Least squares and output error identification algorithms for continuous time systems with unknown time delay operating in open or closed loop

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This is an author-deposited version published in:

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Eprints ID: 7917

To cite this version:

Baysse, Arnaud and Carrillo, Francisco and Habbadi , Abdallah Least squares

and output error identification algorithms for continuous time systems with

unknown time delay operating in open or closed loop. (2012) In: Sysid 2012,

16th IFAC Symposium on System Identification, 11-13 July 2012, Brussels,

Belgium.

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✉♥❦♥♦✇♥ $✐♠❡ ❞❡❧❛② ♦♣❡'❛$✐♥❣ ✐♥ ♦♣❡♥ ♦'

❝❧♦#❡❞ ❧♦♦♣

❆✳ ❇❛②%%❡✱ ❋✳ ❏✳ ❈❛++✐❧❧♦ ❛♥❞ ❆✳ ❍❛❜❜❛❞✐ ❯♥✐✈❡%&✐'( ❞❡ ❚♦✉❧♦✉&❡✱ ■◆1✴❊◆■❚✱ ▲●1✱ ✹✼ ❛✈✳ ❞✬❆③❡%❡✐① ✲ ❇1 ✶✻✷✾ ✲ ✻✺✵✶✻ ❚❛%❜❡& ❈❡❞❡① ❡✲♠❛✐❧✿ ❬❋%❛♥❝✐&❝♦✳❈❛%%✐❧❧♦✱ ❛%♥❛✉❞✳❜❛②&&❡✱ ❆❜❞❛❧❧❛❤✳❍❛❜❜❛❞✐❪❅❡♥✐'✳❢% ❆❜"#$❛❝#✿ ❚❤✐" ♣❛♣❡$ ♣$❡"❡♥#" #✇♦ ♦✛✲❧✐♥❡ ♦✉#♣✉# ❡$$♦$ ✐❞❡♥#✐✜❝❛#✐♦♥ ❛❧❣♦$✐#❤♠" ❢♦$ ❧✐♥❡❛$ ❝♦♥#✐♥✉♦✉"✲#✐♠❡ "②"#❡♠" ✇✐#❤ ✉♥❦♥♦✇♥ #✐♠❡ ❞❡❧❛② ❢$♦♠ "❛♠♣❧❡❞ ❞❛#❛✳ ❚❤❡ ♣$♦♣♦"❡❞ ♠❡#❤♦❞" ✭❢♦$ ♦♣❡♥ ❛♥❞ ❝❧♦"❡❞ ❧♦♦♣ "②"#❡♠"✮ ✉"❡ ❛ ◆♦♥❧✐♥❡❛$ ?$♦❣$❛♠♠✐♥❣ ❛❧❣♦$✐#❤♠ ❛♥❞ ♥❡❡❞" ❛♥ ✐♥✐#✐❛❧✐③❛#✐♦♥ "#❡♣ #❤❛# ✐" ❛❧"♦ ♣$♦♣♦"❡❞ ❢$♦♠ ❛ ♠♦❞✐✜❝❛#✐♦♥ ♦❢ #❤❡ ❨❛♥❣ ❛❧❣♦$✐#❤♠✳ ❙✐♠✉❧❛#✐♦♥"✱ ❛" ✐❧❧✉"#$❛#❡❞ ❜② ▼♦♥#❡✲❈❛$❧♦ $✉♥"✱ "❤♦✇ #❤❛# #❤❡ ♦❜#❛✐♥❡❞ ♣❛$❛♠❡#❡$" ❛$❡ ✉♥❜✐❛"❡❞ ❛♥❞ ✈❡$② ❛❝❝✉$❛#❡✳ ❑❡②✇♦%❞&✿ ✐❞❡♥#✐✜❝❛#✐♦♥ ♦❢ ❝♦♥#✐♥✉♦✉"✲#✐♠❡ "②"#❡♠" ✇✐#❤ #✐♠❡ ❞❡❧❛②✱ ❧❡❛"# "G✉❛$❡" ❛♥❞ ♦✉#♣✉# ❡$$♦$ ✐❞❡♥#✐✜❝❛#✐♦♥ ❛❧❣♦$✐#❤♠"✳ ❖♣❡♥✲❧♦♦♣ ❛♥❞ ❈❧♦"❡❞✲❧♦♦♣ ✐❞❡♥#✐✜❝❛#✐♦♥ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ▼❛♥② ✐♥❞✉"#$✐❛❧ ♣$♦❝❡""❡" ♣♦""❡"" ❛ #✐♠❡ ❞❡❧❛②✱ "✉❝❤ ❛" ❝❤❡♠✐❝❛❧✱ #❤❡$♠❛❧ ♦$ ❜✐♦❧♦❣✐❝❛❧ "②"#❡♠"✳ ■# ✐" ✐♠♣♦$#❛♥# #♦ ✐❞❡♥#✐❢② #❤❡ ✈❛❧✉❡ ♦❢ #❤✐" ❞❡❧❛②✱ ✐♥ ♣❛$#✐❝✉❧❛$ ❢♦$ ❝♦♥#$♦❧✳ ❋♦$ ❡①❛♠♣❧❡✱ #❤❡ ❞❡"✐❣♥ ♦❢ ❛ ❙♠✐#❤ ♣$❡❞✐❝#♦$ ✐♠♣❧✐❡" ❛ ❣♦♦❞ ❦♥♦✇❧❡❞❣❡ ♦❢ #❤❡ #✐♠❡ ❞❡❧❛② ❜❡❝❛✉"❡ #❤✐" ✐♥❢♦$♠❛#✐♦♥ ✐" ❞✐$❡❝#❧② ✐♥❝❧✉❞❡❞ ✐♥ #❤❡ ❝♦♥#$♦❧❧❡$✳ ❚❤❡$❡ ❛$❡ "❡✈❡$❛❧ ♣❛♣❡$" ❝♦♣✐♥❣ ✇✐#❤ #❤❡ ✐❞❡♥#✐✜❝❛#✐♦♥ ♦❢ ♦♣❡♥ ❧♦♦♣ "②"#❡♠" ✇✐#❤ #✐♠❡ ❞❡❧❛②✳ ❆❧♠♦"# ❛❧❧ ❛$❡ ♣$♦♣♦"❡❞ #♦ ✐❞❡♥#✐❢② ✜$"# ❛♥❞ "❡❝♦♥❞ ♦$❞❡$ ❞❡❧❛②❡❞ "②"#❡♠" ❢$♦♠ "#❡♣ "✐❣♥❛❧"✳ ❆♠♦♥❣"# #❤❡ ❡①✐"#✐♥❣ ♠❡#❤♦❞"✱ ♠♦"# ♦❢ #❤❡♠ ✉"❡ ✐♥#❡❣$❛❧" #♦ ♦❜#❛✐♥ ❛ ♣❛$❛♠❡#❡$ ✈❡❝#♦$ #❤❛# ❝♦♥#❛✐♥" #❤❡ #✐♠❡ ❞❡❧❛②✳ ❚❤❡$❡ ❛$❡ ♦✛✲❧✐♥❡ ✭▲✐✉ ❡# ❛❧✳ ✭✷✵✵✼✮❀ ❍✇❛♥❣ ❛♥❞ ▲❛✐ ✭✷✵✵✹✮❀ ❲❛♥❣ ❡# ❛❧✳ ✭✷✵✵✶✮✮ ❛♥❞ ♦♥✲❧✐♥❡ ♠❡#❤♦❞" ✭❆❤♠❡❞ ❡# ❛❧✳ ✭✷✵✵✻✮❀ ●❛$♥✐❡$ ❛♥❞ ❲❛♥❣ ✭✷✵✵✽✱ ❝❤❛♣✳ ✶✶✮✮✳ ❚❤❡ ♣❛$❛♠❡#❡$" ❛$❡ #❤❡♥ ✐❞❡♥#✐✜❡❞ ✉"✐♥❣ ❧❡❛"#✲"G✉❛$❡"✱ ♦$ ✐♥"#$✉♠❡♥#❛❧ ✈❛$✐❛❜❧❡ ♠❡#❤♦❞"✳ ❖#❤❡$ ♠❡#❤♦❞" "❡♣❛$❛#❡ #❤❡ ♠♦❞❡❧ ✐♥ #✇♦ ❞✐✛❡$❡♥# ♣❛$#"✿ #❤❡ ✜$"# ♦♥❡ ❝♦♥#❛✐♥" #❤❡ ❧✐♥❡❛$ ♣❛$❛♠❡#❡$" ❛♥❞ #❤❡ "❡❝♦♥❞ ♦♥❡ #❤❡ ♥♦♥❧✐♥❡❛$ ♣❛$❛♠❡#❡$✱ ✐✳❡✳ #❤❡ #✐♠❡ ❞❡❧❛②✳ ❚❤❡$❡ ❛$❡ ❛❧"♦ ♠❡#❤♦❞" #❤❛# ✉"❡ ♥♦♥❧✐♥❡❛$ ♣$♦❣$❛♠♠✐♥❣ ✭◆▲?✮ #♦ ❡"#✐♠❛#❡ #❤❡ ♥♦♥❧✐♥❡❛$ ♣❛$❛♠❡#❡$✳ ❆♠♦♥❣"# #❤❡"❡✱ ❨❛♥❣ ❡' ❛❧✳ ✭❨❛♥❣ ❡# ❛❧✳ ✭✷✵✵✼✮❀ ●❛$♥✐❡$ ❛♥❞ ❲❛♥❣ ✭✷✵✵✽✱ ❝❤❛♣✳ ✶✷✮✮✱ ♣$♦♣♦"❡❞ ❛♥ ✐♥#❡$❡"#✐♥❣ ♠❡#❤♦❞ ✇❤✐❝❤ ♣❡$♠✐#" #♦ ✐❞❡♥#✐❢② ♠✉❧#✐♣❧❡ ✐♥♣✉# "✐♥❣❧❡ ♦✉#♣✉# ✭▼■❙❖✮ "②"#❡♠" ✇✐#❤ ❧✐##❧❡ ♣$✐♦$ ❦♥♦✇❧❡❞❣❡✱ ✉"✐♥❣ #❤❡ ❧❡❛"#✲"G✉❛$❡"✱ #❤✐" ❛❧❣♦$✐#❤♠ ✇✐❧❧ ❜❡ ❛❞❛♣#❡❞ ❛♥❞ ✉"❡❞ ❢♦$ ✐♥✐#✐❛❧✐③❛#✐♦♥ ♦❢ #❤❡ ♥❡✇ ♦✉#♣✉# ❡$$♦$ ✐❞❡♥#✐✜❝❛#✐♦♥ ❛❧❣♦$✐#❤♠" ✭❢♦$ ♦♣❡♥ ❛♥❞ ❝❧♦"❡❞✲❧♦♦♣ "②"#❡♠"✮✳ ❚❤✐" ♣❛♣❡$✱ ♣$♦♣♦"❡ #✇♦ ❛❧❣♦$✐#❤♠" ✇❤✐❝❤ ♣❡$♠✐# #❤❡ ✐❞❡♥#✐✜❝❛#✐♦♥ ♦❢ ❛❧❧ #❤❡ ♣❛$❛♠❡#❡$" ♦❢ ❛ ❧✐♥❡❛$ ❝♦♥#✐♥✉♦✉"✲ #✐♠❡ "②"#❡♠ ✇✐#❤ ✉♥❦♥♦✇♥ #✐♠❡ ❞❡❧❛②✳ ❚❤❡② ❝❛♥ ❜❡ ❝♦♥"✐❞❡$❡❞ ❛" ❡①#❡♥"✐♦♥" ♦❢ #❤❡ "❡♥"✐#✐✈✐#② ❜❛"❡❞ ✐❞❡♥#✐✲ ✜❝❛#✐♦♥ ❛❧❣♦$✐#❤♠ ♣$❡✈✐♦✉"❧② ♣$♦♣♦"❡❞ ✐♥ ✭❈❛$$✐❧❧♦ ❡# ❛❧✳ ✭✷✵✵✾✮✮✳ ❚❤❡"❡ ❛❧❣♦$✐#❤♠" ❛$❡ ❜♦#❤ ❜❛"❡❞ ♦♥ #❤❡ ♦✉#♣✉# ❡$$♦$ ♠❡#❤♦❞ ❛♥❞ ❝❛♥ ✐❞❡♥#✐❢② ❛ ❧✐♥❡❛$ "#❛❜❧❡ "②"#❡♠ ♦❢ ❛♥② ♦$❞❡$ ✇✐#❤ #✐♠❡ ❞❡❧❛② ✐♥ ❛ ✈❡$② ❛❝❝✉$❛#❡ ♠❛♥♥❡$✳ ❚❤❡ ✜$"# ❛❧❣♦$✐#❤♠ ✇❛" ♣$❡✈✐♦✉"❧② ♣$❡"❡♥#❡❞ ✐♥ ❛♥ ♦♣❡♥ ❧♦♦♣ ❢♦$♠ ✐♥ ✭❇❛②""❡ ❡# ❛❧✳ ✭✷✵✶✶✮✮ ❛♥❞ ✐# ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❈♦♥#✐♥✉♦✉"✲ ❚✐♠❡ ❖✉#♣✉# ❊$$♦$ ❈❚❖❊ ❛❧❣♦$✐#❤♠✳ ❚❤❡ "❡❝♦♥❞ ♣$♦✲ ♣♦"❡❞ ❛❧❣♦$✐#❤♠ ✇✐❧❧ ❜❡ ✉"❡❢✉❧ ❢♦$ #❤❡ ❝❛"❡ ♦❢ ❝❧♦"❡❞ ❧♦♦♣ "②"#❡♠" ❛♥❞ ✐# ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❈♦♥#✐♥✉♦✉"✲❚✐♠❡ ❈❧♦"❡❞✲ ▲♦♦♣ ❖✉#♣✉# ❊$$♦$ ❈❚❈▲❖❊ ❛❧❣♦$✐#❤♠✳ ❚❤❡ ♣❛♣❡$ ✐" ♦$❣❛♥✐③❡❞ ❛" ❢♦❧❧♦✇"✳ ■♥ #❤❡ "❡❝♦♥❞ "❡❝#✐♦♥ #❤❡ ♣$♦❝❡"" ❛♥❞ ♠♦❞❡❧ ❞❡"❝$✐♣#✐♦♥ ❛$❡ ♣$❡"❡♥#❡❞✳ ❚❤❡♥✱ #❤❡ ♥❡✇ ♦✉#♣✉# ❡$$♦$ ♠❡#❤♦❞" ❛$❡ ❞❡#❛✐❧❡❞ ✐♥ #❤❡ #❤✐$❞ "❡❝#✐♦♥ ❢♦$ ♦♣❡♥ ❛♥❞ ❝❧♦"❡❞ ❧♦♦♣ "②"#❡♠"✳ ❙❡❝#✐♦♥ ❢♦✉$ ❞❡"❝$✐❜❡" ❛ ❧❡❛"# "G✉❛$❡" ❛❧❣♦$✐#❤♠ #❤❛# ✇✐❧❧ ❜❡ ✉"❡❞ ❢♦$ ✐♥✐#✐❛❧✐③❛#✐♦♥✳ ❙✐♠✉❧❛#✐♦♥ $❡"✉❧#" ❢♦$ ♦♣❡♥ ❛♥❞ ❝❧♦"❡❞✲❧♦♦♣ "②"#❡♠" ❛$❡ ❞❡"❝$✐❜❡❞ ✐♥ "❡❝#✐♦♥ ✺✳ ❋✐♥❛❧❧②✱ "♦♠❡ ❝♦♥❝❧✉"✐♦♥" ❛$❡ ♣$❡"❡♥#❡❞ ✐♥ "❡❝#✐♦♥ ✻✳ ✷✳ ?❘❖❈❊❙❙ ❆◆❉ ▼❖❉❊▲ ❉❊❙❈❘■?❚■❖◆ ❲❤❡♥ ✉"✐♥❣ "❛♠♣❧❡❞ ❞❛#❛✱ #❤❡ "②"#❡♠ ✇✐#❤ ✐♥♣✉# u(tk) ❛♥❞ ♦✉#♣✉# y(tk)❝❛♥ ❜❡ $❡♣$❡"❡♥#❡❞ ❜② ✭"❡❡ ●❛$♥✐❡$ ❛♥❞ ❲❛♥❣ ✭✷✵✵✽✮✮✿ y(tk) = B(p) A(p)e −Tdp_u(t k) + w(tk) ✭✶✮ ✇❤❡$❡ w(tk)✐" ❛""✉♠❡❞ #♦ ❜❡ ❛ ●❛✉""✐❛♥ ✇❤✐#❡ ♥♦✐"❡✱ p ✐" #❤❡ ❞❡$✐✈❛#✐✈❡ ♦♣❡$❛#♦$ p = d dt✱ Td ✐" #❤❡ #✐♠❡ ❞❡❧❛② ❛♥❞ e−Tdp _{❤❛" #♦ ❜❡ ✉♥❞❡$"#♦♦❞ ❛"✿} e−Tdpu(t) = u(t − Td) ✭✷✮ A(p) ❛♥❞ B(p) ❛$❡ ♣♦❧②♥♦♠✐❛❧" ✇✐#❤ #❤❡ ❢♦❧❧♦✇✐♥❣ "#$✉❝✲ #✉$❡✿

