Hybrid Equations (HyEQ) Toolbox v2.02 A Toolbox for Simulating Hybrid Systems in MATLAB/Simulink R

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1 Hybrid Equaions (HyEQ) Toolbo v. A Toolbo for Simulaing Hybrid Sysems in MATLAB/Simulink R Ricardo G. Sanfelice Universiy of California Sana Cruz, CA 9564 USA David A. Copp Universiy of California Sana Barbara, CA 939 USA Ocober 3, 4 Absrac Pablo Nanez Universidad de Los Andes Colombia This noe describes he Hybrid Equaions (HyEQ) Toolbo implemened in MATLAB/Simulink for he simulaion of hybrid dynamical sysems. This oolbo is capable of simulaing individual and inerconneced hybrid sysems where muliple hybrid sysems are conneced and inerac such as a bouncing ball on a moving plaform, fireflies synchronizing heir flashing, and more. The Simulink implemenaion includes four basic blocks ha define he dynamics of a hybrid sysem. These include a flow map, flow se, ump map, and ump se. The flows and umps of he sysem are compued by he inegraor sysem which is comprised of blocks ha compue he coninuous dynamics of he hybrid sysem, rigger umps, updae he sae of he sysem and simulaion ime a umps, and sop he simulaion. We also describe a lie simulaor which allows for faser simulaion. Conens Inroducion Insallaion 3 3 Lie HyEQ Simulaor: A sand-alone MATLAB code for simulaion of hybrid sysems wihou inpus 3 3. Solver Funcion Evens Deecion Jump Map Sofware Requiremens Configuraion of Solver Iniializaion Posprocessing and Ploing soluions HyEQ Simulaor: A Simulink implemenaion for simulaion of single and inerconneced hybrid sysems wih or wihou inpus 3 4. The Inegraor Sysem CT Dynamics Jump Logic Updae Logic Sop Logic Sofware Requiremens Configuraion of HyEQ Simulaor wih embedded funcions for Windows Configuraion of HyEQ Simulaor wih embedded funcions for Mac/Linu Configuraion of Inegraion Scheme Iniializaion Posprocessing and Ploing soluions Eamples

2 6 Furher Reading 35 7 Acknowledgmens 36 8 References 36 Inroducion To ge sared, a webinar inroducing he HyEQ Toolbo is available a hp:// A free wo-sep regisraion is required by Mahworks. A hybrid sysem is a dynamical sysem wih coninuous and discree dynamics. Several mahemaical models for hybrid sysems have appeared in lieraure. In his paper, we consider he framework for hybrid sysems used in [3,4], where a hybrid sysem H on a sae space R n wih inpu space R m is defined by he following obecs: A se C R n R m called he flow se. A funcion f : R n R m R n called he flow map. A se D R n R m called he ump se. A funcion g : R n R m R n called he ump map. We consider he simulaion in MATLAB/Simulink of hybrid sysems H = (C, f, D, g) wrien as H :, u R m { ẋ = f(, u) (, u) C + = g(, u) (, u) D. () The flow map f defines he coninuous dynamics on he flow se C, while he ump map g defines he discree dynamics on he ump se D. These obecs are referred o as he daa of he hybrid sysem H, which a imes is eplicily denoed as H = (C, f, D, g). We illusrae his framework in a simple, ye rich in behavior, hybrid sysem. Eample. (bouncing ball sysem) Consider a model for a bouncing ball wrien as [ ] f() :=, C := { R γ } () [ ] g() :=, D := { R λ, } (3) where γ > is he graviy consan and λ [, ) is he resiuion coefficien. In his model, we consider he ball o be bouncing on a floor a a heigh of. This model is re-visied as an eample in Secion 3 and Secion 5. The remainder of his noe is organized as follows. In Secion, we describe how o insall he HyEQ Toolbo in MATLAB. In Secion 3, we inroduce he Lie HyEQ Simulaor for solving hybrid sysems wihou inpus. In Secion 4, we inroduce he HyEQ Simulaor implemened in Simulink for solving single and inerconneced hybrid sysems wih inpus. In Secion 5, we work hrough several eamples for he simulaion of single and inerconneced hybrid sysems. In Secion 6, we give direcions o where he simulaor files can be downloaded.

3 Insallaion The following procedure describes how o insall he Hybrid Equaions (HyEQ) Toolbo in MATLAB. This insallaion adds useful.m files o he MATLAB library and several blocks o he Simulink block library. Seps for insallaion:. Download he HyEQ Toolbo from MATLAB Cenral or he auhor s websie a hps://hybrid. soe.ucsc.edu/sofware.. Erac all files and save in any place (ecep he roo folder). 3. Open MATLAB and change he curren folder o he folder where he insall.m is locaed. 4. Type insall in he command window and hi ener o run he file insall.m. 5. Follow he on-screen promps. Mus answer yes o he quesion: Add oolbo permanenly ino your sarup pah (highly recommended)? Y/E/N [Y]: y 6. Once insallaion has finished, close and hen reopen MATLAB. Now he HyEQ Toolbo is ready for use. If you wish o uninsall he HyEQ Toolbo from MATLAB, simply run he bclean.m file inside he HyEQ Toolbo V folder, and follow he on-screen promps. 3 Lie HyEQ Simulaor: A sand-alone MATLAB code for simulaion of hybrid sysems wihou inpus One way o simulae hybrid sysems is o use ODE funcion calls wih evens in MATLAB (see, e.g., []). Such an implemenaion gives fas simulaion of a hybrid sysem. In he lie HyEQ solver, four basic funcions are used o define he daa of he hybrid sysem H as in () (wihou inpus): The flow map is defined in he MATLAB funcion f.m. The inpu o his funcion is a vecor wih componens defining he sae of he sysem. Is oupu is he value of he flow map f. The flow se is defined in he MATLAB funcion C.m. The inpu o his funcion is a vecor wih componens defining he sae of he sysem. Is oupu is equal o if he sae belongs o he se C or equal o oherwise. The ump map is defined in he MATLAB funcion g.m. Is inpu is a vecor wih componens defining he sae of he sysem. Is oupu is he value of he ump map g. The ump se is defined in he MATLAB funcion D.m. Is inpu is a vecor wih componens defining he sae of he sysem. Is oupu is equal o if he sae belongs o D or equal o oherwise. Our Lie HyEQ Simulaor uses a main funcion run.m o iniialize, run, and plo soluions for he simulaion, funcions f.m, C.m, g.m, and D.m o implemen he daa of he hybrid sysem, and HyEQsolver.m which will solve he differenial equaions by inegraing he coninuous dynamics, ẋ = f(), and umping by he updae law + = g(). The ODE solver called in HyEQsolver.m iniially uses he iniial or mos recen sep size, and afer each inegraion, he algorihms in HyEQsolver.m check o see if he soluion is in he se C, D, or neiher. Depending on which se he soluion is in, he simulaion is accordingly rese following he dynamics given in f or g, or he simulaion is sopped. This implemenaion is fas because i also does no sore variables o he workspace and only uses buil-in ODE funcion calls. Time and ump horizons are se for he simulaion using TSPAN = [TSTART TFINAL] as he ime inerval of he simulaion and JSPAN = [JSTART JSTOP] as he inerval for he number of discree umps allowed. The simulaion sops when eiher he ime or ump horizon, i.e. he final value of eiher inerval, is reached. The eample below shows how o use he HyEQ solver o simulae a bouncing ball. 3

