Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Slide 82Slide 83Slide 84Slide 85Slide 86Slide 87Slide 88Slide 89Slide 90Slide 91Slide 92Slide 93Illustrations We use quantitative mathematical models of physical systems to design and analyze control systems. The dynamic behavior is generally described by ordinary differential equations. We will consider a wide range of systems, including mechanical, hydraulic, and electrical. Since most physical systems are nonlinear, we will discuss linearization approximations, which allow us to use Laplace transform methods. We will then proceed to obtain the input–output relationship for components and subsystems in the form of transfer functions. The transfer function blocks can be organized into block diagrams or signal-flow graphs to graphically depict the interconnections. Block diagrams (and signal-flow graphs) are very convenient and natural tools for designing and analyzing complicated control systems Chapter 2: Mathematical Models of Systems ObjectivesIllustrationsIntroduction Six Step Approach to Dynamic System Problems•Define the system and its components•Formulate the mathematical model and list the necessary assumptions•Write the differential equations describing the model•Solve the equations for the desired output variables•Examine the solutions and the assumptions•If necessary, reanalyze or redesign the systemIllustrationsDifferential Equation of Physical SystemsTat( ) Tst( ) 0Tat( ) Tst( ) t( ) st( ) at( )Tat( )= through - variableangular rate difference = across-variableIllustrationsDifferential Equation of Physical Systemsv21Ltidd E12L i2v211ktFdd E12F2k211ktTdd E12T2kP21ItQdd E12I Q2Electrical InductanceTranslational SpringRotational SpringFluid InertiaDescribing EquationEnergy or PowerIllustrationsDifferential Equation of Physical SystemsElectrical CapacitanceTranslational MassRotational MassFluid CapacitanceThermal Capacitancei Ctv21dd E12M v212F Mtv2dd E12M v22T Jt2dd E12J 22Q CftP21dd E12Cf P212q CttT2dd E CtT2IllustrationsDifferential Equation of Physical SystemsElectrical ResistanceTranslational DamperRotational DamperFluid ResistanceThermal ResistanceF b v21 P b v212i1Rv21 P1Rv212T b 21 P b 212Q1RfP21 P1RfP212q1RtT21 P1RtT21IllustrationsDifferential Equation of Physical SystemsM2ty t( )dd2 bty t( )dd k y t( ) r t( )IllustrationsDifferential Equation of Physical Systemsv t( )RCtv t( )dd1L0ttv t( )d r t( )y t( ) K1e1 t sin 1t 1 IllustrationsDifferential Equation of Physical SystemsIllustrationsDifferential Equation of Physical SystemsK21 2.5 210 22y t( ) K2e2 t sin 2t 2 y1 t( ) K2e2 t y2 t( ) K2 e2 t0 1 2 3 4 5 6 7101y t( )y1 t( )y2 t( )tIllustrationsLinear ApproximationsIllustrationsLinear ApproximationsLinear Systems - Necessary conditionPrinciple of SuperpositionProperty of HomogeneityTaylor Serieshttp://www.maths.abdn.ac.uk/%7Eigc/tch/ma2001/notes/node46.htmlIllustrationsLinear Approximations – Example 2.1M 200gm g 9.8ms2 L 100cm 00rad 15 16 T0M g L sin 0 T1 M g L sin T2 M g L cos 0 0 T04 3 2 1 0 1 2 3 41050510T1( )T2( )Students are encouraged to investigate linear approximation accuracy for different values of0IllustrationsThe Laplace TransformHistorical Perspective - Heaviside’s OperatorsOrigin of Operational Calculus (1887)Illustrationsptdd1p0tu1divZ p( )Z p( ) R L pi1R L pH t( )1L p 1RL pH t( )1RRL1pRL21p2RL31p3 ..... H t( )1pnH t( )tnni1RRLtRL2t22RL3t33 .. i1R1 eRL tExpanded in a power seriesv = H(t)Historical Perspective - Heaviside’s OperatorsOrigin of Operational Calculus (1887)(*) Oliver Heaviside: Sage in Solitude, Paul J. Nahin, IEEE Press 1987.IllustrationsThe Laplace TransformDefinitionL f t( )( )0tf t( ) es td= F(s)Here the complex frequency is s j wThe Laplace Transform exists when 0tf t( ) es td this means that the integral convergesIllustrationsThe Laplace TransformDetermine the Laplace transform for the functionsa)f1t( ) 1fort 0F1s( )0tes td=1s es t( )1sb)f2t( ) ea t( )F2s( )0tea t( )es t( )d=1s 1 es a( ) t[ ] F2s( )1s aIllustrationsnote that the initial condition is included in the transformationsF(s) - f(0+)=Ltf t( )dds0tf t( ) es t( )d-f(0+) +=0tf t( ) s es t( )df t( ) es t( )=0vudwe obtainv f t( )anddu s es t( ) dtand, from whichdv df t( )u es t( )whereu v uvd=vudby the use ofLtf t( )dd0ttf t( ) es t( )ddd Evaluate the laplace transform of the derivative of a functionThe Laplace TransformIllustrationsThe Laplace TransformPractical Example - Consider the circuit. The KVL equation is4 i t( ) 2ti t( )dd 0assume i(0+) = 5 AApplying the Laplace Transform, we have0t4 i t( ) 2ti t( )ddes t( )d 0 40ti t( ) es t( )d 20tti t( ) es t( )ddd 04 I s( ) 2 s I s( ) i 0(
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