Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 541

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 543

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 926

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 639

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/scripts/upload.php on line 112

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 608

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 609

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 612

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 620

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php on line 624

Warning: Cannot modify header information - headers already sent by (output started at /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php:541) in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/scripts/upload.php on line 169

Warning: Cannot modify header information - headers already sent by (output started at /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php:541) in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/scripts/upload.php on line 170

Warning: Cannot modify header information - headers already sent by (output started at /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/pmwiki.php:541) in /nfs/bronfs/uwfs/hw00/d27/mesbahi/pmwiki/scripts/upload.php on line 171
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y(in)o(terpretation,)f(includes)h(as)e(its)g(sp)q (ecial)i(case,)f(the)f(BMI)g(problem.)27 b(Since)18 b(the)g(extreme)f (forms)f(of)-1 2659 y(the)g(matrix)f(classes)h(whic)o(h)g(are)f (considered)i(in)g(the)e(c)o(hapter)g(can)h(b)q(e)g(c)o(haracterized)g (b)o(y)f(their)h(rank,)-1 2716 y(the)i(EFP)f(form)o(ulation)g(of)g(the) g(BMI)h(also)f(translates)g(directly)i(to)d(a)h(rank)h(minimization)h (problem)-1 2772 y(o)o(v)o(er)c(a)g(con)o(v)o(ex)g(set)g(of)f(matrices) i([39)o(])e(\(refer)h(to)g Fr(x)p Fw(3.2\).)p eop %%Page: 4 4 4 3 bop 157 -1 a Fw(4)100 b Fn(Mesbahi,)16 b(Saf)o(ono)o(v)h(and)g(P)l (ap)m(a)l(v)l(assilopoulos)p 960 348 187 4 v 960 483 4 136 v 1005 427 a Fv(K)s Fw(\()p Fv(s)p Fw(\))p 1144 483 V 960 486 187 4 v 915 77 277 4 v 915 258 4 181 v 1008 179 a Fv(G)p Fw(\()p Fv(s)p Fw(\))p 1189 258 V 915 261 277 4 v 1189 124 271 4 v 1418 122 a Fl(-)p 1189 214 136 4 v 1322 415 4 203 v 1144 417 181 4 v 1144 415 a(\033)p 648 124 271 4 v 877 122 a(-)p 784 417 181 4 v 782 415 4 203 v 784 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y Fu(K)1276 2691 y Ft(\033)157 2751 y Fw(\(1.15\))p eop %%Page: 5 5 5 4 bop 425 -1 a Fn(Bilinearity)17 b(and)g(Complement)m(arity)h(in)e(R) o(obust)h(Contr)o(ol)100 b Fw(5)-1 107 y(and)653 230 y Fq(S)682 237 y Fu(K)758 230 y Fw(:=)847 158 y Ft(")892 200 y Fv(A)926 207 y Fu(K)p 980 217 2 57 v 1004 200 a Fv(B)1038 207 y Fu(K)p 872 219 224 2 v 893 259 a Fv(C)925 266 y Fu(K)p 980 276 2 57 v 1002 259 a Fv(D)1040 266 y Fu(K)1095 158 y Ft(#)1126 230 y Fv(:)-1140 b Fw(\(1.16\))74 361 y(The)19 b(inclusion)i(of)d(the)h(uncertain)o(t)o(y)g(in)g(this)g (framew)o(ork)f(is)h(no)o(w)f(really)h(an)g(extra)f(b)q(on)o(us;)i(see) -1 417 y(Figure)c(3.)j(The)d(resulting)g(transfer)f(function)h(from)e Fv(u)959 424 y Fs(1)994 417 y Fw(to)g Fv(y)1071 424 y Fs(1)1107 417 y Fw(is)h(simply)l(,)668 518 y Fv(T)695 525 y Fs(\001)739 518 y Fw(:=)25 b Fr(F)845 525 y Fu(U)874 518 y Fr(f)p Fw(\001\()p Fv(s)p Fw(\))p Fv(;)8 b(T)e Fw(\()p Fv(s)p Fw(\))p Fr(g)-1 619 y Fw(pro)o(vided)16 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b(BMI)f(form)o(ulation)h(also)f(eliminates)i (the)f(curv)o(e)g(\014tting)f(step)h(of)f(the)h(traditional)271 2023 y Fv(D)q Fw(-)p Fv(K)h Fw(iteration)d(approac)o(hes)g(to)g Fv(\026)p Fw(/)p Fv(k)930 2030 y Fu(m)963 2023 y Fw(.)225 2135 y Fr(\017)23 b Fw(A)e(broad)g(sp)q(ectrum)h(of)f(robust)g(con)o (trol)g(syn)o(thesis)h(problems)g(can)f(b)q(e)h(form)o(ulated)f(within) 271 2191 y(the)26 b(BMI)h(framew)o(ork.)52 b(These)27 b(include)i(order-constrained)e(con)o(troller)g Fv(\026)p Fw(/)p Fv(k)1725 2198 y Fu(m)1758 2191 y Fw(-syn)o(thesis)271 2247 y(with)g(sp)q(eci\014cations)i(requiring)f(suc)o(h)f(additional)i (prop)q(erties)f(as)e(decen)o(tralized)j(con)o(trol.)271 2304 y(Gain)17 b(sc)o(heduling)j(and)d(sim)o(ultaneous)h Fv(\026)p Fw(/)p Fv(k)1046 2311 y Fu(m)1079 2304 y Fw(-syn)o(thesis)g (for)f(sev)o(eral)g(plan)o(ts)h(also)f(fall)h(in)g(this)271 2360 y(framew)o(ork.)232 2434 y(W)l(e)f(note)g(ho)o(w)o(ev)o(er)f(that) 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h(that)e(the)i(BMI)f(problem,)i(that)d(of)h(\014nding)i(the)f(v)o (ectors)e Fv(x)h Fw(and)h Fv(y)h Fw(suc)o(h)157 728 y(that)262 696 y Ft(P)306 740 y Fu(ij)343 728 y Fv(x)369 735 y Fu(i)383 728 y Fv(y)405 735 y Fu(j)424 728 y Fv(F)453 735 y Fu(ij)506 728 y Fv(>)j Fw(0)e(\(1.1\))f(is)h(equiv)m(alen)o(t)i(to)e(\014nding)i (the)e(corresp)q(onding)h(v)o(ectors)f(suc)o(h)g(that)157 753 y Ft(P)201 796 y Fu(ij)239 785 y Fv(x)265 792 y Fu(i)279 785 y Fv(y)301 792 y Fu(j)319 785 y Fv(F)348 792 y Fu(ij)392 785 y Fv(<)13 b Fw(0,)h(whic)o(h)i(can)f(b)q(e)h(written)g(as,)885 885 y(min)819 923 y FA(k)p Fu(z)q FA(k)p Fs(=1)p Fu(;z)q FA(2)p Fq(R)1008 902 y Fe(p)1033 885 y Fv(x)1059 866 y Fu(T)1087 885 y Fv(G)p Fw(\()p Fv(z)r Fw(\))p Fv(y)e(<)f Fw(0)-1132 b(\(3.43\))157 1008 y(where)782 1064 y([)p Fv(G)p Fw(\()p Fv(z)r Fw(\)])903 1071 y Fu(i;j)955 1064 y Fw(=)12 b Fv(z)1025 1045 y Fu(T)1053 1064 y Fv(F)1082 1071 y Fu(ij)1113 1064 y Fv(z)i Fr(2)f Fq(R)1231 1045 y Fu(n)p FA(\002)p Fu(m)1313 1064 y Fv(:)-1169 b Fw(\(3.44\))157 1143 y(Let)16 b Fv(\032)e Fw(b)q(e)i(a)f(real)h(p)q(ositiv)o(e)g(n)o (um)o(b)q(er)f(suc)o(h)h(that)840 1236 y Fv(\032)c(>)20 b Fw(max)924 1268 y FA(k)p Fu(z)q FA(k)p Fs(=1)1045 1236 y Fv(\033)1071 1243 y Fs(max)1138 1236 y Fw(\()p Fv(G)p Fw(\()p Fv(z)r Fw(\)\))-1112 b(\(3.45\))157 1359 y(and)16 b(let)f Fq(C)e Fr(\022)g Fq(R)449 1340 y Fu(n)p Fs(+)p Fu(m)546 1359 y Fw(b)q(e)j(the)f(con)o(v)o(ex)g(set)608 1451 y Fq(C)d Fw(:=)h Fr(f)p Fv(w)g Fw(:)f Fv(w)847 1433 y Fu(T)875 1451 y Fv(Q)p Fw(\()p Fv(z)r Fw(\))p Fv(w)g Fr(\024)h Fw(1)p Fv(;)23 b(z)14 b Fr(2)f Fq(R)1239 1433 y Fu(p)1259 1451 y Fv(;)8 b Fr(k)p Fv(z)r