4 Eample. (bouncing ball wih Lie HyEQ Solver) Consider he hybrid sysem model for he bouncing ball wih daa given in Eample.. For his eample, we consider he ball o be bouncing on a floor a zero heigh. The consans for he bouncing ball sysem are γ = 9.8 and λ =.8. The following procedure is used o simulae his eample in he Lie HyEQ Solver: Inside he MATLAB scrip run.m, iniial condiions, simulaion horizons, a rule for umps, ode solver opions, and a sep size coefficien are defined. The funcion HyEQsolver.m is called in order o run he simulaion, and a scrip for ploing soluions is included. Then he MATLAB funcions f.m, C.m, g.m, D.m are edied according o he daa given above. Finally, he simulaion is run by clicking he run buon in run.m or by calling run.m in he MATLAB command window. Eample code for each of he MATLAB files run.m, f.m, C.m, g.m, and D.m is given below. funcion run % iniial condiions 3 _ = ; 4 _ = ; 5 = [_;_]; 6 % simulaion horizon 7 TSPAN=[ ]; 8 JSPAN = [ ]; 9 % rule for umps % rule = -> prioriy for umps % rule = -> prioriy for flows rule = ; 3 opions = odese( RelTol,e-6, MaSep,.); 4 % simulae 5 [,,] = HyEQsolver(@f,@g,@C,@D,,TSPAN,JSPAN,rule,opions); 6 % plo soluion 7 figure() % posiion 8 clf 9 subplo(,,),ploflows(,,(:,)) grid on ylabel( ) subplo(,,),ploumps(,,(:,)) 3 grid on 4 ylabel( ) 5 figure() % velociy 6 clf 7 subplo(,,),ploflows(,,(:,)) 8 grid on 9 ylabel( ) 3 subplo(,,),ploumps(,,(:,)) 3 grid on 3 ylabel( ) 33 % plo hybrid arc 34 plohybridarc(,,) 35 label( ) 36 ylabel( ) 37 zlabel( ) 4

5 funcion do = f() % sae 3 = (); 4 = (); 5 % differenial equaions 6 do = [ ; -9.8]; 7 end funcion [value discree] = C() = (); 3 if >= 4 value = ; 5 else 6 value = ; 7 end 8 end funcion plus = g() % sae 3 = (); 4 = (); 5 plus = [- ; -.8*]; 6 end funcion inside = D() = (); 3 = (); 4 if ( <= && <= ) 5 inside = ; 6 else 7 inside = ; 8 end 9 end flows [] flows [] umps [] umps [] (a) Heigh (b) Velociy Figure : Soluion of Eample. 5

6 Figure : Hybrid arc corresponding o a soluion of Eample.: heigh A soluion o he bouncing ball sysem from (, ) = [, ] and wih T SP AN = [ ], JSP AN = [ ], rule =, is depiced in Figure (a) (heigh) and Figure (b) (velociy). Boh he proecion ono and are shown. Figure depics he corresponding hybrid arc for he posiion sae. For MATLAB files of his eample, see Eamples/Eample.. 3. Solver Funcion The solver funcion HyEQsolver solves he hybrid sysem using hree differen funcions as shown below. Firs, he flows are calculaed using he buil-in ODE solver funcion ODE45 in MATLAB. If he soluion leaves he flow se C, he discree even is deeced using he funcion zeroevens as shown in Secion 3... When he sae umps, he ne value of he sae is calculaed via he ump map g using he funcion ump as shown in Secion 3... funcion [ ] = HyEQsolver(f,g,C,D,,TSPAN,JSPAN,rule,opions) %HYEQSOLVER solves hybrid equaions. 3 % Syna: [ ] = HyEQsolver(f,g,C,D,,TSPAN,JSPAN,rule,opions) 4 % compues soluions o he hybrid equaions 5 % 6 % \do{} = f() \in C ˆ+ = g() \in D 7 % 8 % where is he sae, f is he flow map, g is he ump map, C is he 9 % flow se, and D is he ump se. I oupus he sae raecory (,) % -> (,), where is he flow ime parameer and is he ump % parameer. 6

7 % 3 % defines he iniial condiion for he sae. 4 % 5 % TSPAN = [TSTART TFINAL] is he ime inerval. JSPAN = [JSTART JSTOP] is 6 % he inerval for discree umps. The algorihm sop when he firs 7 % sop condiion is reached. 8 % 9 % rule for umps % rule = (defaul) -> prioriy for umps rule = -> prioriy for % flows % 3 % opions - opions for he solver see odese f.e. 4 % opions = odese( RelTol,e-6); 5 % 6 % Eample: Bouncing ball wih Lie HyEQ Solver 7 % 8 % % Consider he hybrid sysem model for he bouncing ball wih daa given in 9 % % Eample.. For his eample, we consider he ball o be bouncing on a 3 % % floor a zero heigh. The consans for he bouncing ball sysem are 3 % % \gamma=9.8 and \lambda=.8. The following procedure is used o 3 % % simulae his eample in he Lie HyEQ Solver: 33 % 34 % % * Inside he MATLAB scrip run_e_.m, iniial condiions, simulaion 35 % % horizons, a rule for umps, ode solver opions, and a sep size 36 % % coefficien are defined. The funcion HyEQsolver.m is called in order o 37 % % run he simulaion, and a scrip for ploing soluions is included. 38 % % * Then he MATLAB funcions f_e_.m, C_e_.m, g_e_.m, D_e_.m 39 % % are edied according o he daa given below. 4 % % * Finally, he simulaion is run by clicking he run buon in 4 % % run_e_.m or by calling run_e_.m in he MATLAB command window. 4 % 43 % % For furher informaion, ype in he command window: 44 % helpview([ Eample.hml ]); 45 % 46 % % Define iniial condiions 47 % _ = ; 48 % _ = ; 49 % = [_; _]; 5 % 5 % % Se simulaion horizon 5 % TSPAN = [ ]; 53 % JSPAN = [ ]; 54 % 55 % % Se rule for umps and ODE solver opions 56 % % 57 % % rule = -> prioriy for umps 58 % % 59 % % rule = -> prioriy for flows 6 % % 6 % % se he maimum sep lengh. A each run of he 6 % % inegraor he opion MaSep is se o 63 % % (ime lengh of las inegraion)*masepcoefficien. 64 % % Defaul value =. 65 % 7

