{"id":"f77ee7c5-9fba-4a8d-a8d9-0cfd47166ea7","arxiv_id":"2411.13252","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A prescribed-performance backstepping controller for non-square strict-feedback systems with actuator faults, using a relaxed controllability condition based on unknown auxiliary Lyapunov matrices.","lead":"This paper proposes a backstepping controller for over-actuated nonlinear systems, such as spacecraft with redundant reaction wheels, that keeps tracking error inside flexible performance bounds even when some actuators lose effectiveness. It relaxes the usual assumptions on the unknown control-gain matrix by assuming only the existence of a hidden auxiliary Lyapunov matrix, and it handles square and non-square systems with the same control structure.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof omits bounds on ∂P_k/∂X_{k-1} despite ˙P_k appearing in h_k; the asserted h_k inequalities do not follow from Assumption 4 for state-dependent P_k.","rationale":"The reader's conditional verdict is appropriate, but the load-bearing issue is not primarily the verifiability of Assumption 3. The real gap is that Assumption 4 gives no control over ∂P_k/∂X_{k-1}, while the proof's h_k bounds require exactly such control. This is a concrete, fixable omission: adding a core-function bound on the state gradient of P_k would restore the recursive argument, at the cost of narrowing the claimed class. Because the fix is localized and does not invalidate the overall backstepping-PPC framework, the verdict remains conditional rather than an outright reject. I partially disagree with the reader's weakest_assumption: Assumption 3's existence is indeed hard to verify, but the missing gradient bound is a more direct threat to the proof of Theorem 1. The concrete test above settles whether the h_k inequalities hold as stated.","tokens_in":18316,"tokens_out":15425,"duration_ms":147173,"concrete_test":"Re-derive Eq. (30) from h2 in Eq. (29) by expanding (1/2)ε2^T ˙P2 ε2 with ˙P2 = ∂P2/∂t + (∂P2/∂x1)(f1+g1x2+d1), and check whether the ∂P2/∂x1 terms can be bounded using only (6)-(7). Then run the scalar example n=1, g1=g2=1, f1=d1=f2=d2=0, P2(x1,t)=x1^3+1, φ21=x1^3+1: verify that for fixed x1 the h2 expression contains a cubic term in ε2, so the inequality h2 ≤ ∥ε2∥^2θ2Φ2 + 5/4 cannot hold for all ε2. If the bound fails, add an explicit gradient bound on P_k(X_{k-1},t) to Assumption 4 and re-check Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 4 bounds only ∥P_k∥ and ∥∂P_k/∂t∥ (Eqs. (6)-(7)), not the state gradient ∂P_k/∂X_{k-1}. Yet the Step-k Lyapunov derivative contains (1/2)ε_k^T ˙P_k ε_k, as seen in h2 in Eq. (29) and the Step-i/Table I recursions, and ˙P_k = ∂P_k/∂t + Σ_{l=1}^{k-1} (∂P_k/∂x_l) ˙x_l. The gradient sum generates terms such as ε_k^T (∂P_k/∂x_{k-1}) g_{k-1} ε_k, which are quadratic — and after substituting x_k = ε_k + a_{k-1}, cubic — in ε_k, with coefficient ∂P_k/∂X_{k-1}. The claimed bound h_k ≤ ∥ε_k∥^2 θ_k Φ_k + const (e.g., (30)) therefore requires a bound on ∂P_k/∂X_{k-1} that is not assumed. Concrete illustration of the missing hypothesis: with n=1, g1=g2=1, f1=d1=f2=d2=0, and P2(x1,t)=x1^3+1, Assumptions 1-4 can be satisfied with φ21=x1^3+1, but ˙P2=3x1^2 x2 and h2 contains 3/2 x1^2 ε2^2(ε2+a1), which is not dominated by ∥ε2∥^2 times the stated Φ2. Thus the recursive boundedness proof of Theorem 1 is incomplete as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a prescribed-performance backstepping controller for strict-feedback MIMO nonlinear systems with possibly non-square control gain matrices and actuator faults. The main contribution is a relaxed controllability condition (Assumption 3) that allows unknown, time-varying auxiliary matrices P_i in the Lyapunov analysis, together with a unified performance function that supports symmetric and asymmetric error bounds without controller redesign. The authors provide a stability theorem (Theorem 1) and two simulation examples (quadrotor and spacecraft).","tokens_in":18687,"tokens_out":9199,"duration_ms":81338,"significance":"If the result holds, it would generalize existing PPC controllability conditions from square systems with symmetric-positive-definite or positive-real gain matrices to non-square, over-actuated systems with intermittent faults, while also removing the need to know or construct the auxiliary matrices P_i. The control law is explicit and does not use ρ or P_i, which is practically attractive, and the paper includes concrete examples and simulations. However, the central Lyapunov recursion in the proof of Theorem 1 currently has a load-bearing gap, so the main claim is not yet established as written.","major_comments":[{"comment":"The proof of Theorem 1 is incomplete because Assumption 4 bounds only ∥P_k∥ and ∥∂P_k/∂t∥, not the state gradient ∂P_k/∂X_{k−1}. Yet the Lyapunov derivative in each backstepping step contains (1/2)ε_k^T Ṗ_k ε_k, with Ṗ_k = ∂P_k/∂t + Σ_{l=1}^{k−1}(∂P_k/∂x_l) ẋ_l; the gradient terms produce terms that are cubic in ε_k and are not covered by the asserted bound h_k ≤ ∥ε_k∥^2 θ_k Φ_k. Concretely, for the system ẋ_1 = x_2, ẋ_2 = u (n=1, N=2, g_1=g_2=1, f_i=d_i=0, ρ=1, A=1), Assumptions 1–4 hold with P_2(x_1,t)=x_1^2+1 and φ_{21}=x_1^2+1, but Ṗ_2 = 2x_1 x_2, so h_2 contains x_1ε_2^2(ε_2+a_1), which cannot be dominated by ∥ε_2∥^2 times a function independent of ε_2. A repair requires an added bound on ∂P_k/∂X_{k−1} (e.g., through a known core function) and a corresponding modification of Φ_k, or a different stability argument that does not rely on the pointwise inequality (30).","section":"§IV (Theorem 1 proof), Eq. (30)"}],"minor_comments":[{"comment":"Assumption 2 states g_i = A b_i for i=1,...,N, but for k=1,...,N−1 the matrices g_k are n×n (as used in (1) and the design), whereas A is n×m with n<m; the decomposition can only apply to g_N. Please revise the assumption to apply only to the final gain g_N.","section":"Assumption 2"},{"comment":"The symbol G_N is used inconsistently: below Eq. (42) it is defined as P_N g_N ρ, while in Assumption 3 and in Eq. (51) it denotes the symmetric matrix P_N g_N ρ A^T + A ρ g_N^T P_N. Please use distinct notation for these two different matrices.","section":"Step N, Eqs. (43), (50)-(51)"},{"comment":"The performance bound in the theorem statement is written as H(−φ_j(t)) < e_j(t) < H(φ_j(t)), omitting the parameters δ_j and δ̄_j from Eq. (3); please correct.","section":"Theorem 1 statement"},{"comment":"In the definition of Υ, the term 2κ_kλ_k/λ_max(P_N) should be 2κ_kλ_k/λ_max(P_k) for k=2,...,N; as written it uses P_N for all intermediate steps.","section":"Eq. (52)"},{"comment":"Remark 6 states that the method 'does not require any feasibility assumptions on P_i', but Assumption 3 is precisely an existence (feasibility) condition on the unknown P_i; please rephrase.","section":"Remark 6"}],"recommendation":"major_revision","confidential_remarks":"The central idea is promising and the paper is within scope, but the missing bound on ∂P_k/∂X_{k−1} is a genuine gap in the proof of Theorem 1. If the authors add a suitable hypothesis and adjust the Φ_k terms accordingly, the contribution could be solid; as it stands, the main theorem is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper is a real extension, not a repackaging: it moves the authors' earlier relaxed-controllability/PPC framework from square systems to over-actuated non-square strict-feedback systems with PLOE actuator faults, and it lets the auxiliary Lyapunov matrices P_i be unknown and absent from the controller. That is useful, and the quadrotor/spacecraft examples support the claimed scope. The backstepping/PPC machinery is standard, and the adaptive core-function treatment of the unknown P_i is sensible.\n\nBut the stress-test note is right, and it hits the main theorem. Assumption 4 bounds ||P_k|| and ||∂P_k/∂t||, not ∂P_k/∂X_{k-1}. Yet h_k contains ε_k^T P_dot_k ε_k, and P_dot_k includes the state-gradient terms. For P_k(X_{k-1},t), those terms bring in x_dot_{k-1}, hence ε_k and the virtual controller, producing cubic-in-ε_k terms. The claimed bound h_k ≤ ||ε_k||^2 θ_k Φ_k + const (e.g., (30)) does not follow. A simple N=2 scalar example with P_2 = x_1^2 + 1 satisfies Assumptions 1–4 but makes h_2 contain x_1 ε_2^2(ε_2 + a_1), which no such bound can dominate for large ||ε_2||. So as written, Theorem 1 is not established. This is fixable—add a bound on the state gradient, or restrict P_k to be time-dependent only—but it narrows the claimed generality and needs to be stated.\n\nLower-severity issues: G_N is overloaded (n×m P_N g_N ρ in Step N, n×n symmetric combination in Assumption 3), and the proof around (50) mixes the two. The output dimension is also sloppy: y* is stated in R^m though e = x_1 − y* ∈ R^n. The 'readily shown' bounds, especially for h_N, should be expanded.\n\nCredit where due: the decomposition g_i = A b_i with known allocation A and unknown b_i is a clean way to handle n < m; the unified symmetric/asymmetric performance via the transformation from [15] is competently embedded; the simulations are relevant and not obviously cherry-picked. No code is shipped, so reproducibility is limited to the reported plots.