with a 10% overshoot.
Sol ) The closed-loop transfer function of the system is
Figure Feedback system for Example 5.4
(5.15)

27. Analysis and Design of Feedback Systems
From Eq. (5.15),
and
Thus,
Since percent overshoot is a function only of , Eq.(5.18) shows that percent overshoot is a function of k.
A 10% overshoot implies that Substituting this value for the damping ratio into Eq. (5.18) and solving for K yields
(5.16)
(5.17)
(5.18)
(5.19)

28. Signal-Flow Graphs Signal flow graphs
Signal-flow graph consists only of branches, which represent systems and nodes, which represent signals
Each signal is the sum of signals flowing into it.
Figure Signal-flow graph components;
a. system;
b. signal;
c. interconnection of systems and signals

29. Signal-Flow Graphs
Ex 5.5 Converting common block diagrams to signal-low graphs
Convert the cascaded, parallel, and feedback forms of the block diagrams shown in Figures 5.4(a), 5.5(a), and 5.6(b), respectively, into signal-flow graphs.
5.4(a)
5.5(a) (b)

30. Signal-Flow Graphs
Sol ) In each case we start by drawing the signal nodes for that system. Next we interconnect the signal nodes with system branches. The signal nodes for the cascaded, parallel, and feedback forms are shown in Figure 5.17(a), (c), and (e), respectively.
Figure Building signal-flow graphs:
a. cascaded system nodes(from Figure 5.4(a));
c. Parallel system nodes (form Figure 5.5(a));
e. feedback system nodes (from Figure 5.6(b))

31. Signal-Flow Graphs
The interconnection of the nodes with branches that represent the subsystems is shown in Figure 5.17(b), (d), and (f) for the cascaded, parallel, and feedback forms, respectively.
Figure Building signal-flow graphs:
b. cascaded system signal-flow graph (from Figure 5.4(a));
d. parallel system signal-flow graph (form Figure 5.5(a));
f. feedback system signal-flow graph (from Figure 5.6(b))

32. Signal-Flow Graphs
Ex 5.6 Converting a block diagram to a signal-flow graph
Convert the block diagram of Figure 5.11 to a signal-flow graph.
Sol ) Begin by drawing the signal nodes, as shown in Figure 5.18(a). Next, interconnect the nodes, showing the direction of signal flow and identifying each transfer function.
Figure Signal-flow graph development :
a. signal nodes

33. Signal-Flow Graphs
The result is shown in Figure 5.18(b). Notice that the negative signs at the summing junctions of the block diagram are represented by the negative transfer Functions of the signal-flow graph.
Figure Signal-flow graph development :
b. signal-flow graph

34. Signal-Flow Graphs
Finally, if desired, simplify the signal-flow graph to the one shown in Figure 5.18(c) by eliminating signals that have a single flow in and a single flow out, such as
Figure Signal-flow graph development :
c. simplified signal-flow graph

35. Mason’s Rule
Signal-flow graphs to single transfer functions that relate the output of a system to it's input
Mason’s formula components
Loop gain
Forward – path gain
Non-touching loops
Non-touching – loop gain

36. Mason’s Rule Loop gain Ends at the same node
Without passing through any other node more than once
Figure Signal-flow graph for demonstrating Mason’s rule
(5.20a)
(5.20b)
(5.20c)
(5.20d)

37. Mason’s Rule Forward – path gain
Gains found by traversing a path from the input node to the output node of the signal-flow graph in the direction of signal flow.
Figure Signal-flow graph for demonstrating Mason’s rule
(5.21a)
(5.21b)

38. Mason’s Rule Non-touching loops
Loops that do not have any nodes in common.
In figure 5.19, loop dose not touch loops , and .
Figure Signal-flow graph for demonstrating Mason’s rule

39. Mason’s Rule Non-touching – loop gain
The product of loop gains from non-touching loops taken two, three, four, or more at a time.
Figure Signal-flow graph for demonstrating Mason’s rule
(5.22a)
(5.22b)
(5.22c)

40. Mason’s Rule MASON’S RULE
The transfer function, , of a system represented by a signal-flow graph is
where
(5.23)

41. Mason’s Rule Ex 5.7 Transfer function via Mason’s rule
Find the transfer function, , for the signal-flow graph in Figure 5.20.
Figure Signal-flow graph for Example 5.7

