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这是matlab解2阶锥工具包

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FB(4)c(W)m(ar)o(nings)11 b(f)o(or)h FA(M)r Fz(A)n(T)r(L)r(A)r(B)k
FB(LMI)11 b(T)l(oolbox)f(users)957 b(33)-29 2248 y FA(4.1)47
b(\223Left\224)12 b(and)f(\223right\224)f(sides)g(of)i(an)f(inequality)
27 b(.)c(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)
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2 2 bop 102 -80 a FC(1)59 b(Pur)o(pose)102 69 y FA(The)8
b(tool)g(described)g(in)h(this)e(report)i(is)f(designed)g(to)g
(associate)g(both)g(ef)o(\002cient)h(Semi-De\002nite)g(Programming)g
(\(SDP\))102 126 y(algorithms)g(and)i(the)g(nice)g(Linear)g(Matrix)g
(Constraint)f(\(LMIs)h(and)g(LMEs\))g(formalism)g(used)g(for)g(control)
f(applica-)102 182 y(tions.)i(This)d(work)h(was)f(inspired)g(by)h(the)f
(observ)o(ation)g(that)g(on)h(the)g(one)g(hand)g(Linear)f(Matrix)i
(Constraints)d(\(LMCs\))102 239 y(ha)o(v)o(e)16 b(a)h(major)g(position)
d(in)h(current)i(academic)g(research)g([6,)f(10,)h(27)o(],)h(and)f(on)f
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(and)g(a)g(pleasant)f(interface)h(for)g(declaring)g(LMC)g(problems)g
(within)e(the)i(most)102 408 y(commonly)e(used)h(software)g(en)n
(vironment:)i(M)r Fz(A)n(T)r(L)r(A)r(B)q FA(.)103 500
y(S)r Fz(E)r FA(D)r Fz(U)r FA(M)r Fz(I)19 b FA(I)r Fz(N)r(T)r(E)r(R)r
(F)o(A)r(C)r(E)g FA(is)14 b(designed)g(as)g(an)h(add-on)f(for)i(M)r
Fz(A)n(T)r(L)r(A)r(B)j FA(and)14 b(allo)o(ws)f(to)i(declare)g(LMC)g
(problems)f(to)102 557 y(be)j(solv)o(ed)g(with)f(the)i(S)r
Fz(E)r FA(D)r Fz(U)r FA(M)r Fz(I)j FA(solv)o(er)c(proposed)f(by)h(Jos)g
(Sturm)h([25)o(].)34 b(The)17 b(major)g(part)h(of)f(this)f(report)h(is)
g(a)102 613 y(description)8 b(of)j(S)r Fz(E)r FA(D)r
Fz(U)r FA(M)r Fz(I)k FA(I)r Fz(N)r(T)r(E)r(R)r(F)o(A)r(C)r(E)f
FA(functions.)f(Before)e(that,)f(we)g(e)o(xpose)g(the)g(choices)f(that)
h(lead)g(us)f(to)h(choose)102 669 y(the)i(S)r Fz(E)r
FA(D)r Fz(U)r FA(M)r Fz(I)j FA(solv)o(er)10 b(and)h(an)h(interface)e
(much)i(alike)e(the)h(LMI)h(Control)e(T)l(oolbox)g(for)i(M)r
Fz(A)n(T)r(L)r(A)r(B)j FA([13].)102 761 y(The)8 b(user)h(mostly)f
(interested)g(in)g(using)g(the)g(interface)h(can)g(skip)f(the)h
(remaining)f(part)h(of)g(this)e(section)h(and)h(go)g(directly)102
818 y(to)h(section)g(2.)102 1032 y Fn(1.1)50 b(Notations)102
1166 y Fw(i)11 b FA(is)f(the)h(imaginary)g(unit)f(equal)h(to)g(the)f
(square)h(root)g(of)g Fp(\000)p FA(1.)102 1222 y Fm(R)131
1206 y Ft(m)p Fr(\002)p Ft(n)210 1222 y FA(\()p Fm(C)255
1206 y Ft(m)p Fr(\002)p Ft(n)324 1222 y FA(\))h(is)f(the)f(set)h(of)h
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1279 y Fm(1)g FA(and)g Fm(0)h FA(are)g(respecti)o(v)o(ely)e(the)h
(identity)e(and)i(the)g(zero)h(matrices)f(of)g(appropriate)f
(dimensions.)102 1335 y(All)g(matrices)h(are)h(written)e(using)g
(capital)h(letters)f(\()p Fw(A)p FA(\))i(while)e(scalars)h(and)g(v)o
(ectors)g(are)h(in)f(lo)o(wercase)g(\()p Fw(a)p FA(\).)111
1417 y(\257)102 1427 y Fw(A)j FA(is)g(the)h(conjugate)e(of)h(the)h
(comple)o(x)f(matrix)h Fw(A)p FA(,)g Fw(A)960 1410 y