Fr(k)k Fw(=)h(1)p Fr(g)-1298 b Fw(\(3.46\))157 1544 y(where)764 1622 y Fv(Q)p Fw(\()p Fv(z)r Fw(\))13 b(=)919 1550 y Ft(")1028 1588 y Fv(I)1161 1570 y Fs(1)p 1161 1577 19 2 v 1161 1604 a Fu(\032)1184 1588 y Fv(G)p Fw(\()p Fv(z)r Fw(\))970 1632 y Fs(1)p 969 1639 V 969 1666 a Fu(\032)992 1650 y Fv(G)1028 1633 y Fu(T)1056 1650 y Fw(\()p Fv(z)r Fw(\))90 b Fv(I)1299 1550 y Ft(#)1331 1622 y Fv(:)-1187 b Fw(\(3.47\))157 1733 y(Notice)18 b(that)f(b)o(y)g(\(3.45\),)e(one)j(has)f(that)g(the)g (matrix)g Fv(Q)p Fw(\()p Fv(z)r Fw(\))f Fv(>)g Fw(0,)h(for)g(all)h Fr(k)p Fv(z)r Fr(k)e Fw(=)g(1.)26 b(It)18 b(follo)o(ws)f(that)157 1790 y(the)e(set)g Fq(C)g Fw(is)h(the)f(in)o(tersection)h(of)f (ellipsoids)j(in)e Fq(R)1066 1772 y Fu(n)p Fs(+)p Fu(m)1148 1790 y Fw(.)232 1846 y(Supp)q(ose)e(that)f Fq(C)g Fw(has)g(diameter)g (strictly)h(greater)e(than)i(2.)k Fq(C)13 b Fw(is)h(the)f(in)o (tersection)i(of)d(an)i(in\014nite)157 1903 y(n)o(um)o(b)q(er)19 b(of)e(ellipsoids)k(and)e(th)o(us)e(its)i(maxim)o(um)f(diameter)g(is)h (ac)o(hiev)o(ed)g(at)e(some)28 b(\026)-33 b Fv(w)1689 1910 y Fs(1)1726 1903 y Fw(=)18 b(\()s(\026)-26 b Fv(x;)10 b Fw(\026)-25 b Fv(y)q Fw(\))17 b(on)157 1959 y(the)e(b)q(oundary)h(of) f Fq(C)g Fw(and,)g(b)o(y)g(symmetry)l(,)f(also)h(at)24 b(\026)-32 b Fv(w)1110 1966 y Fs(2)1142 1959 y Fw(=)13 b Fr(\000)d Fw(\026)-33 b Fv(w)1258 1966 y Fs(1)1277 1959 y Fw(.)20 b(It)c(th)o(us)f(holds)g(that,)774 2052 y(\026)-32 b Fv(w)799 2033 y Fu(T)798 2063 y Fs(1)826 2052 y Fv(Q)p Fw(\()p Fv(z)r Fw(\))9 b(\026)-32 b Fv(w)954 2059 y Fs(1)986 2052 y Fr(\024)13 b Fw(1)p Fv(;)37 b Fr(8)p Fv(z)r(;)23 b Fr(k)p Fv(z)r Fr(k)12 b Fw(=)h(1)-1186 b(\(3.48\))157 2145 y(where)942 2222 y(\026)-32 b Fv(w)966 2229 y Fs(1)998 2222 y Fw(:=)1059 2150 y Ft(")1106 2194 y Fw(\026)-25 b Fv(x)1108 2251 y Fw(\026)f Fv(y)1150 2150 y Ft(#)157 2222 y Fw(\(3.49\))157 2331 y(and)809 2388 y Fr(k)9 b Fw(\026)-32 b Fv(w)865 2395 y Fs(1)884 2388 y Fr(k)12 b Fw(=)h Fr(k)s Fw(\026)-26 b Fv(x)p Fr(k)1039 2369 y Fs(2)1068 2388 y Fw(+)11 b Fr(k)s Fw(\026)-26 b Fv(y)q Fr(k)1183 2369 y Fs(2)1216 2388 y Fv(>)13 b Fw(1)p Fv(:)-1143 b Fw(\(3.50\))157 2467 y(Inequalit)o(y)17 b(\(3.48\))c(is)j(equiv)m(alen)o(t)h(to)606 2578 y Fr(k)s Fw(\026)-26 b Fv(x)o Fr(k)677 2560 y Fs(2)707 2578 y Fw(+)10 b Fr(k)s Fw(\026)-26 b Fv(y)r Fr(k)822 2560 y Fs(2)852 2578 y Fw(+)902 2548 y(2)p 902 2568 24 2 v 902 2610 a Fv(\032)933 2578 y Fw(\026)g Fv(x)956 2560 y Fu(T)984 2578 y Fv(G)p Fw(\()p Fv(z)r Fw(\))s(\026)g Fv(y)14 b Fr(\024)f Fw(1)p Fv(;)22 b Fr(8)p Fv(z)r(;)h Fr(k)p Fv(z)r Fr(k)12 b Fw(=)h(1)-1300 b(\(3.51\))p 157 2682 137 2 v 209 2711 a Ff(2)226 2727 y Fo(It)12 b(is)h(generally)h(accepted)f(in) g(computer)g(science)g(that)g(p)q(olynomial)i(running)f(time)f(of)f(an) g(algorithm)i(is)f(equiv)n(alen)o(t)157 2772 y(to)g(e\016ciency)m(.)p eop %%Page: 13 13 13 12 bop 402 -1 a Fn(Bilinearity)17 b(and)g(Complement)m(arity)h(in)f (R)o(obust)g(Contr)o(ol)99 b Fw(13)-1 107 y(from)15 b(whic)o(h)h(it)f (follo)o(ws)h(that)546 184 y(2)p 546 205 24 2 v 546 246 a Fv(\032)577 215 y Fw(\026)-26 b Fv(x)600 196 y Fu(T)627 215 y Fv(G)p Fw(\()p Fv(z)r Fw(\))s(\026)g Fv(y)14 b Fr(\024)f Fw(1)d Fr(\000)h Fw(\()p Fr(k)s Fw(\026)-26 b Fv(x)o Fr(k)974 196 y Fs(2)1003 215 y Fw(+)11 b Fr(k)s Fw(\026)-26 b Fv(y)r Fr(k)1119 196 y Fs(2)1138 215 y Fw(\))12 b Fv(<)h Fw(0)p Fv(:)-1253 b Fw(\(3.52\))-1 335 y(Th)o(us)512 391 y(\026)-26 b Fv(x)535 372 y Fu(T)563 391 y Fv(G)p Fw(\()p Fv(z)r Fw(\))s(\026)g Fv(y)14 b(<)f Fw(0)p Fv(;)22 b Fw(uniformly)16 b(in)g Fv(z)r(;)22 b Fr(k)p Fv(z)r Fr(k)13 b Fw(=)g(1)p Fv(:)-1285 b Fw(\(3.53\))-1 473 y(and)16 b(consequen)o(tly)g Fr(\000)s Fw(\026)-26 b Fv(x;)11 b Fw(\026)-26 b Fv(y)17 b Fw(satisfy)e(\(1.1\).)74 530 y(On)k(the)f(other)g(hand,)g(supp)q(ose)h(\()p Fr(\000)s Fw(\026)-26 b Fv(x;)10 b Fw(\026)-25 b Fv(y)q Fw(\))18 b(satisfy)g(\(1.1\).)26 b(Without)18 b(loss)g(of)g(generalit)o(y)h(w)o (e)e(ma)o(y)-1 586 y(assume)f Fr(k)s Fw(\026)-26 b Fv(x)o Fr(k)228 570 y Fs(2)258 586 y Fw(+)10 b Fr(k)s Fw(\026)-26 b Fv(y)r Fr(k)373 570 y Fs(2)405 586 y Fw(=)13 b(1.)19 b(It)d(then)f(holds)h(that)411 657 y Ft(")459 700 y Fw(\026)-26 b Fv(x)460 757 y Fw(\026)g Fv(y)502 657 y Ft(#)527 668 y Fu(T)562 729 y Fv(Q)606 657 y Ft(")653 700 y Fw(\026)g Fv(x)654 757 y Fw(\026)g Fv(y)697 657 y Ft(#)763 729 y Fw(=)42 b Fr(k)s Fw(\026)-26 b Fv(x)o Fr(k)911 710 y Fs(2)941 729 y Fw(+)10 b Fr(k)s Fw(\026)-26 b Fv(y)r Fr(k)1056 710 y Fs(2)1085 729 y Fw(+)1136 698 y(2)p 1136 718 V 1136 760 a Fv(\032)1167 729 y Fw(\026)g Fv(x)1190 710 y Fu(T)1218 729 y Fv(G)p Fw(\()p Fv(z)r Fw(\))s(\026)g Fv(y)-1336 b Fw(\(3.54\))763 856 y(=)42 b(1)9 b(+)923 825 y(2)p 923 846 V 923 887 a Fv(\032)954 856 y Fw(\026)-26 b Fv(x)977 837 y Fu(T)1005 856 y Fv(G)p Fw(\()p Fv(z)r Fw(\))s(\026)g Fv(y)14 b(<)f Fw(1)p Fv(;)-1221 b Fw(\(3.55\))-1 982 y(uniformly)13 b(in)e Fv(z)r(;)19 b Fr(k)p Fv(z)r Fr(k)12 b Fw(=)h(1.)18 b(Th)o(us)11 b(the)g(radius)h(of)f(the)g(set)g Fq(C)g Fw(is)g(strictly)h(greater)e(than)h(\()p Fr(k)s Fw(\026)-26 b Fv(x)o Fr(k)1579 965 y Fs(2)1601 982 y Fw(+)r Fr(k)s Fw(\026)g Fv(y)r Fr(k)1708 965 y Fs(2)1727 982 y Fw(\))12 b(=)-1 1038 y(1.)22 b(Since)c Fq(C)e Fw(is)g(hermitian)h (and)f(cen)o(tered)h(ab)q(out)f(the)g(origin,)h(the)f(diameter)g(is)h (precisely)g(t)o(wice)g(the)-1 1095 y(radius.)k(Hence,)16 b(the)f(diameter)g(of)g Fq(C)g Fw(is)h(strictly)f(greater)g(than)g(2.) 