42. Mason’s Rule
Sol) First, identify the forward-path gains. In this example there is only one:
Second, identify the loop gains. There are four, as follows:
Third, identify the non-touching loops taken two at a time. From Eqs. (5.25) and Figure 5.20, we can see that loop 1 dose not touch 2, loop 1 does not touch loop3, and loop 2 does not touch loop 3. Notice that loop 1, 2, and 3 all touch loop 4. Thus, the combinations of non-touching loops taken two at a time are at follows:
(5.24)
(5.25a)
(5.25b)
(5.25c)
(5.25d)

43. Mason’s Rule
Thus, the combinations of non-touching loops taken two at a time are at follows:
Loop 1 and loop 2:
Loop 1 and loop 3:
Loop 2 and loop 3:
Finally, the non-touching loops taken three at a time are as follows:
Loops 1, 2, and 3:
Loops 1, 2, and its definitions, we form and Hence,
(5.26a)
(5.26b)
(5.26c)
(5.27)

44. Mason’s Rule Loops 1, 2, and its definitions, we form and Hence,
We form by eliminating from the loop gains that touch the k th forward path:
Expressions (5.24), (5.28), and (5.29) are now substituted into Eq.(5.23), yielding the transfer function:
(5.28)
(5.29)
(5.30)

45. Signal-Flow Graphs of state Equations
Draw signal-flow graphs from state equations.
(5.31a)
(5.31b)
(5.31c)
(5.31d)
Figure Stages of development of a signal-flow graph for the system of Eqs.3.31:
a. Place nodes;
b. interconnect state variables and derivatives

46. Signal-Flow Graphs of state Equations
Figure Stages of development of a signal-flow graph for the system of Eqs.3.31:
c. form ;
d. form

47. Signal-Flow Graphs of state Equations
Figure Stages of development of a signal-flow graph for the system of Eqs.3.31:
e. form (Continued)

48. Signal-Flow Graphs of state Equations
Figure Stages of development of a signal-flow graph for the system of Eqs.3.31:
f. form output

49. Alternative Representations in State Space
System modeling in state space can take on many representations other than the phase variable form.
Cascade Form
Parallel Form
Controller Canonical Form
Observer Canonical Form

50. Alternative Representations in State Space
Cascade Form
Returning to the system of Figure 3.10(a), the transfer function can be represented
Figure 5.23 shows a block diagram representation of this system formed by cascading each term Eq.(5.32). The output of each first-order system block has been labeled as a state variable. These state variables are not the phase variables.
(5.32)
Figure Representation of Figure 3.10 system as cascaded first-order system

51. Alternative Representations in State Space
The signal flow for each first-order system of Figure 5.23 can be found by transforming each block into an equivalent differential equation. each first-order block is of the form
Cross-multiplying, we get
After taking the inverse Laplace transform, we have
Solving for yields
(5.33)
(5.34)
(5.35)
(5.36)

52. Alternative Representations in State Space
Figure 5.23(a) shows the implementation of Eq.(5.36) as a signal-flow graph Here again, a node was assumed for at the output of an integrator, and its derivative was formed at the input.
Figure a. First-order subsystem;
b. Signal-flow graph for Figure 5.22 system

53. Alternative Representations in State Space
Cascading the transfer functions shown in Figure 5.23(a), we arrive at the system representation show in Figure 5.23(b). Now write the state equations for the new representation of the system. Remember that the derivative of a state variable will be at the input to each integrator:
The output equation is written by inspection from Figure 5.23(b):
(5.37a)
(5.37b)
(5.37c)
(5.38)

54. Alternative Representations in State Space
The state-space representation is completed by rewriting Eqs. (5.37) and (5.38) in vector-matrix form:
Comparing Eqs. (5.39) with Figure 5.23(b), you can form a vivid picture of the meaning of some of the components of the state equation.
(5.39a)
(5.39b)

55. Alternative Representations in State Space
Parallel Form
This form leads to an A matrix that is purely diagonal, provided that no system pole is a repeated root of the characteristic equation.
whereas the previous form was arrived at by cascading the individual first-order subsystems, the parallel form is derived from a partial-fraction expansion of the system transfer function.
performing a partial-fraction expansion on our example system, we get
Equation (5.40) represents the sum of the individual first-order subsystems. To arrive at a signal-flow graph, first solve for
(5.40)
(5.41)

