Consider the differential equation 2 +42 + g(z) = u(t), where g(z) = 2²-el-z. (a) Define a state and rewrite this equation in state-space form. Then, find an equilibrium (Te, ue) where ue = 0. (b) Linearize the state equation about the equilibrium (xe, ue). (c) Compute the matrix exponential eat, where A is the state matrix obtained in part (b).
Consider the differential equation 2 +42 + g(z) = u(t), where g(z) = 2²-el-z. (a) Define a state and rewrite this equation in state-space form. Then, find an equilibrium (Te, ue) where ue = 0. (b) Linearize the state equation about the equilibrium (xe, ue). (c) Compute the matrix exponential eat, where A is the state matrix obtained in part (b).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the differential equation
2+42 + g(z) = u(t),
where g(z) = 2²-el-z.
(a) Define a state x and rewrite this equation in state-space form. Then, find an equilibrium
(Te, ue) where ue = 0.
(b) Linearize the state equation about the equilibrium (xe, Ue).
(c) Compute the matrix exponential eat, where A is the state matrix obtained in part (b).
(d) Consider the linearized state equation derived in part (b), di = Adx + Bou, where dx,
du, A, and B are appropriately defined. Suppose that du = 0 and dx(0) = dro. Solve
this linear differential system for dr(t). What happens to dr(t) as too. Based on
the observed behavior, can you deduce whether the linearized system is asymptotically
stable?
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