Solving a set of equations and inequalities

Question

Karan Juneja 2022-2-1

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1640915-solving-a-set-of-equations-and-inequalities

编辑： Walter Roberson 2022-2-2

Hi, I am working on a problem to solve a set of 2 equations and 2 inequalities I got from applying Hurqitz stability criteria for a controller design algorithm.

I got the system using solve() but how to get a particular solution from the output of this solve() function. Conditions parameter shows some function.

syms Kpx Kix                    % PI controller
% The maxima/minima of this equation is to be solved for Kix and Kpx. Every
% other variable is some constant numerical value.
T = 0.4652;
c3 = T; c2 = 1+1.6*T; c1 = 1.6+4*T; c0 = 4;
d4 = T; d3 = 1+1.6*T; d2 = 1.6+4*T;
ISE4 = (c3*c3*(16*Kix*(4+16*Kpx)*d2 - 16*Kix*16*Kix*d3) + 16*Kix*(4+16*Kpx)*d4*(c2*c2 +2*c1*c3) + 16*Kix*d3*d4*(c1*c1 - 2*c0*c2) + c0*c0*(d2*d3*d4 - (4+16*Kpx)*d4*d4))/(2*16*Kix*d4*((4+16*Kpx)*d2*d3 - 16*Kix*d3*d3- (4+16*Kpx)*(4+16*Kpx)*d4));
ISEKp = diff(ISE4, Kpx) == 0;
ISEKi = diff(ISE4, Kix) == 0;   % Max/Min conditions
ISEKp1 = Kpx < 0.0652;
ISEKp2 = Kpx > -0.0291;         % Stability conditions from Hurwitz Criteria
ISEKi1 = Kix > 0;
% eqns = [ISEKp ISEKi];
% [x1, x2] = vpasolve(eqns, [Kpx, Kix])     % Doesn't take inequalities
% into account
% Kp = double(x1(8,1))
% Ki = double(x2(8,1))
eqns = [ISEKp ISEKi ISEKp1 ISEKp2 ISEKi1];
X = solve(eqns, [Kpx, Kix], Real=true, ReturnConditions=true)   % How to get one solution out of the infinite solutions provided?
X = struct with fields:
           Kpx: (34507100878762346168055722004523112768983738441438576824569961297545565933326311549769537483995417236328125*y^4)/847020229739709285495136549570717729313154390014956487612558202927983338057597417707783758809965409599488 - (632648203064…
           Kix: y
    parameters: y
    conditions: (34507100878762346168055722004523112768983738441438576824569961297545565933326311549769537483995417236328125*y^4)/847020229739709285495136549570717729313154390014956487612558202927983338057597417707783758809965409599488 - (632648203064…

2 个评论
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John D'Errico 2022-2-1

Since you don't show the value of those other incidental variables, we cannot actually show you how to solve your problem, since we cannot run that code. If you want help, is there a good reason why you would make it more difficult to get help?

Karan Juneja 2022-2-1

在 MATLAB Online 中打开

Updated the code with variables. I also tried using the sub() function to get one of the values. But is there any other way, suppose if I don't know this range of values?

eqns = [ISEKp ISEKi ISEKp1 ISEKi1];
X = solve(eqns, [Kpx, Kix], ReturnConditions=true,IgnoreAnalyticConstraints=true)
estimate = 0.14;        % 0.3257 > estimate or the y parameter> 0
Kp = double(subs(X.Kpx, X.parameters, estimate))
Ki = double(subs(X.Kix, X.parameters, estimate))

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Answer 1

Walter Roberson 2022-2-2

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1640915-solving-a-set-of-equations-and-inequalities#answer_886955

编辑：Walter Roberson 2022-2-2

在 MATLAB Online 中打开

syms Kpx Kix % PI controller

% The maxima/minima of this equation is to be solved for Kix and Kpx. Every

% other variable is some constant numerical value.

T = 0.4652;

c3 = T; c2 = 1+1.6*T; c1 = 1.6+4*T; c0 = 4;

d4 = T; d3 = 1+1.6*T; d2 = 1.6+4*T;

ISE4 = (c3*c3*(16*Kix*(4+16*Kpx)*d2 - 16*Kix*16*Kix*d3) + 16*Kix*(4+16*Kpx)*d4*(c2*c2 +2*c1*c3) + 16*Kix*d3*d4*(c1*c1 - 2*c0*c2) + c0*c0*(d2*d3*d4 - (4+16*Kpx)*d4*d4))/(2*16*Kix*d4*((4+16*Kpx)*d2*d3 - 16*Kix*d3*d3- (4+16*Kpx)*(4+16*Kpx)*d4));

ISEKp = diff(ISE4, Kpx) == 0;

ISEKi = diff(ISE4, Kix) == 0; % Max/Min conditions

ISEKp1 = Kpx < 0.0652;

ISEKp2 = Kpx > -0.0291; % Stability conditions from Hurwitz Criteria

ISEKi1 = Kix > 0;

% eqns = [ISEKp ISEKi];

% [x1, x2] = vpasolve(eqns, [Kpx, Kix]) % Doesn't take inequalities

% into account

% Kp = double(x1(8,1))

% Ki = double(x2(8,1))

eqns = [ISEKp ISEKi ISEKp1 ISEKp2 ISEKi1];

X = solve(eqns, [Kpx, Kix], Real=true, ReturnConditions=true) % How to get one solution out of the infinite solutions provided?