8 66 % rule = ; 67 % 68 % opions = odese( RelTol,e-6, MaSep,.); 69 % 7 % % Simulae using he HyEQSolver scrip 7 % % Given he malab funcions ha models he flow map, ump map, 7 % % flow se and ump se (f_e_, g_e_, C_e_, and D_e_ 73 % % respecively) 74 % 75 % [ ] = 76 %,TSPAN,JSPAN,rule,opions); 77 % 78 % % plo soluion 79 % 8 % figure() % posiion 8 % clf 8 % subplo(,,),ploflows(,,(:,)) 83 % grid on 84 % ylabel( ) 85 % 86 % subplo(,,),ploumps(,,(:,)) 87 % grid on 88 % ylabel( ) 89 % 9 % figure() % velociy 9 % clf 9 % subplo(,,),ploflows(,,(:,)) 93 % grid on 94 % ylabel( ) 95 % 96 % subplo(,,),ploumps(,,(:,)) 97 % grid on 98 % ylabel( ) 99 % % % plo hybrid arc % % plohybridarc(,,) 3 % label( ) 4 % ylabel( ) 5 % zlabel( ) 6 % 7 % % plo soluion using ploharc and ploharccolor 8 % 9 % figure(4) % posiion % clf % subplo(,,), ploharc(,,(:,)); % grid on 3 % ylabel( _ posiion ) 4 % subplo(,,), ploharc(,,(:,)); 5 % grid on 6 % ylabel( _ velociy ) 7 % 8 % 9 % % plo a phase plane 8

9 % figure(5) % posiion % clf % ploharccolor((:,),,(:,),); 3 % label( _ ) 4 % ylabel( _ ) 5 % grid on 6 % 7 % % Malab M-file Proec: HyEQ Hybrid Dynamics and Conrol Lab, 9 % hp:// sricardo/inde.php?n=main.sofware 3 % hp://hybridsimulaor.wordpress.com/ 3 % Filename: HyEQsolver.m 3 % % See also ploflows, ploharc, ploharccolor, ploharccolor3d, 34 % plohybridarc, ploumps. 35 % Hybrid Dynamics and Conrol Lab, 36 % Revision:... Dae: 4/3/4 :48: if eis( rule, var ) 4 rule = ; 4 end 4 43 if eis( opions, var ) 44 opions = odese(); 45 end % simulaion horizon 48 sar = TSPAN(); 49 final = TSPAN(end); 5 5 % simulae 5 opions = odese(opions, Evens,@(,) zeroevens(,c,d,rule)); 53 ou = sar; 54 ou =. ; 55 ou = JSPAN(); 56 = ou(end); % Jump if ump is prioriized: 59 if rule == 6 while (<JSPAN(end)) 6 % Check if value i is possible o ump curren posiion 6 insided = D(ou(end,:). ); 63 if insided == 64 [ ou ou ou] = ump(g,,ou,ou,ou); 65 else 66 break; 67 end 68 end 69 end 7 fprinf( Compleed: %3.f%%,); 7 while ( < JSPAN(end) && ou(end) < TSPAN(end)) 7 % Check if i is possible o flow from curren posiion 73 insidec = C(ou(end,:). ); 9

10 74 if insidec == 75 [,] = ode45(@(,) f(),[ou(end) final],ou(end,:)., opions); 76 n = lengh(); 77 ou = [ou; ]; 78 ou = [ou; ]; 79 ou = [ou; *ones(,n) ]; 8 end 8 8 %Check if i is possible o ump 83 insided = D(ou(end,:). ); 84 if insided == 85 break; 86 else 87 if rule == 88 while (<JSPAN(end)) 89 % Check if i is possible o ump from curren posiion 9 insided = D(ou(end,:). ); 9 if insided == 9 [ ou ou ou] = ump(g,,ou,ou,ou); 93 else 94 break; 95 end 96 end 97 else 98 [ ou ou ou] = ump(g,,ou,ou,ou); 99 end end fprinf( \b\b\b\b%3.f%%,ma(*/jspan(end),*ou(end)/tspan(end))); end 3 = ou; 4 = ou; 5 = ou; 6 fprinf( \ndone\n ); 7 end 3.. Evens Deecion funcion [value,iserminal,direcion] = zeroevens(,c,d,rule ) iserminal = ; 3 direcion = -; 4 insidec = C(); 5 if insidec == 6 % Ouside of C 7 value = ; 8 elseif (rule == ) 9 % If prioriy for ump, sop if inside D insided = D(); if insided == % Inside D, inside C 3 value = ; 4 else 5 % ouside D, inside C 6 value = ;

11 7 end 8 else 9 % If inside C and no prioriy for ump or prioriy of ump and ouside % of D value = ; end 3 end 3.. Jump Map funcion [ ou ou ou] = ump(g,,ou,ou,ou) % Jump 3 = +; 4 y = g(ou(end,:). ); 5 % Save resuls 6 ou = [ou; ou(end)]; 7 ou = [ou; y. ]; 8 ou = [ou; ]; 9 end 3. Sofware Requiremens In order o run simulaions using he Lie HyEQ Simulaor, MATLAB R3 or newer is required. 3.3 Configuraion of Solver Before a simulaion is sared, i is imporan o deermine he needed inegraor scheme, zero-cross deecion seings, precision, and oher olerances. Using he defaul seings does no always give he mos efficien or mos accurae simulaions. In he Lie HyEQ Simulaor, hese parameers are edied in he run.m file using opions = odese(reltol,e-6,masep,.);. 3.4 Iniializaion The Lie HyEQ Simulaor is iniialized and run by calling he funcion run.m. Inside run.m, he iniial condiions, simulaion horizons TSPAN and JSPAN, a rule for umps, and simulaion olerances are defined. Afer all of he parameers are defined, he funcion HyEQsolver is called, and he simulaion runs. See below for sample code o iniialize and run he bouncing ball eample, Eample.. % iniial condiions _ = ; 3 _ = ; 4 = [_;_]; 5 % simulaion horizon 6 TSPAN=[,]; 7 JSPAN = [,]; 8 % rule for umps 9 % rule = -> prioriy for umps % rule = -> prioriy for flows rule = ; opions = odese( RelTol,e-6, MaSep,.); 3 % simulae