Ft(T)1000 1427 y FA(is)f(its)g(transpose)f(and)h Fw(A)1397
1410 y Fr(\003)1431 1427 y FA(is)g(its)f(conjugate)h(transpose.)23
b(W)l(e)102 1483 y(remind)16 b(that)f(in)h(the)h(M)r
Fz(A)n(T)r(L)r(A)r(B)j FA(en)n(vironment)c(the)f(conjugate)g(transpose)
g(writes)h(as)g Fx(A')h FA(while)e(the)h(transpose)f(is)102
1540 y(obtained)10 b(by)i Fx(A.')p FA(.)18 b(If)13 b
Fw(A)d Fq(=)h Fw(A)581 1523 y Ft(T)606 1540 y FA(,)i(the)f(matrix)g(is)
f(symmetric)i(and)f(Hermitian)f(if)h Fw(A)f Fq(=)g Fw(A)1503
1523 y Fr(\003)1521 1540 y FA(.)18 b(F)o(or)12 b(real)h(v)o(alued)e
(matrices)102 1596 y(both)f(notions)f(are)j(equi)o(v)o(alent.)102
1653 y('He')f(is)g(the)g(matrix)g(operator)g(such)f(that:)j(He)p
Fp(f)p Fw(A)p Fp(g)d Fq(=)g Fw(A)c Fq(+)g Fw(A)1072 1636
y Fr(\003)1091 1653 y FA(.)102 1745 y(F)o(or)11 b(Hermitian)e
(matrices,)j Fv(>)20 b Fq(\()p Fp(\025)p Fq(\))10 b FA(is)g(the)g(L)t
(\250)-19 b(owner)10 b(partial)f(order)n(,)i(i.e.,)h
Fw(A)d Fv(>)h Fq(\()p Fp(\025)p Fq(\))p Fw(B)g FA(if)g(and)g(only)g(if)
g Fw(A)c Fp(\000)g Fw(B)j FA(is)h(positi)o(v)o(e)102
1801 y(\(semi\))h(de\002nite.)102 1893 y(In)17 b(matrix)h(equalities)e
(and)h(inequalities)f(as)h(well)g(as)h(in)f(optimisation)e(problems,)k
(the)f(decision)e(v)o(ariables)h(and)102 1949 y(unkno)o(wns)9
b(are)j(in)e(bold)g(face)i(\()p FB(x)p FA(\))f(while)g(the)g(data)g(is)
f(written)h(using)e(the)i(usual)g(mathematic)g(fonts)f(\()p
Fw(x)p FA(\).)102 2041 y(Note)17 b(that)g(in)g(the)h(M)r
Fz(A)n(T)r(L)r(A)r(B)k FA(en)n(vironment)16 b(real)i(inte)o(gers)f(are)
h(nicely)f(displayed)f(as)i(follo)o(ws,)g(while)f(comple)o(x)102
2098 y(inte)o(gers)10 b(are)i(displayed)e(as)h(an)o(y)g(other)g(comple)
o(x)g(number:)p 102 2149 1886 2 v 101 2483 2 334 v 127
2205 a FF(>>)21 b(2)127 2255 y(ans)g(=)232 2304 y(2)127
2354 y(>>)g(2+2i)127 2404 y(ans)g(=)190 2454 y(2.0000)h(+)g(2.0000i)p
1986 2483 V 102 2484 1886 2 v 102 2555 a FA(Therefore,)12
b(the)f Fx(num2str)j FA(M)r Fz(A)n(T)r(L)r(A)r(B)i FA(function)10
b(will)g(sometimes)h(appear)h(for)g(nice)f(display)m(.)j(The)d(result)g
(is)g(a)h(string)102 2611 y(that)e(cannot)h(be)g(used)g(for)g
(computation.)p 102 2662 V 101 2848 2 186 v 127 2720
a FF(>>)21 b(num2str\(2+2i\))127 2769 y(ans)g(=)127 2819
y(2+2i)p 1986 2848 V 102 2850 1886 2 v 1035 2951 a FA(2)p
eop
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3 3 bop -98 -80 a Fn(1.2)50 b(LMC)13 b(pr)o(oblems)-98
43 y FA(The)e(academic)h(results)e(on)h(LMIs)h(use)f(the)f(follo)o
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125 y Ft(m)1088 183 y Fj(\345)1087 220 y Ft(j)q Fl(=)p
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181 y Ft(j)1232 174 y Fp(\025)10 b Fm(0)437 b FA(\(1\))-98
316 y(where)16 b(the)g(data)g(are)g(the)g Fw(m)7 b Fq(+)i
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316 y FA(,)i(...,)g Fw(F)992 323 y Ft(m)1035 316 y FA(and)d(the)h
(column)f(v)o(ector)h Fw(c)p FA(,)i(while)d(the)g(opti-)-98
372 y(misation)e(v)o(ariables)h(are)h(gathered)f(in)f(the)h(v)o(ector)h