(3)

A(p) = 1 + a1p+ · · · + anpn ✭✸✮

B(p) = b0+ b1p+ · · · + bmpm; m < n ✭✹✮

y(tk)✐% &❤❡ ❛♠♣❧✐&✉❞❡ ♦❢ &❤❡ ❝♦♥&✐♥✉♦✉%✲&✐♠❡ %✐❣♥❛❧ y(t) ❛&

&✐♠❡ tk = kTs✱ ✇✐&❤ Ts &❤❡ %❛♠♣❧✐♥❣ ♣❡7✐♦❞✳ ❋♦7 &❤❡ 7❡%&

♦❢ &❤❡ ♣7❡%❡♥&❛&✐♦♥✱ &❤❡ %✐❣♥❛❧% ❛7❡ ✇7✐&&❡♥ ❛% ❛ ❢✉♥❝&✐♦♥ ♦❢ &❤❡ &✐♠❡ (t)✳ ❋♦7 ✐♠♣❧❡♠❡♥&❛&✐♦♥ ♣✉7♣♦%❡✱ &❤❡② ❤❛✈❡ &♦ ❜❡ ❝♦♥%✐❞❡7❡❞ ❛& %❛♠♣❧✐♥❣ ✐♥&❡7✈❛❧%✳

✷✳✶ ❖♣❡♥ ❧♦♦♣ )②)+❡♠

▲❡& ✉% ❞❡♥♦&❡ by(t) &❤❡ ❡%&✐♠❛&❡❞ ♦✉&♣✉& ♦❢ &❤❡ ♠♦❞❡❧✱ ❞❡✜♥❡❞ ❛%✿ b y(t) =B(p)b b A(p)e −Tdpb _u(t) _✭✺✮

❲❤❡7❡ bTd ✐% &❤❡ ❡%&✐♠❛&❡❞ &✐♠❡ ❞❡❧❛② ❀ bA(p)❛♥❞ bB(p)❛7❡

♣♦❧②♥♦♠✐❛❧% ♦❢ ❡%&✐♠❛&❡❞ ♣❛7❛♠❡&❡7%✳ ❚❤❡ ♦✉&♣✉& ❝❛♥ ❜❡ ❡①♣7❡%%❡❞ ✐♥ ❛ 7❡❣7❡%%♦7 ❢♦7♠ ❛% ✐& ✇❛% %❤♦✇♥ ✐♥ ❇❛②%%❡ ❡& ❛❧✳ ✭✷✵✶✶✮✳

❚❤❡ ♣7♦♣♦%❡❞ ❛♣♣7♦❛❝❤ ✐❞❡♥&✐✜❡% ❛& &❤❡ %❛♠❡ &✐♠❡ &❤❡ ♣❛7❛♠❡&❡7% bθ ❛♥❞ &❤❡ ♣❛7❛♠❡&❡7 bTd✳ ❚❤✉%✱ ❛ ✈❡❝&♦7 bΘ

✇❤✐❝❤ ❝♦♥&❛✐♥% ❛❧❧ &❤❡ ♣❛7❛♠❡&❡7% ✐% ❞❡✜♥❡❞ ❛% bΘT ₌

_b θT _T_b d ✳ ✷✳✷ ❈❧♦)❡❞✲❧♦♦♣ )②)+❡♠

■❢ &❤❡ ♣7♦❝❡%% ✐% ♦♣❡7❛&✐♥❣ ✐♥ ❝❧♦%❡❞ ❧♦♦♣✱ ✇✐&❤ ❛ ❝♦♥&7♦❧❧❡7 C(p)❞❡✜♥❡❞ ❛%✿

u_{(t) = C(p) [r(t) − y(t)] ;} ✭✻✮ C(p) =R(p)

