Asymptotic Tracking Control of Uncertain MIMO Nonlinear Systems with Less Conservative Controllability Conditions

The paper proves global boundedness and asymptotic tracking for the uncertain MIMO canonical system by (i) introducing the filtered error s, (ii) using the auxiliary matrix α to form A=αg+g^Tα and a quadratic Lyapunov term, (iii) embedding a BL-type Nussbaum gain, and crucially (iv) augmenting the Lyapunov function with a parameter-error term V2=0.5 s^Tαs + (1/(2σ2))\tilde θ^2 so that the non-control terms are absorbed via the adaptation law; Lemma 1 (BL-type Nussbaum) is then applied through inequalities (28)–(29) to bound V2 and ζ, yielding s∈L2∩L∞, ṡ∈L∞, and thus s→0 and e→0 (controller (13)–(15); Assumptions 1–3; Theorem 1; see the control law and inequalities in (11)–(15) and (18)–(29) ). For faults, the same architecture is repeated with ua=ρu+ε and A*=αgρ+ρg^Tα under Assumptions 4–5 (eqs. (30)–(35), Theorem 2) . In contrast, the candidate solution omits the augmented parameter term in the Lyapunov function and never proves θ̂ is bounded. Its Step 4 then argues that ζ bounded implies η and u_a are bounded, but η contains θ̂ and s, which at that point are not yet shown bounded—this is circular. The paper resolves exactly this issue via V2 and the cancellation using ˙̂θ in (27)–(29) . A minor wording issue in the paper is that Assumption 4 states sign-definiteness but implicitly needs uniform bounds to invoke Lemma 1; this is easily fixed. Overall, the paper’s argument is complete and correct; the model’s proof is incomplete at the key parameter-boundedness step.