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Transfer Function Models of Dynamical Processes

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Transfer Function Models of Dynamical Processes Transfer Function Models of Dynamical Processes Process Dynamics and Control 1 LinearLinear SISOSISO ControlControl SystemsSystems General form of a linear SISO control system: this is a underdetermined higher order differential equation the function must be specified for this ODE to admit a well defined solution 2 TransferTransfer FunctionFunction Heated stirred-tank model (constant flow, ) Taking the Laplace transform yields: or letting Transfer functions 3 TransferTransfer FunctionFunction Heated stirred tank example + + + e.g. The block is called the transfer function relating Q(s) to T(s) 4 ProcessProcess ControlControl Time Domain Laplace Domain Process Modeling, Transfer function Experimentation and Modeling, Controller Implementation Design and Analysis Ability to understand dynamics in Laplace and time domains is extremely important in the study of process control 5 TransferTransfer functionsfunctions Transfer functions are generally expressed as a ratio of polynomials Where The polynomial is called the characteristic polynomial of Roots of are the zeroes of Roots of are the poles of 6 TransferTransfer functionfunction Order of underlying ODE is given by degree of characteristic polynomial e.g. First order processes Second order processes Order of the process is the degree of the characteristic (denominator) polynomial The relative order is the difference between the degree of the denominator polynomial and the degree of the numerator polynomial 7 TransferTransfer FunctionFunction Steady state behavior of the process obtained form the final value theorem e.g. First order process For a unit-step input, From the final value theorem, the ultimate value of is This implies that the limit exists, i.e. that the system is stable. 8 TransferTransfer functionfunction Transfer function is the unit impulse response e.g. First order process, Unit impulse response is given by In the time domain, 9 TransferTransfer FunctionFunction Unit impulse response of a 1st order process 10 DeviationDeviation VariablesVariables To remove dependence on initial condition e.g. Compute equilibrium condition for a given and Define deviation variables Rewrite linear ODE 0 or 11 DeviationDeviation VariablesVariables Assume that we start at equilibrium Transfer functions express extent of deviation from a given steady-state Procedure Find steady-state Write steady-state equation Subtract from linear ODE Define deviation variables and their derivatives if required Substitute to re-express ODE in terms of deviation variables 12 ProcessProcess ModelingModeling Gravity tank Fo h F Objectives: height of liquid in tank L Fundamental quantity: Mass, momentum Assumptions: Outlet flow is driven by head of liquid in the tank Incompressible flow Plug flow in outlet pipe Turbulent flow 13 TransferTransfer FunctionsFunctions From mass balance and Newton’s law, A system of simultaneous ordinary differential equations results Linear or nonlinear? 14 NonlinearNonlinear ODEsODEs Q: If the model of the process is nonlinear, how do we express it in terms of a transfer function? A: We have to approximate it by a linear one (i.e.Linearize) in order to take the Laplace. f(x) f(x0) !f (x0) !x x0 x 15 NonlinearNonlinear systemssystems First order Taylor series expansion 1. Function of one variable 2. Function of two variables 3. ODEs 16 TransferTransfer FFunctionunction Procedure to obtain transfer function from nonlinear process models Find an equilibrium point of the system Linearize about the steady-state Express in terms of deviations variables about the steady-state Take Laplace transform Isolate outputs in Laplace domain Express effect of inputs in terms of transfer functions 17 BlockBlock DiagramsDiagrams Transfer functions of complex systems can be represented in block diagram form. 3 basic arrangements of transfer functions: 1. Transfer functions in series 2. Transfer functions in parallel 3. Transfer functions in feedback form 18 BlockBlock DiagramsDiagrams Transfer functions in series Overall operation is the multiplication of transfer functions Resulting overall transfer function 19 BlockBlock DiagramsDiagrams Transfer functions in series (two first order systems) Overall operation is the multiplication of transfer functions Resulting overall transfer function 20 TransferTransfer FunctionsFunctions DC Motor example: In terms of angular velocity In terms of the angle 21 TransferTransfer FunctionsFunctions Transfer function in parallel + + Overall transfer function is the addition of TFs in parallel 22 TransferTransfer FunctionsFunctions Transfer function in parallel + + Overall transfer function is the addition