Text Illustrations in PPTChapter 3: THEORETICAL FOUNDATION AND BACKGROUND MATERIAL: COMPLEX VARIABLES,

DIFFERENTIAL EQUATIONS, LAPLACE TRANSFORM

Automatic Control Systems, 9th Edition

Farid Golnaraghi, Simon Fraser UniversityBenjamin C. Kuo, University of Illinois

ISBN: 978-0-470-04896-2

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Heating system block diagram (simplified).Actual temp. (output)measured by sensor in the thermostat. Simple electronic circuit (comparator) compares temps. Generates error voltage that acts as as switch to open the gas valve to turn out the furnace. Opening windows and doors etc. in room causes heat loss (disturbance). Process of sensing output and comparing with input to generate error signal called Feedback.

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(a) Open loop, dc-motor, speed control system (b) Block diagram; input voltage to the motor, output of (non linear) power amp. Representation issue for NL blocks.

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Common elements in block diagram of most control systems (Compare to Figs.3-1, 3-2

. Comparators (electronic circuit measures error)

. BLOCKS representing transfer functions . Reference sensor . Output sensor . Actuator . Controller . Plant . INPUT or reference signals. OUTPUT signals. Disturbance signal. Feedback loops

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X(s) = G(s) U(s)

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X(s) = G_1(s) G_2(s) U(s)

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G(s) = G_1(s) + G_2(s)

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M(s) = Y(s) = G(s) R(s) 1 + G(s) H(s)

M(s) = Y(s) = G(s) R(s) 1 - G(s) H(s)

negative feedback

positive feedback

x(t) + 2ζωnx(t) + ω2n(t)x(t) = ω2

nu(t)

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2ζω2nsX(s)

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ω2n

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Moving a branch point

H_1(s) G_2(s)

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Moving a comparator from RHS of G_2(s) to its LHS

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Block diagram reduction

first move branch pt. at Y_1 to left of G_2

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1 + G_1G_2H_1

Y (s)R(s)

=G1(s)G2(s)

1 + G1(s)G2(s)H1(s)

Y (s)R(s)

=−G2(s)

1 + G1(s)G2(s)H1(s)

Ytotal = YR|D=0 + YD|R=0

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Block diagram of system undergoing disturbance

We need to determine effects of D(s) on the system

So disturbance interferes with controller signal and adversely affectssystem performance.

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Block diagram for D(s) = 0

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Block diagram for R(s)=0

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Y (s) = [I + G(s)H(s)]−1G(s)R(s)

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Signal Flow Graphs

Basic properties of SFG. Only for linear systems. Equations for SFG algebraic equations. Nodes are used to represent variables. Signals travel along branches only in direction of arrows. Branch from node y_1 to y_2 shows dependence of node y_2 on node y_1

Other SFG termsInput node (source) - A node that only has outgoing branchesOutput node (sink) - A node that only has incoming branches

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Can make y_2 an output node.

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Erroneous operation for SFG of Fig. 3-26(a)

Cannot make y_2 an input node; equation for y_2 different from original SFG.

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4 loops in the SFG of Fig.3-25(d)

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Node y_1 a summing point; also a transmitting point.

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Simplification of SFG

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SFG for feedback system of Fig.3-8

SFG Terms

Forward Path: a path that starts at an input node and ends at an output node and along which no no node is traversed more than once.

Path Gain: The product of the branch gains encountered in traversing a path.

Loop: A path that starts and terminates at the same node and along which no other node is encountered more than once.

Forward-Path Gain: Path gain of a forward path.

Loop Gain: Path gain of a loop.

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What is the gain y7/y1 ?

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