74 1151 y(The)g Fv(\032)p Fw('s)f(used)i(in)g(\(3.47\))d(ma)o(y)h(all)i (b)q(e)g(the)f(same)g(or)f(they)h(ma)o(y)g(b)q(e)g(c)o(hosen)h(to)e (dep)q(end)j(on)d Fv(z)r Fw(.)20 b(All)-1 1208 y(w)o(e)15 b(need)h(is)g(to)f(guaran)o(tee)f(that)575 1338 y Fv(Q)p Fw(\()p Fv(z)r Fw(\))e(=)730 1266 y Ft(")838 1304 y Fv(I)972 1287 y Fs(1)p 971 1294 19 2 v 971 1320 a Fu(\032)994 1304 y Fv(G)p Fw(\()p Fv(z)r Fw(\))780 1348 y Fs(1)p 780 1355 V 780 1382 a Fu(\032)803 1366 y Fv(G)839 1350 y Fu(T)866 1366 y Fw(\()p Fv(z)r Fw(\))91 b Fv(I)1109 1266 y Ft(#)1146 1338 y Fv(>)13 b Fw(0)-1218 b(\(3.56\))-1 1468 y(and)16 b(th)o(us)f(w)o(e)g(can)g(tak)o(e)g Fv(\032)f Fw(to)h(b)q(e)h Fv(z)r Fw(-dep)q(enden)o(t,)g(e.g.,)665 1568 y Fv(\032)c Fw(=)h Fv(\032)p Fw(\()p Fv(z)r Fw(\))e Fv(>)i(\033)917 1575 y Fs(max)985 1568 y Fw(\()p Fv(G)p Fw(\()p Fv(z)r Fw(\)\))p Fv(:)-1130 b Fw(\(3.57\))74 1668 y(Finding)16 b(a)d(di\013eren)o(t)h Fv(\032)g Fw(for)f(eac)o(h)h Fv(z)i Fw(is)f(a)e(lab)q(orious)i(task.)k(A)14 b(single)h(constan)o(t)f Fv(\032)f Fw(that)g(will)j(satisfy)-1 1725 y(\(3.45\))e(for)g(all)i Fv(z)i Fw(can)d(easily)h(b)q(e)g(computed)g(via)f(the)h(matrix)e (inequalit)o(y)508 1825 y Fv(\033)534 1832 y Fs(max)602 1825 y Fw(\()630 1813 y(\026)620 1825 y Fv(Q)o Fw(\))f Fr(\024)g Fv(\033)760 1832 y Fs(max)827 1825 y Fw(\(abs\()p Fv(Q)p Fw(\()p Fv(z)r Fw(\)\)\))e Fr(\024)i Fv(\033)1145 1832 y Fs(max)1213 1825 y Fw(\()p Fv(Q)p Fw(\))-1286 b(\(3.58\))-1 1925 y(where)16 b Fv(ij)s Fw(-th)e(en)o(try)h(of)g(the)g Fv(n)10 b Fr(\002)h Fv(m)k Fw(matrix)786 1914 y(\026)776 1925 y Fv(Q)g Fw(is)h(giv)o(en)f(b)o(y)719 2025 y([)742 2014 y(\026)732 2025 y Fv(Q)p Fw(])781 2032 y Fu(ij)824 2025 y Fw(=)e Fv(\033)898 2032 y Fs(max)965 2025 y Fw(\()p Fv(F)1012 2032 y Fu(ij)1042 2025 y Fw(\))p Fv(:)-1074 b Fw(\(3.59\))74 2125 y(In)16 b(view)g(of)f(Theorem)g(3.1,)e(the)j (follo)o(wing)g(statemen)o(t)e(is)i(ob)o(vious:)k(Consider)15 b(the)h(optimization)-1 2182 y(problem,)725 2238 y(max)743 2263 y Fu(x;y)832 2238 y Fr(k)p Fv(x)p Fr(k)904 2219 y Fs(2)933 2238 y Fw(+)11 b Fr(k)p Fv(y)r Fr(k)1049 2219 y Fs(2)-1 2238 y Fw(\(3.60\))-1 2335 y(sub)s(ject)16 b(to)305 2359 y Ft(")350 2403 y Fv(x)351 2460 y(y)397 2359 y Ft(#)421 2371 y Fu(T)456 2359 y Ft(")564 2398 y Fv(I)697 2380 y Fs(1)p 697 2387 V 697 2414 a Fu(\032)720 2398 y Fv(G)p Fw(\()p Fv(z)r Fw(\))506 2442 y Fs(1)p 506 2449 V 506 2475 a Fu(\032)529 2460 y Fv(G)565 2443 y Fu(T)592 2460 y Fw(\()p Fv(z)r Fw(\))91 b Fv(I)835 2359 y Ft(#)8 b(")912 2403 y Fv(x)913 2460 y(y)959 2359 y Ft(#)996 2431 y Fr(\024)13 b Fw(1)57 b Fr(8)p Fv(z)15 b Fr(2)e Fq(R)1267 2413 y Fu(p)1287 2431 y Fv(;)23 b Fr(k)p Fv(z)r Fr(k)12 b Fw(=)h(1)p Fv(:)-1489 b Fw(\(3.61\))-1 2547 y(Then,)18 b(there)f(exists)h(a)f(feasible)h(pair)g(\()p Fv(x;)8 b(y)r Fw(\))15 b(\(1.1\))h(if)h(and)h(only)f(if)h(the)f(global) h(optim)o(um)f(of)g(\(3.60\){)-1 2603 y(\(3.61\))h(is)h(strictly)h (greater)e(than)i(1;)g(note)f(that)g(since)h(all)g(p)q(oin)o(ts)g(of)f (the)g(form)f(\()p Fv(x;)8 b(y)r Fw(\))18 b(=)i(\()s(^)-26 b Fv(x)o(;)8 b Fw(0\))18 b(or)-1 2659 y(\()p Fv(x;)8 b(y)r Fw(\))14 b(=)i(\(0)p Fv(;)10 b Fw(^)-25 b Fv(y)q Fw(\))16 b(with)h Fr(k)s Fw(^)-26 b Fv(x)p Fr(k)15 b Fw(=)h Fr(k)s Fw(^)-26 b Fv(y)r Fr(k)15 b Fw(=)h(1)g(satisfy)h (\(3.61\),)e(the)i(optim)o(um)g(of)f(\(3.60\))f(is)i(alw)o(a)o(ys)g (greater)-1 2716 y(than)f(or)e(equal)i(to)f(1.)20 b(Actually)l(,)c (instead)g(of)f(solving)h(the)g(problem)g(\(3.60\))d(for)i(its)g (global)h(optim)o(um,)-1 2772 y(all)22 b(w)o(e)e(need)i(is)f(a)f(p)q (oin)o(t)h(\()p Fv(x;)8 b(y)r Fw(\))19 b(where)i Fv(J)t Fw(\()p Fv(x;)8 b(y)r Fw(\))21 b Fv(>)h Fw(1.)36 b(Note)20 b(that)g(\(3.60\){\(3.61\))d(is)k(a)f(nonlinear)p eop %%Page: 14 14 14 13 bop 157 -1 a Fw(14)99 b Fn(Mesbahi,)17 b(Saf)o(ono)o(v)g(and)g(P) l(ap)m(a)l(v)l(assilopoulos)157 107 y Fw(programming)g(problem)i(where) e(a)h(con)o(v)o(ex)f(function)i(is)f(to)f(b)q(e)h(maximized)h(sub)s (ject)f(to)f(an)h(in\014nite)157 164 y(n)o(um)o(b)q(er)g(of)f (quadratic)h(constrain)o(ts)f(\(parameterized)g(b)o(y)h Fv(z)r Fw(\))f(all)h(of)f(whic)o(h)i(are)e(ellipsoids)j(cen)o(tered)157 220 y(at)15 b(the)g(origin.)232 277 y(W)l(e)g(note)h(that)e(the)i(BMI)g (can)f(b)q(e)h(form)o(ulated)f(as)g(follo)o(ws:)21 b(Find)16 b(an)f Fv(n)c Fr(\002)f Fv(m)16 b Fw(matrix)f Fv(N)20 b Fw(of)15 b(rank)157 333 y(1)g(\(i.e.,)g Fv(N)i Fw(=)c Fv(xy)451 317 y Fu(T)478 333 y Fw(\))i(suc)o(h)g(that)g(for)g(all)h Fv(z)r Fw(,)f(with)g Fr(k)p Fv(z)r Fr(k)d Fw(=)h(1,)736 438 y Fq(T)l(r)o Fw(\()p Fv(G)p Fw(\()p Fv(z)r Fw(\))p Fv(N)5 b Fw(\))11 b(=)i Fv(x)1047 419 y Fu(T)1075 438 y Fv(G)p Fw(\()p Fv(z)r Fw(\))p Fv(y)g(<)g Fw(0;)-1132 b(\(3.62\))157 543 y(that)13 b(is,)h(in)g(the)g(Hilb)q(ert)h(space)e (of)g Fv(n)7 b Fr(\002)g Fv(m)14 b Fw(real)g(matrices)f(\014nd)h(a)f(h) o(yp)q(erplane)i(that)e(strictly)h(separates)157 599 y(the)20 b(origin)g(from)e(the)i(set)f Fq(W)i Fw(=)f Fr(f)p Fv(G)p Fw(\()p Fv(z)r Fw(\))e Fr(j)h(k)p Fv(z)r Fr(k)g Fw(=)h(1)p Fr(g)f Fw(and)h(in)g(addition)g(it)g(should)g(hold)h (that)d(the)157 656 y(matrix)d Fv(N)20 b Fw(that)15 b(de\014nes)i(the)f (p)q(erp)q(endicular)i(to)d(this)h(h)o(yp)q(erplane)h(is)f(of)f(rank)h (one.)21 b(In)16 b(the)g(absence)157 712 y(of)f(the)g(restriction)h (rank\()p Fv(N)5 b Fw(\))11 b(=)i(1,)i(the)g(problem)h(ma)o(y)e(b)q(e)i (in)o(terpreted)g(as)f(the)g(LMI,)954 817 y Fv(\017)972 824 y FA(\003)1004 817 y Fw(=)e(min)1062 846 y Fu(\017;N)1136 817 y Fv(\017)-997 b Fw(\(3.63\))157 941 y(sub)s(ject)15 b(to)566 1046 y Fq(T)l(r)p Fw(\()p Fv(N)5 b(G)p Fw(\()p Fv(z)r Fw(\)\))10 b Fr(\024)j Fv(\017;)83 b Fr(8)p Fv(z)r(;)23 b Fr(k)p Fv(z)r Fr(k)12 b Fw(=)h(1)-1044 b(\(3.64\))664 1115 y Fr(j)p Fw([)p Fv(N)5 b Fw(])745 1122 y Fu(ij)774 1115 y Fr(j)12 b(\024)h Fw(1)p Fv(;)82 b Fr(8)p Fv(i)13 b Fw(=)g(1)p Fv(;)8 b(:)g(:)g(:)t(;)g(n)p Fw(;)g Fv(j)14 b Fw(=)f(1)p Fv(;)8 b(:)g(:)g(:)t(;)g(m)p Fw(;)-1340 b(\(3.65\))157 1219 y(a)20 b(solution)h(exists)g(if)f(and)h(only)g(if)f (the)h(minimal)h(cost)d Fv(\017)1181 1226 y FA(\003)1223 1219 y Fv(<)i Fw(0.)35 b(If)20 b(the)h(rank)f(of)g Fv(N)k Fw(is)d(restricted)157 1276 y(to)e(b)q(e)i(less)f(than)f(min)q Fr(f)p Fv(m;)8 b(n)p Fr(g)p Fw(,)19 b(the)h(LMI)g(form)o(ulation)g (fails.)34 b(W)l(e)19 b(notice)i(that)e(the)g(\(3.63\){\(3.65\))157 1332 y(form)o(ulation)h(of)f(the)h(BMI)h(comes)e(remark)m(ably)i(close) f(to)g(an)f(LMI.)h(A)o(t)g(the)g(same)f(time)i(ho)o(w)o(ev)o(er,)157 1389 y(this)f(problem)g(form)o(ulation)f(signi\014es)i(the)e(role)h (that)e(rank)h(restricted)h(LMIs)g(pla)o(y)f(in)h(the)g(con)o(trol)157 1445 y(syn)o(thesis)c(problems)g(in)g(a)f(rather)f(transparen)o(t)g(w)o (a)o(y)l(.)232 1502 y(W)l(e)h(shift)h(our)f(atten)o(tion)f(no)o(w)h(to) g(the)g(cone)g(programming)g(form)o(ulations)g(of)g(the)g(BMI.)157 1627 y Fx(3.2)56 b(Cone)19 b(Programming)d(Approac)n(h)157 1692 y