12 4 [,,] = HyEQsolver(@f,@g,@C,@D,,TSPAN,JSPAN,rule,opions); 3.5 Posprocessing and Ploing soluions The funcion run.m is also used o plo soluions afer he simulaions is complee. See below for sample code o plo soluions o he bouncing ball eample, Eample.. % plo soluion figure() % posiion 3 clf 4 subplo(,,),ploflows(,,(:,)) 5 grid on 6 ylabel( ) 7 subplo(,,),ploumps(,,(:,)) 8 grid on 9 ylabel( ) figure() % velociy clf subplo(,,),ploflows(,,(:,)) 3 grid on 4 ylabel( ) 5 subplo(,,),ploumps(,,(:,)) 6 grid on 7 ylabel( ) 8 % plo hybrid arc 9 plohybridarc(,,) label( ) ylabel( ) zlabel( ) The following funcions are used o generae he plos: ploflows(,,): plos (in blue) he proecion of he raecory ono he flow ime ais. The value of he raecory for inervals [, + ] wih empy inerior is marked wih (in blue). Dashed lines (in red) connec he value of he raecory before and afer he ump. Figure (a) shows a plo creaed wih his funcion. ploumps(,,): plos (in red) he proecion of he raecory ono he ump ime. The iniial and final value of he raecory on each inerval [, + ] is denoed by (in red) and he coninuous evoluion of he raecory on each inerval is depiced wih a dashed line (in blue). Figure (a) shows a plo creaed wih his funcion. plohybridarc(,,): plos (in blue and red) he raecory on hybrid ime domains. The inervals [, + ] indeed by he corresponding are depiced in he plane (in red). Figure shows a plo creaed wih his funcion. ploharc is a funcion for ploing hybrid arcs (n saes). ploharc(,,): plos he raecory versus he hybrid ime domain (, ). If is a mari, hen he ime vecor is ploed versus he rows or columns of he mari, whichever line up. ploharc(,,,sar): plos he raecory versus he hybrid ime domain (, ), and he plo is cu regarding he sar inerval (sar = [ iniial, final ]). ploharc(,,,sar,modificaor): Modificaor is a cell array ha conains he sandard malab ploing modificaors (ype >> help ploharc or >> helpwin ploharc in he command window for more informaion).

13 ploharccolor plos he raecory (vecor) on hybrid ime domain wih color. ploharccolor(,,,l): plos he raecory (vecor) versus he hybrid ime domain (, ). The hybrid arc is ploed wih L daa as color. The inpu vecors,,, L mus have he same lengh. ploharccolor(,,,l,sar): If a specific inerval in is required, sar = [ iniial, final ] mus be provided. (ype >> help ploharccolor or >> helpwin ploharccolor in he command window for more informaion) ploharccolor3d plos an 3D hybrid arc wih color. ploharccolor3d(,,,l) plos he raecory (3 saes) aking ino accoun he hybrid ime domain (, ). The hybrid arc is ploed wih L daa as color. The inpu vecors,,, L mus have he same lengh and mus have hree columns. ploharccolor3d(,,,l,sar) If a specific inerval in is required, sar = [ iniial, final ] mus be provided. ploharccolor3d(,,,l,sar,modificaor) Modificaor is a cell array ha conains he sandard malab ploing modificaors (ype >> help ploharccolor3d or >> helpwin ploharccolor3d in he command window for more informaion). 4 HyEQ Simulaor: A Simulink implemenaion for simulaion of single and inerconneced hybrid sysems wih or wihou inpus The HyEQ Toolbo includes hree main Simulink library blocks ha allow for simulaion of a hybrid sysem H = (C, f, D, g) using eiher eernally defined funcions or embedded MATLAB funcions, and a single hybrid sysem or inerconneced hybrid sysems wih inpus using embedded MATLAB funcions. Figure 3 shows hese blocks in he Simulink Library Browser. Sae HS Time Jumps HS wih eernal funcions Sae Time HS wih embedded funcions HS Jumps u u (in) HSu Sae Time Jumps HS wih embedded funcions and inpus Terminaor Figure 3: MATLAB/Simulink library blocks for Simulink implemenaion. 3

14 This model simulaes a hybrid sysem wih inpu Double Click o Iniialize Double Click o Plo Soluions u do u f flow map f f u u HSu (in) Sae Time Jumps v u C flow se C plus u g ump map g C g 3 Terminaor v u D ump se D D 4 (in) Inegraor Sysem Figure 4: MATLAB/Simulink implemenaion of a hybrid sysem H = (C, f, D, g) wih inpus. Figure 4 shows a Simulink implemenaion for simulaing a hybrid sysem wih inpus using embedded MATLAB funcions. In his implemenaion, four basic blocks are used o define he daa of he hybrid sysem H: The flow map is implemened in an Embedded MATLAB funcion block eecuing he funcion f.m. Is inpu is a vecor wih componens defining he sae of he sysem, and he inpu u. Is oupu is he value of he flow map f which is conneced o he inpu of an inegraor. The flow se is implemened in an Embedded MATLAB funcion block eecuing he funcion C.m. Is inpu is a vecor wih componens and inpu u of he Inegraor sysem. Is oupu is equal o if he sae belongs o he se C or equal o oherwise. The minus noaion denoes he previous value of he variables (before inegraion). The value is obained from he sae por of he inegraor. The ump map is implemened in an Embedded MATLAB funcion block eecuing he funcion g.m. Is inpu is a vecor wih componens and inpu u of he Inegraor sysem. Is oupu is he value of he ump map g. The ump se is implemened in an Embedded MATLAB funcion block eecuing he funcion D.m. Is inpu is a vecor wih componens and inpu u of he Inegraor sysem. Is oupu is equal o if he sae belongs o D or equal o oherwise. In our implemenaion, MATLAB.m files are used. The file iniializaion.m is used o define iniial variables before simulaion. The file posprocessing.m is used o plo he soluions afer a simulaion is complee. These wo.m files are called by double-clicking he Double Click o... blocks a he op of he Simulink Model (see Secion 4.5 for more informaion on hese.m files and heir use). 4. The Inegraor Sysem In his secion we discuss he inernals of he Inegraor Sysem shown in Figure CT Dynamics This block is shown in Figure 6. I defines he coninuous-ime (CT) dynamics by assembling he ime derivaive of he sae [ ]. Saes and are considered saes of he sysem because hey need o be 4

15 f(,u) do f C 4 D 3 g CT dynamics C D ump r Jump logic g(,u) [; ; (:)] updae law IC Updae logic o s [] IC 3 C sop STOP 4 D Sop logic Sop Simulaion Figure 5: Inegraor Sysem updaed hroughou he simulaion in order o keep rack of he ime and number of umps. Wihou and, soluions could no be ploed accuraely. This is given by ṫ =, =, ẋ = f(, u). Noe ha inpu por akes he value of f(, u) hrough he oupu of he Embedded MATLAB funcion block f in Figure 4. Flows Jumps f(,u) do Figure 6: CT dynamics 4.. Jump Logic This block is shown in Figure 7. The inpus o he ump logic block are he oupu of he blocks C and D indicaing wheher he sae is in hose ses or no, and a random signal wih uniform disribuion in [, ]. Figure 7 shows he Simulink blocks used o implemen he Jump Logic. The variable rule defines wheher he simulaor gives prioriy o umps, prioriy o flows, or no prioriy. I is iniialized in iniializaion.m. The oupu of he Jump Logic is equal o one when: he oupu of he D block is equal o one and rule =, he oupu of he C block is equal o zero, he oupu of he D block is equal o one, and rule =, he oupu of he C block is equal o zero, he oupu of he D block is equal o one, and rule = 3, or he oupu of he C block is equal o one, he oupu of he D block is equal o one, rule = 3, and he random signal r is larger or equal han.5. 5

