This paper investigates the design of decentralized controllers for a class of large scale switched nonlinear systems under arbitrary switching laws. A global large scale switched system can be split into a set of smaller interconnected switched Takagi-Sugeno fuzzy subsystems. In this context, to stabilize the overall closedloop system, a set of switched non-ParallelDistributed-Compensation (non-PDC) outputfeedback controllers is considered.
❱❖▲❯▼❊✿ ✷ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❉❡❝❡♥tr❛❧✐③❡❞ ❈♦♥tr♦❧❧❡r ❉❡s✐❣♥ ❢♦r ▲❛r❣❡ ❙❝❛❧❡ ❙✇✐t❝❤❡❞ ❚❛❦❛❣✐✲❙✉❣❡♥♦ ❙②st❡♠s ✇✐t❤ ❍∞ P❡r❢♦r♠❛♥❝❡ ❙♣❡❝✐❢✐❝❛t✐♦♥s 1 ❉❛❧❡❧ ❏❆❇❘■ ✱ ❉❥❛♠❡❧ ❊❞❞✐♥❡ ❈❤♦✉❛✐❜ ❇❊▲❑❍■❆❚ ✱ ❑❡✈✐♥ ●❯❊▲❚❖◆ ◆♦✉r❡❞❞✐♥❡ ▼❆◆❆▼❆◆◆■ 2,∗ ✱ ❋❡r❤❛t ❆❜❜❛s ❯♥✐✈❡rs✐t②✱ ❙❡t✐❢ ✶✱ ❙❡t✐❢✱ ❆❧❣❡r✐❛ ❯♥✐✈❡rs✐t② ♦❢ ❘❡✐♠s ❈❤❛♠♣❛❣♥❡✲❆r❞❡♥♥❡✱ ▼♦✉❧✐♥ ❞❡ ❧❛ ❍♦✉ss❡ ❇P✶✵✸✾✱ ✺✶✻✽✼ ❘❡✐♠s✱ ❋r❛♥❝❡ ✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❑✳ ●❯❊▲❚❖◆ ✭❡♠❛✐❧✿ ❦❡✈✐♥✳❣✉❡❧t♦♥❅✉♥✐✈✲r❡✐♠s✳❢r✮ ✭❘❡❝❡✐✈❡❞✿ ✵✾✲▼❛②✲✷✵✶✽❀ ❛❝❝❡♣t❡❞✿ ✷✶✲❏✉♥❡✲✷✵✶✽❀ ♣✉❜❧✐s❤❡❞✿ ✷✵✲❏✉❧②✲✷✵✶✽✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✽✷✷✳✶✽✼ ❚❤✐s ♣❛♣❡r ✐♥✈❡st✐❣❛t❡s t❤❡ ❞❡s✐❣♥ ♦❢ ❞❡❝❡♥tr❛❧✐③❡❞ ❝♦♥tr♦❧❧❡rs ❢♦r ❛ ❝❧❛ss ♦❢ ❧❛r❣❡ s❝❛❧❡ s✇✐t❝❤❡❞ ♥♦♥❧✐♥❡❛r s②st❡♠s ✉♥❞❡r ❛r❜✐tr❛r② s✇✐t❝❤✐♥❣ ❧❛✇s✳ ❆ ❣❧♦❜❛❧ ❧❛r❣❡ s❝❛❧❡ s✇✐t❝❤❡❞ s②s✲ t❡♠ ❝❛♥ ❜❡ s♣❧✐t ✐♥t♦ ❛ s❡t ♦❢ s♠❛❧❧❡r ✐♥t❡r❝♦♥✲ ♥❡❝t❡❞ s✇✐t❝❤❡❞ ❚❛❦❛❣✐✲❙✉❣❡♥♦ ❢✉③③② s✉❜s②st❡♠s✳ ■♥ t❤✐s ❝♦♥t❡①t✱ t♦ st❛❜✐❧✐③❡ t❤❡ ♦✈❡r❛❧❧ ❝❧♦s❡❞✲ ❧♦♦♣ s②st❡♠✱ ❛ s❡t ♦❢ s✇✐t❝❤❡❞ ♥♦♥✲P❛r❛❧❧❡❧✲ ❉✐str✐❜✉t❡❞✲❈♦♠♣❡♥s❛t✐♦♥ ✭♥♦♥✲P❉❈✮ ♦✉t♣✉t✲ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧❧❡rs ✐s ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ ❧❛tt❡r ✐s ❞❡s✐❣♥❡❞ ❜❛s❡❞ ♦♥ ▲✐♥❡❛r ▼❛tr✐① ■♥❡q✉❛❧✐✲ t✐❡s ✭▲▼■✮ ❝♦♥❞✐t✐♦♥s ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ ♠✉❧t✐♣❧❡ s✇✐t❝❤❡❞ ♥♦♥✲q✉❛❞r❛t✐❝ ▲②❛♣✉♥♦✈✲❧✐❦❡ ❝❛♥❞✐❞❛t❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❝♦♥tr♦❧❧❡rs ♣r♦♣♦s❡❞ ❤❡r❡✐♥ ❛r❡ s②♥t❤❡s✐③❡❞ t♦ s❛t✐s❢② ❍∞ ♣❡r❢♦r♠❛♥❝❡s ❢♦r ❞✐s✲ t✉r❜❛♥❝❡ ❛tt❡♥✉❛t✐♦♥✳ ❋✐♥❛❧❧②✱ ❛ ♥✉♠❡r✐❝❛❧ ❡①✲ ❛♠♣❧❡ ✐s ♣r♦♣♦s❡❞ t♦ ✐❧❧✉str❛t❡ t❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ s✉❣❣❡st❡❞ ❞❡❝❡♥tr❛❧✐③❡❞ s✇✐t❝❤❡❞ ❝♦♥tr♦❧❧❡r ❞❡s✐❣♥ ❛♣♣r♦❛❝❤✳ ❆❜str❛❝t✳ ❑❡②✇♦r❞s ▲❛r❣❡ ❙❝❛❧❡ ❙✇✐t❝❤❡❞ ❋✉③③② ❙②st❡♠✱ ❉❡✲ ❝❡♥tr❛❧✐③❡❞ ♥♦♥✲P❉❈ ❈♦♥tr♦❧❧❡rs✱ ❆r❜✐✲ tr❛r② ❙✇✐t❝❤✐♥❣ ▲❛✇s✳ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ❉✉r✐♥❣ t❤❡ ❧❛st ❢❡✇ ❞❡❝❛❞❡s✱ s❡✈❡r❛❧ ❝♦♠♣❧❡① s②st❡♠s ❛♣♣❡❛r❡❞ t♦ ♠❡❡t t❤❡ s♣❡❝✐✜❝ ♥❡❡❞s ♦❢ t❤❡ ✇♦r❧❞ ♣♦♣✉❧❛t✐♦♥✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ✇❡ ❝❛♥ q✉♦t❡ ❛s ❡①❛♠♣❧❡s ♥❡t✇♦r❦❡❞ ♣♦✇❡r s②st❡♠s✱ ✇❛t❡r tr❛♥s♣♦rt❛t✐♦♥ ♥❡t✇♦r❦s✱ tr❛✣❝ s②st❡♠s✱ ❛s ✇❡❧❧ ❛s ♦t❤❡r