Presentation on theme: "Rules to reduce block diagrams Transfer Function Problem solving"— Presentation transcript:
1 Rules to reduce block diagrams Transfer Function Problem solving BIRLA VISHVAKARMA MAHAVIDYALYAMECHANICAL DEPARTMENTBATCH C-7Subject : Control engineeringCreated by:-Rajesh ValiyaChirag VaniNilesh VishwaniJalay VyasSubmitted to: Prof. S.P. Joshi
2 Interpreting block diagrams Block diagram of a system is a pictorialrepresentation of the functions performed by each component and of the flow signals.Block diagrams are used as schematic representations of mathematical modelsThe various pieces correspond to mathematical entitiesCan be rearranged to help simplify the equations used to model the system
3 Transfer FunctionThe 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.
5 Generic Feedback Control System This is a general model, and may not be the same for every feedback control systemSystems can have additional inputs known as disturbances into or between processesCan combine processes; typically controller and actuator are combineddesired outputoutputcontrolleractuatorplantfeedback
6 Cruise Control System input: desired speed output: actual speed error: desired speed minus measured speeddisturbance: wind, hills, etc.controller: cruise control unitactuator: engineplant: vehicle dynamicssensor: speedometerwind, hillsdesired speedactual speedcruise controlenginevehiclespeedo-meter
7 Toilet Flush Example Float height determines desired water level Flush empties tank, float is lowered and valve opensOpen valve allows water to enter tankFloat returns to desired level and valve closesflushdesired levelactual levelfloatvalvewater tankfloat
8 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.
9 block diagram: example eieoiLet consider the RC circuit:The equations for this circuit are:
11 block representations for Laplace transforms: block diagram: exampleblock representations for Laplace transforms:_+
12 Assembly the elements into a complete block diagram. block diagram: exampleAssembly the elements into a complete block diagram._+
13 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.
14 Components of a block diagram for a linear, time-invariant system
15 a. Cascaded subsystems; b. equivalent transfer function
16 a. Parallel subsystems; b. equivalent transfer function
17 a. Feedback control system; b. simplified model; c a. Feedback control system; b. simplified model; c. equivalent transfer function
18 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
19 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
29 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)
30 Block diagram reduction by moving blocks Problem: Reduce the block diagram shown in figure to a single transfer function
31 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 junctionc) Collapse the summing junctions, add the 2 feedback elements, and combine the last 2 cascade blocksd) Reduce the feedback system to the lefte) finally, Multiple the 2 cascade blocks and obtain final result.
32 References Modern control engineering by K. Ogata Control engineering by Nagrath & GopalModern control system by R.H. BishopControl Engineering by B.S. Manke