Unformatted text preview: Mathematical Models of Systems - II 446 - 3 Prof. Neil A. Duffie University of Wisconsin-Madison Wisconsin- 3 Neil A. Duffie, 1996 All rights reserved. 1 Obtaining a Workable Model Objective is a low-order linear model that lowis valid and makes sense "at a glance". Understand the physical components. Make appropriate assumptions. Use basic principles to formulate model. Linearize where necessary. Write differential and algebraic equations. Combine equations and simplify.
3 Check validity of model. 2 Common Process Types Delay: Integration: First-order: First c(t) = Km(t - D) dc(t) = Km(t) dt dc(t) + c(t) = Km(t) dt Second-order: Second2 1 dc (t) + 2 dc(t) + c(t) = Km(t) n dt n2 dt2 3 3 Assumptions in Spring-Mass Model Springy(t), [m] System
Model spring k [N/m] mass M [kg] M dy (t) + y(t) = 0 k dt2 2 Is the model correct if the spring is compressed too far to the left? Is the model correct if the spring is extended too far to the right?
3 4 Linearity of Spring-Mass Model Springy(t), [m] mass M [kg]
2 M dy (t) + y(t) = 0 k dt2 spring k [N/m] All models incorporate approximation and therefore incorporate some error. There is a range, around an operating point, over which a system's behavior can be approximated by a linear model. 3 5 Linearization For a linear component in a system, if input level x1 produces output y1 and if input level x2 produces output y2, then x1 + x2 produces output y1 + y2 Linear example: dy(t) = K x(t ) dt = K [x(t )]2
6 Nonlinear example: dy(t) dt The real world is nonlinear!
3 Taylor Series Expansion - 1 Variable
f(x) = f(x 0 ) + 1 df 1 d2f (x - x0 ) + (x - x 0 )2 1 dx x 0 ! 2! dx2 x
0 + higher order terms x0 is the operating point about which the expansion made. Over a small range (x - x0), the slope at the operating point is a good approximation of the function: df f(x) f(x 0 ) + (x - x 0 ) dx x0 3 7 Taylor Series Expansion - Multi Variable
f(x, y,...) = f(x 0 , y 0 ,...) + + f y x f (x - x 0 ) x x0 ,y 0 ,... (y - y 0 ) + ...
0 ,y 0 ,... x0, y0, ... is the operating point about which the expansion made. Over a small range (x - x0), (y - y0), ... the slope at the operating point is a good approximation of the function (higher(higherorder terms can be neglected). 3 8 Nonlinear Spring
y(t), [m] mass M [kg] f0 f(y) [N] spring f(y) [N] f0 = f(y 0 ) f(y) f0 + df (y - y 0 ) dy y
0 y0 y [m] 3 df f(y) - f0 (y - y 0 ) dy y
0 f df y dy y
0 9 Component Combination Example Flow control valve Cylinder Mixer + Pipe
cylinder flow q(t) [cm3/s] water concentrated solution
3 valve voltage v(t) [v] + flow control valve Input hydraulic supply return concentration c(t) [%] Output mixtur e pipe length flow rate mixer position dp [m] qp [m/s] y(t) [cm] 10 valve position x(t) [%] water Mixer and Pipe
concentration c(t) [%] Input
pipe length dp [m] concentrated solution flow rate qp [m/s] Delay process
3 d D = qp [s] p c(t) = x(t - D) [%]
11 Input q(t) [cm3/s] Output
x(t) [cm] Hydraulic Cylinder A [cm2] Integration process
3 dx(t) = Kcq(t) cm s dt Kc = 1 A
12 Component Combination Example flow control valve (2nd-order): (2nd2 1 dq (t) + 2 dq(t) + q(t) = K v(t) v n dt n2 dt2 cylinder: dx(t) = Kcq(t) dt pipe: c(t) = x(t - D)
3 13 Input Output Component Combination Example cylinder: pipe: dx(t) c(t) = x(t - D) = Kcq(t) dt dc(t) dx(t - D) = dt dt cylinder + pipe: dc(t) = Kcq(t - D) dt
3 14 dx(t - D) = Kcq( t - D) dt Component Combination Example flow control valve + cylinder + pipe: 3 1 dc (t) + 2 dc2(t) + dc(t) 2 n dt2 dt n dt3 = K v(t - D) K = KcKv Simplify?
3 Third-order process Thirdwith delay (2nd-order+integration (2ndwith delay)
15 Verify Can model be simplified? If the natural frequency, n, is very high compared to the operating frequencies expected and damping is neither very high nor very low ( 1), ( dc(t) Kv(t - D) dt If the delay D is very short compared to response times expected, 3 dc(t) K v(t ) dt 16 Need Laplace Transforms Facilitate combining and manipulating differential equations Facilitate simplification of models Facilitate drawing block diagrams of components and systems Facilitate solving differential equations (finding outputs for various inputs) Facilitate calculating frequency response, analysing stability, etc.
3 Facilitate controller design. 17 ...
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