X = struct with fields:

Kpx: (34507100878762346168055722004523112768983738441438576824569961297545565933326311549769537483995417236328125*y^4)/847020229739709285495136549570717729313154390014956487612558202927983338057597417707783758809965409599488 - (632648203064… Kix: y parameters: y conditions: (34507100878762346168055722004523112768983738441438576824569961297545565933326311549769537483995417236328125*y^4)/847020229739709285495136549570717729313154390014956487612558202927983338057597417707783758809965409599488 - (632648203064…

string(X.conditions)

ans = "(34507100878762346168055722004523112768983738441438576824569961297545565933326311549769537483995417236328125*y^4)/847020229739709285495136549570717729313154390014956487612558202927983338057597417707783758809965409599488 - (10389353489109644537606890544364750875985455986886901586713561389388177805876854668864414635540498558473069375*y^2)/1970169054374563798061687614301489438382397111174788790186810380010489244321971593588305022991979542728409088 - (6326482030647950373666319861024196355951189918432847939302047140666280082419174977438516391877212365740703125*y^3)/246271131796820474757710951787686179797799638896848598773351297501311155540246449198538127873997442841051136 - (7021441003762571396906849433834420747241464620819343818217715727218707127347568050182545324879631345784830457*y)/2462711317968204747577109517876861797977996388968485987733512975013111555402464491985381278739974428410511360 + 10284141242117406867047512558911453927056095857606903324727496881661357853999416489223054215882483933385581789437/18330452881900941577165941563461057734709822722370234903898083775617591929171623706745589933917377665545118154752 < 163/2500 & -291/10000 < (34507100878762346168055722004523112768983738441438576824569961297545565933326311549769537483995417236328125*y^4)/847020229739709285495136549570717729313154390014956487612558202927983338057597417707783758809965409599488 - (10389353489109644537606890544364750875985455986886901586713561389388177805876854668864414635540498558473069375*y^2)/1970169054374563798061687614301489438382397111174788790186810380010489244321971593588305022991979542728409088 - (6326482030647950373666319861024196355951189918432847939302047140666280082419174977438516391877212365740703125*y^3)/246271131796820474757710951787686179797799638896848598773351297501311155540246449198538127873997442841051136 - (7021441003762571396906849433834420747241464620819343818217715727218707127347568050182545324879631345784830457*y)/2462711317968204747577109517876861797977996388968485987733512975013111555402464491985381278739974428410511360 + 10284141242117406867047512558911453927056095857606903324727496881661357853999416489223054215882483933385581789437/18330452881900941577165941563461057734709822722370234903898083775617591929171623706745589933917377665545118154752 & in(y, 'real') & ((3889503364931186315391311562659059508626681930031214187670213233126249714491660752761911232771190967583089971212270400624581571569702044034459084075166513979600687301250146245157387840131537567220636590540952896806265858177988621129955301964230965809*y^2)/10801051397654735969264549761302125540561601513627366742741379932079593203588349979242883592937891313148958732464706151510220746855850232276012834951836641290514412752415558932731142955800972035853762192737733407268186943156079191064434241422611718750 - (40374588249136566629452043465917046096693299415622760437651333125174747950876974148927199606229372572938962941603651774662664618273420812127531323113171608856783722752011896361614510136417731526565402132713811449086080560646724295597725999971523621348546*y)/799952869138803882723655716696438672847843612103026849384283451219644871640762170337676066101962575380094756123167299346225724064011407827942200588620401245578723694475777333455400275164009491405419262399638380475800095477497115088209661005362180419921875 + (13150653199530933189484524389349142473467552929514154883531545163846059670586739946261626072030177516552106834694383252299780139526502248858205831666470855843455032945287444842955186598163655479569531932184428714042311959345976459061417892741773572*y^3)/25922523354371366326234919427125101297347843632705680182579311836991023688612039950182920623050939151557500957915294763624529792454040557462430803884407939097234590605797341438554743093922332886049029262570560177443648663574590058554642179414268125 + y^4 - 287893850481767298624591884489216940217282924012780149574946467096235012614695794458376187274070651746465613137101963135993994678276981942724725334452782600843036073034250608749633053082777984077588606494219123267171994916373376489283/846820863370516630739728246117679264674285450468302143336831798630124189667060762489939981540995959618719472181469702043566550663573292194835823988795643811710799104387143942880033468885440774603209479664780571528174260715819594954604394495304786674000 ~= 0 | ~in(y, 'real')) & (~in(y, 'real') | (15142728523037363746562322002735716030893478256243417298161138309584463514890043086974776842822151749990209251242522416665164633636584594833819773074093439931522874569873698832478848645757669727583659602957198730990847306432894916848343435139273358078*y)/5605071627170301100051944167490236039485865317581495208418297302253174528753723620397556662440276366287802808167049952765530991601816632527544191119731097588901435249155950569676513602968190586482797316444287295618941282909541119468445431624703857421875 + (327557257666691326329465463890405490336879713668331367576218061900464203625275827061870502380060864541507206756679282255967637778516741701204111001915260309185298790492605994838067616163828416771739326039465980792656030516061082738474720579308704109*y^2)/7038267157155618596520922296392802722999972651950434489558165726373606496865435229663767353342726778933544792027485510308109802112660885401675743785788922542012941477421142993821647460942335521254348022674345566144290826134512393054047934394564843750 - (1893640918251017669529939040165146379405804629767878720188130637013793117061350996606375687234228473865902256459468931781330504668383919442574792304081797738220795361434397110658993952053537651336709571291158193553131502392793751803938916163516676*y^3)/5630613725724494877216737837114242178399978121560347591646532581098885197492348183731013882674181423146835833621988408246487841690128708321340595028631138033610353181936914395057317968753868417003478418139476452915432660907609914443238347515651875 + y^4 + 88344240755794687488130686244059155195337972938087348103912678812203952104502023914418832076425398733358119369473697266634159096489313083401327453431339582184128679094916981592115381877897520864295555950780289127614297963706288111/4854652408477554635392771587195732350731775154248446270289346383142496425008839639429014545440054114857119867548972940616657486742183745073921969261659055194031844403958408294425160849409493663803200812888217506763741528081721769940658389721483142000 ~= 0) & (9764844811149744146231378070411615825616264898283630210390215950113091*y)/711173444357079636422148574084292087998259955415451459702917480468750000 - (180587177869066749160684963053363732945126292655783179687840411818*y^2)/1895647186162465066989390008307227405670340379869217132484130859375 - (7023765822910589304615706812702159470193242209300667674394833*y^3)/76785708806580863472987949703583894913228977412423985113281250 - (37370682699467734941240597020444241635402368926180066224428*y^4)/61428567045264690778390359762867115930583181929939188090625 + y^5 + 1556155627217174522033715191849435529216/16825426443624353630394275168732498538665545767568565234375 == 0 & 0 < y"

