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❉❡❝♦♠♣♦s✐çã♦ ❡♠ ❋r❛❝çõ❡s ❙✐♠♣❧❡s
▲✉ís ❇♦r❣❡s ❞❡ ❆❧♠❡✐❞❛
▼❛rç♦ ❞❡ ✷✵✶✷

✶ ■♥tr♦❞✉çã♦
❆ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s r❛❝✐♦♥❛✐s ❡♠ ❢r❛❝çõ❡s s✐♠♣❧❡s ✭t❛♠❜é♠ ❝❤❛♠❛❞❛s ❢r❛❝çõ❡s
♣❛r❝✐❛✐s ♦✉ ❢r❛❝çõ❡s ❡❧❡♠❡♥t❛r❡s✮ é ✉♠❛ ♦♣❡r❛çã♦ ♠❛t❡♠át✐❝❛ ❢r❡q✉❡♥t❡♠❡♥t❡ ❡♠♣r❡❣✉❡✱
♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❝á❧❝✉❧♦ ❞❡ tr❛♥s❢♦r♠❛❞❛s ✭❡s♣❡❝✐❛❧♠❡♥t❡ tr❛♥s❢♦r♠❛❞❛s ✐♥✈❡rs❛s✮ ❞❡
❋♦✉r✐❡r✱ ❞❡ ▲❛♣❧❛❝❡ ❡ ❩✱ ♠❛s t❛♠❜é♠ ❡♠ ❞✐✈❡rs❛s ♦✉tr❛s ❛♣❧✐❝❛çõ❡s✱ ❞❡ q✉❡ é ❡①❡♠♣❧♦
❛ ♣r✐♠✐t✐✈❛çã♦ ❞❡ ❢✉♥çõ❡s r❛❝✐♦♥❛✐s✳ ◆❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❙✐♥❛✐s ❡ ❙✐st❡♠❛s ❞♦ ▼❊❊❈✱ ■❙❚✱
❡st❡ t✐♣♦ ❞❡ ❞❡❝♦♠♣♦s✐çã♦ ✉t✐❧✐③❛✲s❡ s♦❜r❡t✉❞♦ ♥♦ ❝á❧❝✉❧♦ ❞❡ tr❛♥s❢♦r♠❛❞❛s ✐♥✈❡rs❛s
❞❡ ❋♦✉r✐❡r ♦✉ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ❢✉♥çõ❡s r❛❝✐♦♥❛✐s✳ ❊st❡ ❞♦❝✉♠❡♥t♦✱ ♣r❡♣❛r❛❞♦ ♣❛r❛ ♦s
❡st✉❞❛♥t❡s ❞❛ r❡❢❡r✐❞❛ ❞✐s❝✐♣❧✐♥❛✱ ❛❜♦r❞❛ ❛ ❢♦r♠❛ ❞❡ ♣r♦❝❡❞❡r ❛ ❡ss❛ ❞❡❝♦♠♣♦s✐çã♦✳ ❆
✐♥t❡♥çã♦ é ❛♣r❡s❡♥t❛r ❛ ❢♦r♠❛ ❞❡ ♣r♦❝❡❞❡r ❡ ♥ã♦ ❥✉st✐✜❝á✲❧❛✱ ✉♠❛ ✈❡③ q✉❡ s❡ tr❛t❛ ❞✉♠ tó♣✐❝♦ ❥á ❡st✉❞❛❞♦ ♣❡❧♦s ❛❧✉♥♦s ❡♠ ❞✐s❝✐♣❧✐♥❛s ❞❡ ▼❛t❡♠át✐❝❛✳

✷ ❈♦♥❝❡✐t♦ ❞❡ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ❢r❛❝çõ❡s s✐♠♣❧❡s
❈♦♠❡ç❛✲s❡ ♣♦r ✐♥tr♦❞✉③✐r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s✳

•

✕ ❈❤❛♠❛✲s❡ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ❛ q✉❛❧q✉❡r ❢✉♥çã♦ q✉❡ s❡❥❛ ❞❛❞❛
♣❡❧♦ q✉♦❝✐❡♥t❡ ❞❡ ❞♦✐s ♣♦❧✐♥ó♠✐♦s✿

❋✉♥çã♦ r❛❝✐♦♥❛❧

H(s) =
•

N (s)
.
D(s)

✕ ❯♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ ❞✐③✲s❡ ♣ró♣r✐❛ s❡ ♦ ❣r❛✉ ❞♦
♣♦❧✐♥ó♠✐♦ ❞♦ ♥✉♠❡r❛❞♦r ❢♦r ✐♥❢❡r✐♦r ❛♦ ❣r❛✉ ❞♦ ♣♦❧✐♥ó♠✐♦ ❞♦ ❞❡♥♦♠✐♥❛❞♦r✳ P♦r
❡①❡♠♣❧♦✱ ❞❛s ❢✉♥çõ❡s
4s − 1
3s2 + 4s − 1
,
s2 + s − 2 s2 + s − 2
❛ ❞❛ ❡sq✉❡r❞❛ é ♣ró♣r✐❛ ❡ ❛ ❞❛ ❞✐r❡✐t❛ é ✐♠♣ró♣r✐❛✳

❋✉♥çã♦ r❛❝✐♦♥❛❧ ♣ró♣r✐❛

✶

•

❋r❛❝çã♦ s✐♠♣❧❡s

✶

✕ ❈❤❛♠❛r❡♠♦s ❢r❛❝çõ❡s s✐♠♣❧❡s ❛ ❢✉♥çõ❡s ❞❛ ❢♦r♠❛

A
,
(s − p)q
❡♠ q✉❡ s é ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✱ A ❡ p sã♦ ❝♦♥st❛♥t❡s r❡❛✐s ❡ q é ✉♠ ♥ú♠❡r♦
♥❛t✉r❛❧✳
❖ ❢❛❝t♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❡ ❡stá ♥❛ ❜❛s❡ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ❢r❛❝çõ❡s s✐♠♣❧❡s é ♦ s❡❣✉✐♥t❡✿
◗✉❛❧q✉❡r ❢✉♥çã♦ r❛❝✐♦♥❛❧

♣ró♣r✐❛

é ❞❡❝♦♠♣♦♥í✈❡❧ ♥✉♠❛ s♦♠❛ ❞❡

❢r❛❝çõ❡s s✐♠♣❧❡s✳

◆♦t❡✲s❡ q✉❡ ❛ ❞❡❝♦♠♣♦s✐çã♦ s❡ ❛♣❧✐❝❛ ❛♣❡♥❛s ❛ ❢✉♥çõ❡s r❛❝✐♦♥❛✐s ♣ró♣r✐❛s✱ ✐st♦ é✱ ❛
❢✉♥çõ❡s ❡♠ q✉❡ ♦ ❣r❛✉ ❞♦ ♥✉♠❡r❛❞♦r é ✐♥❢❡r✐♦r ❛♦ ❞♦ ❞❡♥♦♠✐♥❛❞♦r✳ P♦r ❡①❡♠♣❧♦✱