S(p) ✭✼✮

✇❤❡7❡ r(t) ✐% &❤❡ 7❡❢❡7❡♥❝❡ %✐❣♥❛❧✳

❯%✐♥❣ &❤❡ ♦✉&♣✉& ❡77♦7 ❛♣♣7♦❛❝❤✱ &❤❡ ❡%&✐♠❛&❡❞ ♦✉&♣✉& ♦❢ &❤❡ ♠♦❞❡❧ by(t)✱ ✐% ❞❡✜♥❡❞ ❛%✿ b y(t) =B(p)b b A(p)e −Tdpb b u(t) ✭✽✮ b u(t) = C(p) [r(t) − by(t)] ✭✾✮ ❚❤❡ ♦✉&♣✉& ❝❛♥ ❜❡ ❡①♣7❡%%❡❞ ✐♥ ❛ 7❡❣7❡%%♦7 ❢♦7♠✿ b y(t) = −ba1by(1)(t) − · · · − banyb(n)(t) +bb0ub(t − bTd) + · · · + bbmub(m)(t − bTd) b y(t) = bθTφ(t)b ✭✶✵✮ ✇✐&❤✿ b θT =ba1 · · · ban bb0 · · · bbm , ✭✶✶✮ b φT_{(t) =} −by(1)(t) · · · −by(n)(t) b u_{(t − b}Td) · · · ˆu(m)(t − bTd) ✭✶✷✮ ✇❤❡7❡ bu(m)_{(t) = p}m_u(t)_b _{❛♥❞ by}(n)_{(t) = p}n_b_y(t)✳ ✸✳ ❚❍❊ ❖❋❋✲▲■◆❊ ❖❯❚S❯❚ ❊❘❘❖❘ ▼❊❚❍❖❉ ❚❤❡ W✉❛❞7❛&✐❝ ❝7✐&❡7✐♦♥ &♦ ❜❡ ♠✐♥✐♠✐③❡❞ ❢7♦♠ N %❛♠♣❧❡❞ ❞❛&❛ ✐%✿ J = N X k=1 (y(tk) − by(tk))2= N X k=1 ε2(tk) ✭✶✸✮

❙✐♥❝❡ &❤✐% ❝7✐&❡7✐♦♥ ✐% ♥♦♥❧✐♥❡❛7 ✐♥ &❤❡ ♣❛7❛♠❡&❡7%✱ ♦♥❡ ♠✉%& ✉%❡ ❛ ◆♦♥▲✐♥❡❛7 S7♦❣7❛♠♠✐♥❣ ✭◆▲S✮ ♠❡&❤♦❞✳ ❚❤❡%❡ ♠❡&❤♦❞% ❛7❡ ❜❛%❡❞ ♦♥ ❛♥ ✐&❡7❛&✐✈❡ ❛♣♣7♦❛❝❤ &♦ ❝♦♥✲ ✈❡7❣❡✳ ❆& &❤❡ ✐&❡7❛&✐♦♥ j + 1✱ &❤❡ ♣❛7❛♠❡&❡7% ❛7❡ ✉♣❞❛&❡❞ ✉%✐♥❣✿

b

Θj+1= bΘj+ ∆ bΘj ✭✶✹✮

✇❤❡7❡ ∆bΘj ✐% &❤❡ ✐♥❝7❡♠❡♥& ♦❢ &❤❡ ♣❛7❛♠❡&❡7% bΘj✳ ❚❤❡

&❤7❡❡ ◆▲S ♠❡&❤♦❞% ♠♦%& ✉%❡❞ ❛7❡ &❤❡ ❣7❛❞✐❡♥& ♠❡&❤♦❞✱ &❤❡ ●❛✉%%✲◆❡✇&♦♥ ♠❡&❤♦❞✱ ❛♥❞ &❤❡ ▲❡✈❡♥❜❡7❣✲▼❛7W✉❛7❞& ♠❡&❤♦❞✱ ✇❤❡7❡ &❤❡ ✐♥❝7❡♠❡♥& ✐% 7❡%♣❡❝&✐✈❡❧② ❞❡%❝7✐❜❡❞ ❛%✿

∆ bΘj= −µJ ′ ( bΘ_j₎ ✭✶✺✮ ∆ bΘj= −µ h J′′( bΘ_j₎i−1_J′_{( b}Θ_j₎ ✭✶✻✮ ∆ bΘj= − h J′′( bΘ_j) + µ❞✐❛❣_J′′_{( b}Θ_j₎i −1 J′( bΘ_j₎✭✶✼✮ ✇❤❡7❡ J′

( bΘ_j₎ ❛♥❞ J′′_{( b}Θ_j₎ ❛7❡ 7❡%♣❡❝&✐✈❡❧② &❤❡ ❣7❛❞✐❡♥& ❛♥❞ &❤❡ ❍❡%%✐❛♥✱ µ ✐% ❛ ♣❛7❛♠❡&❡7 ✇❤✐❝❤ ♣❡7♠✐&% &❤❡ &✉♥✐♥❣ ♦❢ &❤❡ ❛❧❣♦7✐&❤♠%✳ ❚❤❡ ❧❛&&❡7 ❛❧❣♦7✐&❤♠ ✭✶✼✮ ✐% &❤❡ ♠♦%& ♣❡7❢♦7♠❛♥& ❞✉❡ &♦ ✐&% &✉♥✐♥❣ ♣❛7❛♠❡&❡7 µ✳ ❋♦7 ❤✐❣❤ µ ✈❛❧✉❡% &❤❡ ❛❧❣♦7✐&❤♠ ✐% ❝❧♦%❡ &♦ &❤❡ ❣7❛❞✐❡♥& ♠❡&❤♦❞✱ ❢♦7 ❧♦✇ µ ✈❛❧✉❡% &❤❡ ❛❧❣♦7✐&❤♠ ✐% ❝❧♦%❡ &♦ &❤❡ ●❛✉%%✲◆❡✇&♦♥ ♠❡&❤♦❞✳ ❚❤❡ ❣7❛❞✐❡♥& ❛♥❞ &❤❡ ❍❡%%✐❛♥ ❛7❡ ❝♦♠♣✉&❡❞ ✉%✐♥❣ &❤❡ %❡♥%✐&✐✈✐&② ❢✉♥❝&✐♦♥%✳ ❉❡♥♦&✐♥❣ σbθ(t) =

∂by(t) ∂bθ &❤❡

♦✉&♣✉& %❡♥%✐&✐✈✐&② ❢✉♥❝&✐♦♥ ✇✳7✳&✳ bθ✱ ❛♥❞ σbTd(t) = ∂by(t)

∂Tdb

&❤❡ ♦✉&♣✉& %❡♥%✐&✐✈✐&② ❢✉♥❝&✐♦♥ ✇✳7✳&✳ bTd✱ &❤❡ %❡♥%✐&✐✈✐&②

✈❡❝&♦7 σ (t) ✐% ✇7✐&&❡♥ ✐♥ &❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦7♠✿ σT(t) =hσT b θ (t) σbTd(t) i ✭✶✽✮ =_σba1(t) · · · σban(t) σbb0(t) · · · σbbm(t) σbTd(t)

❚❤❡ ❣7❛❞✐❡♥& ❛♥❞ &❤❡ ❍❡%%✐❛♥✱ ✇✐&❤ &❤❡ ●❛✉%%✲◆❡✇&♦♥ ❛♣♣7♦①✐♠❛&✐♦♥✱ ❛7❡ &❤❡♥ 7❡%♣❡❝&✐✈❡❧② ❣✐✈❡♥ ❜②✿

J′_{( b}Θ_{) = −2} N X k=1 ε(tk) σ (tk) ✭✶✾✮ J′′_{( b}Θ_{) ≃ 2} N X k=1 σ(tk) σT(tk) ✭✷✵✮ ✸✳✶ ❙❡♥)✐+✐✈✐+② ❢✉♥❝+✐♦♥) ❝♦♠♣✉+❛+✐♦♥ ❖♣❡♥ ❧♦♦♣ )②)+❡♠ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉&❛&✐♦♥% ❢♦7 ❛♥ ♦♣❡♥ ❧♦♦♣ %②%&❡♠ ✇❡7❡ ❞❡✈❡❧♦♣❡❞ ✐♥ ❛ ♣7❡✈✐♦✉% ♣❛♣❡7 ✭❇❛②%%❡ ❡& ❛❧✳ ✭✷✵✶✶✮✮✳ ❖♥❧② &❤❡ ♠❛✐♥ 7❡%✉❧&% ❛7❡ ♣7❡%❡♥&❡❞ ❤❡7❡✳

(4)

❚❤❡ #❡♥#✐&✐✈✐&✐❡# ♦❢ ❛ ♥✉♠❡-❛&♦- ♣❛-❛♠❡&❡- bbi ❛♥❞ ♦❢ ❛

❞❡♥♦♠✐♥❛&♦- ♣❛-❛♠❡&❡- bai ❛-❡ -❡#♣❡❝&✐✈❡❧② ❝♦♠♣✉&❡❞ ✐♥

&❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ σbbi(t) = pi b A(p)e −Tdpb _u (t) ; σbai(t) = −pi b A(p)y(t)b ✭✷✶✮ ◆♦&✐❝❡ &❤❛& &❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥# σT

b θ (t)❝❛♥ ❜❡ ✇-✐&&❡♥ ✐♥ ❝♦♠♣❛❝& ♥♦&❛&✐♦♥✿ σT b θ (t) = 1 b A(p)φ(t, cb Td)✳ ❙♦ &❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥# σT b θ (t)✐# &❤❡ -❡❣-❡##✐♦♥ ✈❡❝&♦- bφ(t, cTd) ✜❧&❡-❡❞ ❜② 1 b

A(p)✳ ❚❤✐# ✐♠♣❧✐❝✐& ✜❧&❡-✐♥❣ ❣✐✈❡# ❛♥ ♦♣&✐♠❛❧

❡#&✐♠❛&❡ ❛# #❤♦✇♥ ✐♥ ●❛-♥✐❡- ❛♥❞ ❲❛♥❣ ✭✷✵✵✽✮✱ ✇❤❡♥ b

A(p) ⋍ A(p)✳

❚❤❡ #❡♥#✐&✐✈✐&② ♦❢ &❤❡ #✐♠✉❧❛&❡❞ ♦✉&♣✉& ✇✐&❤ -❡#♣❡❝& &♦ &❤❡ &✐♠❡ ❞❡❧❛② bTd ✐# ❝♦♠♣✉&❡❞ ✐♥ &❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡-✿

σbTd(t) = ∂_by(t) ∂ bTd =−p bB(p) b A(p) e −Tdpb _u (t) = −pby(t) ✭✷✷✮ ■& ♠✉#& ❜❡ ♥♦&❡❞ &❤❛& &❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥ σbTd ✐#

-❡❛❧✐③❛❜❧❡ ♦♥❧② ✐❢ m < n✳ ❈❧♦#❡❞✲❧♦♦♣ #②#)❡♠

■♥ &❤✐# ❝❛#❡ &❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥ ♦❢ &❤❡ ❡#&✐♠❛&❡❞ ♦✉&♣✉& ✇✐&❤ -❡#♣❡❝& ♦❢ ♣❛-❛♠❡&❡- ♦❢ &❤❡ ♥✉♠❡-❛&♦- ✐# ❣✐✈❡♥ ❜② σbbi(t) = ∂y(t)_b ∂bbi = ∂ ∂bbi " b B(p)e−Tdpb b A(p) bu(t) # ✭✷✸✮ ❯#✐♥❣ ✭✾✮✱ ✐& ❝❛♥ ❜❡ ✈❡-✐✜❡❞ ❛# ✐♥ ❈❛--✐❧❧♦ ❡& ❛❧✳ ✭✷✵✵✾✮ &❤❛&✿ σbbi(t) = S(p)e−Tdpb _pi b

A(p)S(p) + bB(p)R(p)e−Tdpb u(t)b

σbbi(t) = S(p)e−Tdpb _pi b Pd(p) b u(t); ✭✷✹✮ b Pd(p) = bA(p)S(p) + bB(p)R(p)e−Tdpb ✭✷✺✮

❚❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥ ✇✐&❤ -❡#♣❡❝& &♦ ❛ ♣❛-❛♠❡&❡- ♦❢ &❤❡ ❞❡♥♦♠✐♥❛&♦- ✐# ❝❛❧❝✉❧❛&❡❞ ✐♥ &❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡-✿

σbai(t) = ∂_by(t) ∂_bai = ∂ ∂_bai " b B(p)e−Tdpb b A(p) u(t)b # ✭✷✻✮ ❚❤❡♥✱ ✇✐&❤ ✭✾✮ &❤❡ ❢♦❧❧♦✇✐♥❣ -❡#✉❧& ✐# ♦❜&❛✐♥❡❞✿

σbai(t) =

−S(p)pi b

A(p)S(p) + bB(p)R(p)e−Tdpb y(t)b

σbai(t) = −S(p)pi b Pd(p) b y(t) ✭✷✼✮ ❲✐&❤ bPd(p)❣✐✈❡♥ ❜② ✭✷✺✮✳

❚❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥# σbθ❝❛♥ ❜❡ ✇-✐&&❡♥ ✐♥ &❤❡ ❝♦♠♣❛❝&

♥♦&❛&✐♦♥✿ σbθ(t) = S(p) b Pd(p) b φ(t) ✭✷✽✮

❙♦ &❤❡ #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥ ❝❛♥ ❜❡ #❡❡♥ ❛# ❛ ✜❧&❡-✐♥❣ ✭❜②

S

b

P✮ ♦❢ &❤❡ -❡❣-❡##✐♦♥ ✈❡❝&♦-✳ ❚❤✐# ✐♠♣❧✐❝✐& ✜❧&❡-✐♥❣ ✐# &❤❡

#❛♠❡ &❤❛& ▲❛♥❞❛✉ ▲❛♥❞❛✉ ❛♥❞ ❑❛-✐♠✐ ✭✶✾✾✼✮ ♦❜&❛✐♥❡❞ ✐♥ &❤❡ ❞✐#❝-❡&❡✲&✐♠❡ ❝❛#❡ ❢♦- &❤❡ ❋✲❈▲❖❊ ❛❧❣♦-✐&❤♠ ✇❤✐❝❤ ✐# ✇❡❧❧ ❦♥♦✇♥ &♦ ❤❛✈❡ ❡①❝❡❧❧❡♥& ❝♦♥✈❡-❣❡♥❝❡ ♣-♦♣❡-&✐❡#✳ ❋✐♥❛❧❧②✱ &❤❡ ♦✉&♣✉& #❡♥#✐&✐✈✐&② ❢✉♥❝&✐♦♥ ✇✐&❤ -❡#♣❡❝& &♦ &❤❡ &✐♠❡ ❞❡❧❛② ✐# ❝♦♠♣✉&❡❞ ❜②✿ σbTd(t) = ∂_by(t) ∂ bTd = ∂ ∂ bTd " b B(p) b A(p)e −Tdpb b u(t) # ✭✷✾✮ ❯#✐♥❣ ✭✾✮✱ &❤✐# ❣✐✈❡#✿ σbTd(t) = −p bB(p)S(p) b Pd(p) e−Tdpb u(t)_b ✭✸✵✮ ❲✐&❤ bPd(p)❣✐✈❡♥ ❜② ✭✷✺✮✳

❇❡❝❛✉#❡ &❤❡ &✐♠❡ ❞❡❧❛② ✐# ❝♦♥&❛✐♥❡❞ ✐♥ &❤❡ ❞❡♥♦♠✐♥❛&♦-♦❢ ❛❧❧ &❤❡ #❡♥#✐&✐✈✐&✐❡# ❢✉♥❝&✐♦♥# ✭✷✹✮✱ ✭✷✼✮✱ ✭✸✵✮ ✐& ✐# ♥♦& ♦❜✈✐♦✉# ❤♦✇ &❤❡② ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥&❡❞✳ ❚♦ ❝♦♣❡ ✇✐&❤✱ &❤❡ &❤-❡❡ ✐♠♣❧❡♠❡♥&❛❜❧❡ #&-✉❝&✉-❡# #❤♦✇♥ ✐♥ &❤❡ ✜❣✉-❡ ✶ ❛-❡ ♣-♦♣♦#❡❞✳ ❋♦- ❡①❛♠♣❧❡✱ ✐& ❝❛♥ ❜❡ ❡❛#✐❧② ✈❡-✐✜❡❞ ✭❢-♦♠ #&❛♥❞❛-❞ ♣-♦♣❡-&✐❡# ♦❢ &❤❡ ✜-#& ♣-♦♣♦#❡❞ ❝❧♦#❡❞✲❧♦♦♣ #&-✉❝&✉-❡✮ &❤❛&✿ σbbi(t) = pi b A(p) 1 1 + bB(p) b A(p) R(p) S(p)e−Tdpb e−Tdpb = p i_S(p) b A(p)S(p) + bB(p)R(p)e−Tdpb e −Tdpb _✭✸✶✮ σbbi(t) = pi_S(p) b Pd(p) e−Tdpb ✭✸✷✮ ❚❤✐# ✇✐❧❧ ❜❡ ❛❧#♦ &❤❡ ❝❛#❡ ❢♦- ❛❧❧ &❤❡ ♦&❤❡-# #❡♥#✐&✐✈✐&✐❡# ❢✉♥❝&✐♦♥#✳

(5)