Fw(W)l(e)c(presen)o(t)h(an)f(approac)o(h)g(for)f(solving)i(the)g (BMI)f(based)h(on)f(its)g(connections)i(with)e(certain)h(problems)157 1749 y(de\014ned)19 b(o)o(v)o(er)e(matrix)g(cones.)27 b(These)17 b(problems)i(are,)e(among)g(others,)g(the)g(cone)h (generalization)h(of)157 1805 y(the)e(linear)g(programming)f(\(LP\))g (and)g(the)g(linear)i(complemen)o(tarit)o(y)e(problem)h(\(LCP\))f (\(referred)g(to)157 1862 y(as)c(the)h(Cone-LP)g(and)g(the)f(Cone-LCP)l (,)h(resp)q(ectiv)o(ely\).)20 b(Sp)q(eci\014cally)m(,)15 b(w)o(e)e(sho)o(w)f(that)f(solving)j(a)e(giv)o(en)157 1918 y(BMI)h(is)g(equiv)m(alen)o(t)h(to)d(examining)j(the)e(solution)i (set)e(of)g(a)g(suitably)h(constructed)g(Cone-LP)f(or)g(Cone-)157 1974 y(LCP)l(.)i(In)h(this)g(direction)g(w)o(e)f(end)h(up)f(addressing) h(all)g(of)f(the)g(issues)h(raised)g(ab)q(o)o(v)o(e)e(p)q(ertaining)j (to)d(the)157 2031 y(BMIs)j(and)f(global)h(optimization)g(tec)o (hniques.)232 2087 y(The)f(main)h(results)g(pro)o(v)o(ed)e(in)i(this)g 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Fw(=)e(0\))f(:)63 b Fq(T)l(r)o Fw(\()p Fv(F)1157 1573 y Fu(y)1151 1608 y(i)1178 1595 y Fv(Z)s Fw(\))13 b(=)g(0)p Fv(:)-1328 b Fw(\(3.67\))-1 1674 y(Therefore,)15 b(the)g(BMI)h(\(1.1\))d(has)i(a)g (solution)h(if)g(and)f(only)h(if,)422 1766 y(\()p Fr(9)p Fv(y)f Fr(\025)e Fw(0\))f(\()p Fr(8)p Fv(Z)k Fr(\025)d Fw(0)p Fv(;)8 b(Z)14 b Fr(6)p Fw(=)f(0\))f(:)1007 1725 y Ft(X)1031 1816 y Fu(i)1075 1766 y Fq(T)l(r)o Fw(\()p Fv(F)1181 1743 y Fu(y)1175 1778 y(i)1202 1766 y Fv(Z)s Fw(\))1254 1747 y Fs(2)1287 1766 y Fv(>)g Fw(0)p Fv(:)-1371 b Fw(\(3.68\))-1 1891 y(No)o(w)15 b(let,)268 1995 y Fv(F)297 2002 y Fu(i)324 1995 y Fw(=)e([)511 1923 y Fu(p)385 1951 y Ft(z)p 404 1951 98 5 v 98 w(}|)p 540 1951 V 98 w({)385 1995 y Fq(v)o(ec)h Fv(F)502 2002 y Fu(i)p Fs(1)541 1995 y Fw(0)8 b Fv(:)g(:)g(:)d Fw(0)p Fv(;)801 1923 y Fu(p)675 1951 y Ft(z)p 694 1951 V 98 w(}|)p 830 1951 V 98 w({)675 1995 y Fw(0)j Fq(v)o(ec)14 b Fv(F)823 2002 y Fu(i)p Fs(2)862 1995 y Fv(:)8 b(:)g(:)e Fw(0)p Fv(;)i(:)g(:)g(:)d(;)1173 1923 y Fu(p)1047 1951 y Ft(z)p 1066 1951 99 5 v 99 w(}|)p 1203 1951 V 99 w({)1047 1995 y Fw(0)j(0)g Fv(:)g(:)g(:)t Fq(v)o(ec)14 b Fv(F)1285 2002 y Fu(ip)1317 1995 y Fw(])f Fr(2)g Fq(R)1425 1976 y Fu(p)1443 1964 y Ff(2)1460 1976 y FA(\002)p Fu(p)1505 1964 y Ff(2)-1 1995 y Fw(\(3.69\))-1 2073 y(and)g Fv(Y)e Fw(=diag\()p Fv(y)r Fw(\))k Fr(2)e Fq(SR)426 2055 y Fu(p)p FA(\002)p Fu(p)491 2073 y Fw(.)19 b(Since)14 b(\()p Fv(F)692 2051 y Fu(y)686 2086 y(i)713 2073 y Fw(\))731 2057 y Fu(T)771 2073 y Fw(=)f Fv(F)854 2051 y Fu(y)848 2086 y(i)888 2073 y Fw(=)936 2041 y Ft(P)979 2085 y Fu(j)1005 2073 y Fv(y)1027 2080 y Fu(j)1046 2073 y Fv(F)1075 2080 y Fu(ij)1118 2073 y Fw(\(recalling)h(that)e Fv(F)1440 2080 y Fu(ij)1470 2073 y Fw('s)g(are)h(symmetric)-1 2130 y(matrices\),)668 2186 y Fq(v)o(ec)h Fw(\()p Fv(F)809 2164 y Fu(y)803 2199 y(i)830 2186 y Fw(\))848 2167 y Fu(T)888 2186 y Fw(=)f Fv(F)965 2193 y Fu(i)987 2186 y Fq(v)o(ec)h Fv(Y)d Fw(;)-1126 b(\(3.70\))-1 2264 y(thereb)o(y)l(,)533 2356 y 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2715 y Fu(T)578 2734 y Fr(f)601 2694 y Ft(X)625 2785 y Fu(i)668 2734 y Fv(F)697 2741 y Fu(i)712 2734 y Fw(\()p Fq(v)o(ec)g Fv(Y)c Fw(\)\()p Fq(v)o(ec)k Fv(Y)d Fw(\))1033 2715 y Fu(T)1060 2734 y Fv(F)1095 2715 y Fu(T)1089 2745 y(i)1123 2734 y Fr(g)d Fw(\()p Fq(v)o(ec)13 b Fv(Z)s Fw(\))g Fv(>)g Fw(0)p Fv(:)-1409 b Fw(\(3.72\))p eop %%Page: 16 16 16 15 bop 157 -1 a Fw(16)99 b Fn(Mesbahi,)17 b(Saf)o(ono)o(v)g(and)g(P) l(ap)m(a)l(v)l(assilopoulos)157 107 y Fw(Let)f Fv(X)f Fw(=)e(\()p Fq(v)o(ec)i Fv(Y)10 b Fw(\)\()p Fq(v)o(ec)k Fv(Y)c Fw(\))661 91 y Fu(T)689 107 y Fw(,)k Fv(Y)24 b Fr(2)12 b Fq(R)848 89 y Fu(p)p FA(\002)p Fu(p)913 107 y Fw(,)i(and,)847 197 y Fv(M)5 b Fw(\()p Fv(X)t Fw(\))11 b(=)1033 157 y Ft(X)1057 248 y Fu(i)1101 197 y Fv(F)1130 204 y Fu(i)1144 197 y Fv(X)t(F)1221 178 y Fu(T)1215 208 y(i)1248 197 y Fv(:)-1104 b Fw(\(3.73\))157 321 y(W)l(e)15 b(observ)o(e)g(the)h(follo)o(wing:)213 388 y(1.)22 b(According)28 b(to)f(\(3.69\),)h(for)e(all)j Fv(p)18 b Fr(\002)g Fv(p)27 b Fw(sk)o(ew-symmetric)h(matrices)f Fv(Z)s Fw(,)j Fv(F)1670 395 y Fu(i)1692 388 y Fq(v)o(ec)d Fv(Z)36 b Fw(=)d(0)271 445 y(\()p Fv(i)20 b Fw(=)i(1)p Fv(;)8 b(:)g(:)g(:)t(;)g(n)p Fw(\).)34 b(Consequen)o(tly)l(,)22 b(if)f(there)f(exists)h(a)f(matrix)g Fv(Y)31 b Fr(2)22 b Fq(R)1538 427 y Fu(p)p FA(\002)p Fu(p)1623 445 y Fw(suc)o(h)f(that)e(\(3.72\))271 501 y(holds,)c(then)g(one)g(can)f(assume)h(that)f Fv(Y)25 b Fw(is)15 b Fm(symmetric)p Fw(,)f(since)i Fm(the)g(skew-symmetric)g(p) n(art)g(of)g Fv(Y)271 557 y Fm(do)n(es)d(not)g(c)n(ontribute)g(to)g (the)g(left)g(hand)g(side)g(of)g(the)h(ine)n(quality)e(\(3.72\))p Fw(:)18 b(Let)12 b Fv(Y)23 b Fw(=)13 b Fv(Y)1729 564 y Fs(1)1752 557 y Fw(+)s Fv(Y)1816 564 y Fs(2)1837 557 y Fw(,)f(with)271 614 y Fv(Y)297 621 y Fs(1)330 614 y Fw(and)i Fv(Y)443 621 y Fs(2)476 614 y Fw(b)q(eing)h(the)f(symmetric)f (and)h(sk)o(ew-symmetric)f(part)g(of)g Fv(Y)e Fw(,)i(resp)q(ectiv)o (ely)l(.)21 b(Then)14 b(for)271 670 y(1)e Fr(\024)h Fv(i)f Fr(\024)h Fv(n)p Fw(,)i Fv(F)514 677 y Fu(i)536 670 y Fq(v)o(ec)g Fv(Y)23 b Fw(=)13 b Fv(F)751 677 y Fu(i)765 670 y Fw(\()p Fq(v)o(ec)h Fv(Y)897 677 y Fs(1)928 670 y Fw(+)c Fq(v)o(ec)15 b Fv(Y)1088 677 y Fs(2)1108 670 y Fw(\))d(=)h Fv(F)1215 677 y Fu(i)1237 670 y Fq(v)o(ec)h Fv(Y)1351 677 y Fs(1)1371 670 y Fw(.)213 760 y(2.)22 b(According)i(to)e(\(3.70\),)h(for)g(all)h(matrices)f Fv(Y)37 b Fr(2)26 b Fq(R)1209 742 y Fu(p)p FA(\002)p Fu(p)1274 760 y Fw(,)f Fv(F)1341 767 y Fu(i)1363 760 y Fq(v)o(ec)d Fv(Y)37 b Fw(=)26 b Fq(v)o(ec)c Fv(W)1722 767 y Fu(i)1736 760 y Fw(,)j(for)e(some)271 817 y Fv(W)314 824 y Fu(i)340 817 y Fr(2)13 b Fq(SR)452 799 y Fu(p)p FA(\002)p Fu(p)517 817 y Fw(.)19 b(Therefore)14 b(if)g Fv(X)i Fw(=)d(\()p Fq(v)o(ec)g Fv(Y)e Fw(\)\()p Fq(v)o(ec)h Fv(Y)f Fw(\))1215 800 y Fu(T)1242 817 y Fw(,)j(then)g Fv(M)5 b Fw(\()p Fv(X)t Fw(\))12 b(can)j(b)q(e)f(represen)o(ted)h(b)o (y)l(,)787 906 y Fv(M)5 b Fw(\()p Fv(X)t Fw(\))11 b(=)973 866 y Ft(X)997 957 y Fu(i)1033 906 y Fw(\()p Fq(v)o(ec)j Fv(W)1182 913 y Fu(i)1196 906 y Fw(\)\()p