56. Alternative Representations in State Space
is the sum of three terms. Each term is a first-order subsystem with as the input
Formulating this idea as a signal-flow graph renders the representation shown in Figure 5.24.
Figure Signal-flow representation of Eq.(5.40)

57. Alternative Representations in State Space
By inspection the state variables are the output of each integrator, where the derivatives of the state variables exist at the integrator in put. we write the state equations by summing the signal at the integrator inputs:
The output equation is found by summing the signals that give :
In vector-matrix form Eqs. (5.42) and (5.43) become
(5.42a)
(5.42b)
(5.42c)
(5.43)
(5.44)
(5.45)

58. Alternative Representations in State Space
Controller Canonical Form
Another representation that uses phase variables is called the controller canonical form
For example, consider the transfer function
The phase-variable form was derived in Example 3.5 as
(5.46)
(5.47a)
(5.47b)

59. Alternative Representations in State Space
Where Renumbering the phase variables in reverse order yields
Finally, rearranging Eqs. (5.48) in ascending numerical order yields the controller canonical form as
(5.48a)
(5.48b)
(5.49a)
(5.49b)

60. Alternative Representations in State Space
Figure 5.25 shows the steps we have taken on a signal-flow graph.
Notice that the controller canonical form is obtained simply by renumbering the phase variables in the opposite order. Equations (5.47) can be obtained from Figure 5.25(a), and Eqs. (5.49) from Figure 5.25(b).
Figure Signal-flow graphs for obtaining forms for :
a. phase-variable form;
b. controller canonical form

61. Alternative Representations in State Space
Observer Canonical Form
representation that yields a left companion system matrix as an example, the system modeled by Eq. (5.55) will be represented in this form.
Begin by dividing all terms in the numerator and denominator by the highest power , , and obtain
Cross-multiplying yields
(5.50)
(5.51)

62. Alternative Representations in State Space
Combining terms of like powers of integration gives or
Equation (5.52) or (5.53) can be used to draw the signal-flow graph start with three integrations, as shown in Figure 5.26(a).
(5.52)
(5.53)
Figure Signal-flow graph for observer canonical form variables:
a. planning

63. Alternative Representations in State Space
Identifying the state variables as the outputs of the integrators, we write the following state equations:
The output equation from Figure 5.26(b) is
(5.54a)
(5.54b)
(5.54c)
(5.55)
Figure Signal-flow graph for observer canonical form variables:
b. implementation

64. Alternative Representations in State Space
In vector-matrix form Eqs. (5.54) and (5.55) become
The signal-flow graph of the dual can be obtained from that of the original by reversing all arrows, changing state variables to their derivatives and vice versa, and interchanging and , thus reversing the roles of the input and the output
(5.56a)
(5.56b)

65. Alternative Representations in State Space
Ex 5.8 State-space representation of feedback systems
Represent the feedback control system shown in Figure 5.27 in state space. Model the forward transfer function in cascade from
Sol ) First we model the forward transfer function in cascade from.
The gain of 100, the pole at -2, and the pole at -3 are shown cascaded in Figure 5.28(a).
Next add the feedback and input paths, as shown in Figure 5.28(b).
Figure Feedback control system for Example 5.8

66. Alternative Representations in State Space
Figure Creating a signal-flow graph for the Figure 5.27 system:
a. forward transfer function;
b. complete system

67. Alternative Representations in State Space
Now, by inspection, write the state equations:
But, from Figure 5.28(b),
Substituting Eq. (5.58) into (5.57b), we find the state equations for the system:
The output equation is the same Eq. (5.58), or
In vector-matrix form
(5.57a)
(5.57b)
(5.58)
(5.59a)
(5.59b)
(5.60)
(5.61a)
(5.61b)

68. Alternative Representations in State Space
Using the transfer function as an example, Figure 5.29 compares the aforementioned forms.
Figure State-space forms for Note:

69. Alternative Representations in State Space
Figure State-space forms for Note:

70. Similarity Transformations
Similar systems
The various forms of the state equations
Although their state-space representations are different
it has the same TF, same poles and eigenvalues.
We can make transformations between similar systems without using the TF and signal-flow graphs.
A system represented in state space as
Can be transformed to a similar system,
(5.62a)
(5.62b)
(5.63a)
(5.63b)

71. Similarity Transformations
where, for 2D space,
and
Thus, P is a transformation matrix whose columns are the coordinates of the basis vectors of the space expressed as linear combinations of the space.
(5.63c)
(5.63d)
(5.63e)