✸✳✷ ❆❧❣♦'✐)❤♠ ❞❡.❝'✐♣)✐♦♥ ♦❢ )❤❡ ♣'♦♣♦.❡❞ ❈❚❖❊ ❛♥❞ ❈❚❈▲❖❊ ♠❡)❤♦❞. ❚❤❡ ❛❧❣♦'✐)❤♠ ✐+ )❤❡ +❛♠❡ ❢♦' )❤❡ ♦♣❡♥ ❧♦♦♣ ♦' ❝❧♦+❡❞ ❧♦♦♣ +②+)❡♠✳ ❋♦' )❤❡ ♦♣❡♥ ❧♦♦♣ +②+)❡♠✱ )❤❡ ❛❧❣♦'✐)❤♠ ✇❛+ ♣'❡✈✐♦✉+❧② ♣'❡+❡♥)❡❞ ✐♥ ❇❛②++❡ ❡) ❛❧✳ ✭✷✵✶✶✮✳ ❍❡'❡ )❤❡ ❛❧❣♦'✐)❤♠ ✐+ ♣'❡+❡♥)❡❞ ❢♦' )❤❡ ❝❧♦+❡❞ ❧♦♦♣ +②+)❡♠✳ ✭✶✮ ▲❡) ❥❂✵✳ ❯+❡ ❛♥ ✐♥✐)✐❛❧ ✈❛❧✉❡ ♦❢ bΘ0✱ µ0❛♥❞ J0✳ ✭✷✮ ❙✐♠✉❧❛)❡ )❤❡ ♠♦❞❡❧ ✉+✐♥❣ ✭✽✮ )♦ ♦❜)❛✐♥ by(t)✳ ✭✸✮ ❈♦♠♣✉)❡ )❤❡ +❡♥+✐)✐✈✐)② ❢✉♥❝)✐♦♥ σT_(t k) ✉+✐♥❣ ✭✷✹✮✱ ✭✷✼✮ ❛♥❞ ✭✸✵✮✳ ✭✹✮ ❈♦♠♣✉)❡ )❤❡ ❣'❛❞✐❡♥) ❛♥❞ )❤❡ ❍❡++✐❛♥ ✇✐)❤ ✭✶✾✮ ❛♥❞ ✭✷✵✮✳ ✭✺✮ L❡'❢♦'♠ )❤❡ ❢♦❧❧♦✇✐♥❣✿ ❈♦♠♣✉)❡ b Θj+1= bΘj− h J′′( bΘ_j) + µ❞✐❛❣J′′( bΘ_j₎i −1 J′( bΘ_j)✳ ✭❛✮ ❙✐♠✉❧❛)❡ )❤❡ ♠♦❞❡❧ ✭✽✮ )♦ ♦❜)❛✐♥ )❤❡ ♥❡✇ ❡+)✐✲ ♠❛)❡❞ ♦✉)♣✉) by(t)✳ ✭❜✮ ❈♦♠♣✉)❡ J( bΘ_j+1_{) =} N P k=1 (y (tk) − by(tk))2✳ ✭❝✮ ❈❤❡❝❦ ✐❢ J( bΘ_j+1_{) 6 J( b}Θ_j)✳ ■❢ ♥♦)✱ ❞♦ µ_j+1 ₌ 10µj ❛♥❞ ❣♦ ❜❛❝❦ )♦ ✭❛✮✳ ✭❞✮ ❉♦ µj+1=1₄µj✳ ✭✻✮ ❚❡'♠✐♥❛)❡ )❤❡ ❛❧❣♦'✐)❤♠ ✐❢ )❤❡ ❝♦♥✈❡'❣❡♥❝❡ ❝♦♥❞✐)✐♦♥ ✐+ +❛)✐+✜❡❞✳ ❖)❤❡'✇✐+❡✱ ❧❡) j = j + 1 ❛♥❞ ❣♦ ❜❛❝❦ )♦ +)❡♣ ✷✳ ✹✳ ▲❊❆❙❚ ❙◗❯❆❘❊❙ ❆▲●❖❘■❚❍▼ ❯❙❊❉ ❋❖❘ ■◆■❚■❆▲■❩❆❚■❖◆ ❚❤❡ ♣'♦♣♦+❡❞ ♦✉)♣✉) ❡''♦' ♠❡)❤♦❞ ♥❡❡❞+ ❛♥ ✐♥✐)✐❛❧✐③❛)✐♦♥ +)❡♣ ❢♦' )❤❡ ❡+)✐♠❛)❡❞ ♠♦❞❡❧ ✭✺✮ ♦' ✭✽✮✳ ❚❤✐+ )❛+❦ ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❛♥ ❛❞❛♣)❛)✐♦♥ ♦❢ )❤❡ ♠❡)❤♦❞ ♣'♦♣♦+❡❞ ❜② ❨❛♥❣ ❡) ❛❧✳ ✭✷✵✵✼✮✳ ❚❤✐+ ♠❡)❤♦❞✱ ❜❛+❡❞ ♦♥ )❤❡ +❡♣❛'❛❜❧❡ ❧❡❛+)✲ +_✉❛'❡+ ♠❡)❤♦❞✱ ✐❞❡♥)✐❢② ✜'+)❧② )❤❡ ♥♦♥❧✐♥❡❛' ♣❛'❛♠❡)❡' ✇✐)❤ ❛ ◆▲L ❛❧❣♦'✐)❤♠✱ ❛♥❞ )❤❡♥ )❤❡ ❧✐♥❡❛' ♣❛'❛♠❡)❡'+ ✇✐)❤ )❤❡ ❝❧❛++✐❝❛❧ ❧✐♥❡❛' ❧❡❛+)✲+_✉❛'❡+ ❛❧❣♦'✐)❤♠✳ ❚❤✐+ ❛❧❣♦✲ '✐)❤♠ ✐+ ❝❛❧❧❡❞ ●❙❊L◆▲❙ ❢♦' ●❧♦❜❛❧ ❙❡♣❛'❛❜❧❡ ◆♦♥❧✐♥❡❛' ▲❡❛+)✲❙_✉❛'❡+✳ ❚❤❡ ♥♦♥❧✐♥❡❛' ♣❛'❛♠❡)❡'✱ ✐✳❡✳ )❤❡ )✐♠❡ ❞❡❧❛②✱ ✐+ ✐❞❡♥)✐✜❡❞ ✉+✐♥❣ ❛ ●❛✉++ ◆❡✇)♦♥ ❛❧❣♦'✐)❤♠✳ ■) ✐+ ✇❡❧❧✲❦♥♦✇♥ )❤❛) )❤✐+ ◆▲L ❛❧❣♦'✐)❤♠ ❝❛♥ ♦♥❧② ✐♥+✉'❡+ ❧♦❝❛❧ ❝♦♥✈❡'❣❡♥❝❡✳ ❚❤✉+✱ )♦ ❛✈♦✐❞ ❧♦❝❛❧ ♠✐♥✐♠❛✱ ❨❛♥❣ ❡) ❛❧✳ ✭✷✵✵✼✮ ♣'♦♣♦+❡ )♦ ✉+❡ ❛ '❛♥❞♦♠ ✈❛'✐❛❜❧❡ ❛❞❞❡❞ )♦ )❤❡ ❣'❛❞✐❡♥)✳ ❚❤❡ ❛❧❣♦'✐)❤♠ ♣❡'♠✐)+ )❤❡♥ )♦ ♦❜)❛✐♥ ❛ ❜❡))❡' ❝♦♥✈❡'❣❡♥❝❡✳ ■) ✐+ ♣'♦♣♦+❡❞ ✐♥ )❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥ ❛❞❛♣)❛)✐♦♥ ♦❢ )❤❡ ●❙❊L◆▲❙ ❛❧❣♦'✐)❤♠✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡'❡♥❝❡+ ❝♦♠❡ ❢'♦♠ ✐)+ ✉+❡ ✐♥ ❛ ❙■❙❖ +②+)❡♠✱ )❤❡ ✉+❡ ♦❢ ❛ ❝♦♥)✐♥✉♦✉+✲)✐♠❡ ✜❧)❡' )♦ ❝♦♠♣✉)❡ )❤❡ ❞❡'✐✈❛)✐✈❡+✱ ❛♥❞ )❤❡ ✉+❡ ♦❢ )❤❡ ✜❧)❡'❡❞ ❞❡'✐✈❛)✐✈❡ ❛+ ♦✉)♣✉)✳ ❲❤❡♥ ✉+✐♥❣ )❤❡ ❧❡❛+)✲+_✉❛'❡+ ❛❧❣♦'✐)❤♠✱ )❤❡ ♠♦❞❡❧ ✭✽✮ ❝❛♥ ❜❡ ✇'✐))❡♥ ✐♥ '❡❣'❡++♦' ❢♦'♠ ❛+✿ b y(n)_{(t) = −} 1 ban y_{(t) −}ba1 ban y(1)_{(t) − − · · · −}ban−1 ban y(n−1)(t) +bb0 ban e−Tdpb u(t) + · · · +bbm ban e−Tdpb u(m)(t) b y(n)(t) = bϑTφ(t, bTd) ✭✸✸✮ ✇❤❡'❡ b ϑT =h 1 ban b a1 ban · · · ban−1 ban bb0 ban · · · bbm ban i ✭✸✹✮ φT(t, bTd) =−y(t) −y(1)(t) · · · −y(n−1)(t)

e−Tdpb _u_{(t) · · · e}−Tdpb _u(m)_(t)i ✭✸✺✮ ❇✉)✱ ❜❡❝❛✉+❡ ♦❢ )❤❡ )✐♠❡ ❞❡❧❛②✱ ✭✸✸✮ ✐+ ❛ ♥♦♥❧✐♥❡❛' ♠♦❞❡❧✳ ❚♦ ♠❛❦❡ )❤❡ ✈❡❝)♦' φT_{(t, b}_T d)'❡❛❧✐③❛❜❧❡✱ ✐) ✐+ ♥❡❝❡++❛'② )♦ ❝♦♠♣✉)❡ )❤❡ ❞❡'✐✈❛)✐✈❡+ ♦❢ )❤❡ +✐❣♥❛❧+ u(t) ❛♥❞ y(t)✳ ❚❤✐+ ✐+ ❛❝❤✐❡✈❡❞ ✉+✐♥❣ ❛ ❧♦✇✲♣❛++ ✜❧)❡'✱ ❞❡✜♥❡❞ ❛+✿ F(p) = 1 Λ(p) = 1 (1 + λp)n = 1 1 + λ1p+ . . . + λnpn ✭✸✻✮ ✇❤❡'❡ λ ✐+ ❛ ❝♦♥+)❛♥) ✇❤✐❝❤ ❞❡)❡'♠✐♥❡+ )❤❡ ♣❛++ ❜❛♥❞ ♦❢ )❤❡ ✜❧)❡'✱ ❛♥❞ λ1, . . . , λn ❛'❡ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥) ♦❢ )❤❡ ✜❧)❡'✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ )❤✐+ ♣❛'❛♠❡)❡' ✐♠♣❧✐❡+ ♣'✐♦' ❦♥♦✇❧❡❞❣❡ ♦❢ )❤❡ +②+)❡♠ ❜❛♥❞ ♣❛++✳ ❉❡♥♦)✐♥❣ ✇✐)❤ )❤❡ +✉❜+❝'✐♣) f )❤❡ ✜❧)❡'❡❞ +✐❣♥❛❧+✱ )❤❡ '❡❣'❡++♦' ♠♦❞❡❧ ✭✸✸✮ ❝❛♥ ❜❡ ✇'✐))❡♥✿ b y_f(n)(t) = −ba1 ban y(1)_f (t) + · · · −ban−1 ban y_f(n−1)(t) +bb0 ban e−Tdpb uf(t) + · · · + bbm ban e−Tdpb u(m)_f (t) b y_f(n)(t) = bϑTφf(t, bTd) ✭✸✼✮ ✇❤❡'❡ φT_f(t, bTd) = h −yf(t) · · · −y_f(n−1)(t) e−Tdpb _u f(t) · · · e−Tdpb u (m) f (t) i ✭✸✽✮ ❯+✐♥❣ ✭✸✼✮✱ )❤❡ _✉❛❞'❛)✐❝ ❝'✐)❡'✐♦♥ )❤❛) ❤❛✈❡ )♦ ❜❡ ♠✐♥✐✲ ♠✐③❡❞ ✐+ ✇'✐))❡♥✿ J( bTd) = 1 2 N X k=1 (εf(tk))2 ✭✸✾✮ εf(tk) = yf(n)(tk) − bϑTφf(t, bTd) ✭✹✵✮ ■♥ )❤❡ ❢♦❧❧♦✇✐♥❣✱ ❢♦' )❤❡ ♣'❡+❡♥)❛)✐♦♥ ♦❢ )❤✐+ ❛❧❣♦'✐)❤♠✱ ✐) ✐+ ❡❛+✐❡' )♦ ✉+❡ ♠❛)'✐① ♥♦)❛)✐♦♥✳ ❚❤✉+✱ )❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛'✐❛❜❧❡+ ❛'❡ ❞❡✜♥❡❞✿ Φf( bTd) = φf(1, bTd) · · · φf(N, bTd) T ✭✹✶✮ Z=hy_f(n)(1) · · · yf(n)(N ) iT ✭✹✷✮ E= [ εf(1) · · · εf(N ) ]T ✭✹✸✮ ❲✐)❤ )❤✐+ ♥♦)❛)✐♦♥✱ )❤❡ _✉❛❞'❛)✐❝ ❝'✐)❡'✐♦♥ ✭✸✾✮✱ ✭✹✵✮ ❝❛♥ ❜❡ ✇'✐))❡♥ ❛+✿ J( bTd) = 1 2E T_E _✭✹✹✮ E_{= Z − Φ}f( bTd) bϑ ✭✹✺✮ ❚❤❡ ♣'✐♥❝✐♣❧❡ ♦❢ )❤❡ +❡♣❛'❛❜❧❡ ❛❧❣♦'✐)❤♠ ✐+ )♦ ✜'+)❧② ✐❞❡♥)✐❢② )❤❡ ♥♦♥❧✐♥❡❛' ♣❛'❛♠❡)❡' bTd ✇✐)❤ ❛ ◆▲L ♠❡)❤♦❞