Fq(v)o(ec)g Fv(W)1363 913 y Fu(i)1377 906 y Fw(\))1395 888 y Fu(T)1422 906 y Fv(:)-1164 b Fw(\(3.74\))232 1047 y Fn(Remark)19 b(3.1.)e Fm(Supp)n(ose)g(that)h(the)f(ve)n(ctor)g Fv(y)i Fm(is)e(not)g(r)n(e)n(quir)n(e)n(d)f(to)i(b)n(e)e(nonne)n(gative)g(in)g (the)i(ab)n(ove)157 1103 y(analysis.)23 b(It)17 b(is)f(cle)n(ar)h(that) h(the)g(ab)n(ove)f(steps)f(ar)n(e)i(stil)r(l)e(valid)h(with)h(the)g (obvious)f(mo)n(di\014c)n(ations,)f(and)157 1160 y(that)f(the)g(end)g (r)n(esult)f(would)h(r)n(e)n(ad)g(as)f(fol)r(lows:)20 b(The)14 b(BMI)g(has)h(a)g(solution)f(if)g(and)h(only)f(if)h(ther)n(e)f (exists)157 1216 y(a)j(diagonal)g(matrix)h Fv(Y)10 b Fm(,)17 b(such)g(that)h(for)f(al)r(l)f Fv(Z)h Fr(\025)d Fw(0)p Fv(;)8 b(Z)16 b Fr(6)p Fw(=)e(0)p Fm(,)j(the)g(ine)n(quality)f (\(3.72\))h(holds.)23 b(We)17 b(shal)r(l)157 1273 y(use)f(this)g (observation)g(later)h(in)f Fr(x)p Fm(3.2.2.)232 1329 y Fw(The)22 b(inequalit)o(y)h(\(3.72\))d(can)h(b)q(e)i(in)o(terpreted)f (as)f(requiring)i Fv(M)5 b Fw(\()p Fv(X)t Fw(\))20 b(to)h(b)q(elong)i (to)e(a)g(certain)157 1386 y(matrix)d(class.)30 b(The)19 b(matrices)f(in)h(this)g(class)g(are)f(symmetric)g(\(giv)o(en)h(that)f Fv(X)j Fw(is)e(symmetric\))f(and)157 1442 y(ha)o(v)o(e)j(quadratic)g (forms)f(whic)o(h)i(are)e(p)q(ositiv)o(e)i(o)o(v)o(er)e(the)h Fq(v)o(ec)f Fw(form)g(of)h(the)g(non-zero)g(matrices)g(in)157 1499 y Fq(SR)225 1477 y Fu(p)p FA(\002)p Fu(p)225 1510 y Fs(+)290 1499 y Fw(.)f(This)c(observ)m(ation)g(justi\014es)g(the)f (in)o(tro)q(duction)h(of)f(the)g(follo)o(wing)h(matrix)f(classes.)232 1555 y(Denote)k(b)o(y)h Fr(P)t(S)s(D)q Fw(,)g(the)f(class)h(of)f Fv(p)868 1539 y Fs(2)901 1555 y Fr(\002)13 b Fv(p)972 1539 y Fs(2)1011 1555 y Fw(symmetric)20 b(p)q(ositiv)o(e)h (semi-de\014nite)g(matrices,)g(i.e.,)157 1612 y(matrices)d(for)f(the)h (whic)o(h)h(the)f(quadratic)g(form)f(is)h(nonnegativ)o(e)g(o)o(v)o(er)f (the)h Fq(v)o(ec)f Fw(form)g(of)h(the)g Fv(p)11 b Fr(\002)i Fv(p)157 1668 y Fw(matrices)i(\()p Fq(R)396 1650 y Fu(p)p FA(\002)p Fu(p)461 1668 y Fw(\),)450 1764 y Fr(P)t(S)s(D)e Fw(=)g Fr(f)p Fv(A)g Fr(2)g Fq(SR)794 1745 y Fu(p)812 1733 y Ff(2)829 1745 y FA(\002)p Fu(p)874 1733 y Ff(2)906 1764 y Fw(:)g(\()p Fq(v)o(ec)h Fv(Z)s Fw(\))1090 1745 y Fu(T)1125 1764 y Fv(A)8 b Fw(\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\))e Fr(\025)h Fw(0;)8 b Fv(Z)15 b Fr(2)e Fq(R)1557 1745 y Fu(p)p FA(\002)p Fu(p)1622 1764 y Fr(g)p Fv(:)-1501 b Fw(\(3.75\))157 1853 y(Let)11 b Fr(P)t(S)s(D)337 1860 y Fs(0)367 1853 y Fw(denote)g(the)g(subset)g(of)g Fv(p)789 1837 y Fs(2)809 1853 y Fr(\002)q Fv(p)868 1837 y Fs(2)899 1853 y Fw(symmetric)g(matrices)g(with)g(quadratic)g(forms)f(nonnegativ) o(e)157 1910 y(o)o(v)o(er)k(the)h Fq(v)o(ec)f Fw(form)h(of)f(the)h (symmetric)g Fv(p)9 b Fr(\002)h Fv(p)15 b Fw(matrices)g(\()p Fq(SR)1261 1892 y Fu(p)p FA(\002)p Fu(p)1326 1910 y Fw(\),)f(and)h (with)h(the)f Fq(v)o(ec)f Fw(form)g(of)g(the)157 1966 y(sk)o(ew-symmetric)h(matrices)h(\()p Fq(SKR)796 1948 y Fu(p)p FA(\002)p Fu(p)861 1966 y Fw(\))f(in)h(their)g(n)o(ull)h (space,)e(i.e.,)390 2062 y Fr(P)t(S)s(D)493 2069 y Fs(0)525 2062 y Fw(=)e Fr(f)p Fv(A)g Fr(2)g Fq(SR)754 2043 y Fu(p)772 2031 y Ff(2)789 2043 y FA(\002)p Fu(p)834 2031 y Ff(2)895 2062 y Fw(:)41 b(\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\))1107 2043 y Fu(T)1142 2062 y Fv(A)8 b Fw(\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\))f Fr(\025)g Fw(0;)8 b Fv(Z)15 b Fr(2)e Fq(SR)1604 2043 y Fu(p)p FA(\002)p Fu(p)1669 2062 y Fw(;)949 2131 y Fv(A)p Fw(\()p Fq(v)o(ec)i Fv(W)6 b Fw(\))12 b(=)h(0;)20 b(for)14 b(all)g Fv(W)19 b Fr(2)13 b Fq(SKR)1617 2112 y Fu(p)p FA(\002)p Fu(p)1682 2131 y Fr(g)p Fv(:)-1561 b Fw(\(3.76\))157 2221 y(Clearly)18 b(b)q(oth)g Fr(P)t(S)s(D)g Fw(\(3.75\))d(and)j Fr(P)t(S)s(D)874 2228 y Fs(0)911 2221 y Fw(\(3.76\))d(are)i(closed)h(con)o(v)o(ex)g(cones.)26 b(Moreo)o(v)o(er,)17 b(it)g(can)h(b)q(e)157 2277 y(sho)o(wn)f(that)g (certain)g(essen)o(tial)h(features)f(of)g(the)h Fr(P)t(S)s(D)f Fw(cone)h(can)f(b)q(e)h(generalized)h(for)e(the)g(class)h(of)157 2333 y Fr(P)t(S)s(D)260 2340 y Fs(0)291 2333 y Fw(matrices,)12 b(including)j(rank)c(one)h(decomp)q(osition)h(prop)q(ert)o(y)l(,)f (self-dualit)o(y)l(,)i(and)e(the)f(unit)o(y)i(rank)157 2390 y(of)i(the)g(extreme)g(forms)g([24)o(],)f([39)o(].)232 2446 y(Let)h Fr(C)j Fw(denote)e(the)f(class)g(of)g(symmetric)g(PSD-cop) q(ositiv)o(e)i(matrices,)520 2536 y Fr(C)e Fw(=)e Fr(f)p Fv(A)g Fr(2)g Fq(SR)788 2517 y Fu(p)806 2506 y Ff(2)823 2517 y FA(\002)p Fu(p)868 2506 y Ff(2)900 2536 y Fw(:)f(\()p Fq(v)o(ec)j Fv(Z)s Fw(\))1084 2517 y Fu(T)1119 2536 y Fv(A)8 b Fw(\()p Fq(v)o(ec)13 b Fv(Z)s Fw(\))g Fr(\025)g Fw(0;)20 b Fv(Z)15 b Fr(\025)e Fw(0)p Fr(g)p Fv(:)-1431 b Fw(\(3.77\))157 2626 y(A)14 b(particular)h(subset)g(of)f Fr(C)i Fw(whic)o(h)f(b)q(e)g(useful)h(in)f(our)f(cone)g(form)o (ulations)g(is)h(a)f(subset)h(of)e(matrices)i(in)157 2683 y Fr(C)j Fw(whic)o(h)e(ha)o(v)o(e)f(the)g Fv(p)10 b Fr(\002)g Fv(p)15 b Fw(sk)o(ew-symmetric)h(matrices)f(in)h(their)g(n) o(ull)g(space,)f(i.e.,)602 2772 y Fr(C)626 2779 y Fs(0)658 2772 y Fw(:=)e Fr(f)p Fv(A)f Fr(2)h(C)i Fw(:)e Fv(A)8 b Fq(v)o(ec)14 b Fv(W)19 b Fw(=)13 b(0;)19 b Fv(W)g Fr(2)13 b Fq(SKR)1405 2754 y Fu(p)p FA(\002)p Fu(p)1470 2772 y Fr(g)p Fv(:)-1349 b Fw(\(3.78\))p eop %%Page: 17 17 17 16 bop 402 -1 a Fn(Bilinearity)17 b(and)g(Complement)m(arity)h(in)f (R)o(obust)g(Contr)o(ol)99 b Fw(17)-1 107 y(Finally)l(,)17 b(let)f Fr(B)g Fw(denote)f(the)h(class)f(of)g(symmetric)g (PSD-completely)i(p)q(ositiv)o(e)f(matrices,)279 235 y Fr(B)e Fw(=)f Fr(f)p Fv(A)f Fr(2)h Fq(SR)552 216 y Fu(p)570 204 y Ff(2)587 216 y FA(\002)p Fu(p)632 204 y Ff(2)664 235 y Fw(:)f Fv(A)h Fw(=)807 182 y Fu(t)784 194 y Ft(X)785 286 y Fu(i)p Fs(=1)844 235 y Fw(\()p Fq(v)o(ec)h Fv(Z)981 242 y Fu(i)995 235 y Fw(\)\()p Fq(v)o(ec)g Fv(Z)1150 242 y Fu(i)1164 235 y Fw(\))1182 216 y Fu(T)1209 235 y Fw(;)8 b Fv(Z)1261 242 y Fu(i)1287 235 y Fr(\025)13 b Fw(0)p Fv(;)8 b(t)k Fr(\025)h Fw(1)p Fr(g)p Fv(:)-1515 b Fw(\(3.79\))-1 362 y(One)16 b(can)g(establish)g(the)f(follo)o(wing)h (results.)74 419 y Fn(Lemma)d(3.1)g(\([39]\).)f Fm(The)g(matrix)h (classes)e Fr(B)q Fm(,)j Fr(C)s Fm(,)f(and)g Fr(C)1062 426 y Fs(0)1094 419 y Fm(ar)n(e)f(close)n(d)g(c)n(onvex)g(c)n(ones)g (in)g Fq(SR)1679 401 y Fu(p)1697 389 y Ff(2)1714 401 y FA(\002)p Fu(p)1759 389 y Ff(2)1779 419 y Fm(.)