The paper describes a technique for stability analysis of proportional-integral (PI) controller in linear continuous-time interval control systems. The stability conditions of Kharitonov''s theorem together with related criterions, such as Routh-Hurwitz criterion for continuous-time systems, bring out sets of polynomial inequalities.
❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❋❛st ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ❢♦r Pr♦♣♦rt✐♦♥❛❧✲■♥t❡❣r❛❧ ❈♦♥tr♦❧❧❡r ✐♥ ■♥t❡r✈❛❧ ❙②st❡♠s ❍❛✉ ❍✉✉ ❱❖ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠ ❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❍❛✉ ❍✉✉ ❱❖ ✭❡♠❛✐❧✿ ✈♦❤✉✉❤❛✉❅t❞t✳❡❞✉✳✈♥✮ ✭❘❡❝❡✐✈❡❞✿ ✷✻✲▼❛r❝❤✲✷✵✶✽❀ ❛❝❝❡♣t❡❞✿ ✵✶✲❏✉❧②✲✷✵✶✽❀ ♣✉❜❧✐s❤❡❞✿ ✷✵✲❏✉❧②✲✷✵✶✽✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✽✷✷✳✶✽✹ ❆❜str❛❝t✳ ❚❤❡ ♣❛♣❡r ❞❡s❝r✐❜❡s ❛ t❡❝❤♥✐q✉❡ ❢♦r st❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ♣r♦♣♦rt✐♦♥❛❧✲✐♥t❡❣r❛❧ ✭P■✮ ❝♦♥tr♦❧❧❡r ✐♥ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠s✳ ❚❤❡ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ♦❢ ❑❤❛r✐t♦♥♦✈✬s t❤❡♦r❡♠ t♦❣❡t❤❡r ✇✐t❤ r❡❧❛t❡❞ ❝r✐✲ t❡r✐♦♥s✱ s✉❝❤ ❛s ❘♦✉t❤✲❍✉r✇✐t③ ❝r✐t❡r✐♦♥ ❢♦r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠s✱ ❜r✐♥❣ ♦✉t s❡ts ♦❢ ♣♦❧②✲ ♥♦♠✐❛❧ ✐♥❡q✉❛❧✐t✐❡s✳ ❚❤❡ s❡ts ❛r❡ ✈❡r② ❞✐✣❝✉❧t t♦ s♦❧✈❡ ❞✐r❡❝t❧②✱ ❡s♣❡❝✐❛❧❧② ✐♥ ❝❛s❡ ♦❢ ❤✐❣❤✲♦r❞❡r s②st❡♠s✳ ❉✐r❡❝t t❡❝❤♥✐q✉❡ ✇❛s ✉s❡❞ ❢♦r st❛❜✐❧✐t② ❛♥❛❧②s✐s ✇✐t❤♦✉t s♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐♥❡q✉❛❧✐t✐❡s✳ ❙♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥ ❞✐r❡❝t❧② ♠❛❦❡s ✐ts ❝♦♠♣✉t✐♥❣ s♣❡❡❞ ❧♦✇✳ ■♥ t❤❡ ♣❛♣❡r✱ ❛ s❡t t❤❡♦r②✲ ❜❛s❡❞ t❡❝❤♥✐q✉❡ ✐s ♣r♦♣♦s❡❞ ❢♦r ✜♥❞✐♥❣ r♦❜✉st st❛❜✐❧✐t② r❛♥❣❡ ♦❢ P■ ❝♦♥tr♦❧❧❡r ✇✐t❤♦✉t s♦❧✈✐♥❣ ❛♥② ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ❞✐r❡❝t❧②✳ ❈♦♠♣✉t❛✲ t✐♦♥ r❡s✉❧ts ❝♦♥✜r♠ ❡①♣❡❝t❡❞ ❝♦♠♣✉t✐♥❣ s♣❡❡❞ ♦❢ t❤❡ ♣r♦♣♦s❡❞ t❡❝❤♥✐q✉❡✳ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❙t❛❜✐❧✐t② ❛♥❛❧②s✐s ❛♥❞ ❞❡s✐❣♥ ♦❢ ❝♦♥tr♦❧❧❡rs ❢♦r ♠✉❧t✐♣❧❡ ♠♦❞❡❧ ♦r ✉♥❝❡rt❛✐♥ ♠♦❞❡❧ s②st❡♠s r❡✲ q✉✐r❡ ♠❛♥② ❝♦♠♣❧✐❝❛t❡❞ ♠❡t❤♦❞s ❬✶❪✲❬✹❪✳ ❋♦r ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠s ✇❤✐❝❤ ❛r❡ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❝♦♥tr♦❧ s②st❡♠s ✇✐t❤ ✐♥t❡r✈❛❧ ♣❛r❛♠❡t❡rs✱ r♦❜✉st st❛❜✐❧✐t② ❛♥❛❧✲ ②s✐s ✐s r❡❞✉❝❡❞ ❜② ✉s✐♥❣ ❑❤❛r✐t♦♥♦✈✬s t❤❡♦r❡♠ ❬✺❪✳ ❚❤❡ t❤❡♦r❡♠ ♣r♦✈✐❞❡s ❛♥ ❡❛s②✲✐♠♣❧❡♠❡♥t✐♥❣ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❍✉r✇✐t③ st❛❜✐❧✐t② ♦❢ ❡♥t✐r❡ ❢❛♠✐❧② ♦❢ ❛ s❡t ♦❢ ♣♦❧②♥♦♠✐✲ ❛❧s ✲ ❝❛❧❧❡❞ ✐♥t❡r✈❛❧ ♣♦❧②♥♦♠✐❛❧s ❬✻❪✳ ■♥ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠s✱ ❝❤❡❝❦✐♥❣ st❛❜✐❧✐t② ♦❢ t❤❡ ❢❛♠✐❧② ✐s r❡♣❧❛❝❡❞ ❜② ♦♥❧② ❝❤❡❝❦✐♥❣ st❛❜✐❧✐t② ♦❢ ✹ ♦r ✽ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❝❛s❡ ♦❢ r❡❛❧✲❝♦❡✣❝✐❡♥t ♣♦❧②♥♦♠✐❛❧s ♦r ❝♦♠♣❧❡①✲❝♦❡✣❝✐❡♥t ♣♦❧②♥♦♠✐❛❧s✳ ■♥ ❝❛s❡ ♦❢ ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s✱ ♥✉♠❜❡r ♦❢ ♣♦❧②✲ ♥♦♠✐❛❧s t❤❛t ♠✉st ❜❡ ❝❤❡❝❦❡❞ ❢♦r ❍✉r✇✐t③ st❛✲ ❜✐❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ s②st❡♠ ♦r❞❡r ❛♥❞ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛ s✉♠ ♦❢ ❊✉❧❡r ❢✉♥❝t✐♦♥s ❬✼❪✳ ▼♦st ♦❢ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧❧❡rs ✐♥ t❤❡ ✐♥❞✉str✐❛❧ ♣r♦❝❡ss❡s ❛r❡ P■ ❝♦♥tr♦❧❧❡rs ❬✽❪✲❬✶✵❪ s✉❝❤ ❛s r♦t♦r s♣❡❡❞ ❛❞❛♣t❛t✐♦♥ ♠❡❝❤❛♥✐s♠ ♦❢ ♠♦❞❡❧ r❡❢❡r❡♥❝❡ ❛❞❛♣t✐✈❡ s②st❡♠ ❡st✐♠❛t♦r ✐♥ s♣❡❡❞ s❡♥s♦r❧❡ss ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s✱ ✐♥t❡r✈❛❧ s②s✲ ❝♦♥tr♦❧ ♦❢ ✐♥❞✉❝t✐♦♥ ♠♦t♦r ❬✶✶❪✱ ♣❛r❛♠❡t❡r ❛❞❛♣✲ t❡♠s✱ ♣r♦♣♦rt✐♦♥❛❧✲✐♥t❡❣r❛❧ ✭P■✮ ❝♦♥✲ t✐♦♥s ♦❢ ✐♥❞✉❝t✐♦♥ ♠♦t♦r ❬✶✷✱ ✶✸❪✱ ❢✉③③② ❝♦♥tr♦❧❧❡r tr♦❧❧❡r✱ r♦❜✉st st❛❜✐❧✐t② r❛♥❣❡✱ ❘♦✉t❤✲ ❢♦r ✐♥t❡❧❧✐❣❡♥t ❣❛✉❣❡ ❝♦♥tr♦❧ s②st❡♠ ❬✶✹❪✱ ♣r❡ss✉r❡ ❍✉r✇✐t③ ❝r✐t❡r✐♦♥✳ ❝♦♥tr♦❧ ♦❢ ❤✐❣❤ ♣r❡ss✉r❡ ❝♦♠♠♦♥ r❛✐❧ ✐♥❥❡❝t✐♦♥ s②st❡♠ ❬✶✺❪✳ ❋✐♥❞✐♥❣ r♦❜✉st st❛❜✐❧✐t② r❛♥❣❡s ♦❢ ❑❡②✇♦r❞s ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✶✶ ❱❖▲❯▼❊✿ ✷ t❤❡ P■ ❝♦♥tr♦❧❧❡r ✐s ♥❡❝❡ss❛r② ❜❡❝❛✉s❡ ♦❢ ♣❛r❛♠❡✲ t❡r ✉♥❝❡rt❛✐♥t② ♦❢ t❤❡ ♣r♦❝❡ss❡s✳ ❚❤✐s ✇♦r❦ ❝♦♥✲ s✉♠❡s ❝♦♥s✐❞❡r❛❜❧❡ t✐♠❡ ❢♦r s♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐♥❡q✉❛❧✐t✐❡s r❡❝❡✐✈❡❞ ❢r♦♠ ❑❤❛r✐t♦♥♦✈✬s t❤❡♦r❡♠ ❛♥❞ ❘♦✉t❤✲❍✉r✇✐t③ ❝r✐t❡r✐♦♥ ❬✶✻❪✳ ❉✐r❡❝t t❡❝❤♥✐q✉❡ ✇❛s ✉s❡❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ✐♥✲ ❡q✉❛❧✐t✐❡s ❬✶✼❪✳ ❚❤✐s t❡❝❤♥✐q✉❡ ✐s ❡❛s② t♦ ✉♥❞❡r✲ st❛♥❞✱ t♦ ❝♦♠♣✉t❡✱ ❜✉t ✐t ❞♦❡s ♥♦t ✉t✐❧✐③❡ t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❘♦✉t❤✲❍✉r✇✐t③ ❝r✐t❡r✐♦♥✿ ❝❤❡❝❦✐♥❣ st❛❜✐❧✐t② ✇✐t❤♦✉t s♦❧✈✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦✲ ♠✐❛❧ ❞✐r❡❝t❧②✳ ❚❤❡r❡❢♦r❡✱ ✐ts ❝♦♠♣✉t✐♥❣ s♣❡❡❞ ✐s ❧♦✇ ❜❡❝❛✉s❡ ♦❢ s♦❧✈✐♥❣ ♠❛♥② ♣♦❧②♥♦♠✐❛❧ ❡q✉❛✲ t✐♦♥s✱ ❡s♣❡❝✐❛❧❧② ✐♥ ❝❛s❡ ♦❢ ❤✐❣❤✲♦r❞❡r s②st❡♠s✳ ■♥❝r❡❛s✐♥❣ ❝♦♠♣✉t✐♥❣ s♣❡❡❞ ✐s ♥❡❝❡ss❛r② ❢♦r ✐♠✲ ♣❧❡♠❡♥t❛t✐♦♥ ✐♥t♦ r❡❛❧ ❝♦♥tr♦❧ s②st❡♠s ✇✐t❤ ❞✐❣✲ ✐t❛❧ s✐❣♥❛❧ ♣r♦❝❡ss♦r✳ ■♥ t❤✐s ♣❛♣❡r✱ ❛ t❡❝❤♥✐q✉❡ ❜❛s❡❞ ♦♥ st❡♣s ♦❢ ❝❤❡❝❦✐♥❣ st❛❜✐❧✐t② ✉s✐♥❣ ❘♦✉t❤✲ ❍✉rt✇✐t③ ❝r✐t❡r✐♦♥✱ ✐s ❞❡✈❡❧♦♣❡❞ t♦ ♦✈❡r❝♦♠❡ t❤❡ ❞✐s❛❞✈❛♥t❛❣❡ ♦❢ ❞✐r❡❝t t❡❝❤♥✐q✉❡✳ ❋♦r ✐♠♣❧❡♠❡♥✲ t❛t✐♦♥✱ ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r s♦❧✈✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐♥✲ ❡q✉❛❧✐t② ❬✶✽❪✱ ✐♥t❡rs❡❝t✐♦♥s ✐♥ s❡t t❤❡♦r② ❛r❡ ❞❡✲ s❝r✐❜❡❞✳ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ♣❛♣❡r ❝♦♥s✐sts ♦❢ ✹ s❡❝✲ t✐♦♥s✳ ❑❤❛r✐t♦♥♦✈✬s t❤❡♦r❡♠ ❛♥❞ r♦❜✉st st❛❜✐❧✲ ✐t② ❝♦♥❞✐t✐♦♥s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ✜rst s❡❝t✐♦♥✳ ❚✇♦ t❡❝❤♥✐q✉❡s ❢♦r ✜♥❞✐♥❣ st❛❜✐❧✐t② r❛♥❣❡ ❛r❡ ❞❡✲ s❝r✐❜❡❞ ✐♥ ♥❡①t s❡❝t✐♦♥✳ ❚❤❡ t❤✐r❞ ♦♥❡ ♣r❡s❡♥ts ❝♦♠♣✉t❛t✐♦♥ ❡①❛♠♣❧❡s✳ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❞❡✈❡❧✲ ♦♣♠❡♥ts ❛r❡ ❝❛rr✐❡❞ ♦✉t ✐♥ t❤❡ ❧❛st ♦♥❡✳ ✷✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ K3 (s) = f0+ + f1− s + f2− s2 + f3+ s3 + ✭✹✮ K4 (s) = f0+ + f1+ s + f2− s2 + f3− s3 + ✭✺✮ ✇❤❡r❡ fi ❝♦❡✣❝✐❡♥ts✱ ❢♦r i = 0, 1, , n, ❛r❡ ❜♦✉♥❞❡❞ r❡❛❧ ♥✉♠❜❡rs✱ ❛♥❞ s②♠❜♦❧s − , + r❡s♣❡❝✲ t✐✈❡❧② ❞❡♥♦t❡ ❧♦✇❡r✱ ✉♣♣❡r ❜♦✉♥❞❡rs ♦❢ ❝♦❡✣✲ ❝✐❡♥ts✳ ◆❡①t✱ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ♦❢ ❝❤❡❝❦✐♥❣ r♦❜✉st st❛❜✐❧✐t② ♦❢ ❢❡❡❞❜❛❝❦ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲ t✐♠❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠ ✇✐t❤ ❛ s✐♥❣❧❡ ✐♥♣✉t s✐♥❣❧❡ ♦✉t♣✉t ✭❙■❙❖✮ ♣❧❛♥t G(s)✱ ❛♥❞ ❛ ❝♦♠♣❡♥✲ s❛t♦r ♦r ❛ ❝♦♥tr♦❧❧❡r GC (s) s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳ ❋❛♠✐❧② ♦❢ ✐♥t❡r✈❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ✭❋■❚❋✮ ♦❢ ❋✐❣✳ ✶✿ ❋❡❡❞❜❛❝❦ ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠s✳ t❤❡ ♣❧❛♥t ✐s ❣✐✈❡♥ ❜②✿ G(s) = b0 + b1 s + b2 s2 + + bn sn a0 + a1 s + a2 s2 + + an sn ✭✻✮ ✇❤❡r❡ ❞❡❣r❡❡ n ♦❢ ❞❡♥♦♠✐♥❛t♦r ♦❢ G(s) ✐s ❣✉❛r✲ ❛♥t❡❡❞✱ ❛♥❞ ❝♦❡✣❝✐❡♥ts bi ✱ ❢♦r i = 0, 1, 2, , n ❛r❡ ❧✐♠✐t❡❞ ✐♥ t❤❡✐r ❣✐✈❡♥ r❛♥❣❡s✿ a− i a+ i ✭✼✮ b− i bi b+ i ✭✽✮ ❑❍❆❘■❚❖◆❖❱✬❙ ❚❍❊❖❘❊▼ ❆◆❉ ❘❖❇❯❙❚ ❙❚❆❇■▲■❚❨ ❈❖◆❉■❚■❖◆❙ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ♣❧❛♥t✬s ❋■❚❋ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❢♦❧❧♦✇✿ − + − + n [b− , b+ ] + [b1 , b1 ]s + + [bn , bn ]s G(s) = −0 + ❆ ❢❛♠✐❧② F (s) ♦❢ ✐♥✲ − + + n [a0 , a0 ] + [a1 , a1 ]s + + [a− n , an ]s t❡r✈❛❧ r❡❛❧✲❝♦❡✣❝✐❡♥t ♣♦❧②♥♦♠✐❛❧s ♦❢ ✜①❡❞ ♦r✲ ✭✾✮ ❞❡r n ✐s ❍✉r✇✐t③ st❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ❢♦✉r ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❍✉r✇✐t③ st❛❜❧❡ ❬✺✱ ❆♥❞ ✐ts ❑❤❛r✐t♦♥♦✈ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ❛r❡ ❣✐✈❡♥ ✻❪✳ ❋♦r♠ ♦❢ F (s) ✐s✿ ❜②✿ ❑❤❛r✐t♦♥♦✈✬s t❤❡♦r❡♠✳ F (s) = f0 + f1 s + f2 s2 + + fn sn ✭✶✮ ❛♥❞ ✐ts ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ❛r❡✿ K1 (s) = f0− + f1− s + f2+ s2 + f3+ s3 + K2 (s) = f0− + f1+ s + f2+ s2 + f3− s3 + ✶✶✷ G1 (s) = − + + b− + b1 s + b2 s + b3 s + − − + a0 + a1 s + a2 s + a+ s + ✭✶✵✮ G2 (s) = + + − b− + b1 s + b2 s + b3 s + + + − a− + a1 s + a2 s + a3 s + ✭✶✶✮ ✭✷✮ ✭✸✮ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✶✳ ❉❡s❝r✐♣t✐♦♥ ♦❢ ✉s❡❞ s❡ts ♦♥ ▼❛t❧❛❜ s♦❢t✇❛r❡✳ ❙❡t ❙ ∅ R (−∞, α) (α, +∞) (α, β) ❉❡s❝r✐♣t✐♦♥ ❬✐♥❢ ✐♥❢ ✵❪ ❬✐♥❢ ✐♥❢ ✶❪ [α α 2] [α α 3] [α β 4] G3 (s) = − − + b+ + b1 s + b2 s + b3 s + − − + a+ + a1 s + a2 s + a3 s + ✭✶✷✮ − − K1❴I (s) = (b− kI ) + (b1 kI + a0 )s G4 (s) = + − − b+ + b1 s + b2 s + b3 s + + + − a0 + a1 s + a2 s + a− s + ✭✶✸✮ + + K2❴I (s) = (b− kI ) + (b1 kI + a0 )s ❚❤❡ s②st❡♠ ❤❛s ❢❛♠✐❧② ♦❢ ✐♥t❡r✈❛❧ ❝❤❛r❛❝t❡r✐st✐❝ ❡q✉❛t✐♦♥s ❛s ❢♦❧❧♦✇✿ ✭✶✹✮ + GC (s)G(s) = ❚❤❡ ❝♦♠♣❡♥s❛t♦r ❝❛♥ ❜❡ ♦♥❡ ♦❢ t②♣❡s✿ ❧❡❛❞✱ ❧❛❣✱ ❧❡❛❞✲❧❛❣✱ ♣r♦♣♦rt✐♦♥❛❧ ✭P✮✱ ✐♥t❡❣r❛❧ ✭■✮✱ ❞❡r✐✈❛t✐✈❡ ✭❉✮✱ P■✱ ♣r♦♣♦rt✐♦♥❛❧✲❞❡r✐✈❛t✐✈❡ ✭P❉✮✱ ♣r♦♣♦rt✐♦♥❛❧✲✐♥t❡❣r❛❧✲❞❡r✐✈❛t✐✈❡ ✭P■❉✮✳ ■♥ ❝❛s❡ ♦❢ P✱ ■✱ P■ ❝♦♥tr♦❧❧❡rs✱ t❤❡② ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ r❡✲ s♣❡❝t✐✈❡❧② ❜② ❢♦❧❧♦✇✐♥❣ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s✿ GC (s) = kP ✭✶✺✮ kI s ✭✶✻✮ GC (s) = + + + + (b+ kI + a1 )s + (b3 kI + a2 )s + + − − + (b+ kI + a1 )s + (b3 kI + a2 )s + ✭✷✷✮ ✭✷✸✮ − − K3❴I (s) = (b+ kI ) + (b1 kI + a0 )s − + + + (b− kI + a1 )s + (b3 kI + a2 )s + ✭✷✹✮ + + K4❴I (s) = (b+ kI ) + (b1 kI + a0 )s − − − + (b− kI + a1 )s + (b3 kI + a2 )s + ✭✷✺✮ − − − K1❴P I (s) = (b− kI ) + (b0 kP + b1 kI + a0 )s + + + (b+ kP + b2 kI + a1 )s + + + (b+ kP + b3 kI + a2 )s + ✭✷✻✮ + + + K2❴P I (s) = (b− kI ) + (b0 kP + b1 kI + a0 )s + + + (b+ kP + b2 kI + a1 )s GC (s) = kP + kI s ✭✶✼✮ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ❛r❡ r❡s♣❡❝t✐✈❡❧② ❣✐✈❡♥ ❜② ❊qs✳ ✭✶✽✮✲✭✷✶✮✱ ✭✷✷✮✲✭✷✺✮✱ ✭✷✻✮✲✭✷✾✮ − − − K1❴P (s) = (b− kP + a0 ) + (b1 kP + a1 )s + (b+ kP + a+ )s + (b+ kP + a+ )s − − + (b− kP + b3 kI + a2 )s + ✭✷✼✮ − − − K3❴P I (s) = (b+ kI ) + (b0 kP + b1 kI + a0 )s − − + (b− kP + b2 kI + a1 )s + + + (b+ kP + b3 kI + a2 )s + ✭✷✽✮ + ✭✶✽✮ + + + K4❴P I (s) = (b+ kI ) + (b0 kP + b1 kI + a0 )s − + + K2❴P (s) = (b− kP + a0 ) + (b1 kP + a1 )s + − − + (b+ kP + a2 )s + (b3 kP + a3 )s + ✭✶✾✮ + − − K3❴P (s) = (b+ kP + a0 ) + (b1 kP + a1 )s − + + + (b− kP + a2 )s + (b3 kP + a3 )s + ✭✷✵✮ − − + (b− kP + b2 kI + a1 )s − − + (b− kP + b3 kI + a2 )s + ✭✷✾✮ ❛♥❞ ❝♦♥str❛✐♥ts ❛r❡ r❡s♣❡❝t✐✈❡❧② ❣✐✈❡♥ ❜② ❊qs✳ ✭✸✵✮✱ ✭✸✶✮✲✭✸✷✮✱ ✭✸✸✮✲✭✸✺✮✿ − b− i kP + + b+ i kP + , ∀i = 0, 1, 2, , n ✭✸✵✮ + + + K4❴P (s) = (b+ kP + a0 ) + (b1 kP + a1 )s − − − + (b− kP + a2 )s + (b3 kP + a3 )s + ✭✷✶✮ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ b− kI b+ kI ✭✸✶✮ ✶✶✸ ❱❖▲❯▼❊✿ ✷ − b− i kI + ai−1 + b+ i kI + ai−1 , ∀i = 1, 2, , n ✭✸✷✮ b− kI − b− n kP + an − − b− i−1 kP + bi kI + ai−1 b+ kI ✭✸✸✮ + b+ n kP + an ✭✸✹✮ + + b+ i−1 kP + bi kI + ai−1 , ✭✸✺✮ ∀i = 1, 2, , n ❚❤❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠ ✐s r♦✲ ❜✉st st❛❜❧❡ ✐❢ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❍✉r✲ ✇✐t③ st❛❜❧❡✳ ❘♦✉t❤✲❍✉r✇✐t③ ❝r✐t❡r✐♦♥ ✇❛s ✉s❡❞ t♦ ❝❤❡❝❦ ❍✉r✇✐t③ st❛❜✐❧✐t② ♦❢ s②st❡♠s ❬✶✾❪✲❬✷✷❪✳ ◆❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r st❛❜✐❧✐t② ✐s t❤❛t ❛❧❧ t❤❡ t❡r♠s ♦❢ t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ ❘♦✉t❤ ❛rr❛② ♦r ❛❧❧ t❤❡ ❞❡t❡r♠✐♥❛♥ts ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ♠✐✲ ♥♦rs ♦❢ ❍✉r✇✐t③ ♠❛tr✐① ❤❛✈❡ t❤❡ s❛♠❡ s✐❣♥ ❬✶✽❪✳ ■♥ ♥❡①t s❡❝t✐♦♥s✱ t✇♦ t❡❝❤♥✐q✉❡s ❛r❡ ✉s❡❞ t♦ ✜♥❞ s❡ts SP , SI ♦❢ t✇♦ ♣❛r❛♠❡t❡rs kP , kI t❤❛t ♠❛❦❡ t❤❡ s②st❡♠ st❛❜❧❡✳ ✸✳ ❚❊❈❍◆■◗❯❊❙ ❋❖❘ ❋■◆❉■◆● ❙❚❆❇■▲■❚❨ ❘❆◆●❊ ❆t ✜rst✱ t✇♦ t❡❝❤♥✐q✉❡s ❛r❡ ❞❡s❝r✐❜❡❞ ❢♦r st❛❜✐❧✲ ✐t② ❛♥❛❧②s✐s ✐♥ ❝❛s❡ ♦❢ ❝♦♥tr♦❧❧❡rs ✇✐t❤ ♦♥❡ ♣❛✲ r❛♠❡t❡r kP ♦r kI ✳ ❆ss✉♠❡ t❤❛t ❛❧❧ r♦♦ts ♦❢ t❤❡ t❡r♠s ♦❢ t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ ❘♦✉t❤ ❛rr❛② ♦r t❤❡ ❞❡t❡r♠✐♥❛♥ts ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ♠✐♥♦rs ♦❢ ❍✉r✇✐t③ ♠❛tr✐① ✇❡r❡ ❢♦✉♥❞✳ ❋✐rst ♦♥❡ ✐s t❤❡ ❞✐r❡❝t t❡❝❤✲ ♥✐q✉❡ ✭❉■❚✮ t❤❛t ❞♦❡s ♥♦t s♦❧✈❡ ❛♥② ✐♥❡q✉❛❧✐✲ t✐❡s ✇❤✐❝❤ ❛r❡ ❣❡♥❡r❛t❡❞ ❢r♦♠ st❛❜✐❧✐t② ❝♦♥❞✐✲ t✐♦♥s✳ ❋♦r ❡❛❝❤ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧✱ ♣r♦✲ ❝❡ss❡❞ st❡♣s ♦❢ t❤❡ t❡❝❤♥✐q✉❡ ❛r❡✿ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ✲ ✐♥t❡r✈❛❧ I0 (k < k1 ) : p0 = 2k1 , ✐❢ k1 < 0✱ ♦r p0 = −1✱ ✐❢ k1 ≥ 0❀ ✲ ✐♥t❡r✈❛❧ Ii (ki < k < ki+1 ) : pi = (ki + ki+1 ) /2, ❢♦r i = 1, 2, , l − 1; ✲ ✐♥t❡r✈❛❧ Il (k > kl ) : pl = 2kl , ✐❢ kl > ♦r pl = 1, ✐❢ kl 0✳ • ❙t❡♣ ✸✿ ❢♦r ❡❛❝❤ ✈❛❧✉❡ k = pi ✱ ✜♥❞ ❛❧❧ r♦♦ts ♦❢ ❡❛❝❤ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧✳ ■❢ ❛❧❧ r♦♦ts ❤❛✈❡ ♥❡❣❛t✐✈❡ r❡❛❧ ♣❛rts✱ ✐♥t❡r✈❛❧ Ii s❛t✐s✜❡s st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥✳ ❚❤❡ s❡❝♦♥❞ t❡❝❤♥✐q✉❡ ✐s t❤❡ s❡t t❤❡♦r②✲❜❛s❡❞ ♦♥ t❡❝❤♥✐q✉❡ ✭❙❇❚✮ t❤❛t s♦❧✈❡s t❤❡ ♣♦❧②♥♦♠✐❛❧ ✐♥✲ ❡q✉❛❧✐t✐❡s✱ ❛♥❞ ✉s❡s ❜❛s✐❝ ✐♥t❡rs❡❝t✐♦♥ ✐♥ s❡t t❤❡✲ ♦r②✳ ❉❡s❝r✐♣t✐♦♥ ♦❢ ✉s❡❞ s❡ts ♦♥ ▼❛t❧❛❜ s♦❢t✇❛r❡ ✐s s❤♦✇♥ ✐♥ ❚❛❜✳ ✶✳ ■♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ s❡ts ✐s ✐♠✲ ♣❧❡♠❡♥t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❜❛s✐❝ r✉❧❡s ♦❢ s❡t t❤❡♦r② ✭s❡❡ ❚❛❜✳ ✷✮✳ ❚✇♦ ❝❤❛r❛❝t❡rs m, M ❞❡♥♦t❡ ♠✐♥✱ ♠❛① ❢✉♥❝t✐♦♥s r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡❛❝❤ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧✱ ✐t ✐s ❞❡s❝r✐❜❡❞ ❜② ❢♦❧❧♦✇✐♥❣ st❡♣s✿ • ❙t❡♣ ✶✿ ❛ss✉♠❡ t❤❛t ❡❛❝❤ t❡r♠ ♦❢ t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ ❘♦✉t❤ ❛rr❛② ♦r ❡❛❝❤ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ♠✐♥♦rs ♦❢ ❍✉r✇✐t③ ♠❛tr✐①✱ ✐s ❛ rt❤ ✲♦r❞❡r ♣♦❧②♥♦♠✐❛❧ P (k)✱ ❛♥❞ ❝♦❡❢✲ ✜❝✐❡♥t cr ❛ss♦❝✐❛t❡s ✇✐t❤ k r ✭cr = 0✮✳ ❙♦rt ✐ts ❞✐st✐♥❝t ♦❞❞✲♠✉❧t✐♣❧✐❝✐t② r❡❛❧ r♦♦ts ✐♥ ❛s✲ ❝❡♥❞✐♥❣ ♦r❞❡r✿ k1 < k2 < < kq (q ≤ r) • ❙t❡♣ ✷✿ ♥♦ ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ s♦❧✈❡ t❤❡ ✐♥✲ ❡q✉❛❧✐t② P (k) > ❜② ❛♥ ❛❧❣♦r✐t❤♠ s❤♦✇♥ ✐♥ ❋✐❣✳ ✸✳ • ❙t❡♣ ✸✿ ❛♣♣❧② ✐♥t❡rs❡❝t✐♦♥ t♦ ✜♥❞ r❛♥❣❡ ♦❢ k ✇❤✐❝❤ s❛t✐s✜❡s ❛❧❧ ✐♥❡q✉❛❧✐t✐❡s✳ ❚✇♦ ❞❡s❝r✐❜❡❞ t❡❝❤♥✐q✉❡s ❛r❡ ❛♣♣❧✐❡❞ ❢♦r ❛❧❧ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ✐s ✉s❡❞ t♦ ♦❜t❛✐♥ t❤❡ s❡t SP ♦r t❤❡ s❡t SI t❤❛t s❛t✐s✜❡s t❤❡ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s✳ ■♥ ❝❛s❡ ♦❢ P■ ❝♦♥tr♦❧❧❡r✱ ❛t ✜rst✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ VI ✱ t❤❡ ✜♥❛❧ • ❙t❡♣ ✶✿ s♦rt ✐♥ ❛s❝❡♥❞✐♥❣ ♦r❞❡r ❞✐st✐♥❝t r❡❛❧ ✈❛❧✉❡ VF ✱ t❤❡ ✈❛❧✉❡ ♦❢ ✐♥❝r❡♠❡♥t ∆V ♦❢ ♣❛r❛♠✲ r♦♦ts ♦❢ ❛❧❧ t❤❡ t❡r♠s ♦❢ t❤❡ ✜rst ❝♦❧✉♠♥ ♦❢ ❡t❡r kP ♦r kI ❛r❡ ❣✐✈❡♥✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ✈❛❧✉❡ ♦❢ ❘♦✉t❤ ❛rr❛② ♦r ❛❧❧ t❤❡ ❞❡t❡r♠✐♥❛♥ts ♦❢ t❤❡ kP ♦r kI ✱ t❤❡ s❡t SI ♦r SP ✐s ❢♦✉♥❞ ❜② ❝❤❡❝❦✐♥❣ ♣r✐♥❝✐♣❛❧ ♠✐♥♦rs ♦❢ ❍✉r✇✐t③ ♠❛tr✐①✿ k1 < st❛❜✐❧✐t② ♦❢ ✹ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ✭s❡❡ ❊qs✳ k2 < < kl ✱ ✇❤❡r❡ k ✐s r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ ✭✷✻✮✲✭✷✾✮✮✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ t❤❡s❡ s❡ts SI ♦r SP ❛r❡ t❤❡ ✜♥❛❧ r❡s✉❧ts✳ kP ♦r kI ✳ • ❙t❡♣ ✷✿ s❡❧❡❝t t❤❡ ♣♦✐♥ts k = pi ❢♦r i = 0, 1, 2, , l ❛s ❢♦❧❧♦✇s✿ ✶✶✹ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✷✳ ■♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ s❡ts✳ ❙❡t S1 ❙❡t S2 (−∞, α2 ) (α2 , +∞) (α2 , β2 ) (−∞, α1 ) (−∞, m(α1 , α2 )) (α , α ) ✐❢ α < α 2 ∅, ♦t❤❡r✇✐s❡ (α2 , m(α1 , β2 ), β1 ) ✐❢ α2 < α1 ∅, ♦t❤❡r✇✐s❡ (α1 , +∞) (α , α ) ✐❢ α < α 2 ∅, ♦t❤❡r✇✐s❡ (α1 , β1 ) (α , m(α , β )) ✐❢ α < α 1 ∅, ♦t❤❡r✇✐s❡ (M (α , α ), β ) ✐❢ α < β 2 ∅, ♦t❤❡r✇✐s❡ (M (α1 , α2 ), +∞) (M (α1 , α2 ), β2 ) ✐❢ α1 < β2 ∅, ♦t❤❡r✇✐s❡ ∅, ✐❢ α2 ≥ β1 ♦r α1 ≥ β2 (M (α , α ), m(β , β )), ♦t❤❡r✇✐s❡ 2 ❚❛❜❧❡ ✸✳ ❙❡❧❡❝t❡❞ ♣❧❛♥ts✳ G(s) ♥ ✷ ✸ ✹ ✺ ✻ ✼ [36, 44] + [4.3, 5.7]s [54, 66] + [5.7, 8.3]s + [1, 1]s2 [3.2, 4.3] + [2.3, 3.7]s + [1.1, 1.9]s2 [11.7, 14.9] + [7.5, 9.6]s + [3.3, 5.2]s2 + [1, 1]s3 [7.5, 12.5] + [17, 23]s + [12, 18]s2 + [3.5, 6.5]s3 [10.5, 17.5] + [23, 37]s + [15, 25]s2 + [3, 7]s3 + [1, 1]s4 [46, 54] + [85, 125]s + [90, 110]s2 + [27, 34]s3 + [4, 6]s4 [63, 77] + [150, 198]s + [115, 135]s2 + [52, 58]s3 + [8, 10]s4 + [1, 1]s5 [320, 380] + [554, 574]s + [950, 1050]s2 + [225, 245]s3 + [90, 110]s4 + [10, 12]s5 [340, 400] + [1150, 1250]s + [604, 644]s2 + [470, 530]s3 + [70, 80]s4 + [9, 11]s5 + [1, 1]s6 [329,471]+[706,865]s+[558,643]s2 +[282,319]s3 +[70,83]s4 +[12,15]s5 +[1.0,1.4]s6 [387,521]+[877,1024]s+[711,889]s2 +[326,360]s3 +[89,110]s4 +[13.3,16.7]s5 +[1.2,1.6]s6 +[0.1,0.1]s7 ❚❛❜❧❡ ✹✳ ❙❡ts SP , SI ♦❢ P ❛♥❞ I ❝♦♥tr♦❧❧❡rs✳ ♥ SP ✭P ❝♦♥tr♦❧❧❡r✮ SI ✭I ❝♦♥tr♦❧❧❡r✮ ✷ (−1.325581395348837, +∞) (0, 15.792714212416620) ✸ (−1.136952577372862, +∞) (0, 10.395928891361976) ✹ (−0.071101889488303, +∞) (0, 0.373239166328192) ✺ (−0.857142857142857, +∞) (0, 49.784749592528634) ✻ (−0.020328133920827, +∞) (0, 0.767612236055811) ✼ (−0.079686910635607, +∞) (0, 1.646059600788306) ✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✶✺ ❱❖▲❯▼❊✿ ✷ ✹✳ ❈❖▼P❯❚❆❚■❖◆ R1 = CTmin ♦❢ ❙❇❚ CTmin ♦❢ ❉■❚ ✭✸✾✮ R2 = CTavg ♦❢ ❙❇❚ CTavg ♦❢ ❉■❚ ✭✹✵✮ R3 = CTmax ♦❢ ❙❇❚ CTmax ♦❢ ❉■❚ ✭✹✶✮ ❊❳❆▼P▲❊❙ ❚✇♦ t❡❝❤♥✐q✉❡s ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ♦♥ ▼❛t❧❛❜ s♦❢t✲ ✇❛r❡ ❘✷✵✶✹❛✱ ✈❡rs✐♦♥ ✽✳✸✳✵✳✺✸✷ ✇✐t❤ ♣r♦❝❡ss♦r ■♥t❡❧ ❈♦r❡ ✐✼✲✻✼✵✵❍◗ ❈P❯ ✷✳✻●❍③✱ ✐♥st❛❧❧❡❞ ♠❡♠♦r② ✭❘❆▼✮ ✽✳✵✵ ●❇ ✭✼✳✽✽ ●❇ ✉s❛❜❧❡✮✳ ❍✉r✲ ✇✐t③ ♠❛tr✐① ✐s ✉s❡❞ t♦ ❛✈♦✐❞ t❤❡ ❡rr♦r ❞✉❡ t♦ ♣♦❧②♥♦♠✐❛❧ ❞✐✈✐s✐♦♥ ✐♥ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ ❘♦✉t❤ ❛r✲ r❛②✳ ❆❧❧ ❋■❚❋s ♦❢ s❡❧❡❝t❡❞ ♣❧❛♥ts t❤❛t ❧✐st❡❞ ✐♥ ❚❛❜✳ ✸ ❤❛✈❡ r❡❧❛t✐✈❡ ❞❡❣r❡❡ ♦❢ ✶✳ ❋♦r P✱ ■ ❝♦♥✲ tr♦❧❧❡rs✱ s❡ts SP , SI ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❛♥❞ ❧✐st❡❞ ✐♥ ❚❛❜✳ ✹✳ ❇❡❝❛✉s❡ ❜♦✉♥❞❛r✐❡s αI , βI ♦❢ ❛❧❧ s❡ts SI ❛r❡ ❧✐♠✐t❡❞ ✭s❡❡ ❚❛❜✳ ✹✮✱ s♦ ✈❛❧✉❡s ∆V , ∆I , ∆F ♦❢ ♣❛r❛♠❡t❡r kI ❛r❡ ❝❤♦s❡♥ ❛s ❢♦❧❧♦✇s✿ βI − αI 101 ✭✸✻✮ VI = αI + ∆V ✭✸✼✮ VF = βI − ∆V ✭✸✽✮ ∆V = | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❋✐❣✉r❡ ✷ s❤♦✇s t❤❡s❡ r❛t✐♦s t❤❛t ❛r❡ s♠❛❧❧❡r t❤❛♥ ♦♥❡ ✐♥ ❛❧❧ s✐t✉❛t✐♦♥s✳ ❚❤❡② t❡♥❞ t♦ ❞❡❝r❡❛s❡ ✇✐t❤ t❤❡ ✐♥❝r❡❛s❡ ♦❢ n✱ ❡①❝❡♣t✐♦♥❛❧❧② ❢♦r ❝❤❛♥❣❡s ♦❢ n ❢r♦♠ ✹ t♦ ✺ ❛♥❞ ❢r♦♠ ✻ t♦ ✼✳ ❋♦r t❤❡ ❉■❚✱ ✐♥ ♠♦st ❝❛s❡s✱ t❤❡ ❤✐❣❤❡r ❞❡❣r❡❡ n✱ t❤❡ ❧♦♥❣❡r ❈❚s✱ ❡①❝❡♣t ❢♦r ✈❛❧✉❡s n = 6, 7✳ ❋♦r ❡❛❝❤ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧✱ t❤❡ st❡♣ ✸ ♦❢ t❤✐s t❡❝❤♥✐q✉❡ ✐s ♣❡r❢♦r♠❡❞ (l + 1) t✐♠❡s ✇❤❡r❡ ♣❛r❛♠❡t❡r l ✐s ♥✉♠❜❡r ♦❢ ❞✐st✐♥❝t r❡❛❧ r♦♦ts ♦❢ ❛❧❧ t❤❡ ❞❡t❡r♠✐♥❛♥ts ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ♠✐♥♦rs ♦❢ ❍✉r✇✐t③ ♠❛tr✐①✳ ❋♦r ❉■❚✱ ♣❛r❛♠❡t❡r l✱ ♥✉♠❜❡r ♦❢ ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s nl ✇✐t❤ t❤❡ s❛♠❡ ♣❛✲ r❛♠❡t❡r l✱ ❛♥❞ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t st❡♣ ✸ ✐s ♣❡r❢♦r♠❡❞ ns3 ✱ ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜✳ ✻✳ ■t ❝❛♥ ❡❛s② t♦ s❡❡ t❤❛t t❤❡ ♣❛r❛♠❡t❡r ✇❤✐❝❤ ❛✛❡❝ts ❈❚s ♦❢ ❉■❚ ♠♦st ✐s ns3 ✳ ❊s♣❡❝✐❛❧❧②✱ n ❝❤❛♥❣❡s ❢r♦♠ ✻ t♦ ✼✱ ns3 ❞❡❝r❡❛s❡ ❢r♦♠ ✽✶✹✾ t♦ ✼✷✵✾✱ t❤✐s ♠❛❦❡s ❈❚s s❤♦rt❡r✳ ■♥ ❝❛s❡s ♦❢ n = 4, 5, t❤❡ ✈❛❧✉❡s ♦❢ ns3 ❛r❡ ❡q✉✐✈❛❧❡♥t✱ t❤❡r❡❢♦r❡ ❈❚s ✐♥❝r❡❛s❡ ✐♥✲ s✐❣♥✐✜❝❛♥t❧②✳ ❈♦♠♣✉t✐♥❣ t✐♠❡ ✭❈❚✮ ✐s t❤❡ t✐♠❡ t❤❛t t❤❡ ♣r♦✲ ❝❡ss♦r ❡①❡❝✉t❡s ❛❧❧ st❡♣s ❢♦r ✹ ❑❤❛r✐t♦♥♦✈ ♣♦❧②✲ ♥♦♠✐❛❧s ✇✐t❤ ✶✵✵ ❣✐✈❡♥ ✈❛❧✉❡s ♦❢ kI ✭s❡❡ ❊qs✳ ✭✸✻✮ ✕ ✭✸✽✮✮✳ ❋♦r ❝♦♠♣❛r✐s♦♥ ♦❢ t✇♦ t❡❝❤♥✐q✉❡s✱ t✇♦ ❢✉♥❝t✐♦♥s t✐❝✱ t♦❝ ❛r❡ ✉s❡❞ t♦ ♠❡❛s✉r❡ t❤❡✐r ❈❚✳ ❙t❛t✐st✐❝❛❧❧②✱ t✇♦ t❡❝❤♥✐q✉❡s ❛r❡ r✉♥ ✸✵ t✐♠❡s✱ ❛♥❞ ♠✐♥✐♠✉♠✱ ♠❛①✐♠✉♠✱ ❛✈❡r❛❣❡ ✈❛❧✉❡s ❋♦r ❙❇❚✱ ♦r❞❡r q ♦❢ ✐♥❡q✉❛❧✐t②✱ ♥✉♠❜❡r ♦❢ q t❤ ✲ ♦❢ ❈❚ ✭CTmin , CTmax , CTavg ✮ ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜✳ ❞❡❣r❡❡ ✐♥❡q✉❛❧✐t✐❡s nq ✱ ♥✉♠❜❡r ♦❢ t✐♠❡s t❤❛t st❡♣ ✺✳ ❚❤❡ ❈❚s ♦❢ ❙❇❚ ❛r❡ ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ t❤♦s❡ ✷ ✐s ♣❡r❢♦r♠❡❞ ✭ns2 ✮✱ ♥✉♠❜❡r ♦❢ ✐♥t❡rs❡❝t✐♦♥s ni ♦❢ ❉■❚✳ ❘❛t✐♦s ♦❢ ❈❚s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜✳ ✼✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ns2 = 400(n + 1)✳ ❚❤❡ ❤✐❣❤❡r t❤❡ ns2 ✐s✱ t❤❡ ❧♦♥❣❡r t❤❡ ❈❚s ❛r❡✳ ❇❡s✐❞❡s t❤❛t✱ ❈❚s ✐s s✐❣♥✐✜❝❛♥t❧② ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ♣❛r❛♠❡t❡r ni ✳ ❚❤✐s ✈❛❧✉❡ ♦❢ ni ✭✻✾✺✻✮ ❢♦r n = ✐s ❧❛r❣❡r t❤❛♥ t❤❛t ✭✻✷✵✼✮ ❢♦r n = 7✳ ❚❤✐s ✐♥❝r❡♠❡♥t ♠❛❦❡s ❈❚s ✐♥❝r❡❛s❡ ✐♥s✐❣♥✐✜❝❛♥t❧② ❛❧t❤♦✉❣❤ ❢♦r n = 7✱ ns2 ✭✸✷✵✵✮ ✐s ❧❛r❣❡r t❤❛♥ t❤❛t ✭✷✽✵✵✮ ❢♦r n = 6✳ ❋✐❣✳ ✷✿ ❘❛t✐♦s ♦❢ ❈❚s✳ ✶✶✻ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✺✳ ❱❛❧✉❡s CTmin , CTmax , CTavg ❬♠s❪✳ CTmin ♥ ❉■❚ ❙❇❚ CTavg ❉■❚ CTmax ❙❇❚ ❉■❚ ❙❇❚ ❙♣ ✭P■ ❝♦♥tr♦❧❧❡r✮ ✷ ✺✼✳✾ ✸✵✳✹ ✻✵✳✶ ✸✶✳✹ ✻✹✳✶ ✸✹✳✸ (−1.325581395348837, +∞) ✸ ✼✻✳✺ ✸✺✳✶ ✼✽✳✽ ✸✻✳✸ ✽✷✳✶ ✸✼✳✾ (−1.136952577372862, +∞) ✹ ✶✷✹✳✵ ✺✵✳✻ ✶✷✺✳✹ ✺✶✳✵ ✶✷✼✳✻ ✺✷✳✾ (−0.071101889488303, +∞) ✺ ✶✸✹✳✺ ✺✽✳✹ ✶✸✼✳✵ ✺✾✳✼ ✶✹✻✳✽ ✻✹✳✼ (−0.857142857142857, +∞) ✻ ✸✹✸✳✶ ✽✼✳✷ ✸✹✻✳✻ ✽✽✳✶ ✸✺✷✳✽ ✽✾✳✵ (−0.020328133920827, +∞) ✼ ✸✷✾✳✷ ✾✵✳✽ ✸✸✷✳✶ ✾✶✳✽ ✸✹✷✳✵ ✾✺✳✶ (−0.079686910635607, +∞) ❚❛❜❧❡ ✼✭❛✮✲P❛r❛♠❡t❡rs ♦❢ ❙❇❚✳ n ✷ q ✶ nq ns2 ✸ ni q ✹✵✵ ✶ ✶✷✵✵ ns2 ni ✶✻✵✵ ✷✵✷✷ nq ✶ ✻✺✾ ✷ ✶✶✹✷ ✸ ✶✹✶ ✹ ✺✽ ns2 ni ✷✵✵✵ ✸✸✾✾ ✹✵✵ ✽✵✵ ✸ q ✶✶✼✽ ✷✵✵✵ ✷ ✷ nq ✹ ✷✷ ❚❛❜❧❡ ✼✭❜✮✲P❛r❛♠❡t❡rs ♦❢ ❙❇❚✳ n ✺ q ✵ ✶ nq ns2 ✻ ni ✶✶ ✶✺✾✻ ✷✹✵✵ ✸✷✵✼ q nq ns2 ✼ ni q nq ✶ ✻✾✾ ✶ ✹✵✵ ✷ ✶✵✸✻ ✷ ✹✵✵ ✸ ✾✹✶ ✸ ✹✵✵ ✹ ✶✹✽ ✹ ✶✵✹✹ ✺ ✸✸✵ ✷✽✵✵ ✻✾✺✻ ✷ ✼✽✾ ✺ ✹✵✵ ✻ ✶✻ ✸ ✹ ✻ ✶✺✻ ✼ ✸✵ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ns2 ni ✸✷✵✵ ✻✷✶✼ ✶✶✼ ❱❖▲❯▼❊✿ ✷ ✷ ✸ ✹ ✺ ✻ ✼ ❧ nl ns3 ❧ nl ns3 ❧ nl ns3 ❧ nl ns3 ❧ nl ns3 ❧ nl ns3 ✶✹ ✷✹✽ ✶✺ ✶ ❚❛❜❧❡ ✻✳ P❛r❛♠❡t❡rs ♦❢ ❉■❚ ✭n = 2, ✮ ✳ ✸ ✹✵✵ ✶✻✵✵ ✹ ✸✽✾ ✷✵✷✷ ✻ ✽ ✷✺✾ ✶✶✷ ✸✶✹✵ ✺ ✼ ✶✵ ✸✽✼ ✸✶✽✻ ✶✾ ✷✵ ✸✷✷ ✼ ✽✶✹✾ ✶✻ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✶✾ ✷✹ ✶ ✷✻ ✺ ✶✽ ✻ ✶✼ ✸ ✹ ✼✷✵✾ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ✻ ✶✶ ✶✵ ✷✾ ✾ ✸ ✷✶ ✼✶ ✷✼ ✶✶ ✷✽ ✺ ✷✾ ✷ ✸✵ ✶ ✸✶ ✶ ✳ ❋✐❣✳ ✸✿ ❆❧❣♦r✐t❤♠ ❢♦r s♦❧✈✐♥❣ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✐♥❡q✉❛❧✐t② P (k) > 0✳ ✳ ✶✶✽ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✸✷ ✶ ✸✸ ✺ ✸✹ ✷ ❱❖▲❯▼❊✿ ✷ ✺✳ ❈❖◆❈▲❯❙■❖◆❙ ❚✇♦ t❡❝❤♥✐q✉❡s ✇❛s ❞❡✈❡❧♦♣❡❞ t♦ ✜♥❞ st❛❜✐❧✲ ❬✺❪ ✐t② r❛♥❣❡ ♦❢ ♣r♦♣♦rt✐♦♥❛❧✲✐♥t❡❣r❛❧ ❝♦♥tr♦❧❧❡r ❢♦r ❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠s✳ ❙❡t✲❜❛s❡❞ t❤❡♦r② t❡❝❤♥✐q✉❡ ✉s❡s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ st❛❜✐❧✐t② ❝r✐t❡r✐♦♥s✿ ❝❤❡❝❦✐♥❣ st❛❜✐❧✐t② ✇✐t❤✲ ♦✉t s♦❧✈✐♥❣ ❛♥② ❑❤❛r✐t♦♥♦✈ ♣♦❧②♥♦♠✐❛❧s ❞✐r❡❝t❧②✳ ❬✻❪ ■t ❣✐✈❡s ❝♦♠♣✉t✐♥❣ t✐♠❡ ♠✉❝❤ s❤♦rt❡r t❤❛♥ ❞✐✲ r❡❝t t❡❝❤♥✐q✉❡ ❞♦❡s✱ ❡s♣❡❝✐❛❧❧② ✇✐t❤ ❤✐❣❤✲♦r❞❡r s②st❡♠s✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ♦❜✲ t❛✐♥ ❜♦✉♥❞❛r✐❡s ♦❢ P■✲❜❛s❡❞ ♦r P■❉✲❜❛s❡❞ ✐♥t❡❧✲ ❧✐❣❡♥t ❝♦♥tr♦❧❧❡rs ❢♦r r❡❛❧ s②st❡♠s ❬✸✱ ✷✸❪✳ ❈♦♠✲ ❬✼❪ ❜✐♥❛t✐♦♥ ✇✐t❤ ❤✐❣❤✲❛❝❝✉r❛❝② s②st❡♠ ♦r❞❡r r❡✲ ❞✉❝t✐♦♥ ♠❡t❤♦❞s ❝❛♥ ❞❡❝r❡❛s❡ ❝♦♠♣✉t✐♥❣ t✐♠❡ ❬✷✹❪✳ ❙t❛❜✐❧✐t② ❛♥❛❧②s✐s ❛♥❞ ❞❡s✐❣♥ ♦❢ ❝♦♥tr♦❧❧❡rs ❢♦r ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠s ❝❛♥ ❜❡ ❞♦♥❡ s✐♠✲ ✐❧❛r❧② t♦ t❤❡ ✇♦r❦s ❢♦r s②st❡♠s ✇✐t❤ r❛t✐♦♥❛❧✲ ♦r❞❡r tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ❜② ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ ❬✽❪ s②st❡♠s ✉s✐♥❣ r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ✭❘■▼✮ ✇✐t❤ ❤✐❣❤✲♦r❞❡r ♠♦❞❡❧s ❬✷✺❪✳ ❚❤❡ ♠❛✐♥ ❞r❛✇✲ ❜❛❝❦ ♦❢ t❤✐s ♠❡t❤♦❞ t❤❛t ✐s t❤❡ ✉♥❝❡rt❛✐♥t② ♦❢ ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧ ✐s ♦✈❡r❝♦♠❡ ❜② ❑❤❛r✐t♦♥♦✈✬s ❬✾❪ t❤❡♦r❡♠✳ ❚❤✐s ❝♦♠♣✉t✐♥❣ t❡❝❤♥✐q✉❡ ❝❛♥ ❜❡ ❡①✲ t❡♥❞❡❞ ❢♦r ✜♥❞✐♥❣ st❛❜✐❧✐t② r❛♥❣❡ ♦❢ ❢❡❡❞❜❛❝❦ ❧✐♥❡❛r ❞✐s❝r❡t❡✲t✐♠❡ ✐♥t❡r✈❛❧ ❝♦♥tr♦❧ s②st❡♠s ❬✼❪✱ ❬✶✵❪ ♥♦♥❧✐♥❡❛r s②st❡♠s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❞❡❧❛② ❬✹❪✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ t✐♠❡✲✈❛r②✐♥❣ ❞❡❧❛②✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❙②st❡♠s ❙❝✐❡♥❝❡✱ ✹✼✭✻✮✱ ✶✹✸✸✲✶✹✹✹✳ ❑❤❛r✐t♦♥♦✈✱ ❱✳ ▲✳ ✭✶✾✼✽✮✳ ❆s②♠♣t♦t✐❝ st❛✲ ❜✐❧✐t② ♦❢ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ♣♦s✐t✐♦♥ ♦❢ ❛ ❢❛♠✐❧② ♦❢ s②st❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❉✐✛❡r✲ ❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✶✹✱ ✶✹✽✸✳ ❇❤❛tt❛❝❤❛r②②❛✱ ❙✳ P✳✱ ✫ ❑❡❡❧✱ ▲✳ ❍✳ ✭✶✾✾✺✮✳ ❘♦❜✉st ❝♦♥tr♦❧✿ t❤❡ ♣❛r❛♠❡tr✐❝ ❛♣♣r♦❛❝❤✳ ■♥ ❆❞✈❛♥❝❡s ✐♥ ❈♦♥tr♦❧ ❊❞✉❝❛t✐♦♥ ✶✾✾✹ ✭♣♣✳ ✹✾✲✺✷✮✳ ▼❛♥s♦✉r✱ ▼✳✱ ❑r❛✉s✱ ❋✳✱ ✫ ❆♥❞❡rs♦♥✱ ❇✳ ❉✳ ❖✳ ✭✶✾✽✽✱ ❉❡❝❡♠❜❡r✮✳ ❙tr♦♥❣ ❑❤❛r✐t♦♥♦✈ t❤❡♦r❡♠ ❢♦r ❞✐s❝r❡t❡ s②st❡♠s✳ ■♥ ❉❡❝✐s✐♦♥ ❛♥❞ ❈♦♥tr♦❧✱ ✶✾✽✽✳✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✼t❤ ■❊❊❊ ❈♦♥❢❡r❡♥❝❡ ♦♥ ✭♣♣✳ ✶✵✻✲✶✶✶✮✳ ■❊❊❊✳ ❨✉✱ ❈✳ ❈✳ ✭✷✵✵✻✮✳ ❆✉t♦t✉♥✐♥❣ ♦❢ P■❉ ❝♦♥tr♦❧❧❡rs✿ ❆ r❡❧❛② ❢❡❡❞❜❛❝❦ ❛♣♣r♦❛❝❤✳ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐♥❡ss ▼❡❞✐❛✳ ❖✬❉✇②❡r✱ ❆✳ ✭✷✵✵✾✮✳ ❍❛♥❞❜♦♦❦ ♦❢ P■ ❛♥❞ P■❉ ❝♦♥tr♦❧❧❡r t✉♥✐♥❣ r✉❧❡s✳ ■♠♣❡r✐❛❧ ❈♦❧✲ ❧❡❣❡ Pr❡ss✳ ●♦♣❛❧✱ ▼✳ ✭✷✵✶✷✮✳ ❉✐❣✐t❛❧ ❈♦♥tr♦❧ ❛♥❞ ❙t❛t❡ ❱❛r✐❛❜❧❡ ♠❡t❤♦❞s ❈♦♥✈❡♥t✐♦♥❛❧ ❛♥❞ ◆❡✉r❛❧✲ ❋✉③③② ❈♦♥tr♦❧ ❙②st❡♠✳ ❚❛t❛ ▼❝●r❛✇✲❍✐❧❧ ❊❞✉❝❛t✐♦♥✱ ◆❡✇ ❨♦r❦✳ ❘❡❢❡r❡♥❝❡s ❬✶✶❪ ❖r❧♦✇s❦❛✲❑♦✇❛❧s❦❛✱ ❚✳✱ ✫ ❉②❜❦♦✇s❦✐✱ ▼✳ ✭✷✵✶✵✮✳ ❙t❛t♦r✲❝✉rr❡♥t✲❜❛s❡❞ ▼❘❆❙ ❡st✐✲ ❬✶❪ ❈❤❛❞❧✐✱ ▼✳✱ ▼❛q✉✐♥✱ ❉✳✱ ✫ ❘❛❣♦t✱ ❏✳ ✭✷✵✵✶✱ ♠❛t♦r ❢♦r ❛ ✇✐❞❡ r❛♥❣❡ s♣❡❡❞✲s❡♥s♦r❧❡ss ❙❡♣t❡♠❜❡r✮✳ ❖♥ t❤❡ st❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ✐♥❞✉❝t✐♦♥✲♠♦t♦r ❞r✐✈❡✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♠✉❧t✐♣❧❡ ♠♦❞❡❧ s②st❡♠s✳ ■♥ ❈♦♥tr♦❧ ❈♦♥✲ ♦♥ ■♥❞✉str✐❛❧ ❊❧❡❝tr♦♥✐❝s✱ ✺✼✭✹✮✱ ✶✷✾✻✲✶✸✵✽✳ ❢❡r❡♥❝❡ ✭❊❈❈✮✱ ✷✵✵✶ ❊✉r♦♣❡❛♥ ✭♣♣✳ ✶✽✾✹✲ ✶✽✾✾✮✳ ■❊❊❊✳ ❬✶✷❪ ❇r❛♥❞st❡tt❡r✱ P✳✱ ❈❤❧❡❜✐s✱ P✳✱ P❛❧❛❝❦②✱ P✳✱ ✫ ❙❦✉t❛✱ ❖✳ ✭✷✵✶✶✮✳ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❘❇❋ ❬✷❪ ❈❤❛❞❧✐✱ ▼✳ ✭✷✵✵✻✮✳ ❖♥ t❤❡ ❙t❛❜✐❧✐t② ❆♥❛❧✲ ♥❡t✇♦r❦ ✐♥ r♦t♦r t✐♠❡ ❝♦♥st❛♥t ❛❞❛♣t❛t✐♦♥✳ ②s✐s ♦❢ ❯♥❝❡rt❛✐♥ ❋✉③③② ▼♦❞❡❧s✳ ■♥t❡r♥❛✲ ❊❧❡❦tr♦♥✐❦❛ ■❘ ❡❧❡❦tr♦t❡❝❤♥✐❦❛✱ ✶✶✸✭✼✮✱ ✷✶✲ t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❋✉③③② ❙②st❡♠s✱ ✽✭✹✮✳ ✷✻✳ ❬✸❪ ❆♦✉❛♦✉❞❛✱ ❙✳✱ ❈❤❛❞❧✐✱ ▼✳✱ ❇♦✉❦❤♥✐❢❡r✱ ▼✳✱ ❬✶✸❪ ❱♦✱ ❍✳ ❍✳✱ ❇r❛♥❞➨t❡tt❡r✱ P✳✱ ❉♦♥❣✱ ❈✳ ❙✳ ✫ ❑❛r✐♠✐✱ ❍✳ ❘✳ ✭✷✵✶✺✮✳ ❘♦❜✉st ❢❛✉❧t t♦❧✲ ❚✳✱ ✫ ❚r❛♥✱ ❚✳ ❈✳ ✭✷✵✶✻✮✳ ❙♣❡❡❞ ❡st✐♠❛t♦rs ❡r❛♥t tr❛❝❦✐♥❣ ❝♦♥tr♦❧❧❡r ❞❡s✐❣♥ ❢♦r ✈❡❤✐❝❧❡ ✉s✐♥❣ st❛t♦r r❡s✐st❛♥❝❡ ❛❞❛♣t❛t✐♦♥ ❢♦r s❡♥✲ ❞②♥❛♠✐❝s✿ ❆ ❞❡s❝r✐♣t♦r ❛♣♣r♦❛❝❤✳ ♠❡❝❤❛✲ s♦r❧❡ss ✐♥❞✉❝t✐♦♥ ♠♦t♦r ❞r✐✈❡✳ tr♦♥✐❝s✱ ✸✵✱ ✸✶✻✲✸✷✻✳ ❬✶✹❪ ❚❛♦✱ ●✳ ❖✳ ◆✳ ●✳✱ ✫ ▲❡✐✱ ◗✳ ■✳ ✭✷✵✶✸✮✳ ■♥t❡❧❧✐✲ ❬✹❪ ▲✐✉✱ ❍✳✱ ❙❤✐✱ P✳✱ ❑❛r✐♠✐✱ ❍✳ ❘✳✱ ✫ ❈❤❛❞❧✐✱ ❣❡♥t ❣❛✉❣❡ ❝♦♥tr♦❧ s②st❡♠ ✉s✐♥❣ ❆❘▼ ❛♥❞ ▼✳ ✭✷✵✶✻✮✳ ❋✐♥✐t❡✲t✐♠❡ st❛❜✐❧✐t② ❛♥❞ st❛❜✐❧✐✲ ❢✉③③② P■ ❝♦♥tr♦❧❧❡r✳ ❙t✉❞✐❡s ✐♥ ■♥❢♦r♠❛t✐❝s s❛t✐♦♥ ❢♦r ❛ ❝❧❛ss ♦❢ ♥♦♥❧✐♥❡❛r s②st❡♠s ✇✐t❤ ❛♥❞ ❈♦♥tr♦❧✱ ✷✷✭✶✮✱ ✹✹✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✶✾ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❬✶✺❪ ❲❛♥❣✱ ❍✳✱ ❚✐❛♥✱ ❨✳✱ ✫ ❩❤❡♥❣✱ ❉✳ ✭✷✵✶✻✮✳ ❬✷✶❪ ❘❛♦✱ ❚✳ ❱✳ ❱✳ ▲✳ ◆✳✱ ✫ ❙❛✇✐❝❦✐✱ ❏✳ ❚✳ ✭✷✵✵✹✮✳ ❊❙❖✲❜❛s❡❞ ✐P■ ❈♦♠♠♦♥ ❘❛✐❧ Pr❡ss✉r❡ ❈♦♥✲ ❙t❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ✢♦❛t✐♥❣ ❜❛❧❧ ❜❡❛r✐♥❣✳ tr♦❧ ♦❢ ❍✐❣❤ Pr❡ss✉r❡ ❈♦♠♠♦♥ ❘❛✐❧ ■♥✲ ❚r✐❜♦❧♦❣② tr❛♥s❛❝t✐♦♥s✱ ✹✼✭✹✮✱ ✺✹✸✲✺✹✽✳ ❥❡❝t✐♦♥ ❙②st❡♠✳ ❙t✉❞✐❡s ✐♥ ■♥❢♦r♠❛t✐❝s ❛♥❞ ❬✷✷❪ ❙✐❡✇♥✐❛❦✱ P✳✱ ✫ ●r③❡s✐❦✱ ❇✳ ✭✷✵✶✹✮✳ ❆ ❜r✐❡❢ ❈♦♥tr♦❧✱ ✷✺✭✸✮✱ ✷✼✸✲✷✽✷✳ r❡✈✐❡✇ ♦❢ ♠♦❞❡❧s ♦❢ ❉❈✕❉❈ ♣♦✇❡r ❡❧❡❝✲ ❬✶✻❪ ❘♦✉t❤✱ ❊✳ ❏✳ ✭✶✽✼✼✮✳ ❆ tr❡❛t✐s❡ ♦♥ t❤❡ st❛❜✐❧✲ tr♦♥✐❝ ❝♦♥✈❡rt❡rs ❢♦r ❛♥❛❧②s✐s ♦❢ t❤❡✐r st❛✲ ✐t② ♦❢ ❛ ❣✐✈❡♥ st❛t❡ ♦❢ ♠♦t✐♦♥✿ ♣❛rt✐❝✉❧❛r❧② ❜✐❧✐t②✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr♦♥✐❝s✱ st❡❛❞② ♠♦t✐♦♥✳ ▼❛❝♠✐❧❧❛♥ ❛♥❞ ❈♦♠♣❛♥②✳ r ssỗó ❬✷✸❪ ❨❛♥❣✱ ❳✳✱ ❨✉❛♥✱ ❨✳✱ ▲♦♥❣✱ ❩✳✱ ●♦♥❝❛❧✈❡s✱ ▼✳ ❘✳ ✭✷✵✵✼✮✳ Pr♦♣♦rt✐♦♥❛❧ ❝♦♥tr♦❧❧❡rs✿ ❞✐✲ ❏✳✱ ✫ P❛❧♠❡r✱ P✳ ❘✳ ✭✷✵✶✻✮✳ ❘♦❜✉st st❛❜✐❧✐t② r❡❝t ♠❡t❤♦❞ ❢♦r st❛❜✐❧✐t② ❛♥❛❧②s✐s ❛♥❞ ▼❆❚✲ ❛♥❛❧②s✐s ♦❢ ❛❝t✐✈❡ ✈♦❧t❛❣❡ ❝♦♥tr♦❧ ❢♦r ❤✐❣❤✲ ▲❆❇ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♣♦✇❡r ■●❇❚ s✇✐t❝❤✐♥❣ ❜② ❑❤❛r✐t♦♥♦✈✬s t❤❡✲ ♦♥ ❊❞✉❝❛t✐♦♥✱ ✺✵✭✶✮✱ ✼✹✲✼✽✳ ♦r❡♠✳ ■❊❊❊ ❚r❛♥s✳ P♦✇❡r ❊❧❡❝tr♦♥✳✱ ✸✶✭✸✮✱ ✷✺✽✹✲✷✺✾✺✳ ❬✶✽❪ ❱♦✱ ❍✳ ❍✳✱ ❇r❛♥❞st❡tt❡r✱ P✳✱ ❉♦♥❣✱ ❈✳ ❙✳✱ ❚❤✐❡✉✱ ❚✳ ◗✳✱ ✫ ❱♦✱ ❉✳ ❍✳ ✭✷✵✶✻✮✳ ❆♥ ✐♠✲ ❬✷✹❪ ▼❛♥❣✐♣✉❞✐✱ ❙✳ ❑✳✱ ✫ ❇❡❣✉♠✱ ●✳ ✭✷✵✶✻✮✳ ❆ ♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ▼❆❚▲❆❇ s♦❢t✇❛r❡ ❢♦r st❛✲ ♥❡✇ ❜✐❛s❡❞ ♠♦❞❡❧ ♦r❞❡r r❡❞✉❝t✐♦♥ ❢♦r ❤✐❣❤❡r ❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ♣r♦♣♦rt✐♦♥❛❧ ❝♦♥tr♦❧❧❡rs ♦r❞❡r ✐♥t❡r✈❛❧ s②st❡♠s✳ ✐♥ ❧✐♥❡❛r t✐♠❡ ✐♥✈❛r✐❛♥t ❝♦♥tr♦❧ s②st❡♠s✳ ■♥ ◆❡✇ ❆❞✈❛♥❝❡s ✐♥ ■♥❢♦r♠❛t✐♦♥ ❙②st❡♠s ❬✷✺❪ ◆❣✉②❡♥✱ ❉✳ ◗✳ ✭✷✵✶✼✮✳ ❆♥ ❡✛❡❝t✐✈❡ ❛♣✲ ♣r♦❛❝❤ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r✲ ❛♥❞ ❚❡❝❤♥♦❧♦❣✐❡s ✭♣♣✳ ✻✼✶✲✻✽✵✮✳ ❙♣r✐♥❣❡r✱ ❞❡r s②st❡♠ ✉s✐♥❣ r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞✳ ❈❤❛♠✳ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠✲ ❬✶✾❪ ❆✇❡②❛✱ ❏✳✱ ❖✉❡❧❧❡tt❡✱ ▼✳✱ ✫ ▼♦♥t✉♥♦✱ ❉✳ ♣✉t❛t✐♦♥✱ ✶✭✶✮✱ ✸✾✲✹✼✳ ❨✳ ✭✷✵✵✹✮✳ ❉❡s✐❣♥ ❛♥❞ st❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ ❛ r❛t❡ ❝♦♥tr♦❧ ❛❧❣♦r✐t❤♠ ✉s✐♥❣ t❤❡ ❘♦✉t❤✲ ❍✉r✇✐t③ st❛❜✐❧✐t② ❝r✐t❡r✐♦♥✳ ■❊❊❊✴❆❈▼ ❆❜♦✉t ❆✉t❤♦rs tr❛♥s❛❝t✐♦♥s ♦♥ ♥❡t✇♦r❦✐♥❣✱ ✶✷✭✹✮✱ ✼✶✾✲✼✸✷✳ ❬✷✵❪ ❊❧✲▼❛r❤♦♠②✱ ❆✳ ❆✳✱ ✫ ❆❜❞❡❧✲❙❛tt❛r✱ ◆✳ ❊✳ ❍❛✉ ❍✉✉ ❱❖ ✇❛s ❜♦r♥ ✐♥ ❱✐❡t♥❛♠✳ ❍❡ ✭✷✵✵✹✮✳ ❙t❛❜✐❧✐t② ❛♥❛❧②s✐s ♦❢ r♦t♦r✲❜❡❛r✐♥❣ r❡❝❡✐✈❡❞ ❤✐s ▼✳❙❝✳ ❞❡❣r❡❡ ✐♥ ❆✉t♦♠❛t✐♦♥ ❊♥❣✐✲ s②st❡♠s ✈✐❛ ❘♦✉t❤✲❍✉r✇✐t③ ❝r✐t❡r✐♦♥✳ ❆♣✲ ♥❡❡r✐♥❣ ❢r♦♠ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t② ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ❱✐❡t♥❛♠ ✐♥ ✷✵✵✾ ❛♥❞ P❤✳❉✳ ❞❡❣r❡❡ ♣❧✐❡❞ ❊♥❡r❣②✱ ✼✼✭✸✮✱ ✷✽✼✲✸✵✽✳ ✐♥ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❢r♦♠ ❱❙❇✲❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❖str❛✈❛✱ ❈③❡❝❤ ❘❡♣✉❜❧✐❝ ✐♥ ✷✵✶✼✳ ❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ❝♦♥tr♦❧ t❤❡♦r②✱ ♠♦❞❡r♥ ❝♦♥tr♦❧ ♠❡t❤♦❞s ♦❢ ❡❧❡❝tr✐❝❛❧ ❞r✐✈❡s✱ ❛♥❞ r♦❜♦t✐❝s✳ "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0)." ✶✷✵ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ... ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✶✺ ❱❖▲❯▼❊✿ ✷ ✹✳ ❈❖▼P❯❚❆❚■❖◆ R1 = CTmin ♦❢ ❙❇❚ CTmin ♦❢ ❉■❚ ✭✸✾✮ R2 = CTavg ♦❢ ❙❇❚ CTavg ♦❢ ❉■❚ ✭✹✵✮ R3 = CTmax ♦❢ ❙❇❚ CTmax ♦❢ ❉■❚ ✭✹✶✮ ❊❳❆▼P▲❊❙... ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❚❛❜❧❡ ✺✳ ❱❛❧✉❡s CTmin , CTmax , CTavg ❬♠s❪✳ CTmin ♥ ❉■❚ ❙❇❚ CTavg ❉■❚ CTmax ❙❇❚ ❉■❚ ❙❇❚ ❙♣ ✭P■ ❝♦♥tr♦❧❧❡r✮ ✷ ✺✼✳✾ ✸✵✳✹ ✻✵✳✶ ✸✶✳✹... Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0)." ✶✷✵ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