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372 y FA(.)24 b(The)14 b(formalism)h(underlines)e(that)g(an)-98
429 y(LMI)i(problem)g(is)f(an)g(optimisation)e(problem)j(with)e(a)i
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(constraints)-98 485 y(in)n(v)o(olving)9 b(symmetric)j(matrices)f(that)
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577 y(At)h(this)f(point)g(note)h(that)f(there)h(is)g(another)g(general)
g(formalism)g(for)h(writing)d(LMIs)j([25)o(]:)261 682
y Fw(p)284 664 y Ft(o)r(p)o(t)343 682 y Fq(=)e FA(min)39
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FA(,)i(b)o(uild)e(out)g(of)h Fw(z)g FA(with)e(some)i(stack)g(operator)n
(,)g(is)-98 901 y(positi)o(v)o(e)e(semi-de\002nite.)-98
993 y(W)l(e)k(do)g(not)e(get)h(into)g(more)h(details.)20
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(problems.)19 b(In)13 b(particular)n(,)h(a)f(major)-98
1106 y(dif)o(\002culty)g(is)h(that)g(control)f(problems)h(are)g
(formulated)g(with)g(matrix)g(v)o(ariables)f(while)h(the)g(generic)g
(formulations)-98 1162 y(e)o(xposed)f(abo)o(v)o(e)i(depend)e(on)g(v)o
(ectors)h(of)f(decision)g(v)o(ariables.)21 b(Going)12
b(from)i(one)g(formalism)g(to)f(another)g(may)h(be)-98
1218 y(quite)d(tedious.)-98 1310 y(T)l(ake)g(for)h(e)o(xample)g(the)e
(L)n(yapuno)o(v)g(inequality)m(,)g(it)g(writes)h(as)g(an)g(LMI)h
(constraint:)715 1416 y Fw(A)743 1397 y Ft(T)767 1416
y FB(P)6 b Fq(+)g FB(P)p Fw(A)11 b Fv(<)f Fm(0)-98 1521
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(\(bold)f(face\))i(the)f(LMI)g(writes)g(as:)312 1644
y Fk(\024)357 1680 y Fw(a)380 1687 y FI(11)456 1680 y
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1736 y Fw(a)479 1743 y FI(22)535 1644 y Fk(\025)5 b(\024)609
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y Fk(\024)876 1680 y FB(p)901 1687 y Fi(1)961 1680 y
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1736 y FB(p)986 1743 y Fi(3)1026 1644 y Fk(\025)g(\024)1099
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y FI(12)1099 1736 y Fw(a)1122 1743 y FI(21)1199 1736
y Fw(a)1222 1743 y FI(22)1278 1644 y Fk(\025)1312 1708
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Fw(a)278 2050 y FI(11)354 2043 y Fp(\000)p Fw(a)412 2050
y FI(12)208 2100 y Fp(\000)p Fw(a)266 2107 y FI(12)390
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Fw(a)975 2050 y FI(11)1016 2043 y Fp(\000)h Fw(a)1080
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y FI(21)1344 2100 y Fp(\000)p Fw(a)1402 2107 y FI(21)1479
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b(S)r Fz(E)r FA(D)r Fz(U)r FA(M)r Fz(I)h FA(I)r Fz(N)r(T)r(E)r(R)r(F)o
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i(formulations.)i(A)d(lar)o(ge)h(spectrum)f(of)h(options)d(allo)o(ws)h
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(matrices)i(to)f(an)o(y)g(structured)g(matrix)g(v)o(ariable)g
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(constraints.)836 2951 y(3)p eop
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4 4 bop 170 -80 a Fp(\017)22 b FA(Third,)11 b(v)o(arious)e(types)h(of)h
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