-1 475 y(Mor)n(e)n(over,)k Fr(B)244 459 y FA(\003)277 475 y Fw(=)d Fr(C)s Fm(,)j Fr(C)409 459 y FA(\003)441 475 y Fw(=)d Fr(B)q Fm(,)j Fr(C)j Fm(is)d(solid,)f(and)i Fr(B)g Fm(is)f(p)n(ointe)n(d.)74 532 y Fn(Lemma)i(3.2)e(\([39]\).)g Fm(The)g(extr)n(eme)g(forms)h(of)f Fr(B)h Fm(ar)n(e)g(matric)n(es)f Fw(\()p Fq(v)o(ec)f Fv(Z)s Fw(\)\()p Fq(v)o(ec)g Fv(Z)s Fw(\))1546 515 y Fu(T)1574 532 y Fm(,)h Fv(Z)g Fr(\025)d Fw(0)p Fm(.)-1 588 y Fw(Note)i(that)g Fr(B)f(\022)f(P)t(S)s(D)402 595 y Fs(0)435 588 y Fr(\022)g(P)t(S)s(D)g(\022)g(C)s Fw(,)h(and)i Fr(P)t(S)s(D)891 595 y Fs(0)923 588 y Fr(\022)d(C)995 595 y Fs(0)1028 588 y Fr(\022)g(C)s Fw(.)74 645 y(It)22 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956 y Fw(\()p Fv(M)5 b Fw(\),)13 b(where)i Fv(M)20 b Fw(is)15 b(de\014ned)h(b)o(y)f (\(3.73\),)-1 1012 y(and)h Fr(B)g Fw(is)g(the)f(cone)h(of)f (PSD-completely)h(p)q(ositiv)o(e)g(matrices.)74 1069 y Fm(Pr)n(o)n(of.)24 b Fw([39)o(])g(If)g(the)h(BMI)f(has)g(a)g (solution)g Fv(X)t Fw(,)h(then)g(there)f(exists)g Fv(Y)38 b Fw(=)25 b(diag\()p Fv(y)r Fw(\),)g Fv(y)30 b Fr(\025)d Fw(0,)-1 1125 y Fv(X)c Fw(=)d(\()p Fq(v)o(ec)e Fv(Y)11 b Fw(\)\()p Fq(v)o(ec)18 b Fv(Y)10 b Fw(\))444 1109 y Fu(T)472 1125 y Fw(,)20 b(suc)o(h)f(that)g Fv(M)5 b Fw(\()p Fv(X)t Fw(\))18 b Fr(2)39 b Fw(in)o(t)28 b Fr(C)s Fw(,)19 b(and)h(hence,)h(the)e(EFP)1480 1132 y FA(B)1506 1125 y Fw(\()p Fv(M)5 b Fw(\))19 b(has)g(a)g(so-)-1 1182 y(lution.)74 1295 y(Con)o(v)o(ersely)l(,)g(supp)q(ose)g(that)f(the)h(EFP)767 1302 y FA(B)792 1295 y Fw(\()p Fv(M)5 b Fw(\))18 b(has)g(a)g(solution)i Fv(X)t Fw(.)28 b(Then)19 b(there)g(exists)f Fv(V)28 b Fr(\025)18 b Fw(0,)-1 1351 y(suc)o(h)e(that)e Fv(X)i Fw(=)d(\()p Fq(v)o(ec)o Fv(V)d 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y(W)l(e)15 b(observ)o(e)g(that,)151 1842 y(\()p Fq(v)o(ec)f Fv(Z)s Fw(\))309 1824 y Fu(T)344 1842 y Fr(f)367 1802 y Ft(X)391 1893 y Fu(i)435 1842 y Fv(F)464 1849 y Fu(i)478 1842 y Fw(\()p Fq(v)o(ec)g Fv(Y)d Fw(\)\()p Fq(v)o(ec)n Fv(Y)g Fw(\))784 1824 y Fu(T)811 1842 y Fv(F)846 1824 y Fu(T)840 1854 y(i)874 1842 y Fr(g)d Fw(\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\))234 1959 y(=)f(\()p Fq(v)o(ec)h Fv(Y)d Fw(\))443 1940 y Fu(T)478 1959 y Fr(f)501 1918 y Ft(X)524 2009 y Fu(i)568 1959 y Fv(F)597 1966 y Fu(i)611 1959 y Fw(\()p Fq(v)o(ec)j Fv(Z)s Fw(\)\()p Fq(v)o(ec)g Fv(Z)s Fw(\))927 1940 y Fu(T)955 1959 y Fv(F)990 1940 y Fu(T)984 1970 y(i)1018 1959 y Fr(g)8 b Fw(\()p Fq(v)o(ec)13 b Fv(Y)e Fw(\))234 2075 y(=)i(\()p Fq(v)o(ec)h Fv(V)c Fw(\))443 2056 y Fu(T)470 2075 y Fw(\()p Fv(T)16 b Fr(\012)10 b Fv(T)c Fw(\))627 2056 y Fu(T)662 2075 y Fr(f)685 2034 y Ft(X)709 2125 y Fu(i)752 2075 y Fv(F)781 2082 y Fu(i)796 2075 y Fw(\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\)\()p Fq(v)o(ec)g Fv(Z)s Fw(\))1112 2056 y Fu(T)1139 2075 y Fv(F)1174 2056 y Fu(T)1168 2086 y(i)1202 2075 y Fr(g)8 b Fw(\()p Fv(T)15 b Fr(\012)c Fv(T)6 b Fw(\)\()p Fq(v)o(ec)13 b Fv(V)d Fw(\))234 2191 y(=)j(\(\()p Fv(T)j Fr(\012)10 b Fv(T)c Fw(\)\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\)\))633 2172 y Fu(T)667 2191 y Fr(f)690 2150 y Ft(X)714 2241 y Fu(i)758 2191 y Fv(F)787 2198 y Fu(i)801 2191 y Fw(\()p Fq(v)o(ec)g Fv(V)c Fw(\)\()p Fq(v)o(ec)k Fv(V)9 b Fw(\))1122 2172 y Fu(T)1149 2191 y Fv(F)1184 2172 y Fu(T)1178 2202 y(i)1213 2191 y Fr(g)f Fw(\()p Fv(T)14 b Fr(\012)d Fv(T)6 b Fw(\)\()p Fq(v)o(ec)14 b Fv(Z)s Fw(\))e Fv(>)h Fw(0)-1 2321 y(The)k(last)f (inequalit)o(y)h(follo)o(ws)g(from)e(the)h(fact)f(that)h(if)g Fq(v)o(ec)g Fv(W)k Fw(=)14 b(\()p Fv(T)j Fr(\012)11 b Fv(T)6 b Fw(\))p Fq(v)o(ec)15 b Fv(Z)s Fw(,)h Fv(Z)h Fr(\025)d Fw(0,)i(and)g Fv(T)22 b Fw(is)-1 2377 y(nonsingular,)16 b(then)g Fv(W)j Fr(\025)13 b Fw(0,)h(since)j Fv(W)i Fw(=)12 b Fv(T)6 b(Z)s(T)842 2361 y Fu(T)870 2377 y Fw(.)74 2490 y(Therefore)14 b(the)g(diagonal)g(matrix)g Fv(Y)c 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Fw(in)o(t)24 b Fr(C)s Fm(.)e(This)-1 2772 y(observation)16 b(shal)r(l)g(b)n(e)g(use) n(d)g(in)g(Se)n(ction)f(3.2.3.)p eop %%Page: 18 18 18 17 bop 157 -1 a Fw(18)99 b Fn(Mesbahi,)17 b(Saf)o(ono)o(v)g(and)g(P) l(ap)m(a)l(v)l(assilopoulos)232 107 y Fw(The)22 b(implication)i(of)d (Theorem)g(3.2)g(is)h(that)f(the)h(BMI)f(is)h(equiv)m(alen)o(t)i(to)d (c)o(hec)o(king)h(whether)157 164 y(the)e(image)h(of)e(an)h(extreme)g (form)g(of)f(the)h(matrix)g(cone)h Fr(B)g Fw(under)g(the)f(linear)h (map)f Fv(M)25 b Fw(\(whic)o(h)c(is)157 220 y(constructed)15 b(from)e(the)h(original)i(data)d(of)h(the)g(BMI\),)g(is)h(in)g(the)f (in)o(terior)h(of)e(the)i(dual)g(cone)f Fr(B)1809 204 y FA(\003)1829 220 y Fw(.)20 b(This)157 277 y(equiv)m(alence)h(th)o(us) d(pro)o(vides)h(a)f(rather)g(simple)i(geometric)e(in)o(terpretation)h (of)f(the)g(BMI)h(feasibilit)o(y)157 333 y(problem.)232 390 y(An)c(immediate)h(consequence)h(of)d(the)i(EFP)e(form)o(ulation)h (is)h(the)f(follo)o(wing)h(c)o(haracterization)f(of)157 446 y(the)g(BMI)h(instances)g(for)e(whic)o(h)i(a)f(solution)h(exists.) 232 503 y Fn(Cor)o(ollar)m(y)24 b(3.1)f(\([39]\).)f Fm(The)f(BMI)g(has) h(a)g(solution)f(if)h(the)f(line)n(ar)g(map)i Fv(M)j Fm(\(3.73\))c(is)f Fr(B)q Fm(-)157 559 y(p)n(ositive.)157 615 y(Pr)n(o)n(of.)d Fw([39)o(])e(If)i Fv(M)k Fw(is)17 b Fr(B)q Fw(-p)q(ositiv)o(e,)i(then)e(ev)o(ery)g(\(non-zero\))g (extreme)g(form)f(of)g Fr(B)j Fw(is)e(mapp)q(ed)h(to)e(the)157 672 y(in)o(terior)g(of)f Fr(B)401 655 y FA(\003)433 672 y Fr(\021)e(C)s Fw(.)20 b(Therefore)15 b(the)g(EFP)917 679 y FA(B)942 672 y Fw(\()p Fv(M)5 b Fw(\),)15 b(and)g(consequen)o (tly)h(the)f(BMI,)h(ha)o(v)o(e)e(a)h(solution.)157 1022 y Fq(3.2.3)52 b(Cone-LCP)23 b(F)l(orm)o(ulation)g(of)f(the)h(BMI.)i Fw(In)20 b(this)g(section)h(w)o(e)e(explore)i(the)e(idea)i(of)157 1079 y(establishing)12 b(a)e(connection)i(b)q(et)o(w)o(een)f(the)f(BMI) h(and)f(the)h(class)g(of)f(linear)h(complemen)o(tarit)o(y)g(problems) 157 1135 y(o)o(v)o(er)i(matrix)h(cones)g(\(Cone-LCP\).)g(The)g(EFP)f (form)o(ulation)h(of)g(the)g(BMI)g(discussed)i(in)f Fr(x)p Fw(3.2.2)d(is)j(our)157 1192 y(main)h(to)q(ol)f(in)h(this)g(direction.) 