16 Under hese evens, he oupu of his block, which is conneced o he inegraor eernal rese inpu, riggers a rese of he inegraor, ha is, a ump of H. The rese or ump is acivaed since he configuraion of he rese inpu is se o level hold, which eecues reses when his eernal inpu is equal o one (if he ne inpu remains se o one, muliple reses would be riggered). Oherwise, he oupu is equal o zero. D C NOT AND AND rule OR 3 rule=: ou = D rlule=: ou = D and ~C rule=3: ou = (D and ~C) or (D and C and r>.5) oher : ou = ump 3 r >=.5 Compare To Consan * Mulipor Swich Figure 7: Jump Logic 4..3 Updae Logic This block is shown in Figure 8. The updae logic uses he sae por informaion of he inegraor. This por repors he value of he sae of he inegraor, [ ], a he eac insan ha he rese condiion becomes rue. Noice ha differs from since a a ump, indicaes he value of he sae ha riggers he ump, bu i is never assigned as he oupu of he inegraor. In oher words, D is checked using and if rue, is rese o g(, u). Noice, however, ha u is he same because a a ump, u indicaes he ne evaluaed value of he inpu, and i is assigned as he oupu of he inegraor. The flow ime is kep consan a umps and is incremened by one. More precisely + =, + = +, + = g(, u) where [ ] is he sae ha riggers he ump. 3 g(,u) updae law Figure 8: Updae Logic 4..4 Sop Logic This block is shown in Figure 9. I sops he simulaion under any of he following evens: The flow ime is larger han or equal o he maimum flow ime specified by T. The ump ime is larger han or equal o he maimum number of umps specified by J. The sae of he hybrid sysem is neiher in C nor in D. 6

17 Under any of hese evens, he oupu of he logic operaor conneced o he Sop block becomes one, sopping he simulaion. Noe ha he inpus C and D are roued from he oupu of he blocks compuing wheher he sae is in C or D and use he value of. Flow Horizon, T >= T Jump Horizon, J Logical Operaor >= J OR sop C 3 4 D Logical Operaor NOR Figure 9: Sop Logic 4. Sofware Requiremens In order o run simulaions of single hybrid sysems using eernally defined funcions, MATLAB wih Simulink is required. In order o run simulaions using he HyEQ Simulaor wih embedded MATLAB funcions, MAT- LAB/Simulink and a suppored ANSI, C, or C++ 3-bi compiler mus be insalled. We now briefly describe how o insall necessary compilers for Windows and Mac/Linu. For more informaion on suppored compilers, please visi hp:// 4.. Configuraion of HyEQ Simulaor wih embedded funcions for Windows For 3-bi Windows, he LCC compiler is included wih MATLAB. Firs, open MATLAB and hen locae and choose a compiler for building MEX-files by yping >> me -seup ino he MATLAB command window. Then, follow he promps as shown below. >> me -seup Welcome o me -seup. This uiliy will help you se up a defaul compiler. For a lis of suppored compilers, see hp:// Please choose your compiler for building MEX-files: Would you like me o locae insalled compilers [y]/n? y Selec a compiler: [] Lcc-win3 C.4. [] None Compiler: 7

18 Please verify your choices: Compiler: Lcc-win3 C.4. Are hese correc [y]/n? y Done... For 64-bi Windows, a C-compiler is no supplied wih MATLAB. Before running he HyEQ Toolbo in MATLAB/Simulink, please follow he following seps:. If you don have Microsof.NET Framework 4 on your compuer, download and insall i from hp: // Then download and insall Microsof Windows SDK from hp:// deails.asp?id= Then perform he seps oulined above for 3-bi Windows o seup and insall he compiler. As of Ocober, 3, when insalling he oolbo in Windows 8, please follow he ne seps.. If you don have Microsof.NET Framework 4 on your compuer, download and insall i from hp: // Then download and insall Microsof Windows SDK If you don have Visual C++ SP insalled on your compuer: Download and insall Microsof Windows SDK 7. from hp:// download/deails.asp?displaylang=en&id=44 Apply he following pach from Microsof ono he SDK 7. insallaion: hp://www. microsof.com/en-us/download/deails.asp?displaylang=en&id=44 If you have Visual Visual C++ SP or is redisribuable packages insalled on your compuer: Uninsall he Visual C++ redisribuable packages, boh 64 and 86 versions. This can be done from Conrol Panel / Uninsall Programs Menu. Download and insall Microsof Windows SDK 7. from hp:// download/deails.asp?displaylang=en&id=44 Apply he following pach from Microsof ono he SDK 7. insallaion: hp://www. microsof.com/en-us/download/deails.asp?displaylang=en&id=44 Reinsall he Visual C++ redisribuable packages: 86 version: hp:// 64 version: hp:// 3. Then perform he seps oulined above for 3-bi Windows o seup and insall he compiler. 4.. Configuraion of HyEQ Simulaor wih embedded funcions for Mac/Linu From a erminal window, check ha he file gcc is in he folder /usr/bin. If i is no here, make a symbolic link. You migh require o insall he laes version of Xcode firs. In order o generae a symbolic link for gcc, ha MATLAB can find o compile he simulaion files (see hp:// sysreq/previous_releases.hml), change folder o /usr/bin and hen sudo ln -s gcc gcc-4. Then, i should be possible o seup he gcc compiler in malab as follows: 8

19 >> me -seup Opions files conrol which compiler o use, he compiler and link command opions, and he runime libraries o link agains. Using he mesh -seup command selecs an opions file ha is placed in ~/.malab/r3b and used by defaul for mesh. An opions file in he curren working direcory or specified on he command line overrides he defaul opions file in ~/.malab/r3b. To override he defaul opions file, use he mesh -f command (see mesh -help for more informaion). The opions files available for MEX are: The opions files available for mesh are: : /Applicaions/MATLAB_R3b.app/bin/meops.sh : Templae Opions file for building MEX-files : Ei wih no changes Ener he number of he compiler (-): Overwrie ~/.malab/r3b/meops.sh ([y]/n)?: Y /Applicaions/MATLAB_R3b.app/bin/meops.sh is being copied o /SOME_FOLDER/meops.sh A his poin, i is possible o check if he gcc is properly seup by esing any of he Simulink eamples wih embedded funcions (see Figure 3) (e.g., Eamples.3,.4,.5,.6,.7 or.8). If an error regarding gmake and a warning no such sysrooo direcory: Developer/SDKs/MacOSX.X.sdk is shown when compiling, i is necesary o change some lines in he file meops.sh (copied previously in he folder SOME FOLDER ). Firs, locae he Xcode-SDK in your hard drive. Open a erminal window and eecue he following command find code-selec -prin-pah -name MacOSX.9.sdk which reurns he locaion of MacOSX.9.sdk, denoed here as SDK FOLDER. Now, in he MATLAB command window locae he file meops.sh by yping cd /SOME_FOLDER/ Then, open he file edi meops.sh and edi he lines SDKROOT= /Developer/SDKs/MacOSX.X.sdk o SDKROOT= SDK_FOLDER 9