C = X.conditions;

Y = X.parameters;

assume(~in(Y, 'real'))

C1 = simplify(C)

C1 =

symfalse

assume(Y < 0)

C2 = simplify(C)

C2 =

symfalse

assume(Y > 0)

C3 = simplify(C)

C3 =

assume(Y, 'clear')

string(C3)

ans = "302710189145642068533172720182760090594104211846792536522096694453505821*y + 22046376775069468729086605796613054727946058617878995250790441894531250000*y^5 + 2039032614907334447152225744470705440431276032000000000 == 5815000*y^2*(346798437506210346915400523877169123840791334084220466423244879375*y + 2306471822857774265904693097355543038434989957162675962288915625000*y^2 + 361174355738133498321369926106727465890252585311566359375680823636) & 0 < 466732349284867077941809598559358221148315661199981344170317679822599085688732358865602335451901974959230957031250000*y^4 - 60414090539172582986184068515481026343855426563747332726739359479292253941173909899446571105667999117520898415625000*y^2 - 294307944065742651382957199934845614478849355005496086136331232983795349434140019950439782550127919254257509375000000*y^3 - 32663743549503482138410663566197725316167293416051587442348813563021425556420886569449200851340045020591031285964000*y + 6760973388112952666839400911505106691960094811767423225269331949707393611961444211930844301849674764658090454837677 & 466732349284867077941809598559358221148315661199981344170317679822599085688732358865602335451901974959230957031250000*y^4 - 60414090539172582986184068515481026343855426563747332726739359479292253941173909899446571105667999117520898415625000*y^2 - 294307944065742651382957199934845614478849355005496086136331232983795349434140019950439782550127919254257509375000000*y^3 - 32663743549503482138410663566197725316167293416051587442348813563021425556420886569449200851340045020591031285964000*y + 5680622321385915922635183230608620601720634635067727505620838637181931787635891639714526095119419318495025053591981 < 0 & 751290338139933231840843624813784393767268915525908326223816006793251709869918734963237330272716164837248222417360752222922863216831814472069102860489897297607031512969437367496922804618461148246329002885538603444593787072514245692482485407470111546053743968*y + 5060634090499815796135404219537016527256926398662151066747105866926006081117699511963643916927023175230840855925620445749894437704087573211958062519677819155443993471305186481927143511220706751363862223531195526180757723139375758600677734375 ~= 459385000*y^2*(16438316499413666486855655486686428091834441161892693604414431454807574588233424932827032590037721895690133543367979065374725174408127811072757289583088569804318791181609306053693983247704569349461914915230535892552889949182470573826772365927216965000*y + 32403154192964207907793649283906376621684804540882100228224139796238779610765049937728650778813673939446876197394118454530662240567550696828038504855509923871543238257246676798193428867402916107561286578213200221804560829468237573193302724267835156250*y^2 + 11668510094793558946173934687977178525880045790093642563010639699378749143474982258285733698313572902749269913636811201873744714709106132103377252225499541938802061903750438735472163520394612701661909771622858690418797574533965863389865905892692897427)"

