In practice, the popular way to overcome these difficulties is linearization of the fractional-order system. Here, a systematic approach is proposed for linearizing the transfer function of fractional order systems. This approach is based on the real interpolation method (RIM) to approximate fractional-order transfer function (FOTF) by rational-order transfer function.
❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❆♥ ❡❢❢❡❝t✐✈❡ ❛♣♣r♦❛❝❤ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r s②st❡♠ ✉s✐♥❣ r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ◗✉❛♥❣ ❉✉♥❣ ◆●❯❨❊◆✯ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠ ✯♥❣✉②❡♥q✉❛♥❣❞✉♥❣❅t❞t✳❡❞✉✳✈♥ ✭❘❡❝❡✐✈❡❞✿ ✶✶✲❋❡❜r✉❛r②✲✷✵✶✼❀ ❛❝❝❡♣t❡❞✿ ✸✵✲❆♣r✐❧✲✷✵✶✼❀ ♣✉❜❧✐s❤❡❞✿ ✽✲❏✉♥❡✲✷✵✶✼✮ ❋r❛❝t✐♦♥❛❧✲♦r❞❡r ❝♦♥tr♦❧❧❡rs ❛r❡ r❡❝✲ ♦❣♥✐③❡❞ t♦ ❣✉❛r❛♥t❡❡ ❜❡tt❡r ❝❧♦s❡❞✲❧♦♦♣ ♣❡r❢♦r✲ ♠❛♥❝❡ ❛♥❞ r♦❜✉st♥❡ss t❤❛♥ ❝♦♥✈❡♥t✐♦♥❛❧ ✐♥t❡❣❡r✲ ♦r❞❡r ❝♦♥tr♦❧❧❡rs✳ ❍♦✇❡✈❡r✱ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ♠❛❦❡ t✐♠❡✱ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ❛♥❛❧②s✐s ❛♥❞ s✐♠✉❧❛t✐♦♥ s✐❣♥✐✜❝❛♥t❧② ❞✐✣❝✉❧t✳ ■♥ ♣r❛❝t✐❝❡✱ t❤❡ ♣♦♣✉❧❛r ✇❛② t♦ ♦✈❡r❝♦♠❡ t❤❡s❡ ❞✐❢✲ ✜❝✉❧t✐❡s ✐s ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠✳ ❍❡r❡✱ ❛ s②st❡♠❛t✐❝ ❛♣♣r♦❛❝❤ ✐s ♣r♦♣♦s❡❞ ❢♦r ❧✐♥❡❛r✐③✐♥❣ t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧✲ ♦r❞❡r s②st❡♠s✳ ❚❤✐s ❛♣♣r♦❛❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ✭❘■▼✮ t♦ ❛♣♣r♦①✐♠❛t❡ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ✭❋❖❚❋✮ ❜② r❛t✐♦♥❛❧✲♦r❞❡r tr❛♥s❢❡r ❢✉♥❝t✐♦♥✳ ❚❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞ ✐s ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ ❝♦♠♣❛r❡❞ t♦ ❈❋❊ ❤✐❣❤✲❢r❡q✉❡♥❝② ♠❡t❤♦❞❀ ❈❛r❧s♦♥✬s ♠❡t❤♦❞❀ ▼❛t✲ s✉❞❛✬s ♠❡t❤♦❞❀ ❈❤❛r❡✛✬s ♠❡t❤♦❞❀ ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ❧❡❛st✲sq✉❛r❡s✱ ❢r❡q✉❡♥❝② ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ✭❋■▼✮✳ ❚❤❡ r❡s✉❧ts ♦❢ ❝♦♠♣❛r✐s♦♥ s❤♦✇ t❤❛t✱ t❤❡ ♠❡t❤♦❞ ✐s s✐♠♣❧❡✱ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡❢✲ ✜❝✐❡♥t✱ ✢❡①✐❜❧❡✱ ❛♥❞ ♠♦r❡ ❛❝❝✉r❛t❡ ✐♥ t✐♠❡ ❞♦✲ ♠❛✐♥ t❤❛♥ t❤❡ ❛❜♦✈❡ ❝♦♥s✐❞❡r❡❞ ♠❡t❤♦❞s✳ ❆❜str❛❝t✳ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❝♦♥❝❡♣t ♦❢ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s ❤❛s ❛♣♣❡❛r❡❞ ❧♦♥❣ t✐♠❡ ❛❣♦ ❜✉t ❞✉❡ t♦ ✐ts ❝♦♠♣❧❡①✐t②✱ ✐t ❝♦✉❧❞ ♥♦t ❜❡ ✉s❡❞ ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✳ ■t ✐s ♦♥❧② ✐♥ t❤❡ r❡❝❡♥t ②❡❛rs ✇✐t❤ r❛♣✐❞ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❤❛r❞✲ ✇❛r❡ ❛♥❞ s♦❢t✇❛r❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠♣✉t❡r ❛♥❞ ❡❧❡❝tr♦♥✐❝s ✜❡❧❞s t❤❛t ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s t❤❡♦r② ❤❛s ❜❡❡♥ ✇✐❞❡❧② ✉s❡❞ ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ s❝✐✲ ❡♥❝❡ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣✱ ✐♥❝❧✉❞✐♥❣ ❛❝♦✉st✐❝s ❬✶❪✱ ❬✷❪✱ r♦❜♦t✐❝s ❬✸❪✱ ❬✹❪✱ ❜✐♦♠❡❞✐❝❛❧ ❡♥❣✐♥❡❡r✐♥❣✱ ❝♦♥tr♦❧ s②st❡♠s ❬✺❪✱ ❬✻❪✱ ❬✼❪ ❛♥❞ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ❬✽❪✱ ❬✾❪✳ ■♥ ❢❛❝t✱ ♦♥❡ ❝♦✉❧❞ ❛r❣✉❡ t❤❛t r❡❛❧ ✇♦r❧❞ ♣r♦❝❡ss❡s ❛r❡ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r s②st❡♠s ✐♥ ❣❡♥❡r❛❧ ❬✶✵❪✱ ❬✶✶❪✳ ❋r❛❝t✐♦♥❛❧✲♦r❞❡r ♠♦❞❡❧s ❛r❡ ✐♥✜♥✐t❡ ❞✐♠❡♥✲ s✐♦♥❛❧✱ ❛♥❞ ♠♦r❡ ❛❞❡q✉❛t❡ ❢♦r t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❞②♥❛♠✐❝❛❧ s②st❡♠s t❤❛♥ t❤❡ ✐♥t❡❣❡r✲♦r❞❡r ♠♦❞✲ ❡❧s✳ ■♥ t❡❝❤♥✐❝❛❧ ❧✐t❡r❛t✉r❡✱ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r ❞✐❢✲ ❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛r❡ ♠♦st❧② ❛♥❛❧②③❡❞ ✉s✐♥❣ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ t❡❝❤♥✐q✉❡s ❬✶✵❪✳ ❍♦✇❡✈❡r✱ t❤❡ s✐❣♥❛❧s ✐♥✈♦❧✈❡❞ ✐♥ t❤❡s❡ ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ ❝❤❛r❛❝✲ t❡r✐③❡❞ ❜② ✐rr❛t✐♦♥❛❧ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠✱ s♦ t❤❛t t❤❡ ✐♥✈❡rs❡ tr❛♥s❢♦r♠s ❛r❡ ❣❡♥❡r❛❧❧② ♥♦t ❡❛s✐❧② ❡✈❛❧✉❛t❡❞ ❛♥❞ t❤❡ t✐♠❡✲❞♦♠❛✐♥ ❛♥❛❧②s✐s ❢❛❝❡s ❛ ❧♦t ♦❢ ❞✐✣❝✉❧t✐❡s✳ ❑❡②✇♦r❞s ❆s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ ♦♥❡ ♦❢ t❤❡ ♠❛❥♦r ❞✐✣❝✉❧✲ t✐❡s ✇✐t❤ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r r❡♣r❡s❡♥t❛t✐♦♥ ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❢r❡q✉❡♥❝②✱ ❛♥❞ ❡s♣❡❝✐❛❧❧② t✐♠❡ r❡s♣♦♥s❡s✳ ▼❛♥② st✉❞✐❡s ❤❛✈❡ ❜❡❡♥ ❞♦♥❡ ✐♥ ♦r✲ ❆♣♣r♦①✐♠❛t✐♦♥✱ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠✱ ❞❡r t♦ s✐♠✉❧❛t❡ ❢r❛❝t✐♦♥❛❧ ❝♦♥tr♦❧ s②st❡♠s ♦✈❡r r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞✳ t❤❡ ❧❛st ❞❡❝❛❞❡✳ ❚❤❡ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✸✾ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ♦✉t♣✉t ✐s ♥♦t ♣r❛❝t✐❝❛❧ ❛♥❞ t❤❡r❡ ✐s ♥♦ ❛ ❣❡♥✲ ✐♥t❡❣❡r✲♦r❞❡r ♠♦❞❡❧s ❢♦r ❛tt❛✐♥✐♥❣ ❞❡s✐r❡❞ ❛❝❝✉✲ ❡r❛❧ ♠❡t❤♦❞ ❢♦r ❡st✐♠❛t✐♥❣ ✐t ❬✶✷❪✳ r❛❝② ✐♥ t❤❡ ❞❡s✐r❡❞ ❢r❡q✉❡♥❝② r❛♥❣❡s✳ ❚❤❡r❡ ❛r❡ ■♥ s✉❝❤ ❛❧s♦ s♦♠❡ ♠❡t❤♦❞s ❜❛s❡❞ ♦♥ ▼✐tt❛❣✲▲❡✤❡r ❢✉♥❝✲ ❝❛s❡s✱ ❛ r❡❞✉❝❡❞ ♦r❞❡r ♠♦❞❡❧ ❝❛♥ ❜❡ r❡q✉✐r❡❞ t✐♦♥s✱ ●r✉♥✇❛❧❞✲▲❡t♥✐❦♦✈ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ ❛ ❤✐❣❤ ✐♥t❡❣❡r ♦r❞❡r tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❬✷✹❪✳ ❛♥❞ ●❛♠♠❛ ❢✉♥❝t✐♦♥s ❢♦r ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♠♣✉❧s❡ ❛♥❞ st❡♣ r❡s♣♦♥s❡s ♦❢ ❝♦♠♠❡♥s✉r❛t❡✲ ♦r❞❡r s②st❡♠ ❬✶✸❪✱ ❬✶✹❪✳ ❍♦✇❡✈❡r✱ t❤❡ s♦❧✉✲ t✐♦♥ ♠❡t❤♦❞s ✉s✐♥❣ ▼✐tt❛❣✲▲❡✤❡r ❢✉♥❝t✐♦♥s ❛♥❞ ●❛♠♠❛ ❢✉♥❝t✐♦♥ ❛r❡ t✐♠❡ ❝♦♥s✉♠✐♥❣ ❛♥❞ ❤✐❣❤❧② ✐♥❛❝❝✉r❛t❡✱ ♦❝❝✉rr✐♥❣ ✐♥ s♦❧✈✐♥❣ ❝♦♠♣❧✐❝❛t❡❞ ❛♥❞ ❤✐❣❤ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✳ ❖♥❡ ♣♦ss✐❜❧❡ ❛♣♣r♦❛❝❤ t♦ ♠♦❞❡❧❧✐♥❣ ▼♦st ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ❛r❡ st✉❞✲ ✐❡❞ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✱ ❜❡❝❛✉s❡ ♦❢ t❤❡✐r ❛❝❝✉r❛❝② ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ ♠✐❣❤t ♥♦t r❡❛❝❤ t❤❡ ❞❡s✐r❡❞ ✈❛❧✉❡✳ ❛♣♣r♦❛❝❤ ❢♦r ❚❤✐s ♣❛♣❡r ✐♥tr♦❞✉❝❡s ❛♥ ✐♥✈❡rt✐♥❣ t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠s t♦ r❛t✐♦♥❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ✇✐t❤ ❝♦♠♠❡♥s✉r❛t❡ ♦r❞❡r✳ ❚❤❡ ♣r♦✲ ❢r❛❝✲ ♣♦s❡❞ ❛♣♣r♦❛❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ r❡❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ t✐♦♥❛❧ ♦r❞❡r s②st❡♠ ✐s ❜❛s❡❞ ♦♥ ♥✉♠❡r✐❝❛❧ ❛♣✲ ♠❡t❤♦❞ ❬✷✺❪✱ ❬✷✻❪✱ ✇❤✐❝❤ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② t✇♦ ♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ♥♦♥✲✐♥t❡❣❡r ♦r❞❡r ♦♣❡r❛t♦r ♠❛✐♥ ❢❡❛t✉r❡s✳ ❚❤❡ ✜rst ❢❡❛t✉r❡ ✐♥✈♦❧✈❡s t❤❡ ♦♣✲ ❬✶✺❪✱ ❬✶✻❪✱ ❬✶✼❪✳ ❚❤❡ ♠❡t❤♦❞s ❞❡✈❡❧♦♣✐♥❣ ✐♥t❡❣❡r ❡r❛t♦r ♠❡t❤♦❞✱ ✐♥ ✇❤✐❝❤ t❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ✐♥ ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ❛ttr❛❝t✐✈❡ s✐♥❝❡✱ t❤❡② t❤❡ ✐♠❛❣✐♥❛r② ❞♦♠❛✐♥✱ ✇❤❡r❡ ❝♦♠♣✉t❛t✐♦♥ ❤❛s ❝♦♥✈❡rt t❤❡ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❤❡ ❋❖❚❋s ✐♥t♦ ❝❡rt❛✐♥❧② ♠♦r❡ ❛❞✈❛♥t❛❣❡s t❤❛♥ ✐♥ t❤❡ t✐♠❡ ❞♦✲ ❝❧❛ss✐❝❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s✳ ❚❤❡r❡❢♦r❡ ❛ ❧❛r❣❡ ♠❛✐♥✳ ❚❤❡ s❡❝♦♥❞ ❢❡❛t✉r❡ ✐s t❤❛t t❤❡ ♠♦❞❡❧s ✐♥ ♥✉♠❜❡r ♦❢ ♠❡t❤♦❞s t♦ ❡✈❛❧✉❛t❡ r❛t✐♦♥❛❧ ❛♣♣r♦①✲ t❤❡ ❘■▼ ❛r❡ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❛ r❡❛❧ ✈❛r✐❛❜❧❡✱ ❝♦♠✲ ✐♠❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ❞❡✈❡❧♦♣❡❞✳ ❚❤❡ ♠♦st ♣♦♣✉✲ ♣❛r✐♥❣ ✇✐t❤ ❛ ♠♦❞❡❧ ♣r♦❞✉❝✐♥❣ ✐♥ t❤❡ ✐♠❛❣✐♥❛r② ❧❛r ♦❢ t❤❡s❡ ❛r❡ ❧✐st❡❞✿ ❢r❡q✉❡♥❝② ✐♥t❡r♣♦❧❛t✐♦♥s✱ ❞♦♠❛✐♥ ♦r ✐♥ t❤❡ ❝♦♠♣❧❡① ❞♦♠❛✐♥ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥❛❧ ❡①♣❛♥s✐♦♥ ✭❈❋❊✮ ♠❡t❤♦❞✱ ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞✱ ❈❛r❧s♦♥✬s ♠❡t❤♦❞✱ ▼❛t✲ s✉❞❛✬s ♠❡t❤♦❞✱ ❈❤❛r❡✛ ✬s ♠❡t❤♦❞✱ ❛♥❞ ❧❡❛st✲ sq✉❛r❡ ♠❡t❤♦❞✳ ✷✳ ❘❡❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ✐♥ ❢r❡q✉❡♥❝② ❞♦✲ ♠❛✐♥ ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ❢r❡q✉❡♥❝② ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞s ✭❋■▼✮ ❬✶✽❪✳ ❚❤❡s❡ ♠❡t❤♦❞s r❡q✉✐r❡ s❡♣✲ ❘■▼ ✐s ♦♥❡ ♦❢ t❤❡ ♠❡t❤♦❞s✱ ✇❤✐❝❤ ✇♦r❦s ♦♥ ❛r❛t✐♥❣ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ t❤❡ ❢r❛❝✲ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ ✐♠❛❣✐♥❛r② ❞♦✲ t✐♦♥❛❧ ♦r❞❡r tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ✇❤❡♥ r❡♣❧❛❝✐♥❣ t❤❡ ♠❛✐♥✳ ❢r❡q✉❡♥❝② ✈❛r✐❛❜❧❡s✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ r❡s✉❧ts tr❛♥s❢♦r♠✱ ❚❤❡ ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ r❡❛❧ ✐♥t❡❣r❛❧ ❝♦✉❧❞ ❤❛✈❡ ❤✐❣❤ ❛❝❝✉r❛❝② ✐♥ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✳ ❍♦✇❡✈❡r✱ ✐♥ t✐♠❡ ❞♦♠❛✐♥✱ ❛❝❝✉r❛❝② ✐s ✉♥❝❡rt❛✐♥ ❡s♣❡❝✐❛❧❧② ✇✐t❤ ❧♦✇ ❛♣♣r♦①✐♠❛t❡❞ ♦r❞❡r ❢✉♥❝✲ t✐♦♥✳ t✐♦♥s ❡①♣❛♥s✐♦♥ ✭❈❋❊✮ ❬✶✾❪✱ ♦r ♠♦❞✐✜❡❞ ❈❋❊ s✉❝❤ ❛s ❈❛r❧s♦♥ ♠❡t❤♦❞ ❬✷✵❪✱ ❬✷✶❪✳ ▼❛♥② r❡✲ s❡❛r❝❤❡rs ❤❛✈❡ ❜❡❡♥ ✇♦r❦✐♥❣ ✐♥ t❤✐s ❛r❡❛ ❛♥❞ ❤❛✈❡ ❜❡❡♥ s✉❝❝❡ss❢✉❧ ✐♥ ❞❡✈❡❧♦♣✐♥❣ s♦♠❡ ❛♣✲ ♣r♦①✐♠❛t✐♦♥ t❡❝❤♥✐q✉❡s✱ q✉❡♥❝② ✈❛r✐❛❜❧❡s✳ ❛♣♣❧✐❡❞ t♦ t❤❡ ❢r❡✲ ❚❤❡s❡ ❛r❡ ▼❛ts✉❞❛ ♠❡t❤♦❞ ❬✶✻❪✱ ❈❤❛r❡✛ ✬s ♠❡t❤♦❞ ❬✶✺❪✱ ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞ ❬✷✷❪✱ ❬✷✸❪ ❛♥❞ t❤❡ ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❳✉❡ ❡t ❚❤❡s❡ ♠❡t❤♦❞s ♣r♦❞✉❝❡ ❛♣♣r♦①✐♠❛t❡❞ ✐♥t❡❣❡r ♦r❞❡r ♠♦❞❡❧s ✇❤♦s❡ ❝❤❛r❛❝t❡r✐st✐❝s ✜t ❝❧♦s❡❧② ❡♥♦✉❣❤ t♦ t❤❡ ✐❞❡❛❧ s②st❡♠ ❝❤❛r❛❝t❡r✐s✲ t✐❝s ✐♥ t❤❡ ❞❡s✐r❡❞ ❢r❡q✉❡♥❝② ❜❛♥❞✇✐❞t❤✳ ❖✉t ♦❢ t❤❡s❡✱ s♦♠❡ ♠❡t❤♦❞s ❛♣♣r♦①✐♠❛t❡ ✈❡r② ❤✐❣❤ ✹✵ f (t)e−δ·t dt, δ ∈ (C, ∞), C ≥ 0, ❙♦♠❡ st✉❞✐❡s ❛r❡ ❜❛s❡❞ ♦♥ ❛ ❝♦♥t✐♥✉❡❞ ❢r❛❝✲ ❛❧ ❬✷✹❪✳ ∞ F (δ) = ✭✶✮ F (δ) ✐♥ ❛❝❝♦r✲ f (t) ❛s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❛ss✐❣♥s t❤❡ ✐♠❛❣❡ ❢✉♥❝t✐♦♥ ❞❛♥❝❡ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ r❡❛❧ ✈❛r✐❛❜❧❡ δ✳ ❋♦r♠✉❧❛ ♦❢ ❞✐r❡❝t tr❛♥s✲ ❢♦r♠ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❞✐r❡❝t ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❜② r❡♣❧❛❝✐♥❣ t❤❡ ❝♦♠✲ ♣❧❡① ✈❛r✐❛❜❧❡ s ❢♦r r❡❛❧ δ ✈❛r✐❛❜❧❡✳ ❆♥♦t❤❡r st❡♣ t♦✇❛r❞s t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ✐♥str✉♠❡♥t❛t✐♦♥ ♠❡t❤♦❞ ✐s t❤❡ tr❛♥s✐t✐♦♥ ❢r♦♠ ❝♦♥t✐♥✉♦✉s ❢✉♥❝✲ t✐♦♥s F (δ) t♦ t❤❡✐r ❞✐s❝r❡t❡ ❢♦r♠✱ ✉s✐♥❣ t❤❡ ❝♦♠✲ ♣✉t✐♥❣ r❡s♦✉r❝❡s ❛♥❞ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s✳ ❋♦r t❤❡s❡ ♣✉r♣♦s❡s✱ ❘■▼ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ♥✉✲ ♠❡r✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s {F (δi )}N ✳ ❚❤❡② ❛r❡ ♦❜✲ t❛✐♥❡❞ ❛s ❛ s❡t ♦❢ ✈❛❧✉❡s ♦❢ ❢✉♥❝t✐♦♥ ♥♦❞❡s δi where i ∈ 1, 2, N ✱ F (δ) N ✇❤❡r❡ ✐♥ t❤❡ ✐s t❤❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦❢ ♥✉♠❡r✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✱ ♦r ❢♦r ❝❛❧❧❡❞ ✐ts ❞✐♠❡♥s✐♦♥✳ G (0) = 0, a0 = | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ♦♥❡ ♦❜t❛✐♥❡❞ an δin G(δ1 ) ❙❡❧❡❝t✐♥❣ ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥s δi + + a1 δi G(δi ) − bm δim − −b0 = −G(δi ), i = 1, N , ✐s ❛ ♣r✐♠❛r② st❡♣ ✭✺✮ ✐♥ t❤❡ tr❛♥s✐t✐♦♥ t♦ ❛ ❞✐s❝r❡t❡ ❢♦r♠✱ ✇❤✐❝❤ ❤❛s ❛ ❋♦r ✜①❡❞ s✐❣♥✐✜❝❛♥t ✐♠♣❛❝t ♦♥ t❤❡ ♥✉♠❡r✐❝❛❧ ❝♦♠♣✉t✐♥❣ δi ❜♦t❤ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛✲ ❛♥❞ ❛❝❝✉r❛❝② ♦❢ ♣r♦❜❧❡♠ s♦❧✉t✐♦♥s✳ ❉✐str✐❜✉t✐♦♥ t♦r ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t❤❡ ♦❢ ♥♦❞❡s ✐♥ t❤❡ s✐♠♣❧❡st ✈❛r✐❛♥t ✐s ✉♥✐❢♦r♠✳ ❆♥✲ ✉♥❦♥♦✇♥ ♣r♦❝❡ss ♣❛r❛♠❡t❡rs✳ ♦t❤❡r ✐♠♣♦rt❛♥t ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❘■▼ ✐s ❝r♦ss✲ ❡q✉❛t✐♦♥s ✭✾✮ r❡♣r❡s❡♥ts ❛ ❧✐♥❡❛r s②st❡♠ ♦❢ ❡q✉❛✲ ❝♦♥✈❡rs✐♦♥ ♣r♦♣❡rt②✳ ■t ❞✉❡s t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ t✐♦♥s ❤❛✈✐♥❣ ◆ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✱ ♦♥❡ ♦❜t❛✐♥s ◆ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛r❣✉♠❡♥t δ F (δ) ❢♦r ❧❛r❣❡ ✈❛❧✉❡s ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❊q✳ ✸✳ ✐s ❞❡t❡r♠✐♥❡❞ ♠❛✐♥❧② ❜② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ f (t) ❢♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡ t✳ ■♥ t❤❡ ♦♣♣♦s✐t❡ ❝❛s❡✱ t❤❡ r❡s✉❧t ✐s t❤❡ s❛♠❡✿ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ t❤❡ ❛r❣✉♠❡♥t δ ✐s ❞❡t❡r♠✐♥❡❞ ♠❛✐♥❧② ❜② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❧❛r❣❡ ✈❛❧✉❡s ♦❢ t❤❡ ✈❛r✐❛❜❧❡ F (δ) f (t) ❚❤❡ ♦❜t❛✐♥❡❞ ❊q✳ ✺ ❛r❡ ❝♦♥✈❡♥✐❡♥t❧② r❡✇r✐t✲ t❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① ❢♦r♠✱ ✇❤✐❝❤ ✐s ❡❛s✐❧② s♦❧✈❡❞ ✉s✐♥❣ s♦♠❡ ♦❢ t❤❡ ♠♦❞❡r♥ ❝♦♠♣✉t❡r ❛❧❣❡✲ ❜r❛ ♣❛❝❦❛❣❡s✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐♥tr♦❞✉❝✐♥❣ ❢♦r t✳ M= ✸✳ ❚❤✉s✱ t❤❡ s❡t ♦❢ n δN,1 G(δN,1 ) n δN,2 G(δN,2 ) n δN,N G(δN,N ) ❘❛t✐♦♥❛❧ m δN −n,1 G(δN −n,1 ) − δN −n−1,1 m δN −n,2 G(δN −n,2 ) − δN −n−1,2 m δN −n,N G(δN −n,N ) − δN −n−1,N −1 −1 , −1 ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❋❖❚s ✭✻✮ ❯s✐♥❣ ❘❡❛❧ −G(δ1 ) −G(δ2 ) B= −G(δN ) ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ■♥ t❤✐s ♣❛♣❡r ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♣♣r♦①✲ ✐♠❛t✐♦♥ t❛s❦ ♦❢ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠s✳ ❚❤❡ ❋❖❚❋ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿ G (s) = ✇❤❡r❡ p, q − real numbers K(s) = L(s) p βi i ki s q αi i li s , ♦♥❡ ❡❛s✐❧② ♦❜t❛✐♥s t❤❡ ❞❡s✐r❡❞ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐♥ ♠❛tr✐① ❢♦r♠ ✭✷✮ M · X = B, interger and βi , αi − ✇❤❡r❡ bm sm + · · · + b1 s + b0 B(s) = , A(s) an sn + · · · + a1 s + a0 m ≤ n; m, n X= ✭✸✮ ❛r❡ t❤❡ ✐♥t❡❣❡r✱ ✇❤✐❝❤ s❤♦✉❧❞ ❜❡ ✉s❡❞ t♦ ❛♣♣r♦①✐♠❛t❡ tr❛♥s❢❡r ❢✉♥❝✲ t✐♦♥ G(s) ♦❢ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♦r❞❡r s②st❡♠✳ ❋♦r (G (0) = 0, b0 = 1) ♦r (G (0) = 0, a0 = 1) t❤❡r❡ ❛r❡ N = n + m + r❡❛❧ ❝♦❡✣❝✐❡♥ts ✇❤✐❝❤ s❤♦✉❧❞ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ N ❡q✉❛t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ ♦✈❡r❧❛♣♣✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ ❝❤❛r✲ ❛❝t❡r✐st✐❝s ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐s❝r❡t❡ ♣♦✐♥ts✱ B (δi ) G (δi ) − = 0, i = 1, N , A (δi ) G (δi ) A (δi ) − B (δi ) = 0, i = 1, N , ✭✽✮ ✇❤❡r❡ ❳ ✐s t❤❡ ✈❡❝t♦r ♦❢ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs✱ ▲❡t ✉s ❝♦♥s✐❞❡r r❛t✐♦♥❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥✿ W (s) = ✭✼✮ an an−1 a1 bm b0 ✭✾✮ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♠❡♥t✐♦♥ t❤❛t t❤❡ s❡❧❡❝t❡❞ s❡t ♦❢ ♣♦✐♥ts δ ∈ [ δ1 , δ2 , , δN ] ❝❛♥ ♣r♦❞✉❝❡ ❛ s✐♥❣✉❧❛r ♠❛tr✐① ❢r♦♠ t❤❡ s❡t ♦❢ ❡q✉❛t✐♦♥s✳ ♣♦✐♥ts s❤♦✉❧❞ ❜❡ ✉s❡❞✳ ■t ✐s ❛❧s♦ s✐❣♥✐✜❝❛♥t t♦ ♥♦t❡ t❤❛t ✐t ✐s ❛❧s♦ ♣♦ss✐❜❧❡ t♦ ✉s❡ ♠♦r❡ t❤❛♥ ✭✹✮ ■♥ s✉❝❤ ❛ ❝❛s❡✱ ❛♥♦t❤❡r✱ ♠♦r❡ ❛♣♣r♦♣r✐❛t❡ s❡t ♦❢ ✐♥❝✐❞❡♥t ♣♦✐♥ts ✐♥ t❤❡ s❡❧❡❝t❡❞ s❡t✳ ♥ ❚❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ❝❛♥♥♦t ❜❡ ❢♦✉♥❞ ✐♥ s✉❝❤ ❛ ❝❛s❡✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✶ ❱❖▲❯▼❊✿ ✶ ✹✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❛♥❞ ❉✐s❝✉ss✐♦♥ ❘■▼ ✐s ♦♥❡ ♦❢ t❤❡ ♠❡t❤♦❞s✱ ✇❤✐❝❤ ✇♦r❦s ♦♥ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥s ♦❢ t❤❡ ✐♠❛❣✐♥❛r② ❞♦✲ ♠❛✐♥✳ ❋✐❣✳ ✶✿ ❚✐♠❡ r❡s♣♦♥s❡s ❚❤❡ ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ r❡❛❧ ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠✱ ▲❡t ✉s s❡❧❡❝t s❡✈❡r❛❧ ❋❖❚❋s ❛♥❞ ❝♦♠♣❛r❡ t❤❡✐r ❇♦❞❡ ❝❤❛r❛❝t❡r✐st✐❝s ❛♥❞ r❡s♣♦♥s❡ t♦ ❍❡❛✈✐s✐❞❡ ❡①❝✐t❛t✐♦♥ ✇✐t❤ t❤♦s❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❛✲ t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❞❡t❡r♠✐♥❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ s❡t ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ❊q✳ ✺✳ ■♥ t❤❡s❡ ❡①❛♠✲ ♣❧❡s t❤❡r❡ ❛r❡ ❝♦♠♣❛r✐s♦♥s ❜❡t✇❡❡♥ t❤❡ ❍❡❛✈✐✲ s✐❞❡ r❡s♣♦♥s❡s✱ ❇♦❞❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧s ❛♥❞ t❤❡ ❡①❛❝t ♠♦❞❡❧✳ ✇❤❡r❡ ❤1 ✭t✮ ✲ ❡①❛❝t t✐♠❡ r❡s♣♦♥s❡❀ ❤1R‘ ✭t✮ ✲ t✐♠❡ r❡s♣♦♥s❡ ❜② ❘■▼ ♠❡t❤♦❞❀ ❤1cf e ✭t✮ ✲ t✐♠❡ r❡✲ s♣♦♥s❡ ❜② ❈❋❊ ♠❡t❤♦❞❀ ❤1mat ✭t✮ ✲ t✐♠❡ r❡s♣♦♥s❡ ❜② ▼❛ts✉❞❛✬s ♠❡t❤♦❞❀ ❤1ls ✭t✮ ✲ t✐♠❡ r❡s♣♦♥s❡ ▲❡❛st✲sq✉❛r❡s ♠❡t❤♦❞❀ ❤1car ✭t✮ ✲ t✐♠❡ r❡s♣♦♥s❡ ❜② ❈❛r❧s♦♥✬s ♠❡t❤♦❞✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs ♦❢ t✐♠❡ r❡s♣♦♥s❡s ❛r❡ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳ ✷✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡ ❞❡♠♦♥str❛t❡s ❛ ❝❛s❡ ✇❤❡♥ t❤❡ ♣r♦❝❡ss ♦✉t♣✉ts ✐s ❡q✉❛❧ t♦ t❤❡ ❢r❛❝✲ t✐♦♥❛❧ ❝❛♣❛❝✐t♦r✳ ❤❛s ❛ ❢♦r♠✿ ❚❤❡ ❛❝t✉❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ G1 (s) = 1/s0.5 ❋♦r tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ❝❛♣❛❝✲ ✐t♦r ✇✐t❤ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r ✵✳✺ ✇❡ ❝❛rr② ♦✉t ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ✉s✐♥❣ ❘■▼ ❛♥❞ ❝♦♠♣❛r❡ t♦ ❞✐✛❡r✲ ❡♥t ♠❡t❤♦❞s ❈❋❊ ❤✐❣❤✲❢r❡q✉❡♥❝② ♠❡t❤♦❞❀ ▼❛t✲ s✉❞❛✬s ♠❡t❤♦❞❀ ❈❛r❧s♦♥✬s ♠❡t❤♦❞❀ ❧❡❛st✲sq✉❛r❡s ♠❡t❤♦❞ t❤❡♥ ❡st✐♠❛t❡ t❤❡ ❛❞❡q✉❛❝② ♦❢ ❛♣♣r♦①✐✲ ♠❛t✐♦♥ ♠♦❞❡❧s ❬✶✷❪✳ G1 (0) → (∞) → 0✱ ✇❡ ❝❤♦♦s❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❊q✳ ✸ ✇✐t❤ b0 = 1, a0 = 0✳ ❚❤❡ ♦r❞❡rs ❆❝❝♦r❞✐♥❣ t♦ ❝♦♥s✐❞❡r❡❞ ♠♦❞❡❧ ❊q✳ ✻ ∞ ❛♥❞G1 ♠♦❞❡❧ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡❞ r❛t✐♦♥❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❜❡✐♥❣ ❝♦♥s✐❞❡r❡❞ ❛r❡ m=4 n = ✐♥ ♥✉♠❡r❛t♦r✱ ❛♥❞ ✐♥ ❞❡♥♦♠✐♥❛t♦r✳ ■t ♠❡❛♥s t❤❛t ♥✉♠❜❡r ♦❢ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ✐s ❈♦rr❡s♣♦♥❞✐♥❣ t♦ ✹ th N = n + m = 7✳ ♦r❞❡r ♦❢ t❤❡ r❛t✐♦♥ ❛♣✲ ♣r♦①✐♠❛t❡❞ tr❛♥s❢❡r ❢✉♥❝t✐♦♥✱ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥t ✐s t❤❡ ♥✉♠❜❡r ♦❢ N = 7✳ δi ✐♥ [0.001; 0.005; 0.01; 0.05; 0.1; 1; 5] ✇✐t❤ ✇❡ ❝❤♦♦s❡ ✈❛❧✉❡s ♦❢ t❤❡ ♥♦❞❡s t❤❡ ■♥ t❤❡ ❘■▼ ♥♦❞❡s ❡q✉❛❧❧② s♣❛❝❡❞✳ r❛♥❣❡ ◆ ❂✼ ❋✐❣✳ ✷✿ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡s ∆❤1R‘ ✭t✮ ✲ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ❘■▼ ∆❤1cf e ✭t✮ ✲ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ❈❋❊ ♠❡t❤♦❞❀ ∆❤1mat ✭t✮ ✲ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ▼❛ts✉❞❛✬s ♠❡t❤♦❞❀ ∆❤1ls ✭t✮ ✕ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ▲❡❛st✲sq✉❛r❡s ♠❡t❤♦❞❀ ∆❤1car ✭t✮ ✲ ❡r✲ ✇❤❡r❡ ♠❡t❤♦❞❀ r♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ❈❛r❧s♦♥✬s ♠❡t❤♦❞✳ ❚❛❜✳ ✶✿ ▼❛①✐♠✉♠ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐♥ t✐♠❡ r❛♥❣❡ ❬✵✲✶✺✵❪ ✭s❡❝✮✳ ❚❤❡ r❡s✉❧ts ♦❢ ❛♣♣r♦①✐✲ ♠❛t✐♦♥ ♣r♦❝❡ss ✇✐❧❧ ❜❡ ❛♥❛❧②s❡❞ ✐♥ t❤❡ t✐♠❡ ❛♥❞ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥s ❛♥❞ ❝♦♠♣❛r❡❞ t♦ ♦t❤❡r ♠❡t❤✲ ♦❞s ✇✐t❤ s❛♠❡ ♦r❞❡r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧✳ ✹✷ ❊rr♦r ❘■▼ ❈❋❊ ✵✳✶✺✽ ✶✵✳✻✻ ▼❛ts✉❞❛✬s ▲❙✲ ❈❛r❧s♦♥✬ ♠❡t❤♦❞ ♠❡t❤♦❞ ♠❡t❤♦❞ ✷✳✻✼✹ ✹✳✶✻✽ ✹✳✽✽✾ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❆❝❝♦r❞✐♥❣ t♦ t❤❡ t✐♠❡ r❡s♣♦♥s❡s ♦❢ t❤❡ ❛♣✲ ♣r♦①✐♠❛t❡❞ r❛t✐♦♥❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥✱ t❤❡ t✐♠❡ r❡s♣♦♥s❡ ♦❢ t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❘■▼ ❞❡♠♦♥str❛t❡s s✐❣♥✐✜❝❛♥t❧② ❤✐❣❤❡r ❛❝❝✉r❛❝② t❤❛♥ t❤❡ ♦t❤❡r ♠❡t❤♦❞s ✐♥ ❝♦♥s✐❞❡r❡❞ r❛♥❣❡ ♦❢ t✐♠❡✳ ▼❛①✐♠✉♠ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs ✐♥ ❚❛❜❧❡ ✶✳ s❤♦✇ t❤❛t ♠❛①✐♠✉♠ ❛♣♣r♦①✐♠❛✲ t✐♦♥ ♦❢ t❤❡ ❘■▼ ♠♦❞❡❧ ✐s ❧♦✇❡r t❤❛♥ ▼❛ts✉❞❛✬s ♠❡t❤♦❞✱ ❤❛✈✐♥❣ s❡❝♦♥❞ ♣❧❛❝❡ ❛♥❞ ❈❋❊ ♠❡t❤♦❞✱ ❤❛✈✐♥❣ ❧❡❛st ❛❝❝✉r❛❝② ✐♥ t❤❡ ❡rr♦r ❛❜♦✉t ✶✼ ❛♥❞ ✻✼ t✐♠❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ❇♦❞❡ ♣❧♦ts ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧s ❛r❡ s❤♦✇♥ ✐♥ t❤❡ ❜❡❧♦✇ ✜❣✉r❡s✳ ❚❤❡ ❇♦❞❡ ♣❧♦ts ❋✐❣✳ ✹✿ ❊rr♦rs ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡s ✐❧❧✉str❛t❡ t❤❡ ❧♦❣❛r✐t❤♠✐❝ ♠❛❣♥✐t✉❞❡✱ ♣❤❛s❡ r❡✲ s♣♦♥s❡s✱ ❛♥❞ ❡rr♦rs ♣❧♦ts✱ r❡s♣❡❝t✐✈❡❧②✳ ❋✐❣✳ ✺✿ P❤❛s❡ r❡s♣♦♥s❡s ❋✐❣✳ ✸✿ ▼❛❣♥✐t✉❞❡ r❡s♣♦♥s❡s ❘■▼ ♠❡t❤♦❞❀ ❆r❣1car (ω) ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❈❛r❧s♦♥✬s ♠❡t❤♦❞✳ ✇❤❡r❡✿ ▲1 (ω) ✲ ❡①❛❝t ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡❀ ▲1R‘ (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❘■▼ ♠❡t❤♦❞❀ ▲1cf e (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❈❋❊ ♠❡t❤♦❞❀ ▲1mat (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ▼❛ts✉❞❛✬s ♠❡t❤♦❞❀ ▲1ls (ω)✕ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❧❡❛st✲ sq✉❛r❡s ♠❡t❤♦❞❀ ▲1car (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❈❛r❧s♦♥✬s ♠❡t❤♦❞✳ ∆▲1R‘ (ω) ✲ ❡rr♦r ♦❢ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ∆▲1cf e (ω) ✕ ❡rr♦r ♦❢ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❈❋❊ ♠❡t❤♦❞❀ ∆▲1mat (ω) ✲ ❡rr♦r ✇❤❡r❡✿ ❜② ❘■▼ ♠❡t❤♦❞❀ ♦❢ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ▼❛ts✉❞❛✬s ♠❡t❤♦❞❀ ∆▲1ls (ω)✕ ❡rr♦r ♦❢ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❧❡❛st✲ sq✉❛r❡s ♠❡t❤♦❞❀ ∆▲1car (ω) ✲ ❡rr♦r ♦❢ ♠❛❣♥✐t✉❞❡ ❋✐❣✳ ✻✿ ❊rr♦rs ♦❢ t❤❡ ♣❤❛s❡ r❡s♣♦♥s❡s r❡s♣♦♥s❡❀ ∆❆r❣1cf e (ω) ✕ ❡rr♦r ♦❢ ♣❤❛s❡ r❡s♣♦♥s❡ ∆❆r❣1mat (ω) ✲ ❡rr♦r ♦❢ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ▼❛ts✉❞❛✬s ♠❡t❤♦❞❀ ∆❆r❣1ls (ω)✕ ❡r✲ ❆r❣1cf e (ω) ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❈❋❊ ♠❡t❤♦❞❀ r♦r ♦❢ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❧❡❛st✲sq✉❛r❡s ♠❡t❤♦❞❀ ❆r❣1mat (ω) ∆❆r❣1R‘ (ω) ✲ ❡rr♦r ♦❢ ♣❤❛s❡ ♠❡t❤♦❞❀ ∆❆r❣1car (ω) ✲ ❡rr♦r r❡s♣♦♥s❡ ❜② ❈❛r❧s♦♥✬s ♠❡t❤♦❞✳ ✇❤❡r❡✿ ♠❡t❤♦❞❀ ❆r❣1 (ω) ✲ ✲ ♣❤❛s❡ ❡①❛❝t ✇❤❡r❡✿ ❜② ❈❋❊ ♠❡t❤♦❞❀ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ▼❛ts✉❞❛✬s ❆r❣1ls (ω)✕ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❧❡❛st✲ sq✉❛r❡s ♠❡t❤♦❞❀ ❆r❣1R‘ (ω ✮ ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② r❡s♣♦♥s❡ ❜② ❘■▼ ♦❢ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❈❛r❧s♦♥✬s ♠❡t❤♦❞✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✸ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❋✐❣✳ ✹ ❛♥❞ ❋✐❣✳ ✻ s❤♦✇ t❤❛t t❤❡ ❡rr♦rs ✐♥ ♠❛❣✲ ♥✐t✉❞❡ ❛♥❞ ♣❤❛s❡ r❡s♣♦♥s❡s ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ♣r❡s❡♥t t❤❡ ❧♦✇❡st ✈❛❧✉❡ ✐♥ −3 ❧♦✇ ❢r❡q✉❡♥❝② r❛♥❣❡❞ ❬✶✵ ✱ ✵✳✶❪ ❍③✳ ■♥ ❤✐❣❤❡r ❛♥❞ ❧♦✇❡r ❢r❡q✉❡♥❝② r❡❣✐♠❡✱ r❡s✉❧ts ♦❢ t❤❡ ❘■▼ ✐♥tr♦❞✉❝❡ ❧❡ss ❛❝❝✉r❛❝②✳ ●❡♥❡r❛❧❧② t❤❡ ❘■▼ ✐♥ ❇♦❞❡ ❞✐❛❣r❛♠s ✜t t❤❡ ❡①❛❝t ♠♦❞❡❧ ✐♥ t❤❡ ✇✐❞❡ r❛♥❣❡ ❝♦♠♣❛r✐♥❣ t♦ t❤❡ ♦t❤❡r ♠❡t❤♦❞s✳ ❙❡❝♦♥❞ ♣❛rt ♦❢ ❡①❛♠♣❧❡ ✇❡ ❝♦♠♣❛r❡ ❘■▼ t♦ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✿ ❈❤❛r❡✛ ✬s ♠❡t❤♦❞❀ ❋✐❣✳ ✽✿ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡s ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞ ❛♥❞ ❢r❡q✉❡♥❝② ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞✳ ❚❤❡ ♦r❞❡rs ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡❞ r❛✲ t✐♦♥❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛r❡ ❝❤♦s❡♥ ❤✐❣❤❡r t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡ ✇✐t❤ ❛♥❞ m = n = ✐♥ ❞❡♥♦♠✐♥❛t♦r✳ ✐♥ ♥✉♠❡r❛t♦r ■t ♠❡❛♥s t❤❛t N = n+ b0 = 1✳ ■♥ ✇❡ ❝❤♦♦s❡ ✈❛❧✉❡s ♦❢ t❤❡ ♥♦❞❡s δi [0.001; 0.005; 0.01; 0.05; 0.1; 1; 5; 10; 50] ♥✉♠❜❡r ♦❢ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ✐s m = t❤❡ ❘■▼ ✐♥ r❛♥❣❡ ✇✐t❤ ✇✐t❤ ◆ ❂✾ a0 = ❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❋✐❣✳ ✽✱ t❤❡ ♠❛①✐♠✉♠ ❛♣♣r♦①✲ ✐♠❛t✐♦♥s ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❛♥❞ ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜✳ ✷✳ ❚❛❜✳ ✷✿ ▼❛①✐♠✉♠ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐♥ t✐♠❡ r❛♥❣❡ ❛♥❞ ♥♦❞❡s ❡q✉❛❧❧② s♣❛❝❡❞✳ ❬✵✲✶✺✵❪ ✭s❡❝✮✳ ❘■▼ ❚❤❡ r❡s✉❧ts ❊rr♦r ❈❤❛✳ ❖✉st❛❧♦✉♣✬s ▼❡t❤♦❞ ♠❡t❤♦❞ ✹✳✹✻✼ ✹✳✹✺✸ ✵✳✶✹✷ ❋■▼ ✶✳✶✾✸ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦❝❡ss ✇✐❧❧ ❜❡ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ t✐♠❡ ❛♥❞ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥s✳ ❆s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❋✐❣✳ ✼ ❛♥❞ ❋✐❣✳ ✾✱ ✐t ❝❧❡❛r❧② ❚❤❡ ❡①❛❝t t✐♠❡ r❡s♣♦♥s❡ ♦❢ t❤❡ ❢r❛❝t✐♦♥❛❧ ♦r✲ s❤♦✇s t❤❛t✱ ♥❡✇ ♠❡t❤♦❞ ♣r♦✈✐❞❡s ❛ ✇❡❧❧✲✜tt✐♥❣✳ ❞❡r s②st❡♠✱ ❛s ✇❡❧❧ ❛s t❤♦s❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❈♦♠♣❛r✐♥❣ ❋✐❣✳ ✷ ❛♥❞ ❋✐❣✳ ✽✱ ❞❡t❛✐❧❡❞ ✐♥ ❚❛❜✳ ✶ ♠♦❞❡❧s✱ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❋✐❣✳ ✼✳ ❛♥❞ ❚❛❜✳ ✷✱ ✐t ❧❡❛❞s t♦ t❤❡ ❝♦♥❝❧✉s✐♦♥ t❤❛t ❡rr♦r ■♥ ❛❞❞✐t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs ❛r❡ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✳ ✽✳ th ♦❢ ✹ ✺ th ♦r❞❡r ♠♦❞❡❧ ✭❛❜♦✉t ✵✳✶✺✽✮ ✐s ❤✐❣❤❡r t❤❛♥ ♦r❞❡r ♠♦❞❡❧✱ ❛♣♣r♦①✐♠❛t❡❞ ✭❛❜♦✉t ✵✳✶✹✷✮ ❜② th ♦r❞❡r ♠♦❞❡❧s ❛r❡ th ♠♦r❡ ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ✹ ♦r❞❡r ♠♦❞❡❧✱ ❛♣♣r♦①✲ ❘■▼✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ✺ ✐♠❛t❡❞ ❜② ❘■▼✳ ❚❤❡ ❇♦❞❡ ♣❧♦ts ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧s ❛r❡ s❤♦✇♥ ✐♥ t❤❡ ❋✐❣✳ ✾✕✶✷✳ ❋✐❣✳ ✼✿ ❚✐♠❡ r❡s♣♦♥s❡s ✇❤❡r❡✿ ❤1 ✭t✮ ✲ ❡①❛❝t t✐♠❡ r❡s♣♦♥s❡❀ ❤1R‘ ✭t✮ ✲ t✐♠❡ r❡s♣♦♥s❡ ❜② ❘■▼ ♠❡t❤♦❞❀ ❤1cha ✭t✮ ✲ t✐♠❡ r❡✲ s♣♦♥s❡ ❜② ❈❤❛r❡✛ ❵s ♠❡t❤♦❞❀ ❤1ous ✭t✮ ✲ t✐♠❡ r❡✲ s♣♦♥s❡ ❜② ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ❤1F ✭t✮✲t✐♠❡ r❡✲ s♣♦♥s❡ ❜② ❋■▼✳ ∆❤1R‘ ✭t✮ ✲ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ❘■▼ ∆❤1cha ✭t✮ ✲ ❡rr♦r ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ❈❤❛rr❡❢ ✬s ♠❡t❤♦❞❀ ∆❤1ous ✭t✮ ✲ ❡rr♦r ♦❢ t✐♠❡ r❡✲ s♣♦♥s❡ ❜② ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ∆❤1F ✭t✮ ✕ ❡rr♦r ❋✐❣✳ ✾✿ ▼❛❣♥✐t✉❞❡ r❡s♣♦♥s❡s ♦❢ t✐♠❡ r❡s♣♦♥s❡ ❜② ❋■▼✳ ▲1cha (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❈❤❛rr❡❢ ✬s ✇❤❡r❡ ♠❡t❤♦❞❀ ✹✹ ✇❤❡r❡✿ ▲1 (ω) ✲ ❡①❛❝t ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡❀ ▲1R (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❘■▼ ♠❡t❤♦❞❀ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ♠❡t❤♦❞✱ ▲1ous (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❜② ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ▲1F (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡✲ s♣♦♥s❡ ❜② ❋■▼✳ ❋✐❣✳ ✶✷✿ ❊rr♦rs ♦❢ t❤❡ ♣❤❛s❡ r❡s♣♦♥s❡s ❋✐❣✳ ✶✵✿ ❊rr♦rs ♦❢ t❤❡ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡s ❚❤❡ ❞✐❛❣r❛♠s s❤♦✇ t❤❛t ✐t ❧❡❛❞s t♦ t❤❡ s❛♠❡ ❘■▼ ♠❡t❤♦❞❀ ❛❜♦✈❡ ❝♦♥❝❧✉s✐♦♥✱ t❤❡ ✜t♥❡ss ♦❢ ❘■▼ ♠♦❞❡❧ ✐♥ ❡rr♦r ∆▲1R (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❡rr♦r ❜② ∆▲1cha (ω) ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❜② ❈❤❛rr❡❢ ✬s ♠❡t❤♦❞❀ ∆▲1ous (ω) ✲ ♠❛❣✲ ❇♦❞❡ ❝❤❛r❛❝t❡r✐st✐❝s ✐s ❧♦✇❡r t❤❛♥ t❤❡ ♦t❤❡r ♥✐t✉❞❡ r❡s♣♦♥s❡ ❡rr♦r ❜② ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ♠❡t❤♦❞s ✐♥ t❤❡ r❛♥❣❡ ❬✵✳✶✲✺❪ ❍③✳ ❍♦✇❡✈❡r ❘■▼ ∆▲1F (ω) ✐♥ ❇♦❞❡ ❞✐❛❣r❛♠s ✜t t❤❡ ❡①❛❝t ♠♦❞❡❧ ✐♥ t❤❡ ✇✐❞❡ ✇❤❡r❡✿ ✲ ♠❛❣♥✐t✉❞❡ r❡s♣♦♥s❡ ❡rr♦r ❜② ❋■▼✳ −3 r❛♥❣❡ ❛❜♦✉t ❬✶✵ ✲✶✵❪ ❍③✳ ■♥ ❝♦♠♣❛r✐s♦♥ t♦ ✹ th ♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧ ❜② ❘■▼✱ t❤❡ ❛❝❝✉✲ th r❛❝② ♦❢ ✺ ❘■▼ ♠♦❞❡❧ r❡♣r❡s❡♥ts ♠♦r❡ ❛❝❝✉r❛t❡✳ ❚♦ ❡st✐♠❛t❡ ♦❢ t❤❡ ❘■▼✱ ✇❡ ❝❛rr② ♦✉t ♥✉✲ ♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ✇✐t❤ t②♣✐❝❛❧ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r ✐♥t❡❣r❛t♦r s②st❡♠s✱ ✇❤✐❝❤ ✐s ♠♦♥♦t♦♥✐❝ ✉♥st❛✲ ❜❧❡✳ ■♥ t❤❡ ❡①❛♠♣❧❡s✱ t❤❡r❡ ✇❡r❡ ❝♦♥❞✉❝t❡❞ ✐♥✲ ❞❡♣t❤ ❛♥❛❧②s✐s ♦❢ t❤❡ r❡s✉❧ts ♦❢ s❡✈❡r❛❧ t②♣✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ❛♥❞ t❤❡ ❘■▼ ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ ❛♥❞ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✳ ❚❤❡ ❛❜♦✈❡ r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❘■▼ ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ ✐s s✐❣♥✐✜❝❛♥t❧② ❤✐❣❤❡r ❝♦♠♣❛r✲ ✐♥❣ t♦ ❝♦♥s✐❞❡r❡❞ ♠❡t❤♦❞s✳ ▼❛①✐♠✉♠ ❛♣♣r♦①✐✲ ♠❛t✐♦♥ ❡rr♦r ♦❢ ❘■▼ ♠♦❞❡❧ ✐s s♠❛❧❧❡r ✽✳✺ t✐♠❡s ❋✐❣✳ ✶✶✿ P❤❛s❡ r❡s♣♦♥s❡s ❛♥❞ ✻✼ t✐♠❡s ❝♦♠♣❛r✐♥❣ t♦ ❋■▼ ♠♦❞❡❧ ❛♥❞ ❈❋❊ ❆r❣1 (ω) ✇❤❡r❡✿ ✲ ❡①❛❝t ♣❤❛s❡ r❡s♣♦♥s❡❀ ❆r❣1R (ω) ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❘■▼ ♠❡t❤♦❞❀ ❆r❣1cha (ω) ♠❡t❤♦❞❀ ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❆r❣1ous (ω) ✲ ❜② ♣❤❛s❡ ❈❤❛rr❡❢ ✬s r❡s♣♦♥s❡ ❜② ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ❆r❣1F (ω) ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❜② ❋■▼✳ th ♦r❞❡r ♠♦❞❡❧s ✐s th ♠♦r❡ ✜tt✐♥❣ t❤❛♥ t❤❡ ✹ ♦r❞❡r ♠♦❞❡❧✱ ❛♣♣r♦①✐✲ ♠♦❞❡❧✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ✺ ♠❛t❡❞ ❜② ❘■▼✱ ✐s ❛❜♦✉t ✶✵✪ ♠♦r❡ ❛❝❝✉r❛t❡✳ ■♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ❛s t❤❡ ❇♦❞❡ ❝❤❛r❛❝t❡r✐s✲ t✐❝s✱ t❤❡ ❘■▼ ♠♦❞❡❧s s❤♦✇ ❤✐❣❤❡r ✜tt✐♥❣ t❤❛♥ ❝♦♥s✐❞❡r❡❞ ♠❡t❤♦❞s ✐♥ ❧♦✇ ❛♥❞ ❤✐❣❤ r❛♥❣❡s ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ ❢r❡q✉❡♥❝✐❡s✳ ∆❆r❣1R (ω) ✲ ♣❤❛s❡ ♠❡t❤♦❞❀ ∆❆r❣1cha (ω) ❍♦✇❡✈❡r✱ ♥❡❛r t❤❡ r❡s♣♦♥s❡ ❡rr♦r ❜② ♠❡❞✐✉♠✲❢r❡q✉❡♥❝② r❛♥❣❡ ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ ❢r❡✲ ✲ ♣❤❛s❡ r❡s♣♦♥s❡ q✉❡♥❝② r❛♥❣❡✱ t❤❡ ❘■▼ ✐s ❧❡ss s❛t✐s❢❛❝t♦r② t❤❛♥ ∆❆r❣1ous (ω) ✲ ♦t❤❡r ♠❡t❤♦❞s✳ ●❡♥❡r❛❧❧② t❤❡ ❘■▼ ✐♥ ❇♦❞❡ ❞✐✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❡rr♦r ❜② ❖✉st❛❧♦✉♣✬s ♠❡t❤♦❞❀ ❛❣r❛♠s ✜t t❤❡ ❡①❛❝t ♠♦❞❡❧ ✐♥ t❤❡ ✇✐❞❡ r❛♥❣❡ ∆❆r❣1F (ω) ❝♦♠♣❛r✐♥❣ t♦ t❤❡ ♦t❤❡r ♠❡t❤♦❞s ✇❤❡r❡✿ ❘■▼ ❡rr♦r ❜② ❈❤❛rr❡❢ ✬s ♠❡t❤♦❞❀ ✲ ♣❤❛s❡ r❡s♣♦♥s❡ ❡rr♦r ❜② ❋■▼✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✺ ❱❖▲❯▼❊✿ ✶ ✺✳ ❆✉t♦♠❛t✐❝❛ ❙✐♥✐❝❛✳✱ ❈♦♥❝❧✉s✐♦♥ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ✈♦❧✳ ✸✱ ✐ss✳ ✹✱ ♣♣✳ ✹✸✵✲ ✹✹✶✱ ✷✵✶✻✳ ■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠ ✐s ♣r❡s❡♥t❡❞✳ ❚❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❢❡❛t✉r❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞ ✐s ✐ts ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✣❝✐❡♥❝②✳ ❆♥♦t❤❡r ❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❘■▼ ✐s ✐ts ❤✐❣❤ ❛❝❝✉r❛❝② ✐♥ t❤❡ t✐♠❡ ❞♦✲ ♠❛✐♥✱ ❝♦♠♣❛r✐♥❣ t♦ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ♠❡t❤♦❞s✳ ❚❤❡ ❤✐❣❤❡r ♦r❞❡r ❘■▼ ♠♦❞❡❧s ❛r❡ ♠♦r❡ ❛❝❝✉r❛t❡ t❤❡♥ t❤❡ ❧♦✇❡r ♦r❞❡r ❘■▼ ♠♦❞❡❧s✳ ■♥ ❢❛❝t✱ t❤✐s ❬✺❪ ▼❖❱❆❍❍❊❉✱ ❆✳ ▼✳✱ ❍✳ ❚✳ ❙❍❆❉■❩ ❛♥❞ ❙✳ ❑✳ ❍✳ ❙❆◆■✳ ✏❈♦♠♣❛r✐s♦♥ ♦❢ ❋r❛❝t✐♦♥❛❧ ❖r❞❡r ▼♦❞❡❧❧✐♥❣ ❛♥❞ ■♥t❡❣❡r ❖r❞❡r ▼♦❞✲ ❡❧❧✐♥❣ ♦❢ ❋r❛❝t✐♦♥❛❧ ❖r❞❡r ❇✉❝❦ ❈♦♥✈❡rt❡r ✐♥ ❈♦♥t✐♥✉♦✉s ❈♦♥❞✐t✐♦♥ ▼♦❞❡ ❖♣❡r❛t✐♦♥✑ ❆❞✈❛♥❝❡s ✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐✲ ♥❡❡r✐♥❣✱ ✈♦❧✳ ✶✹✱ ✐ss✳ ✺✱ ♣♣✳ ✺✸✶✲✺✹✷✱ ✷✵✶✻✳ ♠❡t❤♦❞ ✐s ✈❡r② s✐♠♣❧❡ ❜♦t❤ ❝♦♥❝❡♣t✉❛❧❧② ❛♥❞ ❬✻❪ ❙■❘❖❚❆✱ ▲✳✱ ❛♥❞ ❨✳ ❍❆▲❊❱■✱ ✏❋r❛❝t✐♦♥❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧❧②✳ ❚❤❡ ♦❜t❛✐♥❡❞ r❡s✉❧ts ❢r♦♠ t❤❡ ♦r❞❡r ❝♦♥tr♦❧ ♦❢ ✢❡①✐❜❧❡ str✉❝t✉r❡s ❣♦✈❡r♥❡❞ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s ❛r❡ q✉✐t❡ s❛t✐s❢❛❝t♦r②✳ ❜② t❤❡ ❞❛♠♣❡❞ ✇❛✈❡ ❡q✉❛t✐♦♥✑✱ ✐♥ ❚❤❡ ♠❛✐♥ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞ ✐s t❤❛t ✐t ✐s ✉♥❝❡rt❛✐♥ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♠♦❞❡❧ ✐♥ Pr♦❝❡❡❞✲ ✐♥❣ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❈♦♥tr♦❧ ❈♦♥❢❡r❡♥❝❡✳ ❈❤✐❝❛❣♦✱ ❯❙❆✱ ♣♣✳ ✺✻✺✲✺✼✵✱ ✷✵✶✺✳ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✳ ❆♥♦t❤❡r ❧✐♠✐t❛t✐♦♥ ♦❢ t❤❡ ♠❡t❤♦❞ ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❣✉❛r❛♥t❡❡ t❤❡ st❛❜✐❧✲ ❬✼❪ ❏❖❙❍■✱ ▼✳ ▼✳✱ ❱✳ ❆✳ ❱❨❆❲❆❍❆❘❊ ❛♥❞ ✐t② ❛ ♣r✐♦r✐✱ ✐♥ ♦t❤❡r ✇♦r❞s ♥♦ ❝♦♥str❛✐♥ts ♦♥ ▼✳ ❉✳ P❆❚■▲✳ ✏▼♦❞❡❧ ♣r❡❞✐❝t✐✈❡ ❝♦♥tr♦❧ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❡♥❢♦r❝❡❞✳ ❢♦r ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠ ❛ ♠♦❞❡❧✐♥❣ ❛♥❞ ■♥❞❡❡❞✱ t❤❡ ❢♦r♠ ♦❢ t❤❡s❡ ❝♦♥str❛✐♥t ✇♦✉❧❞ ❜❡ s♦ ❝♦♠♣❧✐❝❛t❡❞✱ s♦ t❤❛t t❤❡✐r ✐♥tr♦❞✉❝t✐♦♥ ✇♦✉❧❞ ✐♠♣❛✐r t❤❡ ❡st❛❜✲ ❧✐s❤❡❞ ❡✣❝✐❡♥❝② ♦❢ t❤❡ s♦❧✉t✐♦♥ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r✳ Pr♦❝❡❡❞✲ ✐♥❣ ♦❢ t❤❡ ✹t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❖♥ ❙✐♠✉❧❛t✐♦♥ ❆♥❞ ▼♦❞❡❧✐♥❣ ▼❡t❤♦❞♦❧♦✲ ❣✐❡s✱ ❚❡❝❤♥♦❧♦❣✐❡s ❆♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱✐✲ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❛s❡❞ ❛♥❛❧②s✐s✑✱ ✐♥ ❡♥♥❛✱ ❆✉str✐❛✱ ✷✵✶✹✳ ❬✽❪ ▼❆■❖◆❊✱ ●✳ ✏❈♦♥❝❡r♥✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝✲ ❘❡❢❡r❡♥❝❡s t✐♦♥s r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ♥♦♥✐♥t❡❣❡r ♦r❞❡r ❞✐❣✐t❛❧ ❞✐✛❡r❡♥t✐❛t♦rs✑✱ ❬✶❪ ❇❆❘❇❯✱ ▼✳✱ ❊❞✐t ❑❛♠✐♥s❦② ❛♥❞ ❘✳ ❊✳ ❚r❛❤❛♥ ✏❆❝♦✉st✐❝ ❙❡❛❜❡❞ ❈❧❛ss✐✜❝❛t✐♦♥ ✉s✲ ❙✐❣♥❛❧ Pr♦❝❡ss✱ ■❊❊❊ tr❛♥s❛❝t✐♦♥ ♦❢ ✈♦❧✳ ✶✸✱ ✐ss✳ ✶✷✱ ♣♣✳ ✼✸✺ ✕ ✼✸✽✱ ✷✵✵✼✳ ✐♥❣ ❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❛♥❞ ❚✐♠❡✲ Pr♦✲ ❝❡❡❞✐♥❣s ♦❢ t❤❡ ■❊❊❊ ❖❝❡❛♥✐❝ ❊♥❣✐♥❡❡r✐♥❣ ❙♦❝✐❡t②✳ ❇♦st♦♥✱ ❯❙✱ ♣♣✳ ✹✺✲✺✶✱ ✷✵✵✻✳ ❋r❡q✉❡♥❝② ❚r❛♥s❢♦r♠ ❚❡❝❤♥✐q✉❡s✑✱ ✐♥ ❬✾❪ ◆❊❩❩❆❘■✱ ❍✳✱ ❆✳ ❈❍❆❘❊❋ ❛♥❞ ❉✳ ❇❖❯❈❍❊❘▼❆✳ ❆♥❛❧♦❣ ❈✐r❝✉✐t ■♠♣❧❡♠❡♥✲ t❛t✐♦♥ ♦❢ ❋r❛❝t✐♦♥❛❧ ❖r❞❡r ❉❛♠♣❡❞ ❙✐♥❡ ■❊❊❊ ❏♦✉r♥❛❧ ♦♥ ❊♠❡r❣✐♥❣ ❛♥❞ ❙❡❧❡❝t❡❞ t♦♣✐❝s ✐♥ ❈✐r❝✉✐ts ❛♥❞ ❙②st❡♠s✱ ✈♦❧✳ ✸✱ ✐ss✳ ✸✱ ♣♣✳ ✸✽✻ ✕ ✸✾✸✱ ❛♥❞ ❈♦s✐♥❡ ❋✉♥❝t✐♦♥s✳ ❬✷❪ ❇❆❘❇❯✱ ▼✳✱ ❊❞✐t ❑❛♠✐♥s❦②✱ ❛♥❞ ❘✳ ❊✳ ❚r❛❤❛♥✱ ✏❋r❛❝t✐♦♥❛❧ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❢♦r s♦♥❛r s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✬✱ ✐♥ ❖❝❡❛♥✐❝ ❊♥❣✐♥❡❡r✐♥❣ ❙♦❝✐❡t②✳ t❤❡ ■❊❊❊ ✷✵✶✸✳ ❲❛s❤✐♥❣t♦♥ ❬✶✵❪ ●❖◆❩❆▲❊❩✱ ❊✳ ❆✳✱ ❛♥❞ ■✳ P❊❚❘❆❙✱ ✏❆❞✲ ❉❈✱ ❯❙✱ ✷✵✵✺✳ ✈❛♥❝❡s ✐♥ ❢r❛❝t✐♦♥❛❧ ❝❛❧❝✉❧✉s✲ ❈♦♥tr♦❧ ❛♥❞ ❬✸❪ ❑❊❙❆❘❑❆❘✱ ❆✳ ◆❆❘❆❨❆◆❆❙❆▼❨✳ ❆✳ ❛♥❞ ✏■♥✈❡st✐❣❛t✐♦♥ ❙✳ ♦♥ ❙✉♣❡r✐♦r P❡r❢♦r♠❛♥❝❡ ❜② ❋r❛❝t✐♦♥❛❧ ❈♦♥✲ tr♦❧❧❡r ❢♦r ❈❛rt✲❙❡r✈♦ ▲❛❜♦r❛t♦r② ❙❡t✲❯♣✑✱ ❆❞✈❛♥❝❡s ✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐♥❡❡r✐♥❣✱ ✈♦❧✳ ✶✷✱ ✐ss✳ ✸✱ ♣♣✳ ✷✵✶✲✷✵✾✱ Pr♦❝❝❡❞✲ ✐♥❣ ♦❢ t❤❡ ✶✻th ❈❛r♣❛t❤✐❛♥ ❈♦♥tr♦❧ ❈♦♥❢❡r✲ ❡♥❝❡✳ ❍✉♥❣❛r②✱ ✷✵✶✺✳ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✑✱ ✐♥ ❬✶✶❪ ❉❩■❊▲■◆❙❑■✱ ❆✳✱ ❉✳ ❙■❊❘❖❈■❯❑ ❛♥❞ ●✳ ❙❆❘❲❆❙✳ ❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r ❝❛❧❝✉❧✉s✱ ✷✵✶✹✳ ❆✉t♦♠❛t✐❝s✱ ✈♦❧✳ ✺✽✱ ✐ss✳ ✹✱ ♣♣✳ ✺✽✸✲✺✾✸✱ ✷✵✶✵✳ ❬✹❪ ❈❍❊◆✱ ❍✳ ❛♥❞ ❨✳ ❈❤❡♥✱ ✏❋r❛❝t✐♦♥❛❧✲♦r❞❡r ❣❡♥❡r❛❧✐③❡❞ ♣r✐♥❝✐♣❧❡ ♦❢ s❡❧❢✲s✉♣♣♦rt ✭❋❖●✲ P❙❙✮ ✐♥ ❝♦♥tr♦❧ s②st❡♠ ❞❡s✐❣♥✑✱ ✹✻ ❏♦✉r♥❛❧ ♦❢ ❬✶✷❪ ❆❚❍❊❘❚❖◆✱ ❉✳ P✳✱ ◆✳ ❚❆◆ ❛♥❞ ❆✳ ❨❯❈❊✳ ▼❡t❤♦❞s ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ t✐♠❡ r❡s♣♦♥s❡ ♦❢ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ■❊❚ ❈♦♥tr♦❧ t❤❡✲ ❬✷✶❪ ❈❆❘▲❙❖◆✱ ●✳ ❊✳ ❛♥❞ ❈✳ ❆✳ ❍❆▲■❏❆❍✳ ✈♦❧✳ ✾✱ ✐ss✳ ✻✱ ♣♣✳ ✽✶✼✲ ✏❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❋r❛❝t✐♦♥❛❧ ❈❛♣❛❝✐t♦rs ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠s✱ ♦r② ✫ ❆♣♣❧✐❝❛t✐♦♥s✱ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ✽✸✵✱ ✷✵✶✹✳ ✭✶✲s✮ ❬✶✸❪ ❉❖❘❈❆❑✱ ▲✳✱ ❊✳ ❆✳ ●❖◆❩❆▲❊❩✱ ❏✳ ❚❊❘✲ (1−n) ❜② ❛ ❘❡❣✉❧❛r ◆❡✇t♦♥ Pr♦❝❡ss✧✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈✐r❝✉✐t ❚❤❡♦r②✱✑ ✈♦❧✳ ✸✱ ✐ss✳ ✼✱ ♣♣✳ ✸✶✵✲✸✶✸✱ ✶✾✻✸✳ P❆❑✱ ❏✳ ❱❆▲❙❆ ❛♥❞ ▲✳ P■❱❑❆✳ ■❞❡♥t✐✜❝❛✲ t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r ❞②♥❛♠✐❝❛❧✱ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡✲ ♠❛t✐❝s✱ ✈♦❧✳ ✽✾✱ ✐ss✳ ✷✱ ♣♣✳ ✸✸✺✲✸✺✵✱ ✷✵✶✸✳ ❬✷✷❪ ❖❯❙❚❆▲❖❯P✱ ❆✳✱ ■✳ ❚✳ ❛♥❞ ❚✳ ❱✳ ❨■❖❯▲❚✲ ❙■❙✳ ✏❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ●rü♥✇❛❧❞✕▲❡t♥✐❦♦✈ ▲❊❱❘❖◆✱ ❇✳ ❇❛♥❞ ❈♦♠♣❧❡① ◆♦♥✐♥t❡❣❡r ❉✐✛❡r❡♥t✐❛t♦r✿ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❬✶✹❪ ❘❊❑❆◆❖❙ ❋✳ ▼❆❚❍■❊❯✱ ❋✳ ▼✳ ◆❆◆❖❚✳ ✏❋r❡q✉❡♥❝②✲ ❛♥❞ ■❊❊❊ ❙②♥t❤❡s✐s✱✑ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈✐r❝✉✐t ❚❤❡♦r②✱ ✈♦❧✳ ✹✼✱ ✐ss✳ ✼✱ ♣♣✳ ✷✺✲✸✾✱ ✷✵✵✵✳ ❋r❛❝t✐♦♥❛❧ ❉❡r✐✈❛t✐✈❡ ❢♦r ❋❉❚❉ ▼♦❞❡❧✐♥❣ ♦❢ ❈♦❧❡✕❈♦❧❡ ▼❡❞✐❛✑✳ ♠❛❣♥❡t✐❝s✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ✈♦❧✳ ✺✵✱ ✐ss✳ ✷✱ ♣♣✳ ✶✽✶ ✕ ✶✽✹✱ ❬✷✸❪ ❑❘❆❏❊❲❙❑■✱ ❲✳ ❛♥❞ ❯✳ ❱■❆❘❖✳ ✏❆ ♠❡t❤♦❞ ❢♦r t❤❡ ✐♥t❡❣❡r✲♦r❞❡r ❛♣♣r♦①✐♠❛t✐♦♥ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❋r❛♥❦❧✐♥ ■♥st✐t✉t❡✳ ✷✵✶✸✱ ✈♦❧✳ ✸✺✶✱ ✐ss✳ ②✱ ♣♣✳ ♦❢ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r s②st❡♠s✱✑ ✷✵✶✹✳ ❬✶✺❪ ❖❱■❱■❊❘✱ P✳ ❉✳ ✏❆♣♣r♦①✐♠❛t✐♥❣ ✐rr❛t✐♦♥❛❧ ✺✺✺✕✺✻✹✳ tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ✉s✐♥❣ ▲❛❣r❛❣❡ ✐♥t❡r♣♦❧❛✲ ■❊❊ Pr♦❝❡❡❞✐♥❣s ❉ ✲ ❈♦♥tr♦❧ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✶✸✾✱ ✐ss✳ ✶✱ t✐♦♥ ❢♦r♠✉❧❛✑✱ ♣♣✳ ✾✲✶✸✱ ✶✾✾✷✳ ❬✶✻❪ ❑❘■❙❍◆❆✱ ❇✳ ❚✳ ✏❙t✉❞✐❡s ♦♥ ❢r❛❝t✐♦♥❛❧ ♦r✲ ❞❡r ❞✐✛❡r❡♥t✐❛t♦rs ❛♥❞ ✐♥t❡❣r❛t♦rs ❛ s✉r✲ ✈❡②✑✱ ❙✐❣♥❛❧ Pr♦❝❡ss✱ ✈♦❧✳ ✾✶✱ ✐ss✳ ✸✱ ♣♣✳ ✸✽✻✕ ✹✸✻✱ ✷✵✶✶✳ ❬✶✼❪ ❉❏❖❯❆▼❇■ ❆✳✱ ❆✳ ❈❍❆❘❊❋ ❛♥❞ ❆✳ ❱❖❉❆✱ ✏◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ❛♥❞ ✐❞❡♥t✐✲ ✜❝❛t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ s②st❡♠s ✉s✐♥❣ ❞✐❣✐t❛❧ Pr♦❝❡❡❞✐♥❣ ♦❢ t❤❡ ✷✵✶✸ ❊✉r♦♣❡❛♥ ❝♦♥tr♦❧ ❝♦♥❢❡r❡♥❝❡✳ ❛❞❥✉st❛❜❧❡ ❢r❛❝t✐♦♥❛❧ ♦r❞❡r✑✱ ✐♥ ❩✉r✐❝❤✱ ❙✇✐t③❡r❧❛♥❞✱ ✷✵✶✸ ❬✶✽❪ ❙❊❑❆❘❆✱ ❚✳ ❇✳✱ ▼✳ ❘✳ ❘❆P❆■❈ ❛♥❞ ▼✳ P✳ ▲❆❩❆❘❊❱■❈✳ ❆♥ ❊✣❝✐❡♥t ▼❡t❤♦❞ ❢♦r ❬✷✹❪ ❳❯❊✱ ❉✳✱ ❈❤✳ ❩❍❆❖ ❛♥❞ ❨✳ ◗✳ ❈❍❊◆✳ ✏❆ ♠♦❞✐✜❡❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞ ♦❢ ❋r❛❝✲ Pr♦❝❡❡❞✐♥❣ ♦❢ t❤❡ ✸✵✵✻ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ▼❡❝❤❛tr♦♥✐❝s ❛♥❞ ❆✉t♦♠❛t✐♦♥✳ ▲♦✉②❛♥❣✱ t✐♦♥❛❧ ♦r❞❡r s②st❡♠✱✑ ✐♥ ❈❤✐♥❛✱ ♣♣✳ ✶✵✹✸✲✶✵✹✽✱ ✷✵✵✻✳ ❘❡❛❧ ✐♥t❡r✲ ♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ✐♥ ❝♦♥tr♦❧ ❛✉t♦♠❛t✐♦♥ ✐s✲ s✉❡✳ ✶st ❡❞✳✱ ❚♦♠s❦ ♣♦❧②t❡❝❤♥✐❝ ✉♥✐✈❡rs✐t②✱ ❬✷✺❪ ●❖◆❈❍❆❘❖❱✱ ❱✳ ■✳ ❊❞✐t♦rs✱ ❚♦♠s❦✱ ✷✵✵✾✳ ❬✷✻❪ ❉❯◆●✱ ◆✳ ◗✳ ❛♥❞ ❚✳ ❍✳ ◗✳ ▼■◆❍✳ ✏❆ ■♥✲ ♥♦✈❛t✐♦♥ ■❞❡♥t✐✜❝❛t✐♦♥ ❆♣♣r♦❛❝❤ ♦❢ ❈♦♥tr♦❧ ■❆❊❙ ■♥✲ ❞♦♥❡s✐❛♥ ❏♦✉r♥❛❧ ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✈♦❧✳ ✸✱ ✐ss✳ ✸✱ ♣♣✳ ❙②st❡♠ ❜② ❋r❛❝t✐♦♥❛❧ ❚r❛♥s❢❡r✱✑ ✸✸✻✕✸✹✷✱ ✷✵✶✻✳ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ◆♦♥✲❘❛t✐♦♥❛❧ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥s✳ ❊❧❡❝tr♦♥✐❝s✳ ✷✵✶✸✱ ✈♦❧✳ ✶✼✱ ✐ss✳ ✶✱ ♣♣✳ ✹✵✲✹✹ ❬✶✾❪ ▼❆■❖◆❊✱ ♣r♦①✐♠❛t✐♦♥ ❆❜♦✉t ❆✉t❤♦rs ●✳ ♦❢ ✏❈♦♥t✐♥✉❡❞ t❤❡ ❢r❛❝t✐♦♥s ✐♠♣✉❧s❡ ❛♣✲ r❡s♣♦♥s❡ ❢r❛❝t✐♦♥❛❧✲♦r❞❡r ❞②♥❛♠✐❝ s②st❡♠s ✏✱ ♦❢ ■❊❚ ❈♦♥tr♦❧ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✸✱ ✐ss✳ ✼✱ ♣♣✳ ✺✻✹✲✺✼✸✱ ✷✵✵✽✳ ◗✉❛♥❣ ❉✉♥❣ ◆●❯❨❊◆ ♥❛♠✳ ✇❛s ❜♦r♥ ✐♥ ❱✐❡t✲ ❍❡ r❡❝❡✐✈❡❞ ❤✐s ❇✳❙❝ ❛♥❞ ▼✳❙❝✳ ❛♥❞ ✷✵✶✷✱ r❡s♣❡❝t✐✈❡❧②✳ ❍✐s r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❝♦♥tr♦❧ s②st❡♠✳ ❬✷✵❪ ❙❍❘■❱❆❙❚❆❱❊✱ ◆✳ ❛♥❞ P✳ ❱❆❘❙❍◆❊❨✳ t✐♦♥❛❧ ♦r❞❡r s②st❡♠✱ ❞✐str✐❜✉t❡❞ s②st❡♠✱ ❞❡r s②st❡♠s ✉s✐♥❣ ❈❛r❧s♦♥ ♠❡t❤♦❞✑✱ ✐♥ ❛♥❞ ✐♥❞✉str✐❛❧ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♥❡t✇♦r❦✳ r❡♥❡✇❛❜❧❡ ❡♥❡r❣②✱ ❋r❛❝✲ ♣❛r❛♠❡t❡r ✧❘❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ♦r✲ Pr♦✲ ❝❡❡❞✐♥❣ ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙♦❢t ❈♦♠♣✉t✐♥❣ ❚❡❝❤♥✐q✉❡s ❛♥❞ ■♠♣❧❡♠❡♥✲ t❛t✐♦♥s✳ ❋❛r✐❞❛❜❛❞✱ ■♥❞✐❛✱ ♣♣✳ ✼✻✲✽✵✱ ✷✵✶✺✳ ❢r♦♠ ❚♦♠s❦ P♦❧②t❡❝❤♥✐❝ ❯♥✐✈❡rs✐t②✱ ❘✉ss✐❛ ✐♥ ✷✵✶✵ ❙❈❆❉❆ s②st❡♠s ❍❡ ✐s ✇♦r❦✐♥❣ ❛s ▲❡❝t✉r❡r ✐♥ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠✳ "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✐♦♥ ✭❏❆❊❈✮ ✹✼ ... ❱✐❡t♥❛♠✳ "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... ❳ ✐s t❤❡ ✈❡❝t♦r ♦❢ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs✱ ▲❡t ✉s ❝♦♥s✐❞❡r r❛t✐♦♥❛❧ tr❛♥s❢❡r ❢✉♥❝t✐♦♥✿ W (s) = ✭✼✮ an an−1 a1 bm b0 ✭✾✮ ■t ✐s ✐♠♣♦rt❛♥t t♦ ♠❡♥t✐♦♥ t❤❛t t❤❡ s❡❧❡❝t❡❞ s❡t ♦❢ ♣♦✐♥ts... G (s) = ✇❤❡r❡ p, q − real numbers K(s) = L(s) p βi i ki s q αi i li s , ♦♥❡ ❡❛s✐❧② ♦❜t❛✐♥s t❤❡ ❞❡s✐r❡❞ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✐♥ ♠❛tr✐① ❢♦r♠ ✭✷✮ M · X = B, interger and βi , αi − ✇❤❡r❡ bm