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- 1. Chapter 3: Mathematical ModelingOutline Introduction Types of Models• Theoretical Models• Empirical Models• Semi-empirical Models LTI Systems• State variables Models• Transfer function Models Block diagram algebra Signal flow graph and Mason’s gain formula1
- 2. Introduction• A model is a mathematical representation of aphysical , biological or information system.Models allow us to reason about a system andmake predictions about how a system willbehave.Roughly speaking, a dynamic system is one inwhich the effect of actions do not occurimmediately.A model should capture the essence of thereality that we like to investigateDepending on the questions asked andoperational ranges, a physical system may havedifferent models. 2
- 3. Types of Models• Models can be classified based on how they areobtained.[A] Theoretical (or White Box) Models• Are developed using the physical and chemicallaws of conservation, such as mass balance ,component balance, moment balance andenergy balance.Advantages: provide physical insight into process behavior. applicable over wide ranges of operatingconditionsDisadvantage(s): 3
- 4. Types of Models…[B] Empirical (or Black Box) Models• Are obtained by fitting experimental data.Advantages: easier to develop than theoretical models. applicable over wide ranges of operatingconditionsDisadvantage(s):Typically don‟t extrapolate well!Caution!Empirical models should be used with caution foroperating conditions that were not included in theexperimental data used to fit the model4
- 5. Types of Models…[C] Semi-empirical (or Gray Box) Models• Are a combination of the models in categories (a)& (b).• Used in situation where much physical insight isavailable but certain information( parameter) orunderstanding is lacking.• Those unknown parameter(s) in a theoreticalmodel are calculated from experimental data.Advantages:They incorporate theoretical knowledgeThey can be extrapolated over a wide range ofoperating conditions.Require less development effort5
- 6. Theoretical ( White Box) Models• In this chapter we will be dealing models that aregenerated as a set of linear differential equations63
- 7. Theoretical ( White Box) Models73
- 8. Theoretical ( White Box) Models83
- 9. Theoretical ( White Box) ModelsExample 3.1: Cruise Control for a car• Goals - maintain the speed of a car at aprescribed value in the presence of externaldisturbances (external forces such as windgusts, gravitational forces on a incline, etc).Assume two forces show in the fig.• Fp(t) – the propulsive force from the engine• Fd(t) – a “ disturbance” force from the9
- 10. Theoretical ( White Box) ModelsExample 3.1:Cruise Control for a car…Also assume where is the gas-pedal depression and is a constant.Then from a simple force balance:10
- 11. Theoretical ( White Box) ModelsExample 3.1:Cruise Control for a car…• The corresponding block diagram ( forsimulation)• Closed-loop control for the same(using Pcontroller)11
- 12. Theoretical ( White Box) ModelsExample 3.2(a) Armature Controlled DC Motor12wvaTL
- 13. Theoretical ( White Box) ModelsExample 3.2: Armature Controlled DC Motor…• the back emf (refer the previous schematic) isgiven by(3.1a)• Applying KVL to the armature circuit(3.1b)• Because of const. flux, the torque produced atthe shaft by the armature current is(3.1c)• Assuming J and B for the motor, and TL loadcoupled to the shaft of the motor, the equationbecomes(3.1d) 13
- 14. Theoretical ( White Box) ModelsExample 3.2: Armature Controlled DC Motor…• By substitution of eqns. (3.1c) & (3.1d), we have• Substituting the above result into the armatureeqn.(and assuming constant TL).14
- 15. Theoretical ( White Box) ModelsExample 3.2 :Armature Controlled DC Motor…• or equivalently,Example 3.2(b): Field Controlled DC Motor 15+--+--1/LmkTBRmkbDC Motor Block
- 16. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?Ans: Find the relation between 1 & 3, i.e. , theModel equation relating the two waveforms.16
- 17. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?Typical waveforms in the circuit are:17
- 18. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?•A linearized model of the transformer/rectifiercircuit , with a voltage source (with a waveform asat (2) above) and a series resistor R ( a ThéveninSource)Objective: Find 18
- 19. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?...Solution:• combine series and parallel impedances tosimplify the structure. Draw as an impedance graphWhere19
- 20. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?Solution…• The system output is• Choose mesh loops to contact all branches asshown.The loop equations are:20
- 21. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?Solution…• Or• solving for i2(using Cramer‟s Rule)21
- 22. Theoretical ( White Box) ModelsExample 3.3: How does the circuit shown belowwork?Solution…• And since•Substituting the impedances of the branches22
- 23. Theoretical ( White Box) ModelsExample 3.4: A Bus Suspension System(1/4Bus):Assumptions:• 1/4 model ( one of the four wheels) is used tosimplify the problem to 1D multiple spring-dampersystem.23
- 24. Theoretical ( White Box) ModelsExample 3.4: A Bus Suspension System(1/4Bus)…From the free-body diagram, the dynamic equationsbecome24M1M2
- 25. LTI Systems• The set of ODEs drived so far are not suitable foranalysis and design, hence rearranged to a moresuitable form, i.e., State Space Model & TFModel[1] SS-Model:Def: State VariableA set of characterizing variables which give thetotal information about the system under study atany time provided the initial state & the external 25Set of ODEsState SpaceModelTransfer FunctionModel