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❤❡♥ ❤❡ ❧✐♥❡❛' ♣❛'❛♠❡ ❡'* bϑ✇✐ ❤ ❤❡ ❧✐♥❡❛' ❧❡❛* *,✉❛'❡* ❛❧❣♦'✐ ❤♠✳ ❚♦ *❤♦✇ ❤♦✇ ❤❡ ❣'❛❞✐❡♥ ❛♥❞ ❤❡ ❍❡**✐❛♥ ❛'❡ ❝♦♠♣✉ ❡❞✱ ❤❡ ,✉❛❞'❛ ✐❝ ❝'✐ ❡'✐♦♥ ✭✹✹✮ ♠✉* ❜❡ ✇'✐ ❡♥ ❛* ❛ ❢✉♥❝ ✐♦♥ ♦❢ ❤❡ ✐♠❡ ❞❡❧❛② bTd✳ ❚❤❡ ❡* ✐♠❛ ❡❞ ✈❡❝ ♦' b ϑ✱ ❝❛♥ ❜❡ '❡♣❧❛❝❡❞ ❜② ✐ * ✐❞❡♥ ✐✜❡❞ ✈❛❧✉❡ ✉*✐♥❣ ❤❡ ❧✐♥❡❛' ▲❡❛* ❙,✉❛'❡* ✭▲❙✮ ❛❧❣♦'✐ ❤♠✳ b ϑ=hΦT f( bTd)Φf( bTd) i−1 ΦT f( bTd)Z = Φ†f( bTd)Z ✭✹✻✮ ✇❤❡'❡ Φ† f ✐* ❤❡ ♣*❡✉❞♦✲✐♥✈❡'*❡ ♦❢ Φf✳ ❙♦ ❤❡ ,✉❛❞'❛ ✐❝ ❝'✐ ❡'✐♦♥ ✭✹✹✮ ❝❛♥ ❜❡ ❡①♣'❡**❡❞ ♦♥❧② ❛* ❛ ❢✉♥❝ ✐♦♥ ♦❢ ❤❡ ✐♠❡ ❞❡❧❛②✿ J( bTd) = 1 2 Z − Φf( bTd)Φ†f( bTd)Z 2 2 ✭✹✼✮ ❚♦ ❝♦♠♣✉ ❡ ❤❡ ❣'❛❞✐❡♥ ❛♥❞ ❤❡ ❍❡**✐❛♥✱ ❛ ♥♦ ❛ ✐♦♥ *✐♠✐❧❛' ♦ ◆❣✐❛ ✭✷✵✵✶✮ ✐* ✉*❡❞✿ EbTd= dE d bTd = −dΦf( bTd) d bTd b ϑ_{= −}dΦf( bTd) d bTd Φ†_f( bTd)Z ✭✹✽✮ Ebϑ= dE d bϑ = −Φf( bTd) ✭✹✾✮ ❚❤❡ ❣'❛❞✐❡♥ ❛♥❞ ❤❡ ❍❡**✐❛♥✱ ✇✐ ❤ ❤❡ ●❛✉**✲◆❡✇ ♦♥ ❛♣♣'♦①✐♠❛ ✐♦♥✱ ✇✐❧❧ ❤❛✈❡ ❤❡ ❢♦'♠ ✭❛* *❤♦✇♥ ✐♥ ◆❣✐❛ ✭✷✵✵✶✮✮✿ J′( bTd) = E_TdT_bE ✭✺✵✮ J′′( bTd) = ET_b TdEbTd− E T b TdEbϑ h ET b ϑEbϑ i−1 ET b ϑEbTd ✭✺✶✮ ❨❛♥❣ ❡! ❛❧✳ ♣'♦♣♦*❡ ♦ ✉*❡ ❤❡ ●❛✉**✲◆❡✇ ♦♥ ◆▲N ♠❡ ❤♦❞✱ ✇✐ ❤ ❛ * ♦❝❤❛* ✐❝ ❡'♠ ❛❞❞❡❞ ♦ ❤❡ ❣'❛❞✐❡♥ ✳ ❉❡✲ ♥♦ ✐♥❣ η ❛ ●❛✉**✐❛♥ '❛♥❞♦♠ ✈❛'✐❛❜❧❡✱ ❤❡ ❡✈♦❧✉ ✐♦♥ ♦❢ ❤❡ ❡* ✐♠❛ ❡❞ ✐♠❡ ❞❡❧❛② ✇✐ ❤ ❤❡ ●❛✉**✲◆❡✇ ♦♥ ❛❧❣♦'✐ ❤♠ ✐* ❤❡♥ ✇'✐ ❡♥✿ b Td,j+1= bTd,j− h J′′( bTd) i−1 J′( bTd) − βjη ✭✺✷✮ ✇❤❡'❡ βj ✐* ❤❡ ✈❛'✐❛♥❝❡ ♦❢ ❤❡ '❛♥❞♦♠ ✈❛'✐❛❜❧❡ η✳ ❚❤✐* ✈❛'✐❛♥❝❡ ✐* ❝♦♠♣✉ ❡❞ ✉*✐♥❣ βj = β0J( bTd)✱ ✇✐ ❤ β0 ❛ ♣♦*✐ ✐✈❡ ❝♦♥* ❛♥ ❝❤♦*❡♥ *✉✣❝✐❡♥ ❧② ❧❛'❣❡✳ ❚❤❡ ✐❞❡❛ ✐* ✐❢ ❤❡ ,✉❛❞'❛ ✐❝ ❝'✐ ❡'✐♦♥ ❜❡❝♦♠❡* *♠❛❧❧❡'✱ ❤❡ * ♦❝❤❛* ✐❝ ♣❡' ✉'❜❛ ✐♦♥ βjη ❜❡❝♦♠❡* ✇❡❛❦❡' ❛♥❞ ❤❡ ❛❧❣♦'✐ ❤♠ ❡♥❞ ♦ ❜❡ ❝❧♦*❡' ♦ ❤❡ ❝❧❛**✐❝❛❧ ●❛✉**✲◆❡✇ ♦♥ ❛❧❣♦'✐ ❤♠✳ ❚❤❡ ❛❧❣♦'✐ ❤♠ ✉*❡❞ ❤❛* ❤❡ ❢♦'♠✿ ✭✶✮ ▲❡ ❥❂✵✳ ❙❡ β0✱ ❤❡ ✐♥✐ ✐❛❧ ❡* ✐♠❛ ❡ bTd,0❛♥❞ ❛♥ ✉♣♣❡' ❜♦✉♥❞ ♦❢ ✐♠❡ ❞❡❧❛②* Td✳ ✭✷✮ ❈♦♠♣✉ ❡ ♣'❡✜❧ ❡'❡❞ *✐❣♥❛❧* ❤❛ ❛'❡ ♣❛' ♦❢ Φf( bTd)✱ ✇✐ ❤ ❤❡ ❧♦✇✲♣❛** ✜❧ ❡' ✭✸✻✮✳ ✭✸✮ ❙❡ βj= β0J( bTd)✉*✐♥❣ ✭✹✼✮✳ ✭✹✮ N❡'❢♦'♠ ❤❡ ❢♦❧❧♦✇✐♥❣ ✭❛✮ ❈♦♠♣✉ ❡ ∆ bTd,j+1= − h J′′( bTd,j) i−1 J′( bTd,j) − βjη ✳ ✭❜✮ ❈♦♠♣✉ ❡ bTd,j+1= bTd,j+ ∆ bTd,j+1✳ ✭❝✮ ❈❤❡❝❦ ✐❢ 0 6 bTd,j+1 6Td✳ ■❢ ♥♦ ✱ ❧❡ ∆ bTd,j+1 = 0.5∆ bTd,j+1 ❛♥❞ ❣♦ ❜❛❝❦ ♦ ✭❜✮✳ ✭❞✮ ❈❤❡❝❦ ✐❢ J( bTd,j+1) 6 J( bTd,j)✳ ■❢ ♥♦ ✱ ❧❡ ∆ bTd,j+1= 0.5∆ bTd,j+1 ❛♥❞ ❣♦ ❜❛❝❦ ♦ ✭❜✮✳ ✭✺✮ ❚❡'♠✐♥❛ ❡ ❤❡ ❛❧❣♦'✐ ❤♠ ✐❢ ❤❡ * ♦♣♣✐♥❣ ❝♦♥❞✐ ✐♦♥ ✐* *❛ ✐*✜❡❞✳ ❖ ❤❡'✇✐*❡✱ ❧❡ j = j +1 ❛♥❞ ❣♦ ❜❛❝❦ ♦ * ❡♣ ✸✳ ✭✻✮ ❋✐♥❛❧❧②✱ ❜② *✉❜* ✐ ✉ ✐♥❣ ❤❡ ✜♥❛❧ ✈❛❧✉❡ ♦❢ bTd✱ ❤❡ ❧✐♥❡❛' ♣❛'❛♠❡ ❡' ✈❡❝ ♦' bϑ ❝❛♥ ❜❡ ❡* ✐♠❛ ❡❞ ❜② ❤❡ ▲❙ ♠❡ ❤♦❞ ❣✐✈❡♥ ✐♥ ✭✹✻✮✳ ✺✳ ❙■▼❯▲❆❚■❖◆ ❘❊❙❯▲❚❙ ❈♦♥*✐❞❡' ❛ ❤✐'❞ ♦'❞❡' *②* ❡♠ ✇✐ ❤ ❛ ③❡'♦✱ ❞❡✜♥❡❞ ❜②✿ G(p) = −1 + 2p (p + 2) (p2_{+ 2ξω} np+ ωn2) e−Tdp ✭✺✸✮ G(p) = −b0+ b1p 1 + a1p+ a2p2+ a3p3 e−Tdp ✭✺✹✮ ✇✐ ❤ ξ= 0.2; ωn= √ 3 '❛❞✴*; b0= 0.167; b1= 0.333; a1= 0.731; a2= 0.449; a3= 0.167; Td= 1.3*. ✭✺✺✮ ❚❤❡ *②* ❡♠ ✐* ❡①❝✐ ❡❞ ✇✐ ❤ ❛ N*❡✉❞♦ ❘❛♥❞♦♠ ❇✐♥❛'② ❙✐❣♥❛❧ ✭N❘❇❙✮✳ ❆♥ ❛❞❞✐ ✐✈❡ ✇❤✐ ❡ ♥♦✐*❡ ✐* ❛❞❞❡❞ ♦ ❤❡ ♦✉ ♣✉ ✇✐ ❤ ❛ ❙✐❣♥❛❧ ♦ ◆♦✐*❡ ❘❛ ✐♦ ✭❙◆❘✮ ♦❢ ✶✺ ❞❇✳ ❚❤✐* ❙◆❘ ✐* ❞❡✜♥❡❞ ❛*✿ SN R= 10 log10 var(y0) var_{(y − y}0) ✭✺✻✮ ✇❤❡'❡ y0✐* ❤❡ ♥♦✐*❡❧❡** ♦✉ ♣✉ ❛♥❞ y ✐* ❤❡ ♥♦✐*② ♦✉ ♣✉ ✳ ❚❤❡ *✐❣♥❛❧* ✉*❡❞ ❢♦' ✐❞❡♥ ✐✜❝❛ ✐♦♥ ❛'❡ ♣'❡*❡♥ ❡❞ ✐♥ ✜❣✉'❡ ✷✳ ❋♦' ❤❡ ❝❧♦*❡❞✲❧♦♦♣ *✐♠✉❧❛ ✐♦♥*✱ ❤❡ ❝♦♥ '♦❧❧❡' ✐* ✉♥❡❞ ✉*✲ ✐♥❣ ❤❡ ❞✐'❡❝ ♠❡ ❤♦❞✱ ✇✐ ❤ ❤❡ ♣❛'❛♠❡ ❡' ✈❛❧✉❡* ♦❜ ❛✐♥❡❞ ✇✐ ❤ ❤❡ ❈❚❖❊ ❛❧❣♦'✐ ❤♠✳ ❲✐ ❤ ❤✐* ❛♣♣'♦❛❝❤ ❤❡ ❞❡*✐'❡❞ ❝❧♦*❡❞ ❧♦♦♣ '❛♥*❢❡' ❢✉♥❝ ✐♦♥ ♠♦❞❡❧ GCL ✐* ❝❤♦*❡♥ ♦ ❤❛✈❡ ❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦'♠✿ 0 50 100 150 −1 −0.5 0 0.5 1 time (s) 0 50 100 150 −10 −5 0 5 10 time (s) input u(t) output y(t) reference r(t) input u(t) output y(t)

