Practical determination of the fixed-gain control parameter alpha

Determine the required magnitude of the fixed-gain control parameter alpha in the fixed-gain control signal u_f(t) = −alpha B_i (x(t) − x_0) + alpha B_i ∫_0^t (A_n x(s) + B_n r(s)) ds, so that, with no adaptive learning signal (u_a(t) ≡ 0), the solution of the uncertain closed-loop system dot{x}(t) = A_n x(t) + B_n r(t) + B Λ (u_f(t) + π(x(t), u_n(t))) closely approximates the nominal (ideal) closed-loop system dot{x}_n(t) = A_n x_n(t) + B_n r(t).

Background

Proposition 1 shows that if the fixed-gain control parameter alpha is sufficiently large, the solution to the uncertain system behaves like the nominal closed-loop system. This result holds for both parametric and nonparametric uncertainties without an adaptive learning signal.

However, the authors note that in practice one cannot know how large alpha must be, and analyzing stability to choose alpha without adaptive learning typically requires specific uncertainty structure and bounds, which can be conservative. This highlights an unresolved practical question about selecting alpha to guarantee the desired nominal approximation without relying on potentially conservative assumptions.