20 CFLAGS="-fno-common -arch $ARCHS -isysroo $MW_SDKROOT -mmacos-version-min=$macosx_deployment_target" o CFLAGS="-fno-common -arch $ARCHS -isysroo $MW_SDKROOT -mmacos-version-min=$macosx_deployment_target -Dchar6_=UINT6_T" Replace all he appariions of.x o.9. Finally, resar malab and es any of he aforemenioned Simulink eamples. 4.3 Configuraion of Inegraion Scheme Before a simulaion is sared, i is imporan o deermine he needed inegraor scheme, zero-cross deecion seings, precision, and oher olerances. Using he defaul seings does no always give he mos efficien or mos accurae simulaions. One way o edi hese seings is o open he Simulink Model, selec Simulaion>Configuraion Parameers>Solver, and change he seings here. We have made his simple by defining variables for configuraion parameers in he iniializaion.m file. The las few lines of he iniializaion.m file look like ha given below. %configuraion of solver RelTol = e-8; 3 MaSep =.; In hese lines, RelTol = e-8 and MaSep =. define he relaive olerance and maimum sep size of he ODE solver, respecively. These parameers grealy affec he speed and accuracy of soluions. 4.4 Iniializaion When he block labeled Double Click o Iniialize a he op of he Simulink Model is double-clicked, he simulaion variables are iniialized by calling he scrip iniializaion.m. The scrip iniializaion.m defines he iniial condiions by defining he iniial values of he sae componens, any necessary parameers, he maimum flow ime specified by T, he maimum number of umps specified by J, and olerances used when simulaing. These can be changed by ediing he scrip file iniializaion.m. See below for sample code o iniialize he bouncing ball eample, Eample.3. % iniializaion for bouncing ball eample clear all 3 % iniial condiions 4 = [;]; 5 % simulaion horizon 6 T = ; 7 J = ; 8 % rule for umps 9 % rule = -> prioriy for umps % rule = -> prioriy for flows % rule = 3 -> no prioriy, random selecion when simulaneous condiions rule = ; 3 %configuraion of solver 4 RelTol = e-8; I is imporan o noe ha variables called in he Embedded MATLAB funcion blocks mus be added as inpus and labeled as parameers. This can be done by opening he Embedded MATLAB funcion block selecing Tools>Edi Daa/Pors and seing he scope o Parameer. Afer he block labeled Double Click o Iniialize is double-clicked and he variables iniialized, he simulaion is run by clicking he run buon or selecing Simulaion>Sar.

21 4.5 Posprocessing and Ploing soluions A similar procedure is used o define he plos of soluions afer he simulaion is run. The soluions can be ploed by double-clicking on he block a he op of he Simulink Model labeled Double Click o Plo Soluions which calls he scrip posprocessing.m. The scrip posprocessing.m may be edied o include he desired posprocessing and soluion plos. See below for sample code o plo soluions o he bouncing ball eample, Eample.3. %posprocessing for he bouncing ball eample % plo soluion 3 figure() 4 clf 5 subplo(,,),ploflows(,,) 6 grid on 7 ylabel( ) 8 subplo(,,),ploumps(,,) 9 grid on ylabel( ) % plo hybrid arc plohybridarc(,,) 3 label( ) 4 ylabel( ) 5 zlabel( ) The following funcions are used o generae he plos: ploflows(,,): plos (in blue) he proecion of he raecory ono he flow ime ais. The value of he raecory for inervals [, + ] wih empy inerior is marked wih (in blue). Dashed lines (in red) connec he value of he raecory before and afer he ump. Figure (a) shows a plo creaed wih his funcion. ploumps(,,): plos (in red) he proecion of he raecory ono he ump ime. The iniial and final value of he raecory on each inerval [, + ] is denoed by (in red) and he coninuous evoluion of he raecory on each inerval is depiced wih a dashed line (in blue). Figure (a) shows a plo creaed wih his funcion. plohybridarc(,,): plos (in blue and red) he raecory on hybrid ime domains. The inervals [, + ] indeed by he corresponding are depiced in he plane (in red). Figure shows a plo creaed wih his funcion. ploharc is a funcion for ploing hybrid arcs (n saes). ploharc(,,): plos he raecory versus he hybrid ime domain (, ). If is a mari, hen he ime vecor is ploed versus he rows or columns of he mari, whichever line up. ploharc(,,,sar): plos he raecory versus he hybrid ime domain (, ), and he plo is cu regarding he sar inerval (sar = [ iniial, final ]). ploharc(,,,sar,modificaor): Modificaor is a cell array ha conains he sandard malab ploing modificaors (ype >> help ploharc or >> helpwin ploharc in he command window for more informaion). ploharccolor plos he raecory (vecor) on hybrid ime domain wih color. ploharccolor(,,,l): plos he raecory (vecor) versus he hybrid ime domain (, ). The hybrid arc is ploed wih L daa as color. The inpu vecors,,, L mus have he same lengh.

22 ploharccolor(,,,l,sar): If a specific inerval in is required, sar = [ iniial, final ] mus be provided. (ype >> help ploharccolor or >> helpwin ploharccolor in he command window for more informaion) ploharccolor3d plos an 3D hybrid arc wih color. ploharccolor3d(,,,l) plos he raecory (3 saes) aking ino accoun he hybrid ime domain (, ). The hybrid arc is ploed wih L daa as color. The inpu vecors,,, L mus have he same lengh and mus have hree columns. ploharccolor3d(,,,l,sar) If a specific inerval in is required, sar = [ iniial, final ] mus be provided. ploharccolor3d(,,,l,sar,modificaor) Modificaor is a cell array ha conains he sandard malab ploing modificaors (ype >> help ploharccolor3d or >> helpwin ploharccolor3d in he command window for more informaion). 5 Eamples The eamples below illusrae he use of he Simulink implemenaion above. Eample.3 (bouncing ball wih inpu) For he simulaion of he bouncing ball sysem wih a consan inpu and regular daa given by [ ] f(, u) :=, C := { (, u) R R γ u } (4) [ ] u g(, u) :=, D := { (, u) R R λ u, } (5) where γ > is he graviy consan, u is he inpu consan, and λ [, ) is he resiuion coefficien. The MATLAB scrips in each of he funcion blocks of he implemenaion above are given as follows. An inpu was chosen o be u(, ) =. for all (, ). The consans for he bouncing ball sysem are γ = 9.8 and λ =.8. The following procedure is used o simulae his eample wih HyEQsimulaor.mdl: HyEQsimulaor.mdl is opened in MATLAB/Simulink. The Embedded MATLAB funcion blocks f, C, g, D are edied by double-clicking on he block and ediing he scrip. In each embedded funcion block, parameers mus be added as inpus and defined as parameers by selecing Tools>Edi Daa/Pors, and seing he scope o Parameer. For his eample, gamma and lambda are defined in his way. The iniializaion scrip iniializaion.m is edied by opening he file and ediing he scrip. The flow ime and ump horizons, T and J are defined as well as he iniial condiions for he sae vecor,, and inpu vecor, u, and a rule for umps, rule. The posprocessing scrip posprocessing.m is edied by opening he file and ediing he scrip. Flows and umps may be ploed by calling he funcions ploflows and ploumps, respecively. The hybrid arc may be ploed by calling he funcion plohybridarc. The simulaion sop ime and oher simulaion parameers are se o he values defined in iniializaion.m by selecing Simulaion>Configuraion Parameers>Solver and inpuing T, RelT ol, M asep, ec.. The masked inegraor sysem is double-clicked and he simulaion horizons and iniial condiions are se as desired.

