Treatment of Discontinuities
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- Hubert Doyle
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1 Treatment of Discontinuities Today, we sall look at te problem of dealing wit discontinuities in models. Models from engineering often exibit discontinuities tat describe situations suc as switcing, limiters, dry friction, impulses, or similar penomena. Te modeling environment must deal wit tese problems in special ways, since tey influence strongly te numerical beavior of te underlying differential equation solver. Table of Contents Numerical differential equation solvers Discontinuities in state equations Integration across discontinuities State events Event andling Multi-valued functions Te electrical switc Te ideal diode Friction 1
2 Numerical Differential Equation Solvers All of te differential equation solvers tat are currently on te market operate on polynomial extrapolation. Te value of a state variable x at time t+, were is te current integration step size, is approximated by fitting a polynomial of n t order troug known supporting values of x and dx/dt at te current time t as well as at past instances of time. Te value of te extrapolation polynomial at time t+ represents te approximated solution of te differential equation. In te case of implicit integration algoritms, te state derivative at time t+ is also used as a supporting value. Examples Explicit Euler Integration Algoritm of 1 st Order: x(t+) x(t) + x(t) Implicit Euler Integration Algoritm of 1st Order: x(t+) x(t) + x(t+) 2
3 Discontinuities in State Equations Polynomials are always continuous and continuously differentiable functions. Terefore, wen te state equations of te system: x(t) = f(x(t),t) exibit a discontinuity, te polynomial extrapolation is a very poor approximation of reality. Consequently, integration algoritms wit a fixed step size exibit a large integration error, wereas integration algoritms wit a variable step size must reduce te step size dramatically in te vicinity of te discontinuity. Integration Across Discontinuities An integration algoritm of variable step size reduces te step size at every discontinuity. After passing te discontinuity, te step size is only slowly enlarged again, as te integration algoritm cannot distinguis between a discontinuity on one and and a point of large local stiffness (wit a large absolute value of te derivative) on te oter. Discontinuities t Te step size is constantly to small. Te integration algoritm is at least igly inefficient, if not even inaccurate. 3
4 Te State Event Tese problems can be avoided by telling te integration algoritm explicitly, wen and were discontinuities are contained in te model description. Example: Limiter Function f p f(x) 3 x m α 2 x p x f = f m f = m x f = f p 1 f m m = tg(α) f = if x < xm ten fm else if x < xp ten m*x else fp ; Event Handling I f p f(x) x p x Iteration Tresold x m α x p x t f m x Model switcing x p Event t Step size reduction during process of iteration t 4
5 Event Handling II t t Step size as function of time witout event andling Step size as function of time wit event andling Representation of Discontinuities f = if x < xm ten fm else if x < xp ten m*x else fp ; In Modelica, discontinuities are represented as if-statements. In te process of translation, tese statements are transformed into correct event descriptions (sets of models wit switcing conditions). Te modeler does not need to concern im or erself wit te mecanisms of event descriptions. Tese are idden beind te if-statements. 5
6 Problems Te modeler needs to take into account tat te discontinuous solution is temporarily left during iteration. q = p p = p 1 p 2 ; abs p= if p> 0 ten p else p; q = sqrt(abs p) ; may be dangerous, since abs p can become temporarily negative. p = p 1 p 2 ; abs p=noevent( if p> 0 ten p else p ) ; q = sqrt(abs p) ; solves tis problem. Te noevent Construct p = p 1 p 2 ; abs p=noevent( if p> 0 ten p else p ) ; q = sqrt(abs p) ; Te noevent construct as te effect tat if-statements or Boolean expressions, wic normally would be translated into simulation code containing correct event andling instructions, are anded over to te integration algoritm untouced. Tereby, management of te simulation across tese discontinuities is left to te step size control of te numerical Integration algoritm. 6