of TFs in parallel 23 TransferTransfer FunctionsFunctions Transfer functions in (negative) feedback form Overall transfer function 24 TransferTransfer FunctionsFunctions Transfer functions in (positive) feedback form Overall transfer function 25 TransferTransfer FunctionFunction Example 3.20 26 TransferTransfer FunctionFunction Example A positive feedback loop 27 TransferTransfer FunctionFunction Example 3.20 Two systems in parallel Replace by 28 TransferTransfer FunctionFunction Example 3.20 Two systems in parallel 29 TransferTransfer FunctionFunction Example 3.20 A negative feedback loop 30 TransferTransfer FunctionFunction Example 3.20 Two process in series 31 FirstFirst OrderOrder SystemsSystems First order systems are systems whose dynamics are described by the transfer function where is the system’s (steady-state) gain is the time constant First order systems are the most common behaviour encountered in practice 32 FirstFirst OrderOrder SystemsSystems Examples, Liquid storage Assume: Incompressible flow Outlet flow due to gravity Balance equation: Total Flow In Flow Out 33 FirstFirst OrderOrder SystemsSystems Balance equation: Deviation variables about the equilibrium Laplace transform First order system with 34 FirstFirst OrderOrder SystemsSystems Examples: Cruise control DC Motor 35 FirstFirst OrderOrder SystemsSystems Liquid Storage Tank Speed of a car DC Motor First order processes are characterized by: 1. Their capacity to store material, momentum and energy 2. The resistance associated with the flow of mass, momentum or energy in reaching their capacity 36 FirstFirst OrderOrder SystemsSystems Step response of first order process Step input signal of magnitude M The ultimate change in is given by 37 FirstFirst OrderOrder SystemsSystems Step response 38 FirstFirst OrderOrder SystemsSystems What do we look for? System’s Gain: Steady-State Response Process Time Constant: Time Required to Reach 63.2% of final value What do we need? System initially at equilibrium Step input of magnitude M Measure process gain from new steady-state Measure time constant 39 FirstFirst OrderOrder SystemsSystems First order systems are also called systems with finite settling time The settling time is the time required for the system comes within 5% of the total change and stays 5% for all times Consider the step response The overall change is 40 FirstFirst OrderOrder SystemsSystems Settling time 41 FirstFirst OrderOrder SystemsSystems Process initially at equilibrium subject to a step of magnitude 1 42 FirstFirst orderorder processprocess Ramp response: Ramp input of slope a 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 43 FirstFirst OrderOrder SystemsSystems Sinusoidal response Sinusoidal2 input Asin(ωt) 1.5 1 0.5 AR 0 -0.5 -1 φ -1.5 0 2 4 6 8 10 12 14 16 18 20 44 FirstFirst OrderOrder SystemsSystems 0 Bode Plots 10 High Frequency Asymptote -1 Corner Frequency 10 10 -2 -2 -1 0 1 2 10 10 10 10 10 0 -20 -40 -60 -80 -100 10 -2 10 -1 10 0 10 1 10 2 Amplitude Ratio Phase Shift 45 IntegratingIntegrating SystemsSystems Example: Liquid storage tank Fi h F Laplace domain dynamics If there is no outlet flow, 46 IntegratingIntegrating SystemsSystems Example Capacitor Dynamics of both systems is equivalent 47 IntegratingIntegrating SystemsSystems Step input of magnitude M Slope = Input Output Time Time 48 IntegratingIntegrating SystemsSystems Unit impulse response Input Output Time Time 49 IntegratingIntegrating SystemsSystems Rectangular pulse response Input Output Time Time 50 SecondSecond orderorder SystemsSystems Second order process: Assume the general form where = Process steady-state gain = Process time constant = Damping Coefficient Three families of processes Underdamped Critically Damped Overdamped 51 SecondSecond OrderOrder SystemsSystems Three types of second order process: 1. Two First Order Systems in series or in parallel e.g. Two holding tanks in series 2. Inherently second order processes: Mechanical systems possessing inertia and subjected to some external force e.g. A pneumatic valve 3. Processing system with a controller: Presence of a controller induces oscillatory behavior e.g. Feedback control system 52 SecondSecond orderorder SystemsSystems Multicapacity Second Order Processes Naturally arise from two first order processes in series By multiplicative property of transfer functions 53 TransferTransfer FunctionsFunctions First order systems in parallel + + Overall transfer function a second order process (with one zero) 54 SecondSecond OrderOrder SystemsSystems Inherently second order process: e.g. Pneumatic Valve By Newton’s law p x 55 SecondSecond OrderOrder SystemsSystems Feedback Control Systems 56 SecondSecond orderorder SystemsSystems Second order process: Assume the general form where = Process steady-state
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