- 26. LTI Systems[1] SS-Model…WhereA: system or dynamic matrix,B: input matrix ,C: output matrix,D: direct transfer matrix, 26
- 27. LTI SystemsDifferential equation SS-ModelExample3.5:Derive the state-space model (i.e. find the A,B,Cand D matrices) for each of the following differentialequations. Take u(t) to be the input and y(t) to bethe output.(1)(2)Solution:(1) DefineSo we have the state equations:27
- 28. LTI SystemsDifferential equation SS-ModelExample3.5…And the output equation:The state-space model is then:28
- 29. LTI Systems[2] TF-ModelIn general Transfer function is expressed as ()Differential equation TF modelExample 3.6:Find the transfer function for the system given inexample 3.5Solution:Taking LT on both sides of the equation gives (assuming zero initial conditions)29
- 30. LTI SystemsDifferential equation TF-ModelExample 3.6….Exercise3.1: Find the TF-model for part (2) inexample 3.5Ans:Exercise3.2: Find TF for a bus suspensiondiscussed (refer pp 23-24) using „Matlab SymbolicToolbox „ 30
- 31. Block Diagram Algebra (Interconnection Rules)[1] Series (Cascade) connection:Note: This is only true if the connection of H2(s) toH1(s) doesn‟t alter the output of H1(s)-known as the“no-loading” condition[2] Parallel Connection31++
- 32. Block Diagram Algebra (Interconnection Rules)[3] Associative Rule:[4] Commutative Rule:32++++
- 33. Block Diagram Algebra (Interconnection Rules)[5] The “closed-loop” TF:[5.a] Unity Feedback:From the block diagram:andorRearranging:33+-ReferenceinputerrorController PlantFeedbackpath
- 34. Block Diagram Algebra (Interconnection Rules)[5.b] Feedback with sensor dynamic:Similarly, in this case:But now E(s) is the “indicated error” ( as opposed tothe actual error):So34+-ReferenceinputerrorController PlantIndicatedOutputActualoutput
- 35. Block Diagram Algebra (Interconnection Rules)[5.2] Feedback with sensor dynamic:Or35
- 36. Signal flow graph• is a diagram consisting of nodes that are connectedby several directed branches and is a graphicalrepresentation of a set of linear relationships .•The signal can flow only in the direction of the arrowof the branch and it is multiplied by a factor indicatedalong the branch, which happens to be the coefficientof a model equation(s).Terminologies:Node: A node is a point representing a variable orsignalBranch: A branch is a directed line segment betweentwo nodes. The transmittance is the gain of a branch.Input node: An input node has only outgoing branchesand this represents an independent variable 36
- 37. Signal flow graph…Terminologies…Output node: An output node has only incomingbranches representing a dependent variableMixed node: A mixed node is a node that has bothincoming and outgoing branchesPath: Any continuous unidirectional succession ofbranches traversed in the indicated branch direction iscalled a path.Loop: A loop is a closed pathLoop gain: The loop gain is the product of the branchtransmittances of a loop37
- 38. Signal flow graph…Terminologies…Non-touching loops: Loops are non-touching if they donot have any common node.Forward path: A forward path is a path from an inputnode to an output node along which no node isencountered more than once.Feedback path (loop): A path which originates andterminates on the same node along which no node isencountered more than once is called a feedbackpath.Path gain: The product of the branch gainsencountered in traversing the path is called the pathgain.38
- 39. Signal flow graph…Illustrative example:Q. Write the equations for the system described bythe signal flow graph above. 39Mixednodesinputnodeinputnodeoutput nodex3x1x2x4x3g12 g23g43g321Fig. Signal flow graph
- 40. Signal flow graph…Properties of Signal flow graphs1. A branch indicates the functional dependence ofone variable on another.2. A node performs summing operation on all theincoming signals and transmits this sum to all outgoingbranches40G(s)R(s)Y(s)Y(s)G(s)R(s)R(s)-H(s)Y(s)G(s)E(s)1G(s)H(s)Y(s)E(s)R(s)+-Fig: Block diagrams and corresponding signal flowgraphs
- 41. Manson’s gain formulaIn a control system the transfer function betweenany input and any output may be found by Mason‟sGain formula. Mason‟s gain formula is given byWherewhere41
- 42. Manson’s gain formula…42
- 43. Manson’s gain formulaExample3.7: Find the closed loop transfer functionY(s)/R(s) using gain Manson‟s formula.43R(s) G1(s) G2(s) G3(s) x1(s)x3(s)x4(s)-H2(s)-H1(s)G4(s)Y(s)-1
- 44. Manson’s gain formulaExample 3.7…Here we have two forward paths with gains,And five individual loops with gainsNote for this example there are no non-touchingloops, so ∆ for this graph is44
- 45. Manson’s gain formulaExample 3.7…The value of ∆1is computed in the same way as ∆ byremoving the loops that touch 1st forward path M1• In this example, since path M1 touches all the fiveloops, ∆1 is found as• Proceeding the same way , we find•Therefore, the closed loop transfer functionbetween the input R(s) and output Y(s) is given by,45
- 46. Manson’s gain formulaExample 3.8Find Y(s)/R(s) for the system represented by thesignal flow graph shown below.46G4X2X3G2 G5 X1G3G1X4-H1-H2G6G7R(s) Y(s)
- 47. Manson’s gain formulaExample 3.8…Observe from the signal flow graph , there are threeforward paths between R(s) and Y(s)• The respective forward path gains are:There are four individual loops with gains:47
- 48. Manson’s gain formulaExample 3.8…Since the loops L2 & L4 are the only non-touchingloops in the graph, the determinant ∆ will be givenby:Computing ∆1,which is computed by removing theloops that touch fist forward path M1∆1=1Similarly , ∆2=1 andThus, the closed-loop TF is given by Y(s)/R(s)48

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