7 Multi-valued Functions I Te language constructs tat ave been introduced so far don t suffice to describe multi-valued functions, suc as te dry ysteresis function sown below. f(x) f p x m x p x f m Wen x becomes greater tan x p, f must be switced from f m to f p. Wen x becomes smaller tan x m, f must be switced from f p to f m. Multi-valued Functions II f(x) f p x m x p x f m wen initial() ten reinit(f, fp); end wen; becomes larger wen x > xp or x < xm ten f = if x > 0 ten fp else fm; end wen; is larger Executed at te beginning of te simulation. } Tese statements are only executed, wen eiter x becomes larger tan x p, or wen x becomes smaller tan x m. 7
8 Te Electrical Switc I i u Wen te switc is open, te current is i=0. Wen te switc is closed, te voltage is u=0. 0 = if open ten i else u; Te if-statement in Modelica is a-causal. It is being sorted togeter wit all oter statements. Te Electrical Switc II Possible Implementation: Switc open: s = 1 Switc closed: s = 0 0 = s i + ( 1 s ) u Switc open: SF f = 0 s Sw e f Switc closed: e = 0 SE Te causality of te switc element is a function of te value of te control signal s. 8
9 Te Ideal Diode I u i Switc open i Switc closed u Wen u < 0, te switc is open. No current flows troug. Wen u > 0, te switc is closed. Current may flow. Te ideal diode beaves like a sort circuit. open = u < 0 ; 0 = if open ten i else u; D f e Te Ideal Diode II Since current flowing troug a diode cannot simply be interrupted, it is necessary to sligtly modify te diode model. open = u <= 0 and not i > 0 ; 0 = if open ten i else u; Te variable open must be declared as Boolean. Te value to te rigt of te Boolean expression is assigned to it. 9
10 Te Friction Caracteristic I More complex penomena, suc as friction caracteristics, must be carefully analyzed case by case. Te approac is discussed ere by means of te friction example. R 0 R m f B -R m -R 0 Dry friction Viscous friction v Wen v 0, te friction force is a function of te velocity. Wen v = 0, te friction force is computed suc tat te velocity remains 0. Te Friction Caracteristic II We distinguis between five situations: v = 0 a = 0 v > 0 Sticking: Moving forward: Te friction force compensates te sum of all forces attaced, except if Σf > R 0. Te friction force is computed as: f B = R v v + R m. v < 0 v = 0 a > 0 v = 0 a < 0 Moving backward: Beginning of forward motion: Beginning of backward motion: Te friction force is computed as: f B =R v v R m. Te friction force is computed as: f B =R m. Te friction force is computed as: f B = R m. 10
11 Te State Transition Diagram Te set of events can be described by a state transition diagram. Start v < 0 v > 0 v = 0 v< 0 Σf < R 0 Σf > +R 0 v> 0 Backward motion (v < 0) Backward acceleration (a < 0) Sticking (a = 0) Forward acceleration (a > 0) Forward motion (v > 0) a 0 and not v < 0 a 0 and not v > 0 v 0 v 0 Te Friction Model I model Friction; parameter Real R0, Rm, Rv; parameter Boolean ic=false; Real fb, fc; Boolean Sticking (final start = ic); Boolean Forward (final start = ic), Backward (final start = ic); Boolean StartFor (final start = ic), StartBack (final start = ic); fb = if Forward ten Rv*v + Rm else if Backward ten Rv*v - Rm else if StartFor ten Rm else if StartBack ten -Rm else fc; 0 = if Sticking or initial() ten a else fc; 11
12 Te Friction Model II wen Sticking and not initial() ten reinit(v,0); end wen; Forward = initial() and v > 0 or pre(startfor) and v > 0 or pre(forward) and not v <= 0; Backward = initial() and v < 0 or pre(startback) and v < 0 or pre(backward) and not v >= 0; Te Friction Model III StartFor = pre(sticking) and fc > R0 or pre(startfor) and not (v > 0 or a <= 0 and not v > 0); StartBack = pre(sticking) and fc < -R0 or pre(startback) and not (v < 0 or a >= 0 and not v < 0); Sticking = not (Forward or Backward or StartFor or StartBack); end Friction; 12
13 References I Cellier, F.E. (1979), Combined Continuous/Discrete System Simulation by Use of Digital Computers: Tecniques and Tools, Swiss Federal Institute of Tecnology, ETH Züric, Switzerland. Elmqvist, H., F.E. Cellier, and M. Otter (1993), Objectoriented modeling of ybrid systems, Proc. ESS'93, SCS European Simulation Symposium, Delft, Te Neterlands, pp.xxxi-xli. Cellier, F.E., M. Otter, and H. Elmqvist (1995), Bond grap modeling of variable structure systems, Proc. ICBGM'95, 2 nd SCS Intl. Conf. on Bond Grap Modeling and Simulation, Las Vegas, NV, pp References II Elmqvist, H., F.E. Cellier, and M. Otter (1994), Objectoriented modeling of power-electronic circuits using Dymola, Proc. CISS'94, First Joint Conference of International Simulation Societies, Zuric, Switzerland, pp Glaser, J.S., F.E. Cellier, and A.F. Witulski (1995), Object-oriented switcing power converter modeling using Dymola wit event-andling, Proc. OOS'95, SCS Object-Oriented Simulation Conference, Las Vegas, NV, pp
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