s②st❡♠s ✐♥ ✈❛r✐♦✉s ✜❡❧❞s✳ ●❡♥✲ ❡r❛❧❧② s♣❡❛❦✐♥❣✱ ❡st❛❜❧✐s❤ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❢♦r ❧❛r❣❡ s❝❛❧❡ s②st❡♠s ✐s ❛ ❝♦♠♣❧❡① t❛s❦✱ ❡s♣❡✲ ❝✐❛❧❧② ✇❤❡♥ t❤❡ s②st❡♠ ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛ ✇❤♦❧❡✳ ❍❡♥❝❡✱ t♦ ♦✈❡r❝♦♠❡ t❤❡s❡ ❞✐✣❝✉❧t✐❡s✱ ❛♥ ❛❧t❡r♥❛✲ t✐✈❡ t♦ t❤❡ ❣❧♦❜❛❧ ♠♦❞❡❧❧✐♥❣ ❛♣♣r♦❛❝❤ ❤❛s ❜❡❡♥ ❡①♣❧♦r❡❞✳ ■t ❝♦♥s✐sts ✐♥ ❞❡❝♦♠♣♦s✐♥❣ t❤❡ ♦✈❡r✲ ❛❧❧ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠ ✐♥ ❛ ✜♥✐t❡ s❡t ♦❢ ✐♥t❡r❝♦♥✲ ♥❡❝t❡❞ ❧♦✇✲♦r❞❡r s✉❜s②st❡♠s ❬✶❪✳ ❆♠♦♥❣ t❤❡s❡ ❝♦♠♣❧❡① s②st❡♠s✱ s✇✐t❝❤❡❞ ✐♥✲ t❡r❝♦♥♥❡❝t❡❞ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠ ❤❛✈❡ ❛ttr❛❝t❡❞ ❝♦♥s✐❞❡r❛❜❧❡ ❛tt❡♥t✐♦♥ s✐♥❝❡ t❤❡② ♣r♦✈✐❞❡ ❛ ❝♦♥✲ ✈❡♥✐❡♥t ♠♦❞❡❧❧✐♥❣ ❛♣♣r♦❛❝❤ ❢♦r ♠❛♥② ♣❤②s✐❝❛❧ s②st❡♠s t❤❛t ❝❛♥ ❡①❤✐❜✐t ❜♦t❤ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡ ❞②♥❛♠✐❝ ❜❡❤❛✈✐♦r✳ ■♥ t❤✐s ❝♦♥t❡①t✱ s❡✈✲ ❡r❛❧ st✉❞✐❡s ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ st❛❜✐❧✐t② ❛♥❛❧②s✐s ❛♥❞ st❛❜✐❧✐③❛t✐♦♥ ✐ss✉❡s ❢♦r ❜♦t❤ ❧✐♥❡❛r ❛♥❞ ♥♦♥✲ ❧✐♥❡❛r s✇✐t❝❤❡❞ ✐♥t❡r❝♦♥♥❡❝t❡❞ ❧❛r❣❡✲s❝❛❧❡ s②s✲ t❡♠s ❤❛✈❡ ❜❡❡♥ ❡①♣❧♦r❡❞ ❬✶❪✲❬✽❪✳ ❍❡♥❝❡✱ t❤❡ ♠❛✐♥ ❝❤❛❧❧❡♥❣❡ t♦ ❞❡❛❧ ✇✐t❤ s✉❝❤ ♣r♦❜❧❡♠s ❝♦♥✲ s✐sts ✐♥ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ✇❤♦❧❡ s②st❡♠s ✇✐t❤ ❝♦♥s✐❞❡r❛✲ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✸✾ ❱❖▲❯▼❊✿ ✷ t✐♦♥ t♦ t❤❡ ✐♥t❡r❝♦♥♥❡❝t✐♦♥s ❡✛❡❝ts ❜❡t✇❡❡♥ ✐ts s✉❜s②st❡♠s✳ ◆❡✈❡rt❤❡❧❡ss✱ ❢❡✇ ✇♦r❦s ❜❛s❡❞ ♦♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦♣❡rt② ♦❢ ❚❛❦❛❣✐✲❙✉❣❡♥♦ ✭❚❙✮ ❢✉③③② ♠♦❞❡❧s ❢♦r ♥♦♥❧✐♥❡❛r ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ ❛❝❤✐❡✈❡❞ t♦ ❞❡❛❧ ✇✐t❤ t❤❡ st❛❜✐❧✐③❛t✐♦♥ ♦❢ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❧❛r❣❡✲s❝❛❧❡ s✇✐t❝❤❡❞ ♥♦♥❧✐♥❡❛r s②st❡♠s ❬✸❪✱ ❬✽❪✲❬✶✷❪✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ t❤❛♥ r♦❜✉st ❝♦♥tr♦❧ ❛♣♣r♦❛❝❤❡s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❛ ❧✐♠✐t❛t✐♦♥ ❢♦r s❡✈❡r❛❧ ❡♠❜❡❞❞❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts t❤❡ ❞❡s✐❣♥ ♦❢ ❞❡❝❡♥tr❛❧✲ ✐③❡❞ r♦❜✉st ❝♦♥tr♦❧❧❡rs ❢♦r ❛ ❝❧❛ss ♦❢ s✇✐t❝❤❡❞ ❚❙ ✐♥t❡r❝♦♥♥❡❝t❡❞ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠s ✇✐t❤ ❡①✲ t❡r♥❛❧ ❜♦✉♥❞❡❞ ❞✐st✉r❜❛♥❝❡s✳ ▼♦r❡ s♣❡❝✐✜✲ ❝❛❧❧②✱ t❤❡ ♣r✐♠❛r② ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤✐s ♣❛♣❡r ❝♦♥s✐sts ✐♥ ♣r♦♣♦s✐♥❣ ❛ ▲▼■ ❜❛s❡❞ ♠❡t❤♦❞♦❧✲ ♦❣②✱ ✐♥ t❤❡ ♥♦♥✲q✉❛❞r❛t✐❝ ❢r❛♠❡✇♦r❦✱ ❢♦r t❤❡ ❞❡s✐❣♥ ♦❢ r♦❜✉st ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ ❞❡❝❡♥tr❛❧✐③❡❞ s✇✐t❝❤❡❞ ♥♦♥✲P❉❈ ❝♦♥tr♦❧❧❡rs ❢♦r ❛ ❝❧❛ss ♦❢ ❧❛r❣❡ s❝❛❧❡ s✇✐t❝❤❡❞ ♥♦♥❧✐♥❡❛r s②st❡♠s ✉♥❞❡r ❛r❜✐tr❛r② s✇✐t❝❤✐♥❣ ❧❛✇s✳ ▼♦r❡♦✈❡r✱ t♦ ❞❡❛❧s ✇✐t❤ ❡①t❡r✲ ♥❛❧ ❞✐st✉r❜❛♥❝❡s ❛♣♣❧✐❡❞ ♦♥ t❤❡ ✐♥t❡r❝♦♥♥❡❝t❡❞ ♥♦♥❧✐♥❡❛r s✉❜s②st❡♠s✱ ❛♥ ❝r✐t❡r✐♦♥ ✐s ❝♦♥s✐❞❡r❡❞✳ ■t ❛✐♠s ❛t ❞❡s✐❣♥✐♥❣ ❛ r♦❜✉st ❝♦♥tr♦❧❧❡r✱ ✇❤✐❝❤ ❛tt❡♥✉❛t❡s t❤❡ ❡✛❡❝ts ♦❢ t❤❡ ❞✐st✉r❜❛♥❝❡s✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✈✐❡✇ ❛s ❡①♦❣❡♥♦✉s ✉♥❝♦♥tr♦❧❧❡❞ ✐♥♣✉ts✱ ♦♥ t❤❡ ♦✈❡r❛❧❧ ❝❧♦s❡❞✲❧♦♦♣ ❞②♥❛♠✐❝s✳ ❚❤❡ ♠❛✐♥ ✐♥t❡r❡st ♦❢ ❚✲❙ ♠♦❞❡❧s ✐s t❤❡✐r ❛❜✐❧✲ ✐t② t♦ ❛❝❝✉r❛t❡❧② r❡♣r❡s❡♥t ❛ ♥♦♥❧✐♥❡❛r s②st❡♠ ❛s ✇❡❧❧ ❛s ❛❧❧♦✇✐♥❣ t♦ ❡①t❡♥❞ s♦♠❡ ♦❢ ❧✐♥❡❛r ❝♦♥tr♦❧ ❝♦♥❝❡♣ts t♦ ♥♦♥❧✐♥❡❛r s②st❡♠s✳ ❚♦ st❛❜✐❧✐③❡ ❚✲❙ ♠♦❞❡❧s✱ t❤❡ P❛r❛❧❧❡❧ ❉✐str✐❜✉t❡❞ ❈♦♠♣❡♥s❛t✐♦♥ ✭P❉❈✮ ❝♦♥tr♦❧ s❝❤❡♠❡ ✐s ♦❢t❡♥ ❝♦♥s✐❞❡r❡❞✳ ❚❤❡ ❜❛s✐❝ ♣❤✐❧♦s♦♣❤② ♦❢ s✉❝❤ ❝♦♥tr♦❧ s❝❤❡♠❡ ✐s t♦ ❞❡✲ s✐❣♥ ❛ ❝♦♥tr♦❧❧❡r s❤❛r✐♥❣ t❤❡ s❛♠❡ ❢✉③③② ♠❡♠✲ ❜❡rs❤✐♣ ❢✉♥❝t✐♦♥s str✉❝t✉r❡ ❛s t❤❡ ❚✲❙ ♠♦❞❡❧ t♦ ❜❡ ❝♦♥tr♦❧❧❡❞✳ ▼♦r❡♦✈❡r✱ t♦ r❡❞✉❝❡ t❤❡ ❝♦♥✲ s❡r✈❛t✐s♠ ♦❢ t❤❡ ❞❡s✐❣♥ ❝♦♥❞✐t✐♦♥s✱ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ P❉❈ ❝♦♥trr♦❧❡rs✱ ❝❛❧❧❡❞ ♥♦♥✲P❉❈ ❝♦♥tr♦❧❧❡rs✱ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ✇✐t❤ ♥♦♥✲q✉❛❞r❛t✐❝ ▲②❛♣✉♥♦✈ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ♣❛♣❡r ✐s ♦r❣❛♥✐③❡❞ ❛s ❢✉♥❝t✐♦♥s✱ ♦r ❡①t❡♥❞❡❞ q✉❛❞r❛t✐❝ ♦♥❡s ✭s❡❡ ❡✳❣✳ ❢♦❧❧♦✇s✳ ❙❡❝t✐♦♥ ✷ ♣r❡s❡♥ts t❤❡ ❝♦♥s✐❞❡r❡❞ ❝❧❛ss ❬✶✸✱ ✶✹❪ ❛♥❞ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✮✳ ♦❢ s✇✐t❝❤❡❞ ❚❙ ✐♥t❡r❝♦♥♥❡❝t❡❞ ❧❛r❣❡✲s❝❛❧❡ s②s✲ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❚✲❙ ❢✉③③② s✇✐t❝❤❡❞ ❧❛r❣❡✲ t❡♠✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ♣r♦❜❧❡♠ st❛t❡♠❡♥t✳ ❚❤❡ s❝❛❧❡ s②st❡♠s✱ ❛♥ ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ ❞❡❝❡♥tr❛❧✐③❡❞ ❞❡s✐❣♥ ♦❢ t❤❡ ❞❡❝❡♥tr❛❧✐③❡❞ s✇✐t❝❤❡❞ ♥♦♥✲P❉❈ P❉❈ ❝♦♥tr♦❧❧❡r ❤❛s ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ✐♥ ❬✾❪✳ ■♥ ❝♦♥tr♦❧❧❡rs ✐s ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥ ✸✳ ❆ ♥✉♠❡r✐❝❛❧ t❤❡ s❛♠❡ ✇❛②✱ t❤❡ ❛✉t❤♦rs ♦❢ ❬✶✵❪ ❤❛✈❡ st✉❞✐❡❞ ❡①❛♠♣❧❡ ✐s ♣r♦♣♦s❡❞ t♦ ✐❧❧✉str❛t❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ❞❡s✐❣♥ ♦❢ ❛♥ ❛❞❛♣t✐✈❡ ❢✉③③② ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ t❤❡ ♣r♦♣♦s❡❞ ❛♣♣r♦❛❝❤ ✐♥ s❡❝t✐♦♥ ✹✳ ❚❤❡ ♣❛♣❡r ❝♦♥tr♦❧ ❢♦r ❛ ❝❧❛ss ♦❢ s✇✐t❝❤❡❞ ✉♥❝❡rt❛✐♥ ♥♦♥❧✐♥✲ ❡♥❞s ✇✐t❤ ❝♦♥❝❧✉s✐♦♥s ❛♥❞ r❡❢❡r❡♥❝❡s✳ ❡❛r ❧❛r❣❡✲s❝❛❧❡ s②st❡♠s ✇✐t❤ ✉♥❦♥♦✇♥ ❞❡❛❞ ③♦♥❡s ❛♥❞ ✐♠♠❡❛s✉r❛❜❧❡ st❛t❡s✳ ❘❡❝❡♥t❧②✱ ❛♥ ♦❜s❡r✈❡r✲ ❜❛s❡❞ ❞❡❝❡♥tr❛❧✐③❡❞ ❝♦♥tr♦❧ s❝❤❡♠❡ ✇❛s ❞❡✈❡❧✲ ♦♣❡❞ ✐♥ ❬✶✶❪ ❢♦r ❛ ❝❧❛ss s✇✐t❝❤❡❞ ♥♦♥✲❧✐♥❡❛r ❧❛r❣❡✲ s❝❛❧❡ s②st❡♠s✳ ■♥ t❤❡ s❛♠❡ ❝♦♥t❡①t✱ ❛♥ ❛❞❛♣✲ t✐✈❡ ❢✉③③② ❞❡❝❡♥tr❛❧✐③❡❞ ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ tr❛❝❦✲ ✐♥❣ ❝♦♥tr♦❧ ❤❛s ❜❡❡♥ ❡①♣❧♦r❡❞ ✐♥ ❬✶✷❪ ❢♦r ❛ ❝❧❛ss ♦❢ s✇✐t❝❤❡❞ ♥♦♥❧✐♥❡❛r ❧❛r❣❡✲s❝❛❧❡ s②st❡♠s ✉♥✲ ❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠ ✇❛s ❝♦♠♣♦s❡❞ ♦❢ s✉❜s②st❡♠s ✐♥t❡r❝♦♥♥❡❝t❡❞ ❜② t❤❡✐r ♦✉t♣✉ts✳ ■♥ t❤✐s