Algorithms and Software for LMI Problems in Control Introduction
Algorithms and Software for LMI Problems in Control Introduction
Algorithms and Software for LMI Problems in Control Introduction
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
whereA=[a1:::am]. whichleadsto Asecondpossibility<strong>for</strong>l<strong>in</strong>eariz<strong>in</strong>gtheequationXZ=IistowriteitasZ=X�1,<br />
Z+mXi=1yiAi=0 TrAiX=0;i=1;:::;m<br />
Elim<strong>in</strong>at<strong>in</strong>gZ,weobta<strong>in</strong>X�1XX�1+Z=�Z+X�1: �X�1XX�1+mXi=1yiAi=Z�X�1: TrAiX=0;i=1;:::;m<br />
Specializedtol<strong>in</strong>earprogramm<strong>in</strong>gtheequationsbecome "<br />
TherstSDPmethodswerebasedontheseprimalordualscal<strong>in</strong>gs(see<strong>for</strong>example, �X�2A#"xy#=" AT 0 z�X�1e#: 0<br />
Yoshise[31].Forl<strong>in</strong>earprogramm<strong>in</strong>gtheresult<strong>in</strong>gequations<strong>for</strong>thesearchdirectionshave oneusuallyprefersaprimal-dualsymmetricscal<strong>in</strong>g<strong>in</strong>troducedbyKojima,Mizuno<strong>and</strong> programm<strong>in</strong>g,however,theprimal<strong>and</strong>dualscal<strong>in</strong>gsarerarelyused<strong>in</strong>practice.Instead, Nesterov<strong>and</strong>Nemirovskii[17],Alizadeh[26],<strong>and</strong>V<strong>and</strong>enberghe<strong>and</strong>Boyd[4]).Inl<strong>in</strong>ear<br />
the<strong>for</strong>m Theseequationsareobta<strong>in</strong>edbyl<strong>in</strong>eariz<strong>in</strong>gXZ=Ias " �X�1ZA#"xy#=" AT 0 z�X�1e:#: 0 (8)<br />
scal<strong>in</strong>gcanachieveahigheraccuracythanmethodsbasedonthetheprimalordualscal<strong>in</strong>g Severalresearchershavedemonstratedthatmethodsthatusethisprimal-dualsymmetric XZ+XZ=I�XZ: (9)<br />
(see<strong>for</strong>exampleWright[32]),<strong>and</strong>there<strong>for</strong>ethesymmetricscal<strong>in</strong>gisthebasisofallpractical LP<strong>in</strong>terior-po<strong>in</strong>tmethods. l<strong>in</strong>earization(9)leadstoal<strong>in</strong>earsystemTrAiX=0;i=1;:::;m Theextensionofthissymmetricprimal-dualscal<strong>in</strong>gtoSDPisnotstraight<strong>for</strong>ward:The<br />
�X�1XZ+mXi=1yiAi=Z�X�1 (11) (10)<br />
research<strong>in</strong>SDPhasthere<strong>for</strong>econcentratedonextend<strong>in</strong>gtheprimal-dualsymmetricscal<strong>in</strong>g butun<strong>for</strong>tunatelythesolutionXisnotsymmetric<strong>in</strong>general.Muchofthemostrecent 6
Change language