Post on 30-Nov-2015
Reduction of Multiple Subsystems
Ref: Control System EngineeringNorman Nise : Chapter 5
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Chapter objectives : How to reduce a block diagram of multiple subsystems to
a single block representing the transfer function from input to output
How to analyze and design transient response for a system consisting of multiple subsystems
How to represent in state space a system consisting of multiple subsystems
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1. Block Diagrams for Dynamic SystemsBlock diagram an interconnection of blocks representing basic mathematical operations in such a way that the overall diagram is equivalent to the system’s mathematical model.
In such a diagram, the lines interconnecting the blocks represent the variables describing the system behaviour.
Kx f
A block diagram representing f = Kx
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Summer addition and subtraction of variables x1
x2
x3
++
-
y
A summer representing y = x1 + x2 - x3
Pickoff point input signal distribution to several output point
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Constant has no input, and its output never changes
cy
Gain multiplication of a single by a constant (exp. spring)
Integrator integration with respect to time
u y y y
dtdt
The block diagram for an integrator
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Rules for altering diagram structureTransfer functions which are generally the ratio of two polynomials are often denoted by F(s), G(s) or H(s). When the transfer function is a constant,then that block reduces to a gain block.
Series combination
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Parallel combination
Example 1Evaluate the transfer functions Y(s)/U(s) and Z(s)/U(s) for the block diagram below give the results as rational functions of s
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Equivalent diagrams for the diagram shown in Example 1
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Moving a pick off point a point where an incoming variable in the diagram is directed into more than one block
1 Original diagram, 2 & 3 equivalent diagrams
(2)
(1)
(3)
Moving block to create familiar forms
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Block diagram algebra for pickoff points - equivalent forms for moving a blocka. to the left past a pickoff point;b. to the right past a pickoff point
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Moving a summing junction
Ahead of a block After a block
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Block diagram algebra for summing junctions - equivalent forms for moving a blocka. to the left past asumming junction;b. to the right past asumming junction
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Example 2 Modify the bock diagram in (a) to remove the right summing junction, leaving only the left summing junction
(a) Original diagram, (b), (c) & (d) equivalent diagrams
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Reducing diagrams for feedback systems
G(s) = Y(s)/V(s) forward transfer function
H(s) = Z(s)/Y(s) feedback transfer function
G(s)H(s) open-loop transfer function
T(s) = Y(s)/U(s) closed-loop transfer function
H(s) = 1 unity feedback system
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Block diagram reduction via familiar form
Example 3 reduce the block diagram shown below to a single transfer function
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Steps in solving Example 3:a. collapse summing junctions;b. form equivalent cascaded system in the forward pathand equivalent parallel system in the feedback path;c. form equivalent feedback system and multiply by cascadedG1(s)
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Block diagram reduction by moving blocks
Example 4 reduce the block diagram shown below to a single transfer function
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Steps in the block diagram reduction forExample 4
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Example 5 find the equivalent transfer function T(s)=C(s)/R(s)
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Example 6 Find the closed-loop transfer function for the feedback system below. Compare the locations of the poles of the open-loop and closed-loop transfer function in s-plane.
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Example 7 Find the closed-loop transfer function of the two-loop feedback system in Fig 1. Also express the damping ratio and the un-damped natural frequency of the closed-loop system in terms of the gains a0 and a1.
Equivalent block diagrams
Figure 1
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2. Analysis and Design of Feedback System
Immediate application of the principles of block diagram.
Example 9 find the peak time, percent overshoot and settling time.
Example 10 design the value gain K for the system below so that the system will respond with a 10 % overshoot
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3. Signal-Flow Graphs
Signal flow graphs are alternative to block diagram.A signal flow graph consists only of branches, which represent systems, and nodes, which represent signals.
Signal-flow graph components:a. system;b. signal;c. interconnection of systems and signals
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a. cascaded system nodes; b. cascaded system signal-flow graph;
Converting common block diagrams to signal-flow graphs
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c. parallel system nodes; d. parallel system signal-flow graph;
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e. feedback system nodes; f. feedback system signal-flow graph;
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Example 11 Convert the block diagram in Example 4 to signal-flow graph.
Signal-flow graph development:a. signal nodes;b. signal-flow graph;c. simplified signal-flow graph
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Example 12 Convert the block diagram below to signal-flow graph