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Mathematical models. A system of equations consists of continuous-time ODE (or PDE) and algebraic equations A system of equations consists of discrete-time difference equations and algebraic equations S-domain transfer functions Z-domain Transfer functions State-space models … .
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Mathematical models • A system of equations consists of continuous-time ODE (or PDE) and algebraic equations • A system of equations consists of discrete-time difference equations and algebraic equations • S-domain transfer functions • Z-domain Transfer functions • State-space models • …
Problem Description What we have so far: A system of ODE to describe the process dynamics An equation for control algorithm An equation to describe the control Adjustment via control valve We have to solve this system of equations in order to learn the control performance.
The system of DAE to solve A system of algebra equations Much more easier to solve by algebra and Laplace inversion !!!
What is transfer function model? • A transfer function model in terms of Laplace transformed variables • is used to represent the transfer of input variable(s) to output • variable(s). --- A most convenient way to represent the solution of ODEs. • It consists of dynamic and static parts: • The dynamic part describes the transfer during the transient • stage --- dynamics. • The static part describes the permanent change due to the • transfer --- steady state gain. • Multiplication of input variable by the transfer function gives the output in terms of “s”, which can be converted into a function of time.
E R C Q Laplace transformation It means Q(s) is transfer from E(s) by: a dynamic part Q(s) E(s) and a steady-state gain
Deriving input/output transfer function • Dynamic equation from first principles • Steady state equation • Define deviation variables • Dynamic equation in terms of deviation variable---Linearization • Take Laplace transformation • Final transfer functions
Linearization of Nonlinear Models • Required to derive transfer function. • Good approximation near a given operating point. • Gain, time constants may change with • operating point. • Use 1st order Taylor series. Chapter 4 (4-60) (4-61) Subtract steady-state equation from dynamic equation (4-62)
Properties of Transfer Function Models • Steady-State Gain • The steady-state of a TF can be used to calculate the steady-state change in an output due to a steady-state change in the input. For example, suppose we know two steady states for an input, u, and an output, y. Then we can calculate the steady-state gain, K, from: Chapter 4 For a linear system, K is a constant. But for a nonlinear system, K will depend on the operating condition
Calculation of K from the TF Model: If a TF model has a steady-state gain, then: • This important result is a consequence of the Final Value Theorem • Note: Some TF models do not have a steady-state gain (e.g., integrating process in Ch. 5) Chapter 4
Order of a TF Model • Consider a general n-th order, linear ODE: Chapter 4 Take L, assuming the initial conditions are all zero. Rearranging gives the TF:
Deriving simple transfer functions --- process with single capacitor • Process with single capacitor --- First Order process • Process with two capacitors --- Second Order process • Process with many capacitors --- Higher order process • Infinitive order process --- pure time delay
First Order Thermal Process • Dynamic equation • Steady state equation Subtract the two equations to introduce deviation variables in the dynamic equation.
Thermal Process example ---Developing transfer function between T and Ti
Thermal Process example ---Developing transfer function between T and Ti
First Order Level process • total mass balance
Level process (continue) fi f2 f
Level process (continue)--- Analogy E R C Q C A h E Fi Q R Dh/DF
Level process (continue) --- Analogy Vo atm Ha+atm
Flow process --- Analogy p2 P1 f v2 v1 i R
Capacitance of Gas Process --- Analogy
Capacitance of Gas Process --- Analogy (continued)
Capacitance of Gas Process --- Analogy (continued)
Process with two capacitors (continued) Similarly
2nd order process General 2nd order ODE: Laplace Transform: Chapter 4 2 roots : real roots : imaginary roots
Examples 1. (no oscillation) Chapter 4 2. (oscillation)
From Table 3.1, line 17 Chapter 4
Example 1: Place sensor for temperature downstream from heated tank (transport lag) Distance L for plug flow, Dead time Chapter 4 V = fluid velocity Tank: Sensor: is very small (neglect) Overall transfer function:
Example 2: q0: control, qi: disturbance Use L.T. Chapter 4 (deviation variables) suppose q0 is constant pure integrator (ramp) for step change in qi
