Solving a set of equations and inequalities

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…

[John D'Errico]

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]

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))

Answer 1

[Walter Roberson]

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.

Answer 2

[KSSV]

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

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

Also read about double and vpasolve.

[Karan Juneja]

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.