EL ENG C222 Midterm: eeC222-sp2018-mt1-Arcak-soln

• School: Berkeley
• Department: Electrical Engineering
• Course: EL ENG C222
• Professor: All

Document Summary

Answers without justi cation do not receive full credit: consider the system. 1 + x1 where > 0 is a positive parameter: show that the nonnegative quadrant r2. Solution: we can show that the nonnegative quadrant is positively invariant by in- dividually proving that the half spaces x1 0, x2 0 are both positively invariant. This can be done by proving a) that x1 0 whenever x1 = 0, x2 0 and b) x2 0 whenever x2 = 0, x1 0. 1 + x2 which proves invariance of the x1 0 half space. A symmetric argument can be made for invariance of the x2 0 half space: show that a single equilibrium exists in the nonnegative quadrant. Substituting the expression for x1 into the x2 equation yields: x2 = Multplying both sides by 1 + x2 + yields x2 + x2. The quadratic equation gives us two possible solutions x2 = .