Generalized weighted homogeneity and state dependent time scale for linear controllable systems

Given a matrix R and a Lyapunov function x^TPx, with R ^TP+PR negative definite, we define generalized polar coordinates (ρ, θ) by x=exp(-Rlog(ρ))θ, θ ^TPθ=1. Then we consider a controllable linear system x˙=Ax+Bu stabilized by u=-Kx such that (A-BK)^TP+P(A-BK) is negative definite. We define a globally asymptotically stable vector field F(x)=ρ^-τ exp(-Rlog(ρ))(A-BK)θ. This vector field has a generalized homogeneity property. The interest of this construction appears when τ and R are chosen such that F(x)-Ax is in the Span of B. Indeed in this case we have designed indirectly a control law which, besides asymptotic stability, assigns a generalized homogeneity property to the closed-loop system. This allows us to consider the possibility of getting global asymptotic stability for systems which can be nonlinear perturbations of the linear system. Also, with an appropriate choice of R, we can get that, if σ_i is a pole of (A-BK), then σ_iρ^-τ is a “pole” of the closed loop system. This way we have a technique to modify the time scale as a function of the state