23 The block labeled Double Click o Iniialize is double-clicked o iniialize variables. The simulaion is run by clicking he run buon or selecing Simulaion>Sar. The block labeled Double Click o Plo Soluions is double-clicked o plo he desired soluions flows [] flows [] umps [] umps [] (a) Heigh (b) Velociy funcion do = f(, u, gamma) % sae 3 = (); 4 = (); 5 % flow map: do=f(,u); 6 do = [(); gamma]; funcion v = C(, u) % flow se 3 if (() >= u()) % flow condiion 4 v = ; % repor flow 5 else 6 v = ; % do no repor flow 7 end funcion plus = g(, u, lambda) % ump map 3 plus = [u(); -lambda*()]; Figure : Soluion of Eample.3 funcion v = D(, u) % ump se 3 if (() <= u()) && (() <= ) % ump condiion 4 v = ; % repor ump 5 else 6 v = ; % do no repor ump 7 end A soluion o he bouncing ball sysem from (, ) = [, ] and wih T =, J =, rule =, is depiced in Figure (a) (heigh) and Figure (b) (velociy). Boh he proecion ono and are shown. 3

24 Figure : Hybrid arc corresponding o a soluion of Eample.3: heigh Figure depics he corresponding hybrid arc for he posiion sae. These simulaions reflec he epeced behavior of he bouncing ball model. Noe he only difference beween his eample and he eample of a bouncing ball wihou a consan inpu is ha, in his eample, he ball bounces on a plaform a a heigh of he chosen inpu value. raher han he ground a a value of. For MATLAB/Simulink files of his eample, see Eamples/Eample.3. Eample.4 (alernae way o simulae he bouncing ball) Consider he bouncing ball sysem wih a consan inpu and regular daa as given in Eample.3. This eample shows ha a MATLAB funcion block, such as he ump se D, can be replaced wih operaional blocks in Simulink. Figure shows his implemenaion. The oher funcions and soluions are he same as in Eample.3. For MATLAB/Simulink files corresponding o his alernaive implemenaion, see Eamples/Eample.4. Eample.5 (vehicle following a rack wih boundaries) Consider a vehicle modeled by a Dubins vehicle model raveling along a given rack wih sae vecor = [ξ, ξ, ξ 3 ] wih dynamics given by ξ = u cos ξ 3, ξ = u sin ξ 3, and ξ 3 = ξ 3 +r(q). The inpu u is he angenial velociy of he vehicle, ξ and ξ describe he vehicle s posiion on he plane, and ξ 3 is he vehicle s orienaion angle. Also consider a swiching conroller aemping o keep he vehicle inside he boundaries of a rack given by {(ξ, ξ ) : ξ }. A sae q {, } is used o define he modes of operaion of he conroller. When q =, he vehicle is raveling o he lef, and when q =, he vehicle is raveling o he righ. A logic variable r is defined in order o seer he vehicle back inside he boundary. The sae of he closed-loop sysem is given by := [ξ q]. A model 4

25 This model simulaes a hybrid sysem wih inpu Double Click o Iniialize Double Click o Plo Soluions u do u f flow map f f u u HSu (in) Sae Time Jumps v u C flow se C plus u g ump map g C g 3 Terminaor <= AND D 4 (in) <= Inegraor Sysem Figure : Simulink implemenaion of bouncing ball eample wih operaor blocks of such a closed-loop sysem is given by f(, u) := u cos(ξ 3 ) u sin(ξ 3 ) ξ 3 + r(q) u { 3π, r(q) := 4 if q = π 4 if q = C := { (ξ, u) R 3 {, } R (ξ, q = ) or (ξ, q = ) }, (7) [ ξ if ξ ], q = g(ξ, u) := [ ], (8) ξ if ξ, q = D := { (ξ, u) R 3 {, } R (ξ, q = ) or (ξ, q = ) } (9) The MATLAB scrips in each of he funcion blocks of he implemenaion above are given as follows. The angenial velociy of he vehicle is chosen o be u =, he iniial posiion on he plane is chosen o be (ξ, ξ ) = (, ), and he iniial orienaion angle is chosen o be ξ 3 = π 4 radians. funcion do = f(, u) % sae 3 i = z(saevec); 4 i = i(); %-posiion 5 i = i(); %y-posiion 6 i3 = i(3); %orienaion angle 7 q = i(4); 8 % q = --> going lef 9 % q = --> going righ if q == r = 3*pi/4; elseif q == 3 r = pi/4; (6) 5

26 i i 5 5 flows [] 3 umps [] i (a) Traecory (b) Hybrid arc Figure 3: Soluion of Eample.5 4 else 5 r = ; 6 end 7 % flow map: ido=f(i,u); 8 ido = u*cos(i3); %angenial velociy in -direcion 9 ido = u*sin(i3); %angenial velociy in y-direcion i3do = -i3 + r; %angular velociy qdo = ; do = [ido;ido;i3do;qdo]; funcion v = C(, u) % sae 3 i = z(saevec); 4 i = i(); %-posiion 5 i = i(); %y-posiion 6 i3 = i(3); %orienaion angle 7 q = i(4); 8 % q = --> going lef 9 % q = --> going righ % flow se if ((i < ) && (q == )) ((i > -) && (q == )) % flow condiion v = ; % repor flow 3 else 4 v = ; % do no repor flow 5 end funcion plus = g(, u) % sae 3 i = z(saevec); 4 i = i(); %-posiion 5 i = i(); %y-posiion 6 i3 = i(3); %orienaion angle 7 q = i(4); 8 % q = --> going lef 6

27 9 % q = --> going righ iplus=i; iplus=i; i3plus=i3; 3 qplus=q; 4 % ump map 5 if ((i >= ) && (q == )) ((i <= -) && (q == )) 6 qplus = 3-q; 7 else 8 qplus = q; 9 end plus = [iplus;iplus;i3plus;qplus]; funcion v = D(, u) % sae 3 i = z(saevec); 4 i = i(); %-posiion 5 i = i(); %y-posiion 6 i3 = i(3); %orienaion angle 7 q = i(4); 8 % q = --> going lef 9 % q = --> going righ % ump se if ((i >= ) && (q == )) ((i <= -) && (q == )) % ump condiion v = ; % repor ump 3 else 4 v = ; % do no repor ump 5 end A soluion o he sysem of a vehicle following a rack in {(ξ, ξ ) : ξ }, and wih T = 5, J =, rule =, is depiced in Figure 3(a) (raecory). Boh he proecion ono and are shown. Figure 3(b) depics he corresponding hybrid arc. For MATLAB/Simulink files of his eample, see Eamples/Eample.5. Eample.6 (inerconnecion of hybrid sysems H (bouncing ball) and H (moving plaform)) Consider a bouncing ball (H ) bouncing on a plaform (H ) a some iniial heigh and converging o he ground a zero heigh. This is an inerconnecion problem because he curren saes of each sysem affec he behavior of he oher sysem. In his inerconnecion, he bouncing ball will conac he plaform, bounce back up, and cause a ump in heigh of he plaform so ha i ges closer o he ground. Afer some ime, boh he ball and he plaform will converge o he ground. In order o model his sysem, he oupu of he bouncing ball becomes he inpu of he moving plaform, and vice versa. For he simulaion of he described sysem wih regular daa where H is given by [ ] ξ f (ξ, u, v ) :=, C γ bξ + v := {(ξ, u ) ξ u, u } () [ ] ξ + α g (ξ, u, v ) := ξ, D e ξ + v := {(ξ, u ) ξ = u, u }, y = h (ξ) := ξ () where γ, b, α >, e [, ), ξ = [ξ, ξ ] is he sae, y R is he oupu, u R and v = [v, v ] R are he inpus, and he hybrid sysem H is given by 7

