Rules to reduce block diagrams Transfer Function Problem solving BIRLA VISHVAKARMA MAHAVIDYALYA MECHANICAL DEPARTMENT BATCH C-7 Subject : Control engineering Created by:- 130070119093-Rajesh Valiya 130070119094-Chirag Vani 130070119095-Nilesh Vishwani 130070119096-Jalay Vyas Submitted to: Prof. S.P. Joshi

Interpreting block diagrams Block diagram of a system is a pictorial representation of the functions performed by each component and of the flow signals. Block diagrams are used as schematic representations of mathematical models The various pieces correspond to mathematical entities Can be rearranged to help simplify the equations used to model the system

Transfer Function The transfer function of a linear, time invariant , differential equation system is defined as the ratio of the Laplace transform of output to the Laplace transform of input under the assumption that all initial conditions are zero.

Important comments concerning transfer function

Generic Feedback Control System This is a general model, and may not be the same for every feedback control system Systems can have additional inputs known as disturbances into or between processes Can combine processes; typically controller and actuator are combined desired output output controller actuator plant feedback

Cruise Control System input: desired speed output: actual speed error: desired speed minus measured speed disturbance: wind, hills, etc. controller: cruise control unit actuator: engine plant: vehicle dynamics sensor: speedometer wind, hills desired speed actual speed cruise control engine vehicle speedo-meter

Toilet Flush Example Float height determines desired water level Flush empties tank, float is lowered and valve opens Open valve allows water to enter tank Float returns to desired level and valve closes flush desired level actual level float valve water tank float

procedures for drawing block diagram Write the equations that describe the dynamic behavior for each component. Take Laplace transform of these equations, assuming zero initial conditions. Represent each Laplace-transformed equation individually in block form. Assembly the elements into a complete block diagram.

block diagram: example ei eo i Let consider the RC circuit: The equations for this circuit are:

block diagram: example take Laplace transform:

block representations for Laplace transforms: block diagram: example block representations for Laplace transforms: _ +

Assembly the elements into a complete block diagram. block diagram: example Assembly the elements into a complete block diagram. _ +

block diagram reduction Rules for reduction of the block diagram: Any number of cascaded blocks can be reduced by a single block representing transfer function being a product of transfer functions of all cascaded blocks. The product of the transfer functions in the feedforward direction must remain the same. The product of the transfer functions around the loop mast remain the same.

Components of a block diagram for a linear, time-invariant system

a. Cascaded subsystems; b. equivalent transfer function

a. Parallel subsystems; b. equivalent transfer function

a. Feedback control system; b. simplified model; c a. Feedback control system; b. simplified model; c. equivalent transfer function

Block diagram algebra for summing junctions— equivalent forms for moving a block a. to the left past a summing junction; b. to the right past a summing junction

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

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Block diagram

a. collapse summing junctions; b a. collapse summing junctions; b. form equivalent cascaded system in the forward path and equivalent parallel system in the feedback path; c. form equivalent feedback system and multiply by cascaded G1(s)

Block diagram reduction by moving blocks Problem: Reduce the block diagram shown in figure to a single transfer function

Steps in the block diagram reduction a) Move G2(s) to the left past of pickoff point to create parallel subsystems, and reduce the feedback system of G3(s) and H3(s) b) Reduce parallel pair of 1/G2(s) and unity, and push G1(s) to the right past summing junction c) Collapse the summing junctions, add the 2 feedback elements, and combine the last 2 cascade blocks d) Reduce the feedback system to the left e) finally, Multiple the 2 cascade blocks and obtain final result.

References Modern control engineering by K. Ogata Control engineering by Nagrath & Gopal Modern control system by R.H. Bishop Control Engineering by B.S. Manke