C3c = children(C3).'

C3c = 4×1 cell array

{[22046376775069468729086605796613054727946058617878995250790441894531250000*y^5 + 302710189145642068533172720182760090594104211846792536522096694453505821*y + 2039032614907334447152225744470705440431276032000000000 == 5815000*y^2*(2306471822857774265904693097355543038434989957162675962288915625000*y^2 + 346798437506210346915400523877169123840791334084220466423244879375*y + 361174355738133498321369926106727465890252585311566359375680823636) ]} {[0 < 466732349284867077941809598559358221148315661199981344170317679822599085688732358865602335451901974959230957031250000*y^4 - 294307944065742651382957199934845614478849355005496086136331232983795349434140019950439782550127919254257509375000000*y^3 - 60414090539172582986184068515481026343855426563747332726739359479292253941173909899446571105667999117520898415625000*y^2 - 32663743549503482138410663566197725316167293416051587442348813563021425556420886569449200851340045020591031285964000*y + 6760973388112952666839400911505106691960094811767423225269331949707393611961444211930844301849674764658090454837677 ]} {[466732349284867077941809598559358221148315661199981344170317679822599085688732358865602335451901974959230957031250000*y^4 - 294307944065742651382957199934845614478849355005496086136331232983795349434140019950439782550127919254257509375000000*y^3 - 60414090539172582986184068515481026343855426563747332726739359479292253941173909899446571105667999117520898415625000*y^2 - 32663743549503482138410663566197725316167293416051587442348813563021425556420886569449200851340045020591031285964000*y + 5680622321385915922635183230608620601720634635067727505620838637181931787635891639714526095119419318495025053591981 < 0 ]} {[751290338139933231840843624813784393767268915525908326223816006793251709869918734963237330272716164837248222417360752222922863216831814472069102860489897297607031512969437367496922804618461148246329002885538603444593787072514245692482485407470111546053743968*y + 5060634090499815796135404219537016527256926398662151066747105866926006081117699511963643916927023175230840855925620445749894437704087573211958062519677819155443993471305186481927143511220706751363862223531195526180757723139375758600677734375 ~= 459385000*y^2*(32403154192964207907793649283906376621684804540882100228224139796238779610765049937728650778813673939446876197394118454530662240567550696828038504855509923871543238257246676798193428867402916107561286578213200221804560829468237573193302724267835156250*y^2 + 16438316499413666486855655486686428091834441161892693604414431454807574588233424932827032590037721895690133543367979065374725174408127811072757289583088569804318791181609306053693983247704569349461914915230535892552889949182470573826772365927216965000*y + 11668510094793558946173934687977178525880045790093642563010639699378749143474982258285733698313572902749269913636811201873744714709106132103377252225499541938802061903750438735472163520394612701661909771622858690418797574533965863389865905892692897427)]}

C31 = C3c{1} %equality

C31 =

C32 = C3c{2} %<

C32 =

C33 = C3c{3} %<

C33 =

C34 = C3c{4} %~=

C34 =

C31sol = solve(C31)

C31sol =

cross_check = simplify(subs([C32, C33, C34], Y, C31sol))

cross_check =

Now... for there to be a solution, an entire row of cross_check must be symtrue, in which case the corresponding row of C31sol would be the y value (and so the Kix value). But there are no y values to the polynomial equality C31 that also satisfy all the other conditions.

And what that means is that it turns out that your system has no solutions.

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Answer 2

KSSV 2022-2-2

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1640915-solving-a-set-of-equations-and-inequalities#answer_886780

在 MATLAB Online 中打开

X is a structure. You can access the fields using:

Kpx = X.Spx ;
Kix = X.Kix ;

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Karan Juneja 2022-2-2

编辑：Karan Juneja 2022-2-2

Yes, that I can access but it shows as a sym with X.parameters as y. Kpx and Kix have a term y in them. I need to find the range of y which I can later substitute using the sub() function.

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