a) Open loop system

b) Closed−loop system

❋✐❣✉$❡ ✷✳ ❙✐❣♥❛❧ ✉,❡❞ ❢♦$ ✐❞❡♥0✐✜❝❛0✐♦♥ ✐♥ ❛✮ ♦♣❡♥ ❧♦♦♣ ❛♥❞ ❜✮ ❝❧♦,❡❞✲❧♦♦♣

(7)

GCL= −6 (−1 + 2p) (p + 2) p2_{+ 2 × 1.5 ×}√_{3p + 3} e −Tdp _✭✺✼✮ ❲❤❡'❡ ✐) ❝❛♥ ❜❡ .❡❡♥ )❤❛) GCL ❤❛✈❡ )❤❡ .❛♠❡ ✉♥.)❛❜❧❡ ③❡'♦ ❛♥❞ )❤❡ .❛♠❡ )✐♠❡ ❞❡❧❛② ✈❛❧✉❡ ✭❛. ✐♥ G(p)✮ ❜✉) )❤❡ .)❛)✐❝ ❣❛✐♥ ✐. ♥♦✇ ❡9✉❛❧ )♦ ♦♥❡ ❛♥❞ )❤❡ ❞❡.✐'❡❞ ❝❧♦.❡❞ ❧♦♦♣ ❞❛♠♣✐♥❣ ❢❛❝)♦' ✐. ♥♦✇ ❡9✉❛❧ )♦ ✶✳✺ ✭✇❡❧❧ ❞❛♠♣❡❞ ♣♦❧❡.✮✳ ❚❤❡♥✱ )❤❡ ❝♦♥)'♦❧❧❡' ✐. ❝♦♠♣✉)❡❞ ✉.✐♥❣ )❤❡ ❢♦'♠✉❧❛✿ C(p) = 1 G(p) GCL(p) 1 − GCL(p) ✭✺✽✮ ❚♦ ♦❜)❛✐♥ ❛ ✜♥✐)❡ ❞✐♠❡♥.✐♦♥ ❝♦♥)'♦❧❧❡' C(p)✱ )❤❡ ❢♦❧❧♦✇✐♥❣ .❡❝♦♥❞ ♦'❞❡' C❛❞D ❛♣♣'♦①✐♠❛)✐♦♥ ♦❢ )❤❡ )✐♠❡ ❞❡❧❛② Td✇❛. ✉.❡❞✿ e−Tdp=1 − 1 2Tdp+121 (Tdp) 2 1 + 1 2Tdp+ 1 12(Tdp) 2 ✭✺✾✮ ❲✐)❤ )❤✐. ❛♣♣'♦①✐♠❛)✐♦♥ )❤❡ ❝♦♥)'♦❧❧❡' ❜❡❝♦♠❡. ✿ C(p) = −256 − 353p − 272p 2 − 144p3− 43.9p4− 6p5 236p + 57.5p2_{+ 66p}3_{+ 11.8p}4_{+ p}5 ✭✻✵✮ ✶✵✵ ▼♦♥)❡✲❈❛'❧♦ .✐♠✉❧❛)✐♦♥. ❛'❡ ❝♦♥❞✉❝)❡❞ )♦ ❛♥❛❧②③❡ )❤❡ .)❛)✐.)✐❝❛❧ ♣'♦♣❡')✐❡. ♦❢ )❤❡ ❈❚❖❊ ❛♥❞ ❈❚❈▲❖❊ ♣'♦♣♦.❡❞ ❛❧❣♦'✐)❤♠.✳ ❚❤❡ '❡.✉❧). ❛'❡ ♣'❡.❡♥)❡❞ ✐♥ )❡'♠. ♦❢ )❤❡ ♠❡❛♥ ❛♥❞ )❤❡ .)❛♥❞❛'❞ ❞❡✈✐❛)✐♦♥✳ ❚❤❡ ♣❛'❛♠❡)'✐❝ ❞✐.)❛♥❝❡✱ ❝♦♠♣✉)❡❞ ✉.✐♥❣ D = s Pn+m+1 l=1 P100 i=1 Θl−bΘl,i Θl 2 ✐. ❛❧.♦ ♣'❡.❡♥)❡❞✳ ✇❤❡'❡ i ✐. )❤❡ ✐♥❞❡① ♦❢ )❤❡ ▼♦♥)❡✲❈❛'❧♦ .✐♠✉❧❛)✐♦♥✱ ❛♥❞ l )❤❡ ✐♥❞❡① ♦❢ )❤❡ ♣❛'❛♠❡)❡' ✈❡❝)♦' Θ✳ ❚❤❡ .❛♠♣❧✐♥❣ ♣❡'✐♦❞ ❝❤♦.❡♥ ✐. Ts = 10 ♠.✳ ❚❤❡ ✐♥✐)✐❛❧ ♣❛'❛♠❡)❡' ❢♦' )❤❡ ♣'♦♣♦.❡❞ ❛❧❣♦'✐)❤♠ ✐. µ = 0.1❀ ❢♦' )❤❡ ●❙❊C◆▲❙ ♠❡)❤♦❞✱ )❤❡ ✐♥✐)✐❛❧ ✈❛'✐❛♥❝❡ ♦❢ )❤❡ .)♦❝❤❛.)✐❝ ✈❛'✐❛❜❧❡ ✐. β0 = 105✱ ❛. ♣'♦♣♦.❡❞ ✐♥ ❨❛♥❣ ❡) ❛❧✳ ✭✷✵✵✼✮✳ ❚❤❡ ✐♥✐)✐❛❧ ✈❛❧✉❡ ♦❢ Td✐. ✺ . ❛♥❞ )❤❡ ♣❛'❛♠❡)❡' ❢♦' )❤❡ ✜❧)❡' ✭✸✻✮ ✐. ❝❤♦.❡♥ ❛. λ = 0.2 ✇❤✐❝❤ ❝♦''❡.♣♦♥❞. ❛♣♣'♦①✐♠❛)❡❧② )♦ )❤❡ ❜❛♥❞♣❛.. ♦❢ )❤❡ .②.)❡♠✳ ❚❤❡ .②.)❡♠ ✐. ✜'.)❧② ✐❞❡♥)✐✜❡❞ ✐♥ ♦♣❡♥ ❧♦♦♣✳ ❚❤❡ .✐♠✲ ✉❧❛)✐♦♥ ♣'♦❝❡❞✉'❡ ✐. )♦ ♦❜)❛✐♥ ❛♥ ✐♥✐)✐❛❧ ❡.)✐♠❛)❡ ♦❢ )❤❡ ♣❛'❛♠❡)❡'. ✉.✐♥❣ )❤❡ ●❙❊C◆▲❙ ♠❡)❤♦❞✳ ❚❤❡♥✱ )❤❡ ♣'♦✲ ♣♦.❡❞ ♦✉)♣✉) ❡''♦' ♠❡)❤♦❞. .)❛') ✇✐)❤ )❤✐. ✐♥✐)✐❛❧ ❡.)✐♠❛)❡✳ ❚❤❡ .✐♠✉❧❛)✐♦♥ '❡.✉❧). ❛'❡ .✉♠♠❛'✐③❡❞ ✐♥ )❛❜❧❡ ✶✳ ❲✐)❤ ❈❚❖❊ ❛♥❞ ❈❚❈▲❖❊ )❤❡ ♣❛'❛♠❡)❡' ❡.)✐♠❛)✐♦♥ 9✉❛❧✐)② ✐. ✈❡'② .✐♠✐❧❛' ❛♥❞ )❤❡ ❡.)✐♠❛)❡❞ .②.)❡♠ ❛'❡ ✉♥❜✐❛.❡❞✳ ✻✳ ❈❖◆❈▲❯❙■❖◆ ❚❤✐. ♣❛♣❡' ❤❛. ♣'❡.❡♥)❡❞ )✇♦ )✐♠❡ ❞♦♠❛✐♥ ♦✉)♣✉) ❡'✲ '♦' ❛❧❣♦'✐)❤♠. ❝❛❧❧❡❞ ❈❚❖❊ ❛♥❞ ❈❚❈▲❖❊ )♦ ✐❞❡♥)✐❢② ❛ ❝♦♥)✐♥✉♦✉.✲)✐♠❡ ❧✐♥❡❛' .②.)❡♠ ✇✐)❤ )✐♠❡ ❞❡❧❛②✱ ✐♥ ♦♣❡♥ ♦' ❝❧♦.❡❞✲❧♦♦♣ ✇✐)❤ ✈❡'② ❣♦♦❞ .)❛)✐.)✐❝❛❧ ♣'♦♣❡')✐❡.✳ ■) ❤❛. ❜❡❡♥ ❛❧.♦ ♣'♦♣♦.❡❞ ❛ ♠♦❞✐✜❡❞ ✈❡'.✐♦♥ ♦❢ )❤❡ ●❙❊C◆▲❙ )♦ ✐♥✐)✐❛❧✐③❡ )❤❡♠✳ ❚❛❜❧❡ ✶✳ ❙✐♠✉❧❛)✐♦♥ '❡.