21 b(In)16 b(particular)g(w)o(e)e(shall)j(pro)o(vide)e(a)g(pro)q(of)g (of)g(Theorem)g(3.3.)232 1248 y(Our)21 b(motiv)m(ation)h(for)e(the)h (Cone-LCP)h(approac)o(h)e(is)i(t)o(w)o(ofold.)36 b(First,)22 b(the)f(complemen)o(tarit)o(y)157 1305 y(form)o(ulation)28 b(enables)h(one)f(to)f(address)h(the)g(t)o(w)o(o)f(imp)q(ortan)o(t)g (structural)h(issues)g(\(2\))f(and)h(\(3\))157 1361 y(men)o(tioned)19 b(in)g(the)f(op)q(ening)h(paragraph)e(of)g Fr(x)p Fw(3.)28 b(The)18 b(basic)h(idea)f(is)h(to)e(use)h(the)g(ric)o(h)h(theory)e (that)157 1417 y(has)22 b(b)q(een)h(dev)o(elop)q(ed)h(for)d(the)h (linear)i(complemen)o(tarit)o(y)e(problems)h(o)o(v)o(er)e(the)h(last)g (few)g(decades)157 1474 y(to)f(examine)g(the)g(prop)q(erties)h(of)f (the)g(BMI.)g(Additionally)l(,)j(w)o(e)d(shall)h(sho)o(w)f(that)f(the)h (Cone-LCP)157 1530 y(that)j(arises)g(from)g(the)g(BMI)h(can)f(b)q(e)h (form)o(ulated)f(on)h(the)f Fr(P)t(S)s(D)h Fw(cone,)h(rather)e(than)g (the)h(less)157 1587 y(\(computationally\))e(understo)q(o)q(d)h(matrix) e(cone)i Fr(B)g Fw(of)e Fr(x)p Fw(3.2.2.)42 b(The)23 b(complemen)o(tarit)o(y)g(problem)157 1643 y(o)o(v)o(er)i(the)g Fr(P)t(S)s(D)h Fw(cone)f(also)g(has)h(the)f(adv)m(an)o(tage)f(of)h(b)q (eing)i(readily)f(amenable)g(to)f(an)g(in)o(terior)157 1700 y(p)q(oin)o(t)20 b(approac)o(h)e([30)o(].)31 b(Ho)o(w)o(ev)o(er,) 19 b(one)g(has)g(to)f(note)h(that)f(a)h(\\general")g(Cone-LCP)g(is)g(a) g(di\016cult)157 1756 y(computational)14 b(problem,)g(as)f(is)h(the)g (LCP)f(itself.)21 b(One)14 b(of)f(the)h(main)g(adv)m(an)o(tages)e(of)i (the)f(Cone-LCP)157 1813 y(form)o(ulation)i(ho)o(w)o(ev)o(er,)f(is)i (the)f(abilit)o(y)i(of)d(recognizing)j(e\016cien)o(tly)f(solv)m(able)h (instances.)232 1869 y(In)25 b(this)g(section)f(w)o(e)g(shall)i(assume) 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y Fs(1)1856 2208 y Fw(.)29 b(As)157 2264 y(the)21 b(next)f(prop)q (osition)h(states,)f(the)g(cone)h Fr(P)t(S)s(D)g Fw(can)f(substitute)h (the)f(cone)h Fr(P)t(S)s(D)1655 2271 y Fs(0)1695 2264 y Fw(in)g(the)g(ab)q(o)o(v)o(e)157 2321 y(statemen)o(t.)232 2377 y Fn(Pr)o(oposition)d(3.2)i(\([39)o(]\).)f Fm(Ther)n(e)e(exists)h (an)g(extr)n(eme)g(form)h(of)f(the)h(matrix)f(c)n(one)g Fr(P)t(S)s(D)q Fm(,)g Fv(X)t Fm(,)157 2434 y(such)i(that)h Fv(M)5 b Fw(\()p Fv(X)t Fw(\))18 b Fr(2)h Fw(in)o(t)d Fr(C)s Fm(,)k(if)g(and)g(only)f(if)h(ther)n(e)g(exists)f(an)h(extr)n (eme)g(form)g(of)g(the)h(matrix)f(c)n(one)157 2490 y Fr(P)t(S)s(D)260 2497 y Fs(0)280 2490 y Fm(,)c Fv(Y)10 b Fm(,)16 b(such)h(that)g Fv(M)5 b Fw(\()p Fv(Y)10 b Fw(\))i Fr(2)h Fw(in)o(t)j Fr(C)s Fm(.)157 2547 y Fw(Let)h(us)g(denote) g(b)o(y)j(\026)-27 b Fv(p)15 b Fw(=)g Fv(p)p Fw(\()p Fv(p)c Fw(+)g(1\))p Fv(=)p Fw(2)16 b(the)g(dimension)i(of)f(the)f (space)h(of)f(symmetric)h Fv(p)11 b Fr(\002)g Fv(p)17 b Fw(matrices.)157 2603 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b(Saf)o(ono)o(v)g(and)g(P) l(ap)m(a)l(v)l(assilopoulos)184 107 y Fy([1])k(B.)12 b(D.)f(O.)g(Anderson)h(and)g(S.)f(V)m(ongpanitlerd,)f Fb(Network)i(A)o(nalysis)g(and)i(Synthesis)f(-)g(A)f(Mo)n(dern)h (Systems)250 157 y(The)n(ory)i(Appr)n(o)n(ach)p Fy(,)f(Pren)o (tice-Hall,)f(Englew)o(o)q(o)q(d)h(Cli\013s,)f(NJ,)h(1973.)184 207 y([2])21 b(G.)d(J.)f(Balas,)i(J.)f(C.)35 b(Do)o(yle,)18 b(K.)g(Glo)o(v)o(er,)g(A.)f(P)o(ac)o(k)n(ard,)i(and)f(R.)f(Smith,)g Fj(\026)p Fb(-Analysis)h(and)i(Synthesis)250 257 y(T)m(o)n(olb)n(ox)15 b(\()p Fj(\026)p Fb(-To)n(ols\))p Fy(,)e(Math)o(w)o(orks,)g(Natic)o(k,) g(MA,)h(1991.)184 307 y([3])21 b(E.)15 b(Beran,)g(L.)f(V)m(anderb)q (erghe,)i(and)e(S.)g(P)m(.)g(Bo)o(yd,)h Fb(A)g(glob)n(al)g(BMI)h (algorithm)f(b)n(ase)n(d)h(on)g(the)g(gener)n(alize)n(d)250 356 y(Benders)g(De)n(c)n(omp)n(osition)p Fy(,)e(Pro)q(ceedings)h(of)e (the)i(Europ)q(ean)f(Con)o(trol)g(Conference,)h(1997.)184 406 y([4])21 b(A.)14 b(Berman,)f Fb(Cones,)i(Matric)n(es,)f(and)h 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bop 402 -1 a Fn(Bilinearity)17 b(and)g(Complement)m(arity)h(in)f (R)o(obust)g(Contr)o(ol)99 b Fw(25)93 107 y Fb(c)n(onditions)20 b(and)g(state-sp)n(ac)n(e)f(formulas)p Fy(,)g(Pro)q(ceedings)i(of)d (the)h(American)g(Con)o(trol)f(Conference,)j(San)93 157 y(F)m(rancisco,)13 b(CA,)h(June)h(1993.)5 207 y([30])22 b(M.)k(Ko)r(jima,)i(S.)e(Shindoh,)j(and)e(S.)f(Hara,)k Fb(Interior-p)n(oint)c(metho)n(ds)h(for)f(the)h(monotone)h(line)n(ar)93 257 y(c)n(omplementarity)13 b(pr)n(oblem)f(in)i(symmetric)e(matric)n (es)p Fy(,)f(T)m(ec)o(hnical)h(rep)q(ort,)h(Departmen)o(t)f(of)f (Information)93 307 y(Science,)k(T)m(oky)o(o)d(Institute)j(of)e(T)m(ec) o(hnology)m(,)f(Japan,)h(1994.)5 356 y([31])22 b(F.)9 b(C.)f(Lee,)j(H.)e(Flashner,)h(and)f(M.)g(G.)g(Safono)o(v,)f Fb(Positivity)i(emb)n(e)n(dding)i(for)e(nonc)n(olo)n(c)n(ate)n(d)h(and) h(nonsquar)n(e)93 406 y(\015exible)j(systems)p Fy(,)e(Applied)h (Mathematics)f(and)h(Computation,)d(70)i(\(2)h(&)g(3\))g(\(1995\),)e (pp.)i(233{246.)5 456 y([32])22 b(F.)h(C.)g(Lee,)j(H.)d(Flashner,)j (and)d(M.)g(G.)f(Safono)o(v,)i Fb(A)o(n)g(LMI)g(appr)n(o)n(ach)g(to)g (p)n(ositivity)f(emb)n(e)n(dding)p Fy(,)93 506 y(Pro)q(ceedings)15 b(of)e(the)i(American)e(Con)o(trol)g(Conference,)i(Seattle,)f(W)-5 b(A,)14 b(June)g(1995.)5 556 y([33])22 b(F.)c(C.)f(Lee,)j(H.)e (Flashner,)h(and)f(M.)g(G.)f(Safono)o(v,)h Fb(Positivity-b)n(ase)n(d)g (c)n(ontr)n(ol)g(system)h(synthesis)g(using)93 606 y(alternating)14 b(LMIs)g Fy(Pro)q(ceedings)i(of)d(the)i(American)e(Con)o(trol)g (Conference,)i(Seattle,)f(W)-5 b(A,)13 b(June)i(1995.)5 655 y([34])22 b(S-M.)9 b(Liu,)h Fb(Par)n(al)r(lel)g(A)o(lgorithms)h (for)f(Glob)n(al)h(Non-Convex)i(Minimization)e(with)g(Applic)n(ations)g (to)g(R)n(obust)93 705 y(Contr)n(ol)j(Design)h(via)g(BMIs)p Fy(,)f(Ph.D.)f(Dissertation,)g(USC,)h(August)g(1995.)5 755 y([35])22 b(S-M.)9 b(Liu)h(and)g(G.)g(P)m(.)f(P)o(apa)o(v)n (assilop)q(oulos,)g Fb(Numeric)n(al)i(exp)n(erienc)n(e)g(with)g(p)n(ar) n(al)r(lel)g(algorithms)g(for)g(solving)93 805 y(the)k(BMI)g(pr)n (oblem)p Fy(,)e(Pro)q(ceedings)i(of)f(the)g(IF)-5 