st✉❞②✱ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ✷✳ P❘❖❇▲❊▼ ✇❤♦❧❡ ❝❧♦s❡❞✲❧♦♦♣ s②st❡♠ ❛♥❞ t❤❡ tr❛❝❦✐♥❣ ♣❡r✲ ❙❚❆❚❊▼❊◆❚ ❆◆❉ ❢♦r♠❛♥❝❡ ✇❡r❡ ❛❝❤✐❡✈❡❞ ❜② ✉s✐♥❣ t❤❡ ▲②❛♣✉♥♦✈ ❢✉♥❝t✐♦♥ ❛♥❞ ✉♥❞❡r ❝♦♥str❛✐♥❡❞ s✇✐t❝❤✐♥❣ s✐❣✲ P❘❊▲■▼■❆❘■❊❙ ♥❛❧s ✇✐t❤ ❞✇❡❧❧ t✐♠❡✳ ❍♦✇❡✈❡r✱ s✉❝❤ ❛♣♣r♦❛❝❤❡s ♠❛② ❜❡ r❡str✐❝t✐✈❡ s✐♥❝❡ t❤❡② ❛r❡ ✉♥s✉✐t❛❜❧❡ ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✐✳❡✳ ✇❤❡♥ t❤❡ s✇✐t❝❤✐♥❣ s❡✲ q✉❡♥❝❡s ❛r❡ ❛r❜✐tr❛r② ♦r ✉♥❦♥♦✇♥✳ ▼♦r❡♦✈❡r✱ ▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❝❧❛ss ♦❢ ♥♦♥❧✐♥❡❛r ❤②❜r✐❞ s②s✲ ♥♦t❡ t❤❛t ❛❞❛♣t✐✈❡ ❝♦♥tr♦❧ ❛♣♣r♦❛❝❤❡s ❛r❡ ❜❛s❡❞ t❡♠s S ❝♦♠♣♦s❡❞ ♦❢ n ❝♦♥t✐♥✉♦✉s t✐♠❡ s✇✐t❝❤❡❞ ♦♥❡ ♣❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥s✳ ❚❤❡r❡❢♦r❡ t❤❡② ♦❢t❡♥ ♥♦♥❧✐♥❡❛r s✉❜s②st❡♠ Si r❡♣r❡s❡♥t❡❞ ❜② s✇✐t❝❤❡❞ r❡q✉✐r❡ ♠♦r❡ ♦♥❧✐♥❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝❛♣❛❜✐❧✐t✐❡s ❚❙ ♠♦❞❡❧s✳ ❚❤❡ n st❛t❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ✇❤♦❧❡ ✐♥t❡r❝♦♥♥❡❝t❡❞ s✇✐t❝❤❡❞ ❢✉③③② s②st❡♠ S ✶✹✵ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ ❛r❡ ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s❀ ❢♦r i = 1, 2, , n✿ rji x˙ i (t) = ji =1 sji =1 n α=1,α=i wα Fi,α,hj xα (t) + Bhj wα (t) hsji hkji Ysji ,kji sji =1 kji =1 mi yi (t) = rji Yhj ,hj = ξji (t) hsji (zji (t)) w Ahj xi (t) + Bhj ui (t) + Bhj wi (t) · + ❛♥❞ rji mi | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ▼♦r❡♦✈❡r✱ ❢♦r ♠❛tr✐❝❡s ♦❢ ❛♣♣r♦♣r✐❛t❡ ❞✐♠❡♥✲ dXhj i ❛♥❞ s✐♦♥s ✇❡ ✇✐❧❧ ❞❡♥♦t❡✿ X˙ hj = −1 ξji (t) Chj xi (t) d Xhj dt −1 i X˙ hj = ✳ ❆s ✉s✉❛❧✱ ❛ st❛r (∗ ) dt ✭✶✮ ✐♥❞✐❝❛t❡s ❛ tr❛♥s♣♦s❡ q✉❛♥t✐t② ✐♥ ❛ s②♠♠❡tr✐❝ ♠❛tr✐① ❛♥❞ sym (G) = G + GT ✳ ❚❤❡ t✐♠❡ t ✇✐❧❧ ❜❡ ♦♠✐tt❡❞ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t②✳ ❍♦✇✲ ✇❤❡r❡ xi (t) ∈ Rηi ✱ yi (t) ∈ Rρi ✱ ui (t) ∈ Rυi r❡♣✲ ❡✈❡r✱ ♦♥❡ ❞❡♥♦t❡s tj→j + t❤❡ s✇✐t❝❤✐♥❣ ✐♥st❛♥ts r❡s❡♥t r❡s♣❡❝t✐✈❡❧② t❤❡ st❛t❡✱ t❤❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤❡ ith s✉❜s②st❡♠ ❜❡t✇❡❡♥ t❤❡ ❝✉rr❡♥t ♠♦❞❡ ✭♦✉t♣✉t✮ ❛♥❞ t❤❡ ✐♥♣✉t ✈❡❝t♦rs ❛ss♦❝✐❛t❡❞ t♦ t❤❡ j ✭❛t t✐♠❡ t✮ ❛♥❞ t❤❡ ✉♣❝♦♠✐♥❣ ♠♦❞❡ j + ✭❛t t✐♠❡ it❤ s✉❜s②st❡♠✳ wi (t) ∈ Rυi ✐s ❛♥ ✉♥❝♦♥tr♦❧❧❛❜❧❡ + t ✮✱ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡✿ t✐♠❡✲✈❛r②✐♥❣ L2 ✲♥♦r♠ ❜♦✉♥❞❡❞ ❡①t❡r♥❛❧ ❞✐st✉r✲ t❤ ❜❛♥❝❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ i s✉❜s②st❡♠✳ mi ✐s ξj (t) = ξj (t+ ) = ❛♥❞ ✭✸✮ t❤❡ ♥✉♠❜❡r ♦❢ s✇✐t❝❤✐♥❣ ♠♦❞❡s ♦❢ t❤❡ it❤ s✉❜✲ ξj + (t) = ξj + (t+ ) = s②st❡♠✳ rji ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❢✉③③② r✉❧❡s ❛ss♦❝✐✲ ❛t❡❞ t♦ t❤❡ it❤ s✉❜s②st❡♠ ✐♥ t❤❡ jith ♠♦❞❡❀ ❢♦r ■♥ t❤❡ s❡q✉❡❧✱ ✇❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ t❤❡ r♦❜✉st i = 1, , n✱ ji = 1, , mi ❛♥❞ sji = 1, , rji ✱ ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ ❞✐st✉r❜❛♥❝❡ ❛tt❡♥✉❛t✐♦♥ ❢♦r t❤❡ Asji ∈ Rηi ×ηi ✱ Bsji ∈ Rηi ×υi ✱ Bswj ∈ Rηi ×υi ❝♦♥s✐❞❡r❡❞ ❝❧❛ss ♦❢ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠ S ✳ ❋♦r i ❛♥❞ Clji ∈ Rρi ×ηi ❛r❡ ❝♦♥st❛♥t ♠❛tr✐❝❡s ❞❡✲ t❤❛t ♣✉r♣♦s❡✱ ❛ s❡t ♦❢ ❞❡❝❡♥tr❛❧✐③❡❞ ♦✉t♣✉t✲ s❝r✐❜✐♥❣ t❤❡ ❧♦❝❛❧ ❞②♥❛♠✐❝s ♦❢ ❡❛❝❤ ♣♦❧②t♦♣❡s❀ ❢❡❡❞❜❛❝❦ s✇✐t❝❤❡❞ ♥♦♥✲P❉❈ ❝♦♥tr♦❧ ❧❛✇s ✐s ♣r♦✲ Bswjα ∈ Rηi ×υα ❛♥❞ Fi,α,sji ∈ Rηi ×ηα ❡①♣r❡ss t❤❡ ♣♦s❡❞ ❛s❀ ❢♦r i = 1, , n✿ i ✐♥t❡r❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ s✉❜s②st❡♠s✳ zji (t) ❛r❡ mi t❤❡ ♣r❡♠✐s❡s ✈❛r✐❛❜❧❡s ❛♥❞ hsji (zji (t)) ❛r❡ ♣♦s✐✲ −1 ξji (t) Khj Xhj yi (t) ✭✹✮ ui (t) = t✐✈❡ ♠❡♠❜❡rs❤✐♣ ❢✉♥❝t✐♦♥s s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥✈❡① j =1 ji =1 s✉♠ ♣r♦♣r✐❡t✐❡s i rji sji =1 hsji (zji (t)) = 1❀ ξji (t) ✐s ✇❤❡r❡ t❤❡ ♠❛tr✐❝❡s t❤❡ s✇✐t❝❤✐♥❣ r✉❧❡s ♦❢ t❤❡ i s✉❜s②st❡♠✱ ❝♦♥✲ s✐❞❡r❡❞ ❛r❜✐tr❛r② ❜✉t ❛ss✉♠❡❞ t♦ ❜❡ r❡❛❧ t✐♠❡ ❛✈❛✐❧❛❜❧❡✳ ❚❤❡s❡ ❛r❡ ❞❡✜♥❡❞ s✉❝❤ t❤❛t t❤❡ ❛❝✲ t✐✈❡ s②st❡♠ ✐♥ t❤❡ lith ♠♦❞❡ ❧❡❛❞ t♦✿ th ξji (t) = ✐❢ ji = li ξji (t) = ✐❢ ji = li ◆♦t❛t✐♦♥s✿ Ghj = hsji Gsji sji =1 hsji (zji (t)) Kkji , Khj = kji =1 rji Xhj = ✭✷✮ ■♥ ♦r❞❡r t♦ ❧✐❣❤t❡♥ t❤❡ ♠❛t❤❡♠❛t✐✲ , ❝❛❧ ❡①♣r❡ss✐♦♥✱ ♦♥❡ ❛ss✉♠❡s t❤❡ s❝❛❧❛r N = n−1 th t❤❡ ✐♥❞❡① i ❛ss♦❝✐❛t❡❞ t♦ t❤❡ i s✉❜s②st❡♠ t♦ ❞❡✲ ♥♦t❡ t❤❡ ♠♦❞❡ ji ✳ ❚❤❡ ♣r❡♠✐s❡ ❡♥tr✐❡s zji ✇✐❧❧ ❜❡ ♦♠✐tt❡❞ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t✐❡s ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ✐s ❡♠♣❧♦②❡❞ ❢♦r ❢✉③③② ♠❛tr✐✲ ❝❡s✿ rji rji hsji (zji (t))Xs9j i sji =1 ❛r❡ t❤❡ ❢✉③③② ❣❛✐♥s t♦ ❜❡ s②♥t❤❡s✐③❡❞ ✇✐t❤ Xs9j = Xs9j i T i > 0✳ ❘❡♠❛r❦ ✶✿ ❲❤❡♥ ❛ ❧❛r❣❡ s❝❛❧❡ s②st❡♠ ✐s ❝♦♥s✐❞❡r❡❞ ❛s ❛ ✇❤♦❧❡✱ ✐✳❡✳ ❛ ❤✐❣❤✲♦r❞❡r s②s✲ t❡♠✱ t❤❡ s✐③❡ ♦❢ t❤❡ ❞❡❝✐s✐♦♥ ♠❛tr✐❝❡s ✭❝♦♥tr♦❧ ❣❛✐♥s✱ ▲②❛♣✉♥♦✈ ♠❛tr✐❝❡s✳ ✳ ✳ ✮ ✐♥ t❤❡ ▲▼■ ❝♦♥❞✐✲ t✐♦♥s ✐♥❝r❡❛s❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st t♦ ❝❤❡❝❦ ✇❤❡t❤❡r ❛ s♦❧✉t✐♦♥ ❡①✐sts✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❛✈❛✐❧✲ ❛❜❧❡ ❝♦♥✈❡① ♦♣t✐♠✐s❛t✐♦♥ t♦♦❧s ♠❛② ❢❛✐❧ t♦ ✜♥❞ ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ▲▼■ ♣r♦❜❧❡♠ ✭✉♥❢❡❛s✐❜✐❧✐t② ♦r ❝♦♠♣✉t❛t✐♦♥❛❧ ❝r❛s❤❡s✮✳ ❚❤✐s ✐s ♠❛✐♥❧② ✇❤② t❤❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✹✶ ❱❖▲❯▼❊✿ ✷ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠s ✐♥t♦ ❧♦✇❡r✲ ♦r❞❡r ✐♥t❡r❝♦♥♥❡❝t❡❞ s✉❜s②st❡♠s ❝❛♥ ❜❡ ❝♦♥s✐❞✲ ❡r❡❞ ❛s ❛ ❣♦♦❞ ❛❧t❡r♥❛t✐✈❡✳ ■♥❞❡❡❞✱ ✐♥ t❤✐s ❝❛s❡✱ ❞❡❝❡♥tr❛❧✐③❡❞ ❝♦♥tr♦❧❧❡rs ❞❡s✐❣♥ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❡❛❝❤ ❧♦✇❡r✲♦r❞❡r s✉❜s②st❡♠✱ ✐✳❡✳ ✇✐t❤ ❧♦✇❡r✲ s✐③❡❞ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ❛♥❞ ▲▼■s✱ ❤❡❧♣✐♥❣ t♦ r❡✲ ❞✉❝❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✇♦r❦❧♦❛❞ ♦❢ t❤❡ ❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s✳ ❙✉❜st✐t✉t✐♥❣ ✭✹✮ ✐♥t♦ ✭✶✮✱ ♦♥❡ ❡①♣r❡ss❡s t❤❡ ♦✈❡r❛❧❧ ❝❧♦s❡❞✲❧♦♦♣ ❞②♥❛♠✐❝s Scl ❛s✱ ❢♦r i = 1, , n✿ −1 mi A + B K X C x hj hj hj hj i hj x˙ i = ξj n + Fi,α,hj xα j=1 α=1,α=i ❣❛✐♥ ❝♦♥tr♦❧❧❡rs Khj ❛♥❞ t❤❡ s②st❡♠✬s ♠❛tr✐❝❡s −1 Bhj Khj (Xhj ) Chj ❛r❡ ♣r❡s❡♥t✳ ❍❡♥❝❡✱ ✐♥ ✈✐❡✇ ♦❢ t❤❡ ✇❡❛❧t❤ ♦❢ ✐♥t❡r❝♦♥♥❡❝t✐♦♥s ❝❤❛r❛❝✲ t❡r✐③✐♥❣ ♦✉r s②st❡♠✱ t❤❡s❡ ❝r♦ss✐♥❣ t❡r♠s ❧❡❛❞ s✉r❡❧② t♦ ✈❡r② ❝♦♥s❡r✈❛t✐✈❡ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❞❡✲ s✐❣♥ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❝♦♥tr♦❧❧❡r✳ ■♥ ♦r❞❡r t♦ ❞❡✲ −1 ❝♦✉♣❧❡ t❤❡ ❝r♦ss✐♥❣ t❡r♠s Bhj Khj (Xhj ) Chj ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✭✺✮✱ ❛♥❞ t♦ ♣r♦✈✐❞❡ ▲▼■ ❝♦♥❞✐t✐♦♥s✱ ✇❡ ✉s❡ ❛♥ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt② ❝❛❧❧❡❞ t❤❡ ❞❡s❝r✐♣t♦r r❡❞✉♥❞❛♥❝② ❬✶✸❪✲❬✶✻❪✳ ■♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ ❝❧♦s❡❞✲❧♦♦♣ ❞②♥❛♠✐❝s ✭✺✮ ❝❛♥ ❜❡ ❛❧t❡r♥❛t✐✈❡❧② ❡①♣r❡ss❡❞ ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ ❢r♦♠ t❤❡ ♦✉t♣✉t ❡q✉❛t✐♦♥ ♦❢ ✭✶✮ ❛♥❞ t❤❡ ❝♦♥✲ tr♦❧ ❧❛✇ ✭✹✮✱ ✇❡ ✐♥tr♦❞✉❝❡ ♥✉❧❧ t❡r♠s s✉❝❤ t❤❛t✱ ❢♦r i = 1, , n✿ ✭✺✮ ❚❤✉s✱ t❤❡ ♣r♦❜❧❡♠ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s st✉❞② ❝❛♥ ❜❡ r❡s✉♠❡❞ ❛s ❢♦❧❧♦✇s✿ | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ 0y˙ i = −yi + Chj xi ✭✼✮ ❛♥❞✿ Pr♦❜❧❡♠ ✶✿ −1 ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❞❡s✐❣♥ t❤❡ = ui − Khj Xhj yi ✭✽✮ ❝♦♥tr♦❧❧❡rs ✭✹✮ s✉❝❤ t❤❛t t❤❡ ❝❧♦s❡❞✲❧♦♦♣ ✐♥t❡r✲ ❝♦♥♥❡❝t❡❞ ❧❛r❣❡✲s❝❛❧❡ s✇✐t❝❤❡❞ ❚❙ s②st❡♠ ✭✺✮ ❚❤❡♥✱ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❛✉❣♠❡♥t❡❞ st❛t❡ ✈❡❝✲ t♦rs x ˜Ti = xTi yiT uTi , x ˜Tα = xTα yαT uTα ❛♥❞ s❛t✐s✜❡s ❛ r♦❜✉st ❍∞ ♣❡r❢♦r♠❛♥❝❡✳ T w ˜i,α = wiT wαT ✱ t❤❡ ❝❧♦s❡❞✲❧♦♦♣ ❉❡✜♥✐t✐♦♥ ✶✿ ❚❤❡ s✇✐t❝❤❡❞ ✐♥t❡r❝♦♥♥❡❝t❡❞ ❞✐st✉r❜❛♥❝❡s ❞②♥❛♠✐❝s ♦❢ t❤❡ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠ ✭✶✮ ✉♥❞❡r t❤❡ ❧❛r❣❡✲s❝❛❧❡ s②st❡♠ ✭✶✮ ✐s s❛✐❞ t♦ ❤❛✈❡ ❛ r♦❜✉st ♥♦♥✲P❉❈ ❝♦♥tr♦❧❧❡r ✭✹✮ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ❛s ❍∞ ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ ♣❡r❢♦r♠❛♥❝❡ ✐❢ t❤❡ ❢♦❧❧♦✇✲ ❢♦❧❧♦✇s✱ ❢♦r i = 1, , n✿ ✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ Ex ˜˙ i = A˜hj,hj x ˜i ❈♦♥❞✐t✐♦♥ ✶ ✭❙t❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥✮✿ ❲✐t❤ ③❡r♦ ❞✐st✉r❜❛♥❝❡s ✐♥♣✉t ❝♦♥❞✐t✐♦♥ wi ≡ 0✱ ❢♦r i = 1, , n ✱ t❤❡ ❝❧♦s❡❞✲❧♦♦♣ ❞②♥❛♠✐❝s ✭✺✮ ✐s st❛❜❧❡✳ n wα ˜hj F˜i,α,hj x ˜α + B w ˜i,α + ✭✾✮ α=1,α=i ❈♦♥❞✐t✐♦♥ ✷ ✭❘♦❜✉st♥❡ss ❝♦♥❞✐t✐♦♥✮✿ ❋♦r ❛❧❧ ♥♦♥✲③❡r♦ wi ∈ L2 [0, ∞)✱ ✉♥❞❡r ③❡r♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ xi (t0 ) ≡ 0✱ ✐t ❤♦❧❞s t❤❛t ❢♦r i = 1, , n✱ +∞ xTi xi dt Ji = +∞ n wiT wi + ςi2 wαT wα dt α=1,α=i ✭✻✮ I E = 0 0 Fi,α,hj 0 0 , F˜i,α,hj = 0 0 0 0 w N Bhj wα wα ˜hj Bhj B = 0 Ahj Bhj −1 γ A˜hj,hj = −I Khj Xhj Chj −I ✇❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ s❝❛❧❛rs ✇❤✐❝❤ r❡♣r❡s❡♥ts ◆♦t❡ t❤❛t t❤❡ s②st❡♠ ✭✾✮ ✐s ❛ ❧❛r❣❡ s❝❛❧❡ s✇✐t❝❤❡❞ t❤❡ ❞✐st✉r❜❛♥❝❡ ❛tt❡♥✉❛t✐♦♥ ❧❡✈❡❧ ❛ss♦❝✐❛t❡❞ t♦ ❞❡s❝r✐♣t♦r✳ ❍❡♥❝❡✱ ✐t ✐s ✇♦rt❤ ♣♦✐♥t✐♥❣ ♦✉t t❤❛t th t❤❡ i s✉❜s②st❡♠✳ t❤❡ ♦✉t♣✉t✲❢❡❡❞❜❛❝❦ st❛❜✐❧✐③❛t✐♦♥ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❋r♦♠ t❤❡ ❝❧♦s❡❞✲❧♦♦♣ ❞②♥❛♠✐❝s ✭✺✮✱ ✐t ❝❛♥ s②st❡♠ ✭✶✮ ❝❛♥ ❜❡ ❝♦♥✈❡rt❡❞ ✐♥t♦ t❤❡ st❛❜✐❧✐③❛✲ ❜❡ s❡❡♥ t❤❛t s❡✈❡r❛❧ ❝r♦ss✐♥❣ t❡r♠s ❛♠♦♥❣ t❤❡ t✐♦♥ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❛✉❣♠❡♥t❡❞ s②st❡♠ ✭✾✮✳ ςi2 ✶✹✷ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✷ ❘❡♠❛r❦ ✷✿ ■t ♠❛② ❜❡ ❤❛r❞ t♦ ✇♦r❦ ✇✐t❤ t❤❡ ✜rst ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝❧♦s❡❞✲❧♦♦♣ ❞②♥❛♠✐❝s ✭✺✮✱ ❞✉❡ t♦ t❤❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣ t❡r♠s✳ ❍♦✇✲ ❡✈❡r✱ t❤❡ ❣♦❛❧ ♦❢ ♦✉r st✉❞② ❝❛♥ ♥♦✇ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❛✉❣♠❡♥t❡❞ ❝❧♦s❡❞✲❧♦♦♣ ❞②✲ ♥❛♠✐❝s ✭✾✮ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❞❡s❝r✐♣t♦r ❢♦r♠✳ ■♥ t❤✐s ❝♦♥t❡①t✱ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ♦❢ ❉❡✜♥✐t✐♦♥ ✶✱ ❣✐✈❡♥ ❜② ❡q✉❛t✐♦♥ ✭✻✮✱ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✿ +∞ +∞ y˜iT Q˜ yi dt NI ✇✐t❤ Ξ = 0 I 0 ▲❡♠♠❛ ✶✳ ❬✶✼❪✿ ▲❡t ✉s ❝♦♥s✐❞❡r t✇♦ ♠❛tr✐❝❡s A ❛♥❞ B ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❞✐♠❡♥s✐♦♥s ❛♥❞ ❛ ♣♦s✲ ✐t✐✈❡ s❝❛❧❛r τ ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t② ✐s ❛❧✇❛②s s❛t✐s✜❡❞✿ ✸✳ −µji →ji + Xk1j i Xk1j ✭✶✶✮ τ AT A + τ −1 B T B Γsji lji kji (∗) X kji I ▲▼■ ❇❛s❡❞ ✇✐t❤ ❆ss✉♠❡ t❤❛t ❢♦r ❡❛❝❤ s✉❜s②s✲ t❡♠ i ♦❢ ✭✶✮✱ t❤❡ ❛❝t✐✈❡ ♠♦❞❡ ✐s ❞❡♥♦t❡❞ ❜② ji ❛♥❞✱ ❢♦r ji = 1, , mi ❛♥❞ sji = 1, , rji ✱ h˙ sji (z (t)) λsji ✳ ❚❤❡ ♦✈❡r❛❧❧ ✐♥t❡r❝♦♥♥❡❝t❡❞ s✇✐t❝❤❡❞ ❚❙ s②st❡♠ ✭✶✮ ✐s st❛❜✐❧✐③❡❞ ❜② ❛ s❡t ♦❢ n ❞❡❝❡♥tr❛❧✐③❡❞ s✇✐t❝❤❡❞ ♥♦♥✲P❉❈ ❝♦♥tr♦❧ ❧❛✇s ✭✹✮ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❉❡✜♥✐t✐♦♥ ✶✱ ✐❢ t❤❡r❡ ❡①✐sts✱ ❢♦r ❛❧❧ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ i = 1, , n✱ ji = 1, , mi ✱ ji + = 1, , mi ✱ sji = 1, , rji ✱ kji = 1, , rji ✱ kj1i = 1, , rji ❛♥❞ lji = 1, , rji ✱ t❤❡ ♠❛tr✐❝❡s > 0✱ = ∗ ∗ ∗ −I Γsji lji kji w N Bkj = 0 Xk5j i T wα Bkj rji λlji Xl1j + Wsji kji kj Φsji lji kji kj = i i , i lji =1 ❚❤❡♦r❡♠ ✶✳ = −I i ■♥ t❤✐s s❡❝t✐♦♥✱ t❤❡ ♠❛✐♥ r❡s✉❧t ❢♦r t❤❡ ❞❡s✐❣♥ ♦❢ ❛ r♦❜✉st ❞❡❝❡♥tr❛❧✐③❡❞ s✇✐t❝❤❡❞ ♥♦♥✲P❉❈ ❝♦♥✲ tr♦❧❧❡r ✭✹✮ ❡♥s✉r✐♥❣ t❤❡ ❝❧♦s❡❞✲❧♦♦♣ st❛❜✐❧✐t② ♦❢ ✭✺✮ ❛♥❞ t❤❡ ❍∞ ❞✐st✉r❜❛♥❝❡ r❡❥❡❝t✐♦♥ ♣❡r❢♦r✲ ♠❛♥❝❡ ✭✶✵✮ ✐s ♣r❡s❡♥t❡❞✳ ■t ✐s s✉♠♠❛r✐③❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳ Xk5j i 0✱ Ws1j sj kj ✱ Kkji ✱ ❛♥❞ t❤❡ i i i i i s❝❛❧❛rs✱ τ1,i , , τi−1,i , τi+1,i , , τn,i ✭❡①❝❡♣t❡❞ τi,i ✇❤✐❝❤ ❞♦♥✬t ❡①✐st s✐♥❝❡ t❤❡r❡ ✐s ♥♦ ✐♥t❡r❛❝t✐♦♥ ❜❡✲ t✇❡❡♥ ❛ s✉❜s②st❡♠ ❛♥❞ ❤✐♠s❡❧❢✮✱ s✉❝❤ t❤❛t t❤❡ ▲▼■s ❞❡s❝r✐❜❡❞ ❜② ✭✶✷✮✱ ✭✶✸✮✱ ✭✶✹✮ ❛♥❞ ✭✶✺✮ ❛r❡ s❛t✐s✜❡❞✳ n ςi2 | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ Xkji Xkj i = 0 Xk5j i 0 Xk9j i I = (−1) · diag τ1,i I X kji = Xkji τi−1,i I ··· Xkji τi+1,i I Xkji ··· τn,i I Xkji > 0❀ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✹✸ ❱❖▲❯▼❊✿ ✷ = Γsji lji kji | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ sym Xk1j ATsj i i T +τi,α Fi,α,sji Fi,α,s ji −Φsji lji kji kj (∗) (∗) Γsji lji kji −sym(Xk5j ) (∗) (∗) i Xk9j i i T Bsji +Csji Xk1j Klji i T −sym(Xk9j ) i Pr♦♦❢✳ ❚❤❡ ♣r❡s❡♥t ♣r♦♦❢ ✐s ❞✐✈✐❞❡❞ ✐♥ t✇♦ ♣❛rts ❚❤❛t ✐s t♦ s❛②✿ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❈♦♥❞✐t✐♦♥s ✶ ❛♥❞ ✷ ❣✐✈❡♥ ✐♥ ❉❡✜♥✐t✐♦♥ ✶✳ Xhj+ −1 − µj→j + Xhj −1 ✭✷✵✮ P❛rt ✶ ✭❙t❛❜✐❧✐t② ❈♦♥❞✐t✐♦♥ ✶✱ ❉❡✜♥✐t✐♦♥ ✶✮✿ ▲❡❢t ❛♥❞ r✐❣❤t ♠✉❧t✐♣❧②✐♥❣ ❜② Xhj ✱ t❤❡♥ ✉s✐♥❣ ❲✐t❤ ③❡r♦ ❞✐st✉r❜❛♥❝❡s ✐♥♣✉t ❝♦♥❞✐t✐♦♥ w ˜i,α ≡ ❙❝❤✉r ❝♦♠♣❧❡♠❡♥t✱ ✭✷✵✮ ✐s ❡q✉✐✈❛❧❡♥t t♦✿ 0✱ ❢♦r i = 1, , n✳ ▲❡t ✉s ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ 1 −µj→j + Xhj Xhj ♠✉❧t✐♣❧❡ s✇✐t❝❤❡❞ ♥♦♥✲q✉❛❞r❛t✐❝ ▲②❛♣✉♥♦✈✲❧✐❦❡ 1 Xhj −Xhj + ❝❛♥❞✐❞❛t❡ ❢✉♥❝t✐♦♥❛❧✿ n mi ξji vji (xi ) > 0, V (x1 , x2 , , xn ) = i=1 ji =1 ✭✶✻✮ ◆♦✇✱ ❧❡t ✉s ❞❡❛❧ ✇✐t❤ ✭✶✼✮✱ ✇✐t❤ t❤❡ ❛❜♦✈❡ ❞❡✲ ✜♥❡❞ ♥♦t❛t✐♦♥s✱ ✐t ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s✱ ∀t = tj→j + ✿ n ✇❤❡r❡ vj i = x ˜Ti E(Xhj ) rji −1 −1 ˜i + x sym ˜˙xTi E(Xhj ) x ˜Ti E X˙ hj x ˜i ✭✷✶✮ −1 x ˜i i=1 < −1 hsji Xsji x ˜Ti E sji =1 ✭✷✷✮ ❙✉❜st✐t✉t✐♥❣ ✭✾✮ ✐♥t♦ ✭✷✷✮✱ ✇❡ ❝❛♥ ✇r✐t❡✱ ∀t = tj→j + ✿ ❛♥❞ ✇✐t❤ EXhj = Xhj E > 0✱ Xhj = −1 −1 ˜ T ˙ n ˜i sym (Xhj ) Ahj,hj + E Xhj x ˜i 1T x Xhj Xhj diag Xhj ✱ Xhj = Xhj ✳ n −1 T T ❚❤❡ ❛✉❣♠❡♥t❡❞ s②st❡♠ ✭✾✮✱ ❛♥❞ ✐♠♣❧✐❝✐t❧② t❤❡ i=1 + ˜i sym x ˜α Fi,α,hj (Xhj ) x α=1,α=i ❝❧♦s❡❞✲❧♦♦♣ ✐♥t❡r❝♦♥♥❡❝t❡❞ s✇✐t❝❤❡❞ s②st❡♠ ✭✺✮✱ ✐s ❛s②♠♣t♦t✐❝❛❧❧② st❛❜❧❡ ✐❢✿ 0✱ ❧❡❢t ❛♥❞ r✐❣❤t ♠✉❧t✐♣❧②✐♥❣ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ✭✷✺✮ r❡s♣❡❝t✐✈❡❧② ❜② Xhj ✱ t❤❡ ✐♥❡q✉❛❧✐t② ✭✷✺✮ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s✿ sym A˜hj,hj Xhj + EXhj X˙ hj −1 Xhj n −1 T τi,α F˜i,α,hj F˜i,α,hj + τα,i Xhj Xhj + α=1,α=i ✭✷✻✮ < | ■❙❙❯❊✿ ✷ | ✷✵✶✽ | ❏✉♥❡ ❙✉❜st✐t✉t✐♥❣ ✭✾✮ ✐♥t♦ ✭✷✽✮✱ ✇❡ ❝❛♥ ✇r✐t❡✱ ∀t = tj→j + ✿ −1 sym A˜Thj,hj (Xhj ) x ˜Ti ˜i −1 x ˙ n +Q + E Xhj −1 T T sym(˜ x F (X ) x ˜ ) hj i α i,α,hj n i=1 T −ςi2 w ˜i,α Ξw ˜i,α + α=1,α=i ˜ wα )T (Xhj )−1 x ˜i ) +sym(w ˜ T (B i,α hj ✭✷✾✮