Lect 4 - Reduction of Multiple Subsystems

Reduction of Multiple Subsystems

Ref: Control System EngineeringNorman Nise : Chapter 5

RM 1511 – AUTOMATIC CONTROL

Multiple subsystems - 2

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



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



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



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



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



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



Equivalent diagrams for the diagram shown in Example 1



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



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



Moving a summing junction

Ahead of a block After a block



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



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



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



Block diagram reduction via familiar form

Example 3 reduce the block diagram shown below to a single transfer function



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)



Block diagram reduction by moving blocks

Example 4 reduce the block diagram shown below to a single transfer function



Steps in the block diagram reduction forExample 4



Example 5 find the equivalent transfer function T(s)=C(s)/R(s)



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.



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



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



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



a. cascaded system nodes; b. cascaded system signal-flow graph;

Converting common block diagrams to signal-flow graphs



c. parallel system nodes; d. parallel system signal-flow graph;



e. feedback system nodes; f. feedback system signal-flow graph;



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



Example 12 Convert the block diagram below to signal-flow graph

Lect 4 - Reduction of Multiple Subsystems

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