b(A)o(C)14 b(W)m(orld)f(Congress,)h(July)g(1996.)5 855 y([36])22 b(J.)14 b(H.)g(Ly)m(,)f(M.)h(G.)f(Safono)o(v,)f(and)i(R.)g(Y.)g (Chiang,)e Fb(On)k(c)n(omputation)g(of)f(multivariable)f(stability)g (mar)n(gin)93 904 y(using)20 b(gener)n(alize)n(d)f(Pop)n(ov)h (multipliers)f(|)h(LMI)f(appr)n(o)n(ach)p Fy(,)i(Pro)q(ceeding)f(of)f (the)g(American)f(Con)o(trol)93 954 y(Conference,)d(July)e(1994.)5 1004 y([37])22 b(J.)i(H.)g(Ly)m(,)h(K.)f(C.)g(Goh,)h(and)f(M.)g(G.)f (Safono)o(v,)i Fb(LMI)g(appr)n(o)n(ach)g(to)f(multiplier)f Fj(k)1528 1010 y Fh(m)1559 1004 y Fj(=\026)p Fb(-synthesis)p Fy(,)93 1054 y(Pro)q(ceedings)15 b(of)e(the)i(American)e(Con)o(trol)g (Conference,)i(Seattle,)f(W)-5 b(A,)14 b(June)g(1995.)5 1104 y([38])22 b(J.)13 b(H.)h(Ly)m(,)f Fb(A)h(Multiplier)g(Appr)n(o)n (ach)h(to)g Fj(\026)p Fb(/)p Fj(k)806 1110 y Fh(m)852 1104 y Fb(Synthesis)p Fy(,)f(Ph.D.)f(Dissertation,)h(USC,)f(F)m (ebruary)h(1995.)5 1153 y([39])22 b(M.)11 b(Mesbahi)g(and)h(G.)e(P)m(.) 22 b(P)o(apa)o(v)n(assilop)q(oulos,)10 b Fb(A)i(c)n(one)i(pr)n(o)n(gr)n (amming)e(appr)n(o)n(ach)h(to)g(the)f(biline)n(ar)g(matrix)93 1203 y(ine)n(quality)j(and)g(its)f(ge)n(ometry)p Fy(,)g(Mathematical)e (Programmi)o(ng,)f(77)i(\(2\))h(\(1997\),)f(pp.)g(247{272.)5 1253 y([40])22 b(M.)14 b(Mesbahi)h(and)g(G.)f(P)m(.)29 b(P)o(apa)o(v)n(assilop)q(oulos,)13 b Fb(On)j(the)g(r)n(ank)g (minimization)g(pr)n(oblem)f(over)h(a)g(p)n(ositive)93 1303 y(semi-de\014nite)c(line)n(ar)g(matrix)g(ine)n(quality)p Fy(,)f(IEEE)h(T)m(ransactions)f(on)h(Automatic)d(Con)o(trol,)i(42)g (\(2\))g(\(1997\),)93 1353 y(pp.)i(239{243.)5 1403 y([41])22 b(M.)c(Mesbahi,)h Fb(Matrix)g(Ine)n(qualities,)h(Stability)f(The)n (ory,)h(Interior)e(Point)i(Metho)n(ds,)g(and)g(Distribute)n(d)93 1452 y(Computation)p Fy(,)14 b(Ph.D.)e(Dissertation,)i(USC,)f(June)i (1996.)5 1502 y([42])22 b(M.)d(Mesbahi,)j(G.)d(P)m(.)g(P)o(apa)o(v)n (assilop)q(oulos,)g(and)h(M.)g(G.)f(Safono)o(v,)h 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(on)d(Decision)h(and)g(Con)o(trol,)f(Brigh)o(ton,)g(England,)g(1991.)5 1950 y([46])22 b(A.)f(P)o(ac)o(k)n(ard,)j(K.)e(Zhou,)i(P)m(.)d(P)o (andey)m(,)j(J.)e(Leonhardson,)i(and)e(G.)f(Balas,)i Fb(Optimal,)h(c)n(onstant)f(I/O)93 2000 y(similarity)10 b(sc)n(aling)i(for)f(ful)r(l-information)g(and)i(state-fe)n(e)n(db)n (ack)f(c)n(ontr)n(ol)f(pr)n(oblems)p Fy(,)f(Systems)h(and)f(Con)o(trol) 93 2050 y(Letters,)15 b(19)e(\(1992\),)g(pp.)g(271{280.)5 2100 y([47])22 b(A.)11 b(P)o(ac)o(k)n(ard)h(and)f(J.)h(C.)f(Do)o(yle,)g Fb(The)i(c)n(omplex)g(structur)n(e)n(d)g(singular)f(value)p Fy(,)g(Automatica,)e(21)i(\(1\),)f(\(1993\))93 2150 y(pp.)i(71{109.)5 2200 y([48])22 b(R.)13 b(T.)g(Ro)q(c)o(k)n(afellar,)f Fb(Convex)j(A)o(nalysis)p Fy(,)e(Princeton,)i(1970.)5 2249 y([49])22 b(M.)j(G.)f(Safono)o(v)g(and)i(M.)e(A)o(thans,)29 b Fb(Gain)c(and)i(phase)f(mar)n(gin)f(for)g(multilo)n(op)g(LQG)g(r)n(e) n(gulators)p Fy(,)93 2299 y(Pro)q(ceedings)i(of)e(the)h(IEEE)h (Conference)g(on)e(Decision)h(and)f(Con)o(trol,)j(Clearw)o(ater)e(Beac) o(h,)j(FL.,)93 2349 y(Decem)o(b)q(er)14 b(1976.)5 2399 y([50])22 b(M.)15 b(G.)g(Safono)o(v)f(and)i(M.)f(A)o(thans,)h Fb(A)h(multilo)n(op)f(gener)n(alization)g(of)h(the)f(cir)n(cle)g (criterion)g(for)g(stability)93 2449 y(mar)n(gin)e(analysis)p Fy(,)g(IEEE)g(T)m(ransactions)g(on)g(Automatic)e(Con)o(trol)i(26)f (\(2\),)g(\(1981\))g(pp.)h(415{422.)5 2498 y([51])22 b(M.)9 b(G.)h(Safono)o(v,)f Fj(L)399 2483 y Fa(1)446 2498 y Fb(optimal)i(sensitivity)g(vs.)g(stability)f(mar)n(gin)p Fy(,)g(Pro)q(ceedings)i(of)e(the)g(IEEE)h(Conference)93 2548 y(on)i(Decision)h(and)g(Con)o(trol,)f(Albuquerque,)h(NM,)f(Decem)o (b)q(er)i(1983.)5 2598 y([52])22 b(M.)16 b(G.)f(Safono)o(v)g(and)i(P)m (.)e(H.)h(Lee,)i Fb(A)f(multiplier)e(metho)n(d)j(for)e(c)n(omputing)i (r)n(e)n(al)f(multivariable)f(stability)93 2648 y(mar)n(gin)p Fy(,)d(Pro)q(ceedings)i(of)e(the)i(IF)-5 b(A)o(C)14 b(W)m(orld)f (Congress,)h(July)g(1993.)5 2698 y([53])22 b(M.)16 b(G.)g(Safono)o(v)g (and)h(R.)f(Y.)g(Chiang,)g Fb(R)n(e)n(al/c)n(omplex)j Fj(K)1033 2704 y Fh(m)1064 2698 y Fb(-synthesis)f(without)f(curve)h (\014tting)p Fy(,)f(in)g(C.T.)93 2747 y(Leondes,)25 b(Editor,)g(Con)o (trol)d(and)g(Dynamic)f(Systems,)j(56,)g(Academic)e(Press,)k(New)e(Y)m (ork)e(\(1993\),)p eop %%Page: 26 26 26 25 bop 157 -1 a Fw(26)99 b Fn(Mesbahi,)17 b(Saf)o(ono)o(v)g(and)g(P) l(ap)m(a)l(v)l(assilopoulos)250 107 y Fy(pp.)d(303{324.)163 157 y([54])21 b(M.)i(G.)g(Safono)o(v,)h(K.)f(C.)g(Goh,)h(and)g(J.)f(H.) g(Ly)m(,)h Fb(Contr)n(ol)f(system)h(synthesis)g(via)f(biline)n(ar)g (matrix)250 207 y(ine)n(qualities)p Fy(,)13 b(Pro)q(ceedings)j(of)d (the)i(American)e(Con)o(trol)g(Conference)i(Baltimore,)d(MD,)h(June)i (1994.)163 257 y([55])21 b(M.)14 b(G.)f(Safono)o(v)f(and)i(G.)e(P)m(.)h (P)o(apa)o(v)n(assilop)q(oulos,)f Fb(The)i(diameter)h(of)f(an)i (interse)n(ction)e(of)g(el)r(lipsoids)h(and)250 307 y(BMI)g(r)n(obust)f (synthesis)p Fy(,)f(Pro)q(ceedings)h(of)f(the)h(IF)-5 b(A)o(C)13 b(Symp)q(osium)d(on)j(Robust)g(Con)o(trol)f(Design,)h(Rio)f (de)250 356 y(Janeiro,)i(Brazil,)f(Septem)o(b)q(er)i(1994.)163 406 y([56])21 b(I.)d(W.)f(Sandb)q(erg,)h Fb(On)h(the)f Fj(L)740 412 y Fg(2)759 406 y Fb(-b)n(ounde)n(dness)i(of)e(solutions)h (of)f(nonline)n(ar)h(functional)g(e)n(quations)p Fy(,)g(Bell)250 456 y(System)14 b(T)m(ec)o(hnical)f(Journal,)g(43)h(\(4\))g(P)o(art)g (I)q(I)g(\(1964\),)e(pp.)i(1581{1599.)163 506 y([57])21 b(J.)16 b(Sto)q(er)g(and)f(C.)g(Witzgall,)f Fb(Convexity)i(and)h (Optimization)f(in)g(Finite)g(Dimension)p Fy(,)g(Springer-V)m(erlag,) 250 556 y(1970.)163 606 y([58])21 b(O.)i(T)m(ok)o(er)g(and)f(H.)604 595 y(\177)598 606 y(Ozba)o(y,)j Fb(On)e(the)g(NP-har)n(dness)h(of)f (solving)g(biline)n(ar)f(matrix)h(ine)n(qualities)g(and)250 655 y(simultane)n(ous)d(stabilization)e(with)h(static)f(output)i(fe)n (e)n(db)n(ack)p Fy(,)g(Pro)q(ceedings)f(of)f(the)h(American)f(Con)o (trol)250 705 y(Conference,)d(1995.)163 755 y([59])21 b(S.)11 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(tice-Hall,)h(Englew)o(o)q(o)q(d)250 1153 y(Cli\013s,)e(NJ,)h(1996.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF