[SOLVED] Nonlinear-Control-Homework 8

Description

1. Consider the one-form _ω = (1+2x₁+x₂)dx1 + x₁dx₂. Is it exact ? If yes, compute α(x₁, x₂) such that ω =dα(x₁, x₂).

2. Consider the one-form ω = x₁dx₂ + x₂dx₃ + x₃dx₄   . Is it integrable ? Compute dω ∧ dω ∧ω.

Do there exist two functions ϕ₁and ϕ₂ such that ω∈ span{dϕ₁,dϕ₂} ?  If yes, compute ϕ₁and ϕ₂ .

3. The nonholonomic integrator

Check the accessibility of the following nonholonomic integrator:

x&1 =u1 x&2 =u2

x&3 =x1u 2 −x2u1

Is it fully (means input-state) linearizable by static state feedback ?

Is it fully linearizable by dynamic state feedback ? à Hint: try to find 2 output functions with relative degree 1 and such that the decoupling matrix has rank 1 only !