Adaptive MIMO controller design for Chaos synchronization in coupled josephson junctions via fuzzy neural networks

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Ngày đăng: 11/02/2020, 16:58

In this paper, we have discussed the synchronization between coupled Josephson Junctions which experience di erent chaotic oscillations. Due to potential high-frequency applications, the shunted nonlinear resistive-capacitive-inductance junction (RCLSJ) model of Josephson junction was considered in this paper. ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❆❞❛♣t✐✈❡ ▼■▼❖ ❈♦♥tr♦❧❧❡r ❉❡s✐❣♥ ❢♦r ❈❤❛♦s ❙②♥❝❤r♦♥✐③❛t✐♦♥ ✐♥ ❈♦✉♣❧❡❞ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥s ✈✐❛ ❋✉③③② ◆❡✉r❛❧ ◆❡t✇♦r❦s ❚❛t✲❇❛♦✲❚❤✐❡♥ ◆●❯❨❊◆✯ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠ ✯♥❣✉②❡♥t❛t❜❛♦t❤✐❡♥❅t❞t✳❡❞✉✳✈♥ ✭❘❡❝❡✐✈❡❞✿ ✶✻✲❋❡❜r✉❛r②✲✷✵✶✼❀ ❛❝❝❡♣t❡❞✿ ✷✾✲❆♣r✐❧✲✷✵✶✼❀ ♣✉❜❧✐s❤❡❞✿ ✽✲❏✉♥❡✲✷✵✶✼✮ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❤❛✈❡ ❞✐s❝✉ss❡❞ t❤❡ s②♥❝❤r♦♥✐③❛t✐♦♥ ❜❡t✇❡❡♥ ❝♦✉♣❧❡❞ ❏♦s❡♣❤✲ s♦♥ ❏✉♥❝t✐♦♥s ✇❤✐❝❤ ❡①♣❡r✐❡♥❝❡ ❞✐✛❡r❡♥t ❝❤❛♦t✐❝ ♦s❝✐❧❧❛t✐♦♥s✳ ❉✉❡ t♦ ♣♦t❡♥t✐❛❧ ❤✐❣❤✲❢r❡q✉❡♥❝② ❛♣♣❧✐❝❛t✐♦♥s✱ t❤❡ s❤✉♥t❡❞ ♥♦♥❧✐♥❡❛r r❡s✐st✐✈❡✲ ❝❛♣❛❝✐t✐✈❡✲✐♥❞✉❝t❛♥❝❡ ❥✉♥❝t✐♦♥ ✭❘❈▲❙❏✮ ♠♦❞❡❧ ♦❢ ❏♦s❡♣❤s♦♥ ❥✉♥❝t✐♦♥ ✇❛s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ♣❛✲ ♣❡r✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ t❤❡ s②♥❝❤r♦♥✐③❛t✐♦♥✱ ❛♥ ❛❞❛♣t✐✈❡ ▼■▼❖ ❝♦♥tr♦❧❧❡r ✐s ❞❡✈❡❧♦♣❡❞ t♦ ❞r✐✈❡ t❤❡ st❛t❡s ♦❢ t❤❡ s❧❛✈❡ ❝❤❛♦t✐❝ ❥✉♥❝t✐♦♥ t♦ ❢♦❧✲ ❧♦✇ t❤❡ st❛t❡s ♦❢ t❤❡ ♠❛st❡r ❝❤❛♦t✐❝ ❥✉♥❝t✐♦♥✳ ❚❤❡ ❞❡✈❡❧♦♣❡❞ ❝♦♥tr♦❧❧❡r ❤❛s t✇♦ ♣❛rts✿ t❤❡ ❢✉③③② ♥❡✉r❛❧ ❝♦♥tr♦❧❧❡r ❛♥❞ t❤❡ s❧✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦❧❧❡r✳ ❚❤❡ ❢✉③③② ♥❡✉r❛❧ ❝♦♥tr♦❧❧❡r ❡♠♣❧♦②s ❛ ❢✉③③② ♥❡✉✲ r❛❧ ♥❡t✇♦r❦ t♦ s✐♠✉❧❛t❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ✐❞❡❛❧ ❢❡❡❞❜❛❝❦ ❧✐♥❡❛r✐③❛t✐♦♥ ❝♦♥tr♦❧❧❡r✱ ✇❤✐❧❡ t❤❡ s❧✐❞✲ ✐♥❣ ♠♦❞❡ ❝♦♥tr♦❧❧❡r ✐s ✉s❡❞ t♦ ❡♥s✉r❡ t❤❡ r♦❜✉st✲ ♥❡ss ♦❢ t❤❡ ❝♦♥tr♦❧❧❡❞ s②st❡♠ ❛♥❞ r❡❞✉❝❡ t❤❡ ✉♥✲ ❞❡s✐r❡❞ ❡✛❡❝ts ♦❢ t❤❡ ❡st✐♠❛t❡ ❡rr♦rs✳ ■♥ ❛❞❞✐✲ t✐♦♥✱ t❤❡ ▲②❛♣✉♥♦✈ ❝❛♥❞✐❞❛t❡ ❢✉♥❝t✐♦♥ ✐s ❛❧s♦ ❣✐✈❡♥ ❢♦r ❢✉rt❤❡r st❛❜✐❧✐t② ❛♥❛❧②s✐s✳ ❚❤❡ ♥✉♠❡r✲ ✐❝❛❧ s✐♠✉❧❛t✐♦♥s ❛r❡ ❝❛rr✐❡❞ ♦✉t t♦ ✈❡r✐❢② t❤❡ ✈❛✲ ❧✐❞✐t② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❝♦♥tr♦❧ ❛♣♣r♦❛❝❤✳ ❆❜str❛❝t✳ ❑❡②✇♦r❞s ❈❤❛♦s ❙②♥❝❤r♦♥✐③❛t✐♦♥❀ ❈❤❛♦t✐❝ ❙②st❡♠s❀ ❋✉③③② ◆❡✉r❛❧ ◆❡t✇♦r❦s❀ ❏♦s❡♣❤s♦♥ ❏✉♥❝✲ t✐♦♥✳ ✽✵ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❙✐♥❝❡ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥ ✭❏❏✮ ♣♦ss❡ss❡s t❤❡ ❛❞✲ ✈❛♥❝❡❞ ❝❤❛r❛❝t❡r✐st✐❝s s✉❝❤ ❛s ✉❧tr❛✲❧♦✇ ♥♦✐s❡✱ ❧♦✇ ♣♦✇❡r ❝♦♥s✉♠♣t✐♦♥ ❛♥❞ ❤✐❣❤ ✇♦r❦✐♥❣ ❢r❡✲ q✉❡♥❝② ❬✶❪✱ ❏❏ ❤❛s r❡❝❡✐✈❡❞ ♠✉❝❤ ❛tt❡♥t✐♦♥ ❢r♦♠ ♠❛♥② r❡s❡❛r❝❤❡rs✳ ❚❤❡♥ ❞✐✛❡r❡♥t ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ t♦ r❡♣r❡s❡♥t ❏❏ ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✺❪✳ ❆♠♦♥❣ ♠❛♥② t②♣❡s ♦❢ ❏❏ ♠♦❞❡❧s✱ t✇♦ t②♣❡s ♦❢ ❏❏ ♠♦❞❡❧s ❤❛✈❡ ❛ttr❛❝t❡❞ ♠♦r❡ r❡s❡❛r❝❤❡rs ❞✉❡ t♦ t❤❡✐r ❡①❛❝t ♠♦❞❡❧✐♥❣ ✐♥ ❏❏ ❜❡❤❛✈✐♦rs✳ ❚❤❡s❡ ♠♦❞❡❧s ❛r❡ t❤❡ s❤✉♥t❡❞ ❧✐♥❡❛r r❡s✐st✐✈❡✲ ❝❛♣❛❝✐t✐✈❡ ❥✉♥❝t✐♦♥ ✭❘❈❙❏✮ ❛♥❞ t❤❡ s❤✉♥t❡❞ ♥♦♥❧✐♥❡❛r r❡s✐st✐✈❡✲ ❝❛♣❛❝✐t✐✈❡✲✐♥❞✉❝t❛♥❝❡ ❥✉♥❝✲ t✐♦♥ ✭❘❈▲❙❏✮✳ ❚❤❡ ❘❈❙❏ ♠♦❞❡❧ ✐s t❤❡ s❡❝♦♥❞ ♦r❞❡r s②st❡♠ ✇❤✐❧❡ t❤❡ ❘❈▲❙❏ ♠♦❞❡❧ ✐s t❤❡ t❤✐r❞ ♦r❞❡r s②st❡♠✳ ❚❤❡ ❘❈▲❙❏ ♠♦❞❡❧ ✐s ❢♦✉♥❞ t♦ ❜❡ ♠♦r❡ ❛❝❝✉r❛t❡ ✐♥ ❤✐❣❤ ❢r❡q✉❡♥❝② ❛♣♣❧✐❝❛t✐♦♥s ❬✸❪✱ ❬✹❪✳ ❇❡❝❛✉s❡ t❤❡ ❘❈▲❙❏ ♠♦❞❡❧ ✐s t❤❡ t❤✐r❞ ♦r❞❡r s②st❡♠✱ t❤✐s ♠♦❞❡❧ ❝❛♥ ❡①❤✐❜✐t ❝❤❛♦s ❡✈❡♥ ✇✐t❤ ❡①t❡r♥❛❧ ❞❝ ❝✉rr❡♥t ♦♥❧②✳ ❚❤❡ ❝❤❛♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❘❈▲❙❏ ♠♦❞❡❧ ❤❛s ❜❡❡♥ ❡①t❡♥s✐✈❡❧② st✉❞✲ ✐❡❞ ❜② ❉❛♥❛✱ ❡t ❛❧✳ ❬✺❪✳ ❆❢t❡r✇❛r❞✱ t❤❡r❡ ❤❛✈❡ ❜❡❡♥ s♦♠❡ ❝♦♥tr♦❧ ♠❡t❤♦❞s ❞❡✈❡❧♦♣❡❞ t♦ ❝♦♥✲ tr♦❧ ♦r s②♥❝❤r♦♥✐③❡ ❘❈▲❙❏ ♠♦❞❡❧ ♦❢ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥ s✉❝❤ ❛s ♥♦♥❧✐♥❡❛r ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧ ❬✻❪✱ ❜❛❝❦st❡♣♣✐♥❣ ❝♦♥tr♦❧ ❬✼❪✱ ❬✽❪✱ ❞❡❧❛② ❧✐♥❡❛r ❢❡❡❞✲ ❜❛❝❦ ❝♦♥tr♦❧ ❬✾❪✱ t✐♠❡ ❞❡❧❛② ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧ ❬✶✵❪ ❛♥❞ s❧✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦❧ ❬✶✶❪❀ ❤♦✇❡✈❡r✱ s♦♠❡ s❤♦rt❝♦♠✐♥❣s ❡①✐st✳ ❚❤❡ ♥♦♥❧✐♥❡❛r ❜❛❝❦st❡♣♣✐♥❣ ♠❡t❤♦❞ ❤❛s q✉✐t❡ ❝♦♠♣❧✐❝❛t❡❞ ♣r♦❝❡❞✉r❡ t♦ ❞❡✲ s✐❣♥ t❤❡ ❝♦♥tr♦❧❧❡r ✇❤✐❧❡ ❝❤♦♦s✐♥❣ t❤❡ t✐♠❡ ❞❡✲ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ❧❛② ✐s ♣r♦❜❧❡♠❛t✐❝ ✐♥ ❞❡❧❛② ❧✐♥❡❛r ❢❡❡❞❜❛❝❦✳ ❚❤❡ ❝❤❛tt❡r✐♥❣ ♣❤❡♥♦♠❡♥♦♥ ✐s ❛ ❞r❛✇❜❛❝❦ ♦❢ t❤❡ s❧✐❞✲ ✐♥❣ ♠♦❞❡ ♠❡t❤♦❞✳ ▼♦r❡♦✈❡r✱ t❤❡s❡ ❝♦♥tr♦❧ t❡❝❤✲ ♥✐q✉❡s ❛❧♠♦st r❡q✉✐r❡ t❤❡ ❡①❛❝t ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s t♦ ❞❡s✐❣♥ t❤❡ ❝♦♥tr♦❧❧❡rs✳ ❚❤✐s r❡q✉✐r❡✲ ♠❡♥t ❜❡❝♦♠❡s t❤❡ s✐❣♥✐✜❝❛♥t ❧✐♠✐t❛t✐♦♥ ✐♥ ❞❡s✐❣♥ ❛ ♥♦♥❧✐♥❡❛r ❝♦♥tr♦❧❧❡r ✇❤❡♥ t❤❡ s②st❡♠ ♣❛r❛♠✲ ❡t❡rs ❛r❡ ✉♥❦♥♦✇♥ ♦r t❤❡ s②st❡♠ ✐s ❡✛❡❝t❡❞ ❜② ✉♥❝❡rt❛✐♥t✐❡s✳ ◆♦✇❛❞❛②s✱ ❢✉③③② ❧♦❣✐❝ ❛♥❞ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❛r❡ ✉s❡❞ ❛s t❤❡ ♣♦✇❡r t♦♦❧s ❢♦r ♠♦❞❡❧❧✐♥❣ ❛♥❞ ❝♦♥✲ tr♦❧❧✐♥❣ ❤✐❣❤❧② ✉♥❝❡rt❛✐♥✱ ♥♦♥❧✐♥❡❛r ❛♥❞ ❝♦♠♣❧❡① s②st❡♠s ❬✶✷❪✱ ❬✶✸❪✱ ❬✶✹❪✱ ❬✶✺❪✱ ❬✶✻❪✳ ■♥ t❤✐s st✉❞②✱ t❤❡ ❝❤❛♦s s②♥❝❤r♦♥✐③❛t✐♦♥ ♦❢ ❝♦✉♣❧❡❞ ❘❈▲❙❏ ♠♦❞❡s ✐s ❡①♣❡❝t❡❞✳ ❚❤❡ s②♥❝❤r♦♥✐③❛t✐♦♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✇❤❡♥ t❤❡ s❧❛✈❡ ❢♦❧❧♦✇s t❤❡ ♠❛st❡r ❛s ❝❧♦s❡ ❛s ♣♦ss✐❜❧❡✳ ❇❛s❡❞ ♦♥ ❢✉③③② ♥❡✉r❛❧ ♥❡t✲ ✇♦r❦s✱ ✇❡ ❞❡✈❡❧♦♣ ❛ ▼■▼❖ ❝♦♥tr♦❧❧❡r t❤❛t ❝❛♥ ❢♦r❝❡ t❤❡ st❛t❡s ♦❢ s❧❛✈❡ t♦ tr❛❝❦ t❤❡ st❛t❡s ♦❢ ♠❛st❡r ✇✐t❤ ③❡r♦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ st❛t❡ ❡rr♦rs✳ ■♥ t❤✐s ♠❛♥♥❡r t❤❡ ❝❤❛♦s s②♥❝❤r♦♥✐③❛t✐♦♥ ✐s ♦❜✲ t❛✐♥❡❞✳ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ♦❢ ❘❈▲❙❏ ✐s ❞❡s❝r✐❜❡❞✳ ❚❤❡ ▼■▼❖ ❢✉③③② ♥❡✉r❛❧ ❝♦♥tr♦❧❧❡r ❞❡s✐❣♥ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✸ ✇✐t❤ t❤❡ ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥s ❛r❡ ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✹✳ ❋✐♥❛❧❧②✱ t❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s ❣✐✈❡♥ ❙❡❝t✐♦♥ ✺✳ ✷✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ βC ❛♥❞ βL ❝♦rr❡s♣♦♥❞ t♦ ❝❛♣❛❝✐t✐✈❡ ❛♥❞ ✐♥❞✉❝✲ t❛♥❝❡ ❝♦♥st❛♥ts r❡s♣❡❝t✐✈❡❧② ❛♥❞ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ♠♦❞❡❧ ♣❛r❛♠❡t❡rs✳ iz st❛♥❞s ❢♦r t❤❡ ❡①t❡r✲ ♥❛❧ ❝✉rr❡♥t ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❞❝ ❝♦♠♣♦♥❡♥t ♦♥❧②✳ ❚❤❡ ♥♦♥❧✐♥❡❛r ❞❛♠♣✐♥❣ t❡r♠ g(z2 ) ✐s ❛♣♣r♦①✐✲ ♠❛t❡❞ ✇✐t❤ ❝✉rr❡♥t ✈♦❧t❛❣❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ ❥✉♥❝t✐♦♥ r❡s✐st❛♥❝❡s ❛♥❞ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ st❡♣ ❢✉♥❝t✐♦♥✿ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❘❈▲❙❏ ♠♦❞❡❧ ✇❛s ❡①t❡♥✲ s✐✈❡❧② st✉❞✐❡❞ ✐♥ ❬✺❪✳ ❚❤✐s st✉❞② ❞❡♠♦♥str❛t❡❞ t❤❛t t❤❡ ❘❈▲❙❏ ♠♦❞❡❧ ❝❛♥ ♣r♦❞✉❝❡ ❝❤❛♦t✐❝ ♦s❝✐❧✲ ❧❛t✐♦♥s ✇❤❡♥ t❤❡ ❡①t❡r♥❛❧ ❞❝ ❝✉rr❡♥t ❛♥❞ t❤❡ ♣❛✲ r❛♠❡t❡rs ❢❛❧❧ ✐♥t♦ ❛ ❝❡rt❛✐♥ ❛r❡❛✳ ❋♦r ❡①❛♠♣❧❡s✱ t❤❡ ❥✉♥❝t✐♦♥ ✐♥ ❊q✳ ✭✶✮ ✇✐t❤ ③❡r♦ ✐♥✐t✐❛❧ st❛t❡s ❡①❤✐❜✐ts ❝❤❛♦s ✇❤❡♥ βC = 0.707✱ βL = 2.6✱ iz = 1.2 ❛♥❞ ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳ ❚❤❡ ❘❈▲❙❏ ▼♦❞❡❧ ♦❢ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥ ■♥ ❤✐❣❤ ❢r❡q✉❡♥❝② ❛♣♣❧✐❝❛t✐♦♥✱ t❤❡ ❘❈▲❙❏ ♠♦❞❡❧ ♦❢ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥ ✐s ❢♦✉♥❞ ♠♦r❡ ❛❝❝✉r❛❝② ❛♥❞ ❛♣♣r♦♣r✐❛t❡ t❤❛♥ ♦t❤❡rs ❬✸❪✱ ❬✹❪✳ ■♥ t❤❡ ❞✐✲ ♠❡♥s✐♦♥❧❡ss ❢♦r♠✱ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ♦❢ ❘❈▲❙❏ ✐s ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s ❬✺❪✿ z1 = z2 , [iz − g(z2 )z2 − sin (z1 − z3 )] , z2 = βC z3 = [z2 − z3 ] , βL ❋✐❣✳ ✶✿ ❈❤❛♦t✐❝ ♠♦t✐♦♥ ✐♥ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥✳ ✭✶✮ ❘❡♠❛r❦ ✶✳ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ ❏❏ ♠✉❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡✐r ❝✐r❝✉✐t ♣❛r❛♠❡t❡rs✱ ✐♥❝❧✉❞✐♥❣ βL ❛♥❞ βC ✱ ❛♥❞ t❤❡ ❡①t❡r♥❛❧ ❉❈ ❝✉rr❡♥t iz ✳ ❚❤❡ ❏❏ ✇❤❡r❡ st❛t❡ ✈❛r✐❛❜❧❡s z1 , z2 ❛♥❞ z3 r❡♣r❡s❡♥t s❤♦✇s t❤❡ ❝❤❛♦t✐❝ ❜❡❤❛✈✐♦rs ✇❤❡♥ t❤❡s❡ ♣❛r❛♠✲ t❤❡ ♣❤❛s❡ ❞✐✛❡r❡♥❝❡✱ ❥✉♥❝t✐♦♥ ✈♦❧t❛❣❡ ❛♥❞ ❝✉r✲ ❡t❡rs ❢❛❧❧ ✐♥t♦ t❤❡ ❝❤❛♦t✐❝ r❡❣✐♦♥✳ ❚❤✐s ❝❤❛♦t✐❝ r❡♥t t❤r♦✉❣❤ s❤✉♥t❡❞ ✐♥❞✉❝t❛♥❝❡✱ r❡s♣❡❝t✐✈❡❧②✳ r❡❣✐♦♥ ❝❛♥ ❜❡ r❡❢❡rr❡❞ ✐♥ ❋✐❣s✳ ✾✲✶✵ ♦❢ ❬✹❪✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✽✶ ❱❖▲❯▼❊✿ ✶ ✸✳ ❙②♥❝❤r♦♥✐③❛t✐♦♥ ♦❢ t❤❡ ❈♦✉♣❧❡❞ ❘❈▲❙❏ ▼♦❞❡❧s ✸✳✶✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❉✉❡ t♦ t❤❡ r❡❧❛t✐✈❡ ❞❡❣r❡❡ ♦❢ t❤❡ s②st❡♠ ❣✐✈❡♥ ❜② ❊q✳ ✭✸✮ r1 = r2 = r3 = ✱ t❤❡ ♦✉t♣✉ts ♦❢ t❤❡ s❧❛✈❡ s②st❡♠ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s✿ Pr♦❜❧❡♠ st❛t❡♠❡♥t ❛♥❞ y = f (x) + g(x)u ♣r❡❧✐♠✐♥❛r✐❡s ✭✹✮ ❈♦♥s✐❞❡r t❤❡ ❘❈▲❙❏ ❝❤❛♦t✐❝ s②st❡♠ ❞❡✜♥❡❞ ✐♥ ◆♦✇✱ ✇❡ ❞❡✜♥❡ t❤❡ ❡rr♦rs ❜❡t✇❡❡♥ t❤❡ ❞❡♣❡♥✲ ❊q✳ ✭✶✮ ❛s t❤❡ ♠❛st❡r s②st❡♠ ✇✐t❤ ✇❤✐❝❤ t❤❡ ❞❡♥t ✈❛r✐❛❜❧❡s ♦❢ ♠❛st❡r ❛♥❞ s❧❛✈❡ ❛s✿ s❧❛✈❡ s②st❡♠ ♥❡❡❞ t♦ ❜❡ s②♥❝❤r♦♥✐③❡❞✳ ❈♦♥s✐❞❡r t❤❡ s❡❝♦♥❞ ❘❈▲❙❏ ❝❤❛♦t✐❝ s②st❡♠ t❤❛t ❝♦♥t❛✐♥s t❤❡ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ✐♥✐t✐❛❧ ❝♦♥✲ ❞✐t✐♦♥s ❛♥❞ ❡①t❡r♥❛❧ ❝✉rr❡♥t ❛s ❢♦❧❧♦✇s✿ x1 = x2 + u1 , x2 = [ix − g(x2 )x2 − sin(x1 ) − x3 ] + u2 , βC x3 = [x2 − x3 ] + u3 , βL ✭✷✮ ✇❤❡r❡ u1 ✱u2 ❛♥❞ u3 ❛r❡ ❝♦♥tr♦❧ s✐❣♥❛❧s✳ ❍❡r❡✱ t❤❡ ❛✐♠ ♦❢ t❤❡s❡ ❝♦♥tr♦❧ s✐❣♥❛❧s ✐s t♦ ❢♦r❝❡ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ s❧❛✈❡ s②st❡♠ ❞❡s❝r✐❜❡❞ ❜② ❊q✳ ✭✷✮ t♦ ❢♦❧❧♦✇ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ♠❛s✲ t❡r s②st❡♠ ❣✐✈❡♥ ❜② ❊q✳ ✭✶✮ ❛s ❝❧♦s❡ ❛s ♣♦ss✐✲ ❜❧❡✳ ❚❤✉s✱ ♦♥❡✲✇❛② s②♥❝❤r♦♥✐③❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❘❈▲❙❏ ❝❤❛♦t✐❝ s②st❡♠s ✇✐❧❧ ❜❡ ❛❝❤✐❡✈❡❞✳ ❙✐♥❝❡ ❛❧❧ st❛t❡ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ s❧❛✈❡ s②st❡♠ ❛r❡ ❝♦♥s✐❞✲ ❡r❡❞ ❛s ♦✉t♣✉ts✱ t❤❡ s❧❛✈❡ s②st❡♠ ✇✐t❤ ❝♦♥tr♦❧ ✐♥♣✉ts ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ✐♥ t❤❡ ▼■▼❖ ❢♦r♠ ❛s✿ x = f (x) + g(x)u y = h(x) ✇❤❡r❡ ✭✸✮ e = y − yd ✭✺✮ ✇❤❡r❡ e = [e1 e2 e3 ]T ❛♥❞ yd = [z1 z2 z3 ]T ✳ ❚❤❡♥✱ ✐♥ ♦r❞❡r t♦ ♠❡❡t t❤❡ ❝♦♥tr♦❧ ♦❜❥❡❝t✐✈❡✱ ✇❡ ✉s❡ t❤❡ ✐♥♣✉t✲♦✉t♣✉t ❧✐♥❡❛r✐③❛t✐♦♥ t❡❝❤♥✐q✉❡ ❛♥❞ t❤❡ ♥♦♥❧✐♥❡❛r ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧❧❡r ❝❛♥ ❜❡ ♦❜✲ t❛✐♥❡❞ ❛s ❬✶✼❪✿ u∗ = g−1 (x)[−f (x) + v(t)] ✭✻✮ ✇❤❡r❡ v(t) ✐s t❤❡ ♥❡✇ ✐♥♣✉t ✈❛r✐❛❜❧❡ ❛♥❞ ✐t ✐s ❣✐✈❡♥ ❛s✿ v(t) = yd − ke ✭✼✮ ✇❤❡r❡ k = diag(k1 , k2 , k3 ) ✐s ♣♦s✐t✐✈❡ ❞❡✜♥❡❞ ♠❛tr✐①✳ ❙✉❜st✐t✉t✐♥❣ ❊q✳ ✭✻✮ ✐♥t♦ ❊q✳ ✭✹✮✱ ✇❡ ❝❛♥ ❣❡t✿ y = v(t) ✭✽✮ ❙✉❜st✐t✉t✐♥❣ ❊q✳ ✭✼✮ ✐♥t♦ ❊q✳ ✭✽✮✱ ❛♥❞ ✉s✐♥❣ ❊q✳ ✭✺✮ ✐♠♣❧✐❡s t❤❛t✿ e + ke = ✭✾✮ ❚❤❡ ❡q✉❛t✐♦♥ ✐♥ ❊q✳ ✭✾✮ ✐♠♣❧✐❡s t❤❛t ej ✇✐t❤ j = 1, 2, ❝♦♥✈❡r❣❡s t♦ ③❡r♦ ❡①♣♦♥❡♥t✐❛❧❧②✳ ❍♦✇✲ ❡✈❡r✱ t❤❡ ✐❞❡❛❧ ♥♦♥❧✐♥❡❛r ❝♦♥tr♦❧❧❡r ✐♥ ❊q✳ ✭✻✮ ❝❛♥ ♥♦ ❧♦♥❣❡r ❜❡ ✉s❡❞ ✇❤❡♥ f (x) ❛♥❞ g(x) ✐♥ ❊q✳ ✭✸✮ ❝❤❛♥❣❡ t❤❡✐r ✈❛❧✉❡s ❛♥❞ ❜❡❝♦♠❡ ✉♥❦♥♦✇♥ ❞✉❡ t♦ ♣❛r❛♠❡t❡r ♣❡rt✉r❜❛t✐♦♥ ❛♥❞ ♥♦✐s❡ ❞✐st✉r❜❛♥❝❡✳ ■♥ ♦r❞❡r t♦ ❜②♣❛ss t❤✐s ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ ❛ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦ ✇❛s ✉s❡❞ t♦ ❞✐r❡❝t❧② ❛♣♣r♦①✐♠❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ ❝♦♥tr♦❧ s✐❣♥❛❧s✳ ✽✷ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ✸✳✷✳ ❉❡s✐❣♥❡❞ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦ ❙✐♥❝❡ ❢✉③③② ❧♦❣✐❝ ❛♥❞ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❤❛✈❡ ❡①✲ ❤✐❜✐t❡❞ t❤❡ s✉♣❡r✐♦r ❛❜✐❧✐t✐❡s ✐♥ ♠♦❞❡❧✐♥❣ ❛♥❞ ❝♦♥tr♦❧❧✐♥❣ t❤❡ ❤✐❣❤❧② ✉♥❝❡rt❛✐♥✱ ✐❧❧✲❞❡✜♥❡❞ ❛♥❞ ❝♦♠♣❧❡① s②st❡♠s✱ ✇❡ ❡♠♣❧♦② ❛ ❢✉③③② ♥❡✉r❛❧ ♥❡t✲ ✇♦r❦ ✇❤✐❝❤ ❝♦♠❜✐♥❡s t❤❡ ❛❞✈❛♥t❛❣❡♦✉s ♠❡r✐ts ♦❢ ❛ ❢✉③③② ❧♦❣✐❝ s②st❡♠ ❛♥❞ ❛ ♥❡✉r❛❧ ♥❡t✇♦r❦ t♦ ❛♣✲ ♣r♦①✐♠❛t❡ t❤❡ ♥♦♥❧✐♥❡❛r ❝♦♥tr♦❧ ❧❛✇s u1 ✱ u2 ❛♥❞ u3 ✳ ❚❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✳ ✷✳ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ t❤❡ ♦✉t♣✉t ❧❛②❡r✱ ✸ ♥♦❞❡s ❛❝t ❢♦r t❤❡ ✈❛❧✉❡s ♦❢ ❝♦♥tr♦❧ s✐❣♥❛❧s u1 ✱ u2 ❛♥❞ u3 ❛t t✐♠❡ t✳ ✸✳✸✳ ❆❞❛♣t✐✈❡ ❢✉③③② ❝♦♥tr♦❧❧❡r ❞❡s✐❣♥ ❲❤❡♥ f (x) ❛♥❞ g(x) ❛r❡ ✉♥❦♥♦✇♥✱ t❤❡ ✐❞❡❛❧ ❝♦♥✲ tr♦❧ ❧❛✇ ✐♥ ❊q✳ ✭✻✮ ❝❛♥♥♦t ❜❡ ❞❡t❡r♠✐♥❡❞✱ ❛♥❞ t❤❡r❡❢♦r❡ t❤✐s ❝♦♥tr♦❧ ❧❛✇ ❝❛♥♥♦t ❜❡ ✉s❡❞✳ ❚♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠✱ ✇❡ ❞❡✈❡❧♦♣❡❞ ❛ ❢✉③③② ♥❡✉✲ r❛❧ ♥❡t✇♦r❦ t♦ ❞✐r❡❝t❧② ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥♦♥❧✐♥✲ ❡❛r ❝♦♥tr♦❧ ❧❛✇✳ ■♥ ♦r❞❡r t♦ ❡♥s✉r❡ ♦✉r ❞❡s✐❣♥ ♣r♦♣❡r✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s✳ ❆ss✉♠♣t✐♦♥ ✶✳ ❚❤❡ s❝❛❧❛r ♠❛tr✐① g(x) ✐s ♣♦s✐t✐✈❡ ❞❡✜♥❡❞✱ t❤❡♥ ✐t ❡①✐sts s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥✲ st❛♥ts g, g ∈ R s✉❝❤ t❤❛t gI ≤ g(x) ≤ gI✳ ❆ss✉♠♣t✐♦♥ ✷✳ ❚❤❡ r❛t❡ ♦❢ ✈❛r✐❛t✐♦♥ ♦❢ ❣✭①✮ ✐s ❜♦✉♥❞❡❞✱ t❤❛t ✐s✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t D ∈ R s✉❝❤ t❤❛t | g(x) |≤ DI✳ ▲❡t t❤❡ ❢✉③③② ♥❡✉r❛❧ ❝♦♥tr♦❧❧❡r uf ❜❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✐❞❡❛❧ ❝♦♥tr♦❧❧❡r ❣✐✈❡♥ ✐♥ ❊q✳ ✭✻✮✳ uf ✐s ♦♥❧✐♥❡ ❡st✐♠❛t❡❞ ❜② ❛ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦ ❛s ❢♦❧❧♦✇s✿ ❋✐❣✳ ✷✿ ❙tr✉❝t✉r❡ ♦❢ t❤❡ ❞❡s✐❣♥❡❞ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦✳ ❚❤✐s ♥❡t✇♦r❦ str✉❝t✉r❡ ❤❛s ❢♦✉r ❧❛②❡rs✿ ✐♥♣✉t ❧❛②❡r✱ ♠❡♠❜❡rs❤✐♣ ❧❛②❡r✱ r✉❧❡ ❧❛②❡r ❛♥❞ ♦✉t♣✉t ❧❛②❡r✳ ◆♦❞❡s ✐♥ t❤❡ ✐♥♣✉t ❧❛②❡r ❛r❡ ✸ st❛t❡ ✈❛r✐✲ ❛❜❧❡s ♦❢ t❤❡ s❧❛✈❡ ❝❤❛♦t✐❝ ❏❏ ❛♥❞ t❤❡✐r ✈❛❧✉❡s ❛r❡ ❞✐r❡❝t❧② tr❛♥s♠✐tt❡❞ t♦ t❤❡ ♠❡♠❜❡rs❤✐♣ ❧❛②❡r✳ ❲❤❡♥ ✾ ❢✉③③② r✉❧❡s ✇❡r❡ ✉s❡❞ ❢♦r ♥❡t✇♦r❦ ❞❡✲ s✐❣♥✱ t❤❡ ♠❡♠❜❡rs❤✐♣ ❧❛②❡r ❤❛s 3×9 ♥♦❞❡s✳ ❊❛❝❤ ♥♦❞❡ ♣❡r❢♦r♠s ❛ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥ ❛♥❞ ❡♠✲ ♣❧♦②s ❛ ●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥ t♦ ❝❛❧❝✉❧❛t❡ ✐ts ✈❛❧✉❡✳ ❚❤❡ r✉❧❡ ❧❛②❡r ❤❛s ✾ ♥♦❞❡s ❛♥❞ ❡❛❝❤ ♥♦❞❡ ❝♦r✲ r❡s♣♦♥❞s t♦ ❛♥ ❡❧❡♠❡♥t ψ(x) ♦❢ t❤❡ ❢✉③③② ❜❛s✐s ✈❡❝t♦r ψ(x) ❛♥❞ ♣❡r❢♦r♠s ❛ ❢✉③③② r✉❧❡✳ ❚❤✉s✱ ✐♥ t❤❡ r✉❧❡ ❧❛②❡r✱ ❛❧❧ ♥♦❞❡s ❞❡♥♦t❡ t❤❡ ❢✉③③② r✉❧❡ s❡t✳ ❚❤❡ ♦✉t♣✉t ❧❛②❡r ✐s ❝♦♥♥❡❝t❡❞ t♦ t❤❡ r✉❧❡ ❧❛②❡r t❤r♦✉❣❤ ✇❡✐❣❤t✐♥❣ ❢❛❝t♦rs✱ θij ✇✐t❤ i = ✾✱ j = ✸✳ ❚❤❡ ✇❡✐❣❤t✐♥❣ ❢❛❝t♦rs θij ❛r❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ✇❡✐❣❤t✐♥❣ ✈❡❝t♦r θ(t)✳ ❚❤❡s❡ ❢❛❝t♦rs ❛r❡ t❤❡ ♣❛✲ r❛♠❡t❡rs ♦❢ t❤❡ ♥❡t✇♦r❦s ❛♥❞ t❤❡② ✇✐❧❧ ❜❡ t✉♥❡❞ ❜② ❞❡s✐❣♥❡❞ ❛❞❛♣t✐✈❡ ❧❛✇s ❣✐✈❡♥ ✐♥ ❊q✳ ✭✶✶✮✳ ■♥  θ11  ✳✳ ✇❤❡r❡  ✳ θ91 uf = θT (t)ψ(x), ✭✶✵✮  θ13 ✳✳  ✳  ✐s ✇❡✐❣❤t✐♥❣ ♠❛tr✐① ♦❢ θ93 ✇❤✐❝❤ ❡❛❝❤ ❡♥tr② ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ❧✐♥❦ ❜❡✲ t✇❡❡♥ ❘✉❧❡ ❧❛②❡r ❛♥❞ ❖✉t♣✉t ❧❛②❡r ✐♥ t❤❡ ❝❤♦s❡♥ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦✳ ψ(x) = [ψ1 ψ1 ]T ✐s ❢✉③③② ❜❛s✐❝ ✈❡❝t♦r ♦❢ ✇❤✐❝❤ ❡❛❝❤ ❡❧❡♠❡♥t ψi ✇✐t❤ i = ✾ ✐s ❞❡✜♥❡❞ ❛s✿ µAij (x) ϕi (x) = j=1   i=1 , µAij (x) j=1 ✇❤❡r❡ t❤❡ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥s µAij (x) ✬s ❡♠✲ ♣❧♦② ●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥ t♦ ❝❛❧❝✉❧❛t❡ t❤❡✐r ✈❛❧✉❡s✳ ❚❤❡ ❛❞❛♣t✐✈❡ ❧❛✇ ✇❤✐❝❤ ❛❧❧♦✇s t❤❡ ✇❡✐❣❤t✐♥❣ ♠❛tr✐① θ(t) t♦ ✈❛r② s♦ t❤❛t t❤❡ ❢✉③③② ♥❡✉r❛❧ ❝♦♥✲ tr♦❧❧❡r uf r❡❛❝❤❡s t❤❡ ✐❞❡❛❧ ❝♦♥tr♦❧❧❡r u∗ ✐s ❝❤♦✲ s❡♥ ❛s✿ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✽✸ ❱❖▲❯▼❊✿ ✶ θ˙ij = −wij ψi ej with i = 9, j = 3, ✭✶✶✮ V =− | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ eT ke ≤ g ✭✶✹✮ ❙✐♥❝❡ t❤❡ ❝❤♦s❡♥ ▲②❛♣✉♥♦✈ ❝❛♥❞✐❞❛t❡ ❢✉♥❝t✐♦♥ ✇❤❡r❡ wij s ❛r❡ ♣♦s✐t✐✈❡ ❢❛❝t♦rs ✇❤✐❝❤ ❣♦✈❡r♥ t❤❡ ✐s ♣♦s✐t✐✈❡ ❛♥❞ ✐ts t✐♠❡ ❞❡r✐✈❛t✐✈❡ ✐s ❧❡ss t❤❛♥ r❛t❡ ♦❢ ❛❞❛♣t✐♦♥✳ ③❡r♦✱ t❤❡ ❝♦♥tr♦❧❧❡❞ s②st❡♠ ✐s st❛❜❧❡✳ ❙✐♥❝❡ t❤❡ ❞❡s✐❣♥❡❞ ❢✉③③② ♥❡✉r❛❧ ♥❡t✇♦r❦ ❤❛s t❤❡ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ✉♥✐ts ✐♥ t❤❡ ❤✐❞❞❡♥ ❧❛②❡r✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs ❛r❡ ✉♥❛✈♦✐❞❛❜❧❡✳ ❲❡ ✹✳ ◆✉♠❡r✐❝❛❧ ❙✐♠✉❧❛t✐♦♥s ❛ss✉♠❡ t❤❛t✱ t❤❡s❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs ❛r❡ ❜♦✉♥❞❡❞ ❜② ❛ ❦♥♦✇♥ ✈❡❝t♦r δ = [δ1 δ2 δ3 ]✳ ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ❛r❡ ❣✐✈❡♥ t♦ ❚❤❡♥ ❛ s❧✐❞✐♥❣ ♠♦❞❡ ❝♦♥tr♦❧❧❡r us ✐s ❛❞❞❡❞ t♦ ✈❡r✐❢② t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✳ ■♥ ♦r❞❡r t♦ ❞❡♠♦♥✲ r❡❞✉❝❡ t❤❡ ✉♥❞❡s✐r❛❜❧❡ ❡✛❡❝ts ♦❢ t❤❡ ❛♣♣r♦①✐♠❛✲ str❛t❡ t❤❡ ♣r♦❝❡❞✉r❡✱ ✇❡ ❦❡❡♣ t❤❡ ③❡r♦ ✐♥✐t✐❛❧ t✐♦♥ ❡rr♦rs✳ ❚❤❡ ❢♦r♠✉❧❛ ♦❢ us ✐s ❣✐✈❡♥ ❛s✿ ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡ ❡①t❡r♥❛❧ ❝✉rr❡♥t iz = 1.2 ❢♦r t❤❡ ♠❛st❡r s②st❡♠✳ ❚❤❡♥ ✇❡ ❝❤♦♦s❡ t❤❡ ❞✐✛❡r❡♥t ✈❛❧✉❡s ❢♦r t❤❡ s❧❛✈❡ s②st❡♠✱ t❤❛t ✐s D ✭✶✷✮ [x1 [x2 [x3 ]T = [1 1]T ❛♥❞ ix = 1.135✳ us = −diag(sgn(e))(δ + | e |), 2g ❚❤❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs βL = 2.6 ❛♥❞ βC = 0.707 ❛r❡ ❝❤♦s❡♥ ❛♥❞ ✜①❡❞ ❢♦r ❜♦t❤ ♠❛st❡r ❛♥❞ s❧❛✈❡✳ ✇❤❡r❡ | | ❞❡♥♦t❡s t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ❡①t❡r♥❛❧ ❞❝ ❝✉r✲ ❋r♦♠ ❊q✳ ✭✶✵✮ ❛♥❞ ❊q✳ ✭✶✷✮✱ t❤❡ t♦t❛❧ ❝♦♥✲ r❡♥ts ❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ♠❛st❡r ❛♥❞ s❧❛✈❡ ♣r♦❞✉❝❡ ❝❤❛♦t✐❝ ♦s❝✐❧❧❛t✐♦♥s ❞✐✛❡r❡♥t❧②✳ tr♦❧❧❡r ✐s ❛❝❤✐❡✈❡❞ ❛s✿ ❋✐rst✱ t❤❡ ❝♦✉♣❧❡❞ s②st❡♠s ❛r❡ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✇✐t❤♦✉t ❝♦♥tr♦❧ s✐❣♥❛❧s✳ ❉✉❡ t♦ t❤❡ u = uf + us ❞✐✛❡r❡♥t ❝❤❛♦t✐❝ ♠♦t✐♦♥s ❜❡t✇❡❡♥ ♠❛st❡r ❛♥❞ D s❧❛✈❡✱ t❤❡ ❝❤❛♦t✐❝ ♦s❝✐❧❧❛t✐♦♥s ❛r❡ r❡✢❡❝t❡❞ ✐♥t♦ T = θ (t)ψ(x) − diag(sgn(e))(δ + |e|) 2g st❛t❡ ❡rr♦rs ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✸✳ ✭✶✸✮ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦✉♣❧❡❞ ❘❈▲❙❏ ♠♦❞❡❧s ❝❛♥ ❜❡ s②♥❝❤r♦♥✐③❡❞ ✇✐t❤ t❤❡ ❝♦♥tr♦❧ ❧❛✇ ✐♥ ❊q✳ ✭✶✸✮ ❛♥❞ t❤❡ ❛❞❛♣t✐✈❡ ♠❡❝❤❛♥✐s♠ ✐♥ ❊q✳ ✭✶✶✮✳ ▼♦r❡♦✈❡r✱ ❢♦r st❛❜✐❧✐t② ❛♥❛❧②s✐s✱ t❤❡ ▲②❛♣✉♥♦✈ ❛♣♣r♦❛❝❤ ❝❛♥ ❜❡ ✉s❡❞✳ ❋✐rst✱ t❤❡ ▲②❛♣✉♥♦✈ ❝❛♥✲ ❞✐❞❛t❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❢♦❧❧♦✇s✿ V = T −1 e g e+ 2 i=1 j=1 ˜T θ θij , wij ij ✇❤❡r❡ θ˜ij ✐s ❛ ♣❛r❛♠❡t❡r ❡rr♦r ❜❡t✇❡❡♥ t❤❡ ❝✉r✲ ∗ r❡♥t ♣❛r❛♠t❡r θij ❛♥❞ ♦♣t✐♠❛❧ ♣❛♣❛♠❡t❡r θij ✳ ∗ ◆♦t✐❝❡ t❤❛t t❤❡ ♦♣t✐♠❛❧ ♣❛♣❛♠❡t❡r θij ✐s ❛♥ ❛r✲ t✐✜❝✐❛❧ ❝♦♥st❛♥t q✉❛♥t✐t② ✐♥tr♦❞✉❝❡❞ ♦♥❧② ❢♦r ❛♥✲ ❛❧②t✐❝❛❧ ♣✉r♣♦s❡ ❛♥❞ ✐t ✐s ♥♦t ♥❡❡❞❡❞ ❢♦r ✐♠✲ ♣❧❡♠❛♥t❛t✐♦♥✳ ❚❛❦✐♥❣ s♦♠❡ ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✲ ❋✐❣✳ ✸✿ ❚❤❡ st❛t❡ ❡rr♦rs ❜❡t✇❡❡♥ ♠❛st❡r ❛♥❞ s❧❛✈❡ ✇✐t❤✲ ✉❧❛t✐♦♥s ❛♥❞ ✐♥❝♦r♣♦r❛t✐♥❣ t❤❡ ❝♦♥tr♦❧ ❧❛✇ ✐♥ ♦✉t ❝♦♥tr♦❧ ❡✛❡❝ts✳ ❊q✳ ✭✶✸✮ ❛♥❞ t❤❡ ❛❞❛♣t✐✈❡ ❧❛✇ ✐♥ ❊q✳ ✭✶✶✮✱ ♦♥❡ ❙❡❝♦♥❞✱ t❤❡ ❝♦✉♣❧❡❞ s②st❡♠s ❛r❡ ❝♦♥tr♦❧❧❡❞ ❝❛♥ ❣❡t t❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ✇❤✐❝❤ ✐s ❧❡ss t❤❛♥ ❜② t❤❡ ▼■▼❖ ❢✉③③② ♥❡✉r❛❧ ❝♦♥tr♦❧❧❡r✳ ■♥ t❤✐s ③❡r♦ ❛s✿ ✽✹ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ | wij = 5000 , D = , δj = 05 g = σ = wij wij δj , j = − δj , j = − g g | | ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✶ | ✷✵✶✼ | ❏✉♥❡ ❬✶✷❪ ❲❆◆●✱ ▲✳ ❳✳ ❆ ❈♦✉rs❡ ✐♥ ❋✉③③② ❙②st❡♠s ❛♥❞ ❈♦♥tr♦❧✱ Pr❡♥t✐❝❡ ❍❛❧❧ P❚❘✱ ✶✾✾✼✳ ✾✵✱ ♣♣✳ ✶✺✵✲✶✺✷✱ ✻✴✷✽✴ ✶✾✽✷✳ ❬✸❪ ❲❍❆◆✱ ❈✳ ❇✳ ❛♥❞ ❈✳ ❏✳ ▲❖❇❇✳ ✧❈♦♠✲ ♣❧❡① ❞②♥❛♠✐❝❛❧ ❜❡❤❛✈✐♦r ✐♥ ❘❈▲✲s❤✉♥t❡❞ ❬✶✸❪ ◆●❯❨❊◆✱ ❚✳✲❇✳✲❚✳✱ ❚✳✲▲✳ ▲■❆❖ ❛♥❞ ❏✳✲ ❏♦s❡♣❤s♦♥ ❥✉♥❝t✐♦♥s✱✧ ❆♣♣❧✐❡❞ ❙✉♣❡r❝♦♥✲ ❏✳ ❨❆◆✳ ✧❆❞❛♣t✐✈❡ tr❛❝❦✐♥❣ ❝♦♥tr♦❧ ❢♦r ❛♥ ❞✉❝t✐✈✐t②✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥✱ ✈♦❧✳ ✺✱ ♣♣✳ ✉♥❝❡rt❛✐♥ ❝❤❛♦t✐❝ ♣❡r♠❛♥❡♥t ♠❛❣♥❡t s②♥✲ ✸✵✾✹✲✸✵✾✼✱ ✶✾✾✺✳ ❝❤r♦♥♦✉s ♠♦t♦r ❜❛s❡❞ ♦♥ ❢✉③③② ♥❡✉r❛❧ ♥❡t✲ ✇♦r❦s✱✧ ❏♦✉r♥❛❧ ♦❢ ❱✐❜r❛t✐♦♥ ❛♥❞ ❈♦♥tr♦❧✱ ❬✹❪ ❈❆❲❚❍❖❘◆❊✱ ❆✳ ❇✳✱ ❈✳ ❇✳ ❲❍❆◆ ❛♥❞ ❏✉❧② ✽✱ ✷✵✶✸✳ ❈✳ ❏✳ ▲❖❇❇✱ ✧❈♦♠♣❧❡① ❞②♥❛♠✐❝s ♦❢ r❡s✐s✲ t✐✈❡❧② ❛♥❞ ✐♥❞✉❝t✐✈❡❧② s❤✉♥t❡❞ ❏♦s❡♣❤s♦♥ ❬✶✹❪ ◆●❯❨❊◆✱ ❚✳✲❇✳✲❚✳✱ ❚✳✲▲✳ ▲■❆❖✱ ❍✳✲❍✳ ❥✉♥❝t✐♦♥s✱✧ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ P❤②s✐❝s✱ ✈♦❧✳ ❑❯❖✱ ❛♥❞ ❏✳✲❏✳ ❨❆◆✳ ✧❆♥ ■♠♣r♦✈❡❞ ❆❞❛♣✲ ✽✹✱ ♣♣✳ ✶✶✷✻✲✶✶✸✷✱ ✶✾✾✽✳ t✐✈❡ ❚r❛❝❦✐♥❣ ❈♦♥tr♦❧❧❡r ♦❢ P❡r♠❛♥❡♥t ▼❛❣✲ ♥❡t ❙②♥❝❤r♦♥♦✉s ▼♦t♦r✱✧ ❆❜str❛❝t ❛♥❞ ❆♣✲ ❬✺❪ ❉❆◆❆✱ ❙✳ ❑✳✱ ❉✳ ❈✳ ❙❊◆●❯P❚❆ ❛♥❞ ❑✳ ♣❧✐❡❞ ❆♥❛❧②s✐s✱ ✈♦❧✳ ✷✵✶✹✱ ♣✳ ✶✷✱ ✷✵✶✹✳ ❉✳ ❊❉❖❍✳ ✧❈❤❛♦t✐❝ ❞②♥❛♠✐❝s ✐♥ ❏♦s❡♣❤✲ s♦♥ ❥✉♥❝t✐♦♥✱✧ ❈✐r❝✉✐ts ❛♥❞ ❙②st❡♠s ■✿ ❋✉♥✲ ❬✶✺❪ ◆●❯❨❊◆✱ ❚✳✲❇✳✲❚✳✱ ❚✳✲▲✳ ▲■❆❖ ❛♥❞ ❏✳✲❏✳ ❞❛♠❡♥t❛❧ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ■❊❊❊ ❨❆◆✳ ✧❆❞❛♣t✐✈❡ ❙❧✐❞✐♥❣ ▼♦❞❡ ❈♦♥tr♦❧ ♦❢ ❚r❛♥s❛❝t✐♦♥s ♦♥✱ ✈♦❧✳ ✹✽✱ ♣♣✳ ✾✾✵✲✾✾✻✱ ✷✵✵✶✳ ❈❤❛♦s ✐♥ P❡r♠❛♥❡♥t ▼❛❣♥❡t ❙②♥❝❤r♦♥♦✉s ▼♦t♦r ✈✐❛ ❋✉③③② ◆❡✉r❛❧ ◆❡t✇♦r❦s✱✧ ▼❛t❤❡✲ ❬✻❪ ❆✳ ❯❈❆❘✱ ❑✳ ❊✳ ▲❖◆◆●❘❊◆ ❛♥❞ ❊✳✲ ♠❛t✐❝❛❧ Pr♦❜❧❡♠s ✐♥ ❊♥❣✐♥❡❡r✐♥❣✱ ✈♦❧✳ ✷✵✶✹✱ ❲✳ ❇❆■✱ ✧❈❤❛♦s s②♥❝❤r♦♥✐③❛t✐♦♥ ✐♥ ❘❈▲✲ ♣✳ ✶✶✱ ✷✵✶✹✳ s❤✉♥t❡❞ ❏♦s❡♣❤s♦♥ ❥✉♥❝t✐♦♥ ✈✐❛ ❛❝t✐✈❡ ❝♦♥✲ tr♦❧✱✧ ❈❤❛♦s✱ ❙♦❧✐t♦♥s ✫ ❋r❛❝t❛❧s✱ ✈♦❧✳ ✸✶✱ ❬✶✻❪ ◆●❯❨❊◆✱ ❚✳✲❇✳✲❚✳✱ ❚✳✲▲✳ ▲■❆❖ ❛♥❞ ❏✳✲ ♣♣✳ ✶✵✺✲✶✶✶✱ ✶✴✷✵✵✼✳ ❏✳ ❨❆◆✳ ✧■♠♣r♦✈❡❞ ❆❞❛♣t✐✈❡ ❙❧✐❞✐♥❣ ▼♦❞❡ ❈♦♥tr♦❧ ❢♦r ❛ ❈❧❛ss ♦❢ ❯♥❝❡rt❛✐♥ ◆♦♥❧✐♥✲ ❬✼❪ ❍❆❘❇✱ ❆✳ ▼✳ ❛♥❞ ❇✳ ❆✳ ❍❆❘❇✳ ✧❈♦♥✲ ❡❛r ❙②st❡♠s ❙✉❜❥❡❝t❡❞ t♦ ■♥♣✉t ◆♦♥❧✐♥❡❛r✲ tr♦❧❧✐♥❣ ❈❤❛♦s ✐♥ ❏♦s❡♣❤s♦♥✲❏✉♥❝t✐♦♥ ❯s✐♥❣ ✐t② ✈✐❛ ❋✉③③② ◆❡✉r❛❧ ◆❡t✇♦r❦s✱✧ ▼❛t❤❡♠❛t✲ ◆♦♥❧✐♥❡❛r ❇❛❝❦st❡♣♣✐♥❣ ❈♦♥tr♦❧❧❡r✱✧ ❆♣✲ ✐❝❛❧ Pr♦❜❧❡♠s ✐♥ ❊♥❣✐♥❡❡r✐♥❣✱ ✈♦❧✳ ✷✵✶✺✱ ♣✳ ♣❧✐❡❞ ❙✉♣❡r❝♦♥❞✉❝t✐✈✐t②✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ✶✸✱ ✷✵✶✺✳ ♦♥✱ ✈♦❧✳ ✶✻✱ ♣♣✳ ✶✾✽✽✲✶✾✾✽✱ ✷✵✵✻✳ ❬✽❪ ❍❆❘❇✱ ❆✳ ❛♥❞ ❇✳ ❍❆❘❇✱ ✧❈❤❛♦s ❙②♥❝❤r♦✲ ❬✶✼❪ ❙▲❖❚■◆❊✱ ❏✳ ❏✳ ❊✳ ❛♥❞ ❲✳ ▲■✳ ❆♣♣❧✐❡❞ ◆♦♥✲ ❧✐♥❡❛r ❈♦♥tr♦❧✱ P❡❛rs♦♥ ❊❞✉❝❛t✐♦♥ ❚❛✐✇❛♥ ♥✐③❛t✐♦♥ ✐♥ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥s✱✧ ❏♦✉r♥❛❧ ▲t❞✱ ✷✵✵✺✳ ♦❢ ❙✉♣❡r❝♦♥❞✉❝t✐✈✐t② ❛♥❞ ◆♦✈❡❧ ▼❛❣♥❡t✐s♠✱ ✈♦❧✳ ✷✺✱ ♣♣✳ ✶✻✹✼✲✶✻✺✸✱ ✷✵✶✷✳ ❬✾❪ ❋❊◆●✱ ❨✳ ▲✳ ❛♥❞ ❑✳ ❙❍❊◆✳ ✧❈♦♥tr♦❧✲ ❧✐♥❣ ❝❤❛♦s ✐♥ ❘❈▲✲s❤✉♥t❡❞ ❏♦s❡♣❤s♦♥ ❥✉♥❝✲ t✐♦♥ ❜② ❞❡❧❛②❡❞ ❧✐♥❡❛r ❢❡❡❞❜❛❝❦✱✧ ❈❤✐♥❡s❡ P❤②s✐❝s ❇✱ ✈♦❧✳ ✶✼✱ ✷✵✵✽✳ ❬✶✵❪ ❳❯✱ ❙✳✱ ❨✳ ❚❆◆●✱ ❍✳ ❙❯◆✱ ❩✳ ❩❍❖❯ ❛♥❞ ❨✳ ❨❆◆●✳ ✧❈❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ❛♥t✐❝✐♣❛t✲ ✐♥❣ ❝❤❛♦t✐❝ s②♥❝❤r♦♥✐③❛t✐♦♥ ♦❢ ❘❈▲✲s❤✉♥t❡❞ ❏♦s❡♣❤s♦♥ ❥✉♥❝t✐♦♥s✱✧ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r✲ ♥❛❧ ♦❢ ◆♦♥✲▲✐♥❡❛r ▼❡❝❤❛♥✐❝s✱ ✈♦❧✳ ✹✼✱ ♣♣✳ ✶✶✷✹✲✶✶✸✶✱ ✷✵✶✷✳ ❬✶✶❪ ❈❍❊◆✱ ❉✳✲❨✳✱ ❲✳✲▲✳ ❩❍❆❖✱ ❳✳✲❨✳ ▼❆ ❛♥❞ ❘✳✲❋✳ ❩❍❆◆●✳ ✧❈♦♥tr♦❧ ❛♥❞ ❙②♥❝❤r♦✲ ♥✐③❛t✐♦♥ ♦❢ ❈❤❛♦s ✐♥ ❘❈▲✲❙❤✉♥t❡❞ ❏♦s❡♣❤✲ s♦♥ ❏✉♥❝t✐♦♥ ✇✐t❤ ◆♦✐s❡ ❉✐st✉r❜❛♥❝❡ ❯s✐♥❣ ❖♥❧② ❖♥❡ ❈♦♥tr♦❧❧❡r ❚❡r♠✱✧ ❆❜str❛❝t ❛♥❞ ❆♣♣❧✐❡❞ ❆♥❛❧②s✐s✱ ✈♦❧✳ ✷✵✶✷✱ ♣✳ ✶✹✱ ✷✵✶✷✳ ✽✻ ❆❜♦✉t ❆✉t❤♦rs r❡❝❡✐✈❡❞ ❛ P❤✳❉✳ ❞❡❣r❡❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❈♦♥tr♦❧ ❛♥❞ ❈♦♠♠✉♥✐❝❛✲ t✐♦♥ ❢r♦♠ ◆❛t✐♦♥❛❧ ❈❤❡♥❣ ❑✉♥❣ ❯♥✐✈❡rs✐t②✱ ❚❛✐✇❛♥✱ ♦♥ ❉❡❝❡♠❜❡r ✷✵✶✹✳ ❙✐♥❝❡ ✷✵✶✻✱ ❤❡ ❤❛s ❜❡❡♥ ❛ ❧❡❝t✉r❡r ❛t t❤❡ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❱✐❡t♥❛♠✳ ❍✐s ❝✉rr❡♥t r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ❛✉t♦♠❛t✐❝ ❝♦♥tr♦❧✱ ❡♠❜❡❞❞❡❞ s②st❡♠s ❛♥❞ r♦❜♦t✐❝s✳ ❚❛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✐♦♥ ✭❏❆❊❈✮ ... 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✐♦♥s ❛♥❞ ❡①t❡r♥❛❧ ❝✉rr❡♥t ❛s ❢♦❧❧♦✇s✿ x1 = x2 + u1 , x2 = [ix − g(x2 )x2 − sin(x1 ) − x3 ] + u2 , βC x3 = [x2 − x3 ] + u3 , βL ✭✷✮ ✇❤❡r❡ u1 ✱u2 ❛♥❞ u3 ❛r❡ ❝♦♥tr♦❧ s✐❣♥❛❧s✳... ♠❡♥s✐♦♥❧❡ss ❢♦r♠✱ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ♦❢ ❘❈▲❙❏ ✐s ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s ❬✺❪✿ z1 = z2 , [iz − g(z2 )z2 − sin (z1 − z3 )] , z2 = βC z3 = [z2 − z3 ] , βL ❋✐❣✳ ✶✿ ❈❤❛♦t✐❝ ♠♦t✐♦♥ ✐♥ ❏♦s❡♣❤s♦♥ ❏✉♥❝t✐♦♥✳ ✭✶✮
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Xem thêm: Adaptive MIMO controller design for Chaos synchronization in coupled josephson junctions via fuzzy neural networks, Adaptive MIMO controller design for Chaos synchronization in coupled josephson junctions via fuzzy neural networks

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