72. Similarity Transformations
Ex 5.9 Similarity transformations on state equations
Given the system represented in state space by Eqs. (5.64),
Transform the system to a new set of state variables, z, where the new state variables are related to the original state variables, x, as follows:
(5.64a)
(5.64b)
(5.65a)
(5.65b)
(5.65c)

73. Similarity Transformations
Sol ) Expressing Eqs. (5.65) in vector-matrix form,
Using Eqs. (5.63) as a guide,
(5.66)
(5.67)

74. Similarity Transformations
Therefore, the transformed system is
One major advantage of finding these similar systems is apparent in the transformation to a system that has a diagonal matrix.
(5.68)
(5.69)
(5.70)
(5.71)

75. Similarity Transformations
Diagonalizing a System Matrix
Parallel form of a signal-flow graph can yield a diagonal system matrix
Rather than using partial fraction and signal flow graph
, will be a diagonal matrix
Definitions
Eigenvector:
The eigenvectors of the matrix A are all vectors, , which under the transformation A become multiples of themselves; that is,
where are constants.
(5.72)

76. Similarity Transformations
Figure shows this definition of eigenvectors. If Ax is not collinear with x after the transformation, as in Figure 5.30(a), x is not an eigenvector.
If Ax is collinear with x after the transformation, as in Figure 5.30(b), x is an eigenvector.
Figure To be an eigenvector, the transformation Ax must be collinear with x;.
thus, in (a), x is not an eigenvector; in (b), it is

77. Similarity Transformations
Eigenvalue:
The eigenvalues of the matrix A are the values of that satisfy Eq. (5.72) for
To find the eigenvectors, we rearrange Eq. (5.72). Eigenvectors, , satisfy
Solving for by pre-multiplying both sides by yields
Since , a nonzero solution exists if
From which , the eigenvalues, can be found.
(5.73)
(5.74)
(5.75)

78. Similarity Transformations
Ex 5.10 Finding eigenvectors
Find the eigenvectors of the matrix
Sol ) The eigenvectors, , satisfy Eq. (5. 73). First, use to find the eigenvalues, , for Eq.(5.73):
from which the eigenvalues are , and -4.
(5.76)
(5.77)

79. Similarity Transformations
Using Eq. (5.72) successively with each eigenvalue, we have or
from which Thus,
Using the other eigenvalue, -4, we have
(5.78)
(5.79a)
(5.79b)
(5.80)
(5.81)

80. Similarity Transformations
Using Eqs. (5.80) and (5.81), one choice of eigenvectors is
(5.82)

81. Similarity Transformations
System Matrix
We now show that if the eigenvectors of the matrix A are chosen as the basis vectors of a transformation, P, the resulting system matrix will be diagonal.
Let the transformation matrix P consist of the eigenvectors of A, .
Since are eigenvectors, , which can be written equivalently as a set of equations expressed by
where D is a diagonal matrix consisting of , the eigenvalues, along the diagonal, and P is as defined in Eq. (5.83). Solving Eq. (5.84) for D by pre-multiplying by , we get
which is the system matrix of Eq. (5.63).
(5.83)
(5.84)
(5.85)

82. Similarity Transformations
Ex 5.11 Diagonalizing a system in state space
Given the system of Eq. (8.85), find the diagonal system that is similar.
Sol ) First find the eigenvalues and the eigenvectors. This step was performed in Example 5.10 Next form the transformation matrix P, whose columns consist of the eigenvectors.
Finally, form the similar system’s system matrix, input matrix, and output matrix, respectively.
(5.86a)
(5.86b)
(5.87)

83. Similarity Transformations
Given the system of Eqs. (8.85), find the diagonal system that is similar.
Substituting Eqs. (5.88) into Eqs. (5.63), we get
(5.88a)
(5.88b)
(5.88c)
(5.89a)
(5.89b)

84. Similarity Transformations
Similar systems
In this section, we learned how to move between different state-space representation of the same system via matrix transformations rather than transfer function manipulation and signal-flow graphs.
These different representations are called similar.
The characteristics of similar systems are that the transfer functions relating the output to the input are the same, as are the eigenvalues and poles.
A particularly useful transformation was converting a system with distinct, real eigenvalues to a diagonal system matrix.

85. Intelligent Robotics LaboratoryH A N K Y O U
Homework: Divide by 8
Intelligent Robotics Laboratory