Robust Feedback Stabilization of Linear MIMO Systems Using Generalized Homogenization

Abstract

A robust nonlinear control is designed for stabilizing linear MIMO systems. The presented control law homogenizes a linear system (without its transformation to a canonical form) with a specified degree and stabilizes it in a finite time (or with a fixed-time attraction to any compact set containing the origin) if the degree of homogeneity is negative (positive). The tuning procedure is formalized in LMI form. Performance of the approach is illustrated by numerical and experimental examples.

Generalized homogenization of linear controllers: Theory and experiment

“…In [31] it is shown that the system (14) can be homogeneously stabilized with a degree µ = 0 if and only if the pair {A, B} is controllable (or, equivalently, rank(B, AB, ..., A n−1 B) = n. We refer the reader to [32] for more details about controllability of linear plants. The following theorem is the corollary of a more general theorem proved [20] for evolution system in Hilbert spaces (see also [31] for more details about the finite dimensional case).…”

“…where the canonical homogeneous norm If A 0 is nilpotent, then (16) has a solution with respect to G d [31], such that G d is anti-Hurwitz matrix. The feasibility of (17) is proven in [18] and refined in [31]. The proof of the latter theorem follows from the following computations…”

“…In order to apply the proposed method to design homogeneous PID for the nonlinear quadrotor model (30) and (31), the following will firstly present the linear PID controller for the linearized model of (30) and (31), and finally get the homogeneous PID for the nonlinear model (30) and (31) from the linear PID controller. The original linear controller for z-subsystem contains the integrator.…”

On Generalized Homogenization of Linear Quadrotor Controller

“…In [20], it is shown that the system (9) can be homogeneously stabilized with a degree µ = 0 if and only if the pair {A, B} is controllable (or, equivalently, rank(B, AB, ..., A n−1 B) = n. We refer the reader to [21] for more details about controllability of linear plants. The following theorem is the corollary of a more general theorem proven in [17] for evolution system in Hilbert spaces (see also [20] for more details about the finite dimensional case).…”

“…where F T is assumed to be close to mg and the angle θ, φ are assumed to be close to 0. Denoting ζ = (x, y,ẋ,ẏ, θ, −φ,θ, −φ) from (21) and (20) we derivė…”