✉❧). ♦❢ )❤❡ ✶✵✵ ▼♦♥)❡✲ ❈❛'❧♦ '✉♥. ❛"❛♠❡%❡" ●❙❊ ◆▲❙ ❈❚❖❊ ❈❚❈▲❖❊ bb0✭✲✵✳✶✻✼✮ ✲✵✳✶✺✷✸ ✲✵✳✶✻✻✻ ✲✵✳✶✻✻✽ ±✵✳✵✵✶✵ ±✵✳✵✵✵✺ ±✵✳✵✵✵✺ bb1✭✵✳✸✸✸✮ ✵✳✷✽✸✺ ✵✳✸✸✸✸ ✵✳✸✸✸✸ ±✵✳✵✵✶✻ ±✵✳✵✵✵✼ ±✵✳✵✵✶✵ ba1✭✵✳✼✸✶✮ ✵✳✹✸✼✺ ✵✳✼✸✵✾ ✵✳✼✸✵✼ ±✵✳✵✵✸✾ ±✵✳✵✵✸✷ ±✵✳✵✵✹✻ ba2✭✵✳✹✹✾✮ ✵✳✹✸✻✵ ✵✳✹✹✽✽ ✵✳✹✹✽✺ ±✵✳✵✵✶✷ ±✵✳✵✵✼ ±✵✳✵✵✶✼ ba3✭✵✳✶✻✼✮ ✵✳✵✼✶✽ ✵✳✶✻✻✼ ✵✳✶✻✻✺ ±✵✳✵✵✷✵ ±✵✳✵✵✶✷ ±✵✳✵✵✷✶ b Td✭✶✳✸✮ ✶✳✸✾✽✺ ✶✳✷✾✾✾ ✶✳✸✵✵✺ ±✵✳✵✵✽✶ ±✵✳✵✵✷✹ ±✵✳✵✵✸✼ ❉ ✼✳✷ ✵✳✶ ✵✳✶✺ ❆✈❡"❛❣❡ ♦❢ ✐%❡"❛%✐♦♥D ✷✸ ✶✵ ✶✶ ❘❊❋❊❘❊◆❈❊❙ ❆❤♠❡❞✱ ❙✳✱ ❍✉❛♥❣✱ ❇✳✱ ❛♥❞ ❙❤❛❤✱ ❙✳ ✭✷✵✵✻✮✳ C❛'❛♠❡)❡' ❛♥❞ ❞❡❧❛② ❡.)✐♠❛)✐♦♥ ♦❢ ❝♦♥)✐♥✉♦✉.✲)✐♠❡ ♠♦❞❡❧. ✉.✐♥❣ ❛ ❧✐♥❡❛' ✜❧)❡'✳ ❏♦✉#♥❛❧ ♦❢ (#♦❝❡++ ❈♦♥-#♦❧✱ ✶✻✭✹✮✱ ✸✷✸ ✕ ✸✸✶✳ ❇❛②..❡✱ ❆✳✱ ❈❛''✐❧❧♦✱ ❋✳❏✳✱ ❛♥❞ ❍❛❜❜❛❞✐✱ ❆✳ ✭✷✵✶✶✮✳ ❚✐♠❡ ❞♦♠❛✐♥ ✐❞❡♥)✐✜❝❛)✐♦♥ ♦❢ ❝♦♥)✐♥✉♦✉.✲)✐♠❡ .②.)❡♠. ✇✐)❤ )✐♠❡ ❞❡❧❛② ✉.✐♥❣ ♦✉)♣✉) ❡''♦' ♠❡)❤♦❞ ❢'♦♠ .❛♠♣❧❡❞ ❞❛)❛✳ ■♥ ✶✽-❤ ■❋❆❈ ❲♦#❧❞ ❈♦♥❣#❡++✱ ✈♦❧✉♠❡ ✶✽✳ ▼✐❧❛♥♦✱ ■)❛❧②✳ ❈❛''✐❧❧♦✱ ❋✳❏✳✱ ❇❛②..❡✱ ❆✳✱ ❛♥❞ ❍❛❜❜❛❞✐✱ ❆✳ ✭✷✵✵✾✮✳ ❖✉)♣✉) ❡''♦' ✐❞❡♥)✐✜❝❛)✐♦♥ ❛❧❣♦'✐)❤♠. ❢♦' ❝♦♥)✐♥✉♦✉.✲)✐♠❡ .②.✲ )❡♠. ♦♣❡'❛)✐♥❣ ✐♥ ❝❧♦.❡❞✲❧♦♦♣✳ ■♥ ✶✺-❤ ■❋❆❈ ❙②♠♣♦+✐✉♠ ♦♥ ❙②+-❡♠ ■❞❡♥-✐✜❝❛-✐♦♥✱ ❙❨❙■❉ ✷✵✵✾✱ ✈♦❧✉♠❡ ✶✺ ♦❢ ❙②+-❡♠ ■❞❡♥-✐✜❝❛-✐♦♥✳ ❙) ▼❛❧♦✱ ❋'❛♥❝❡✳ ●❛'♥✐❡'✱ ❍✳ ❛♥❞ ❲❛♥❣✱ ▲✳ ✭❡❞.✳✮ ✭✷✵✵✽✮✳ ■❞❡♥-✐✜❝❛-✐♦♥ ♦❢ ❈♦♥-✐♥✉♦✉+✲-✐♠❡ ▼♦❞❡❧+ ❢#♦♠ ❙❛♠♣❧❡❞ ❉❛-❛✳ ❆❞✈❛♥❝❡. ✐♥ ■♥❞✉.)'✐❛❧ ❈♦♥)'♦❧✳ ❙♣'✐♥❣❡'✳ ❍✇❛♥❣✱ ❙✳❍✳ ❛♥❞ ▲❛✐✱ ❙✳❚✳ ✭✷✵✵✹✮✳ ❯.❡ ♦❢ )✇♦✲.)❛❣❡ ❧❡❛.)✲.9✉❛'❡. ❛❧❣♦'✐)❤♠. ❢♦' ✐❞❡♥)✐✜❝❛)✐♦♥ ♦❢ ❝♦♥)✐♥✉♦✉. .②.)❡♠. ✇✐)❤ )✐♠❡ ❞❡❧❛② ❜❛.❡❞ ♦♥ ♣✉❧.❡ '❡.♣♦♥.❡.✳ ❆✉-♦♠❛-✐❝❛✱ ✹✵✭✾✮✱ ✶✺✻✶ ✕ ✶✺✻✽✳ ▲❛♥❞❛✉✱ ■✳❉✳ ❛♥❞ ❑❛'✐♠✐✱ ❆✳ ✭✶✾✾✼✮✳ ❘❡❝✉'.✐✈❡ ❛❧❣♦'✐)❤♠. ❢♦' ✐❞❡♥)✐✜❝❛)✐♦♥ ✐♥ ❝❧♦.❡❞ ❧♦♦♣✿ ❆ ✉♥✐✜❡❞ ❛♣♣'♦❛❝❤ ❛♥❞ ❡✈❛❧✉❛)✐♦♥✳ ❆✉-♦♠❛-✐❝❛✱ ✸✸✭✽✮✱ ✶✹✾✾ ✕ ✶✺✷✸✳ ▲✐✉✱ ▼✳✱ ❲❛♥❣✱ ◗✳●✳✱ ❍✉❛♥❣✱ ❇✳✱ ❛♥❞ ❍❛♥❣✱ ❈✳❈✳ ✭✷✵✵✼✮✳ ■♠♣'♦✈❡❞ ✐❞❡♥)✐✜❝❛)✐♦♥ ♦❢ ❝♦♥)✐♥✉♦✉.✲)✐♠❡ ❞❡❧❛② ♣'♦✲ ❝❡..❡. ❢'♦♠ ♣✐❡❝❡✇✐.❡ .)❡♣ )❡.).✳ ❏♦✉#♥❛❧ ♦❢ (#♦❝❡++ ❈♦♥-#♦❧✱ ✶✼✭✶✮✱ ✺✶ ✕ ✺✼✳ ◆❣✐❛✱ ▲✳ ✭✷✵✵✶✮✳ ❙❡♣❛'❛❜❧❡ ♥♦♥❧✐♥❡❛' ❧❡❛.)✲.9✉❛'❡. ♠❡)❤✲ ♦❞. ❢♦' ❡✣❝✐❡♥) ♦✛✲❧✐♥❡ ❛♥❞ ♦♥✲❧✐♥❡ ♠♦❞❡❧✐♥❣ ♦❢ .②.)❡♠. ✉.✐♥❣ ❦❛✉)③ ❛♥❞ ❧❛❣✉❡''❡ ✜❧)❡'.✳ ❈✐#❝✉✐-+ ❛♥❞ ❙②+-❡♠+ ■■✿ ❆♥❛❧♦❣ ❛♥❞ ❉✐❣✐-❛❧ ❙✐❣♥❛❧ (#♦❝❡++✐♥❣✱ ■❊❊❊ ❚#❛♥+✲ ❛❝-✐♦♥+ ♦♥✱ ✹✽✭✻✮✱ ✺✻✷ ✕✺✼✾✳ ❲❛♥❣✱ ◗✳●✳✱ ●✉♦✱ ❳✳✱ ❛♥❞ ❩❤❛♥❣✱ ❨✳ ✭✷✵✵✶✮✳ ❉✐'❡❝) ✐❞❡♥)✐✜❝❛)✐♦♥ ♦❢ ❝♦♥)✐♥✉♦✉. )✐♠❡ ❞❡❧❛② .②.)❡♠. ❢'♦♠ .)❡♣ '❡.♣♦♥.❡.✳ ❏♦✉#♥❛❧ ♦❢ (#♦❝❡++ ❈♦♥-#♦❧✱ ✶✶✭✺✮✱ ✺✸✶ ✕ ✺✹✷✳ ❨❛♥❣✱ ❩✳❏✳✱ ■❡♠✉'❛✱ ❍✳✱ ❑❛♥❛❡✱ ❙✳✱ ❛♥❞ ❲❛❞❛✱ ❑✳ ✭✷✵✵✼✮✳ ■❞❡♥)✐✜❝❛)✐♦♥ ♦❢ ❝♦♥)✐♥✉♦✉.✲)✐♠❡ .②.)❡♠. ✇✐)❤ ♠✉❧)✐♣❧❡ ✉♥❦♥♦✇♥ )✐♠❡ ❞❡❧❛②. ❜② ❣❧♦❜❛❧ ♥♦♥❧✐♥❡❛' ❧❡❛.)✲.9✉❛'❡. ❛♥❞ ✐♥.)'✉♠❡♥)❛❧ ✈❛'✐❛❜❧❡ ♠❡)❤♦❞.✳ ❆✉-♦♠❛-✐❝❛✱ ✹✸✭✼✮✱ ✶✷✺✼ ✕ ✶✷✻✹✳