28 g (η, u, v ) := [ f (η, u, v ) := [ ] η α η e η + v η η η + v ], C := {(η, u ) η u, η } (), D := {(η, u ) η = u, η }, y = h (η) := η (3) where α >, e [, ), η = [η, η ] R is he sae, y R is he oupu, and u R and v = [v, v ] R are he inpus. Therefore, he inerconnecion may be defined by he inpu assignmen u = y, u = y. (4) The signals v and v are included as eernal inpus in he model in order o simulae he effecs of environmenal perurbaions, such as a wind gus, on he sysem. The MATLAB scrips in each of he funcion blocks of he implemenaion above are given as follows. The consans for he inerconneced sysem are γ =.8, b =., and α, α =.. This model simulaes he inerconnecion of wo hybrid sysems; a bouncing ball and a moving plaform. Double Click o Iniialize Double Click o Plo Soluions More Info u v v HS_ sae u 3 sae HS_ v v Figure 4: MATLAB/Simulink implemenaion of inerconneced hybrid sysems H and H For hybrid sysem H : funcion do = f(, u) % sae 3 i = (); 4 i = (); 5 %inpu 6 y = u(); 8

29 .8 i, ea flows [].5 i, ea flows [] Figure 5: Soluion of Eample.6: heigh and velociy 7 v = u(); 8 v = u(3); 9 % flow map %do=f(,u); ido = i; ido = -.8-.*i+v; 3 do = [ido;ido]; funcion v = C(, u) % sae 3 i = (); 4 i = (); 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 if (i >= y) % flow condiion v = ; % repor flow else v = ; % do no repor flow 3 end funcion plus = g(, u) % sae 3 i = (); 4 i = (); 9

30 .8.5 i.6 i flows [] flows [].5 i.6 i umps [] umps [] (a) Heigh (b) Velociy Figure 6: Soluion of Eample.6 for sysem H 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 %ump map iplus=y+.*iˆ; iplus=.8*abs(i)+v; plus = [iplus;iplus]; funcion v = D(, u) % sae 3 i = (); 4 i = (); 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 % ump se if (i <= y) % ump condiion v = ; % repor ump else 3 v = ; % do no repor ump 4 end For hybrid sysem H : funcion do = f(, u) % sae 3 ea = (); 4 ea = (); 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 % flow map 3

31 ea ea flows [] umps [] ea ea flows [] umps [] (a) Heigh (b) Velociy Figure 7: Soluion of Eample.6 for sysem H eado = ea; eado = -ea-*ea+v; do = [eado;eado]; funcion v = C(, u) % sae 3 ea = (); 4 ea = (); 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 % flow se if (ea <= y) % flow condiion v = ; % repor flow else 3 v = ; % do no repor flow 4 end funcion plus = g(, u) % sae 3 ea = (); 4 ea = (); 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 % ump map eaplus = y-.*abs(ea); eaplus = -.8*abs(ea)+v; plus = [eaplus;eaplus]; funcion v = D(, u) % sae 3

32 3 ea = (); 4 ea = (); 5 %inpu 6 y = u(); 7 v = u(); 8 v = u(3); 9 % ump se if (ea >= y) % ump condiion v = ; % repor ump else 3 v = ; % do no repor ump 4 end A soluion o he inerconnecion of hybrid sysems H and H wih T = 8, J =, rule =, is depiced in Figure 5. Boh he proecion ono and are shown. A soluion o he hybrid sysem H is depiced in Figure 6(a) (heigh) and Figure 6(b) (velociy). A soluion o he hybrid sysem H is depiced in Figure 7(a) (heigh) and Figure 7(b) (velociy). These simulaions reflec he epeced behavior of he inerconneced hybrid sysems. For MATLAB/Simulink files of his eample, see Eamples/Eample.6. Eample.7 (biological eample: synchronizaion of wo fireflies) Consider a biological eample of he synchronizaion of wo fireflies flashing. The fireflies can be modeled mahemaically as periodic oscillaors which end o synchronize heir flashing unil hey are flashing in phase wih each oher. A sae value of τ i = corresponds o a flash, and afer each flash, he firefly auomaically reses is inernal imer (periodic cycle) o τ i =. The synchronizaion of he fireflies can be modeled as an inerconnecion of wo hybrid sysems because every ime one firefly flashes, he oher firefly noices and umps ahead in is inernal imer τ by ( + ε)τ, where ε is a biologically deermined coefficien. This happens unil evenually boh fireflies synchronize heir inernal imers and are flashing simulaneously. Each firefly can be modeled as a hybrid This model simulaes inerconneced hybrid sysems. Double Click o Iniialize Double Click o Plo Soluions u (in) Hybrid Sysem sae ime umps u (in) Hybrid Sysem sae ime umps Figure 8: Inerconnecion Diagram for Eample.7 3

33 sysem given by f i (τ i, u i ) :=, (5) C i := { (τ i, u i ) R τ i } { (τ i, u i ) R u i } (6) { ( + ε)τi ( + ε)τ g i (τ i, u i ) := i < (7) ( + ε)τ i D i := { (τ i, u i ) R τ i = } { (τ i, u i ) R u i = }. (8) The inerconnecion diagram for his eample is simpler han in he previous eample because now no eernal inpus are being considered. The only even ha affecs he flashing of a firefly is he flashing of he oher firefly. The inerconnecion diagram can be seen in Figure au.5 au flows [] flows [].5 au.5 au umps [] umps [] (a) Soluion for sysem H (b) Soluion for sysem H For hybrid sysem H i, i =, : funcion audo = f(au, u) % flow map 3 audo = ; Figure 9: Soluion of Eample.7 funcion v = C(au, u) % flow se 3 if ((au > ) && (au < )) ((u > ) && (u <= )) % flow condiion 4 v = ; % repor flow 5 else 6 v = ; % do no repor flow 7 end funcion auplus = g(au, u) % ump map 3 if (+e)*au < 4 auplus = (+e)*au; 5 elseif (+e)*au >= 6 auplus = ; 7 else 8 auplus = au; 9 end 33