Pitch Plane Trajectory Tracking Control for Sounding Rockets via Adaptive Feedback Linearization

The paper does not claim stability of the outer-loop PI–PD for arbitrary positive gains; it explicitly selects gains via an LQR design on the augmented state, which by construction yields a Hurwitz closed-loop (see the definition of v_out and the resulting LTI error dynamics in Eqs. (59)–(62), and the proof text of Theorem 2 stating the characteristic polynomial is Hurwitz) . The model’s critique about “any positive gains” attacks a claim the paper does not make. Regarding estimator convergence, the inner-loop adaptation estimates additive constant (or slowly varying) disturbances (fax, τa) directly with integral laws (41)–(42); this is exactly the setting of Lemma 1 (Eqs. (36)–(37)), which ensures convergence of both the tracking errors and the disturbance-estimation errors without requiring persistence of excitation, under the stated assumptions (cf. Theorem 1 and its proof) . With these inner estimates converging and the aerodynamic moment modeling used in the indirect estimate (57), the horizontal force estimation error i f̃_az also vanishes, yielding asymptotic stability of the outer loop for |θ|<π/2 as claimed (Theorem 2), and likewise for the full cascade (Theorem 3) .