\n\nMy bottom line: this deserves a serious referee, but not acceptance in the current form. The missing gradient bound is load-bearing, and the authors should either prove it under a stated assumption or narrow the class of admissible P_k. If the fix is clean, the paper is a solid methods contribution for prescribed-performance control of over-actuated systems.","headline":"A genuinely useful extension of relaxed-controllability PPC to non-square systems with actuator faults, but the main theorem's proof has a load-bearing gap: Assumption 4 does not bound the state gradient of the auxiliary matrices, so the claimed h_k bounds do not follow.","tokens_in":19225,"tokens_out":6028,"would_cite":false,"duration_ms":60664,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C40","93C10","93C35","93B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that one adaptive backstepping control law can keep tracking errors within prescribed asymmetric bounds for non-square strict-feedback systems with unknown time-varying gains and actuator faults, under a relaxed…","keywords":["non-square MIMO nonlinear systems","prescribed performance control","relaxed controllability condition","actuator faults","adaptive backstepping","over-actuated systems","unknown time-varying control gain"],"falsifier":"For the reaction-wheel spacecraft with $A=D$, choose $\\rho(t)$ varying within $(0,1]^4$ so that for every positive definite $P(t)$ the matrix $P g_N\\rho A^T+A\\rho g_N^T P$ is indefinite at some time, then run the controller (46); if the tracking error leaves the funnel while Assumptions 1–4 are satisfied, the central claim is false.","tokens_in":18095,"feed_emoji":"🎛️","tokens_out":11524,"duration_ms":112862,"temperature":0.7,"pith_summary":"This paper aims to show that prescribed-performance tracking — keeping the output error inside time-varying asymmetric bounds — can be achieved for MIMO strict-feedback systems whose gain matrices are non-square, unknown, and time-varying, even when actuators partially lose effectiveness. It replaces classical controllability conditions such as \"gain matrix plus its transpose is positive definite\" with the existence of unknown auxiliary matrices $P_i$ that make certain symmetrized products uniformly sign-definite, and it builds an adaptive backstepping controller that never needs to know or invert those matrices. This matters because standard conditions fail for over-actuated plants such as spacecraft with reaction wheels, where four inputs drive three outputs and faults can destroy $g\\rho+\\rho g^T$ positive definiteness. If the theorem is right, the payoff is a single controller structure that delivers symmetric or asymmetric error funnels by tuning a few parameters, with no fault detector and no redesign when performance requirements change.","feed_headline":"One backstepping law enforces prescribed error bounds despite faults","feed_subtitle":"A relaxed controllability condition lets one law handle unknown gains and intermittent actuator faults.","key_machinery":"Two mechanisms carry the argument. The first is the matrix decomposition $g_i(X_i,t)=A b_i(X_i,t)$ with a known allocation matrix $A=[I_n,\\Lambda_1,\\dots,\\Lambda_{m-n}]$ for the over-actuated case $n<m$; this turns the non-square final gain into a square-like object and makes the actual control $u=-A^T\\|A\\|^{-1}(\\kappa_N\\varepsilon_N+\\hat\\theta_N\\Phi_N\\varepsilon_N)$ well defined without estimating $b_N$. The second is the error transformation $s_j=\\zeta_j/((\\delta_j+\\zeta_j)(\\bar\\delta_j-\\zeta_j))$ with $\\zeta_j=\\eta_j/\\varphi_j$ and $\\eta_j=e_j/\\sqrt{e_j^2+l_j^2}$, paired with the non-monotonic performance function $H(\\varphi)=l\\varphi^p/\\sqrt{1-\\varphi^2}$; boundedness of $s_j$ forces the tracking error to respect the funnel $H(-\\delta_j\\varphi_j)<e_j<H(\\bar\\delta_j\\varphi_j)$. The backstepping recursion uses $W\\varepsilon_1$ feedback at the first step to absorb the performance-induced scaling $W$, and \"core functions\" bound the unknown nonlinearities, the auxiliary matrices, and their derivatives so that adaptive estimators compensate them without knowing $P_i$.","core_discovery":"The central claim is Theorem 1: under Assumptions 1–4, the control law (46) together with virtual controllers (21), (32), (38) and adaptive laws (22), (33), (39), (47) keeps all closed-loop signals bounded and guarantees $H(-\\delta_j\\varphi_j(t))<e_j(t)<H(\\bar\\delta_j\\varphi_j(t))$ for each output channel, despite partial-loss-of-effectiveness actuator faults. The enabling relaxation is Assumption 3: rather than assuming $g_i$ or $g_i+g_i^T$ is positive definite, the paper assumes there exists an unknown symmetric positive definite $P_i$ (diagonal for the first block, depending only on $X_{i-1}$ for later blocks) such that $P_i g_i+g_i^T P_i$ for $i<N$, and $P_N g_N\\rho A^T+A\\rho g_N^T P_N$ for the last block, are uniformly positive definite with known sign. The paper demonstrates by two numerical examples that this condition can hold when the classical ones fail, and it embeds the unknown $P_i$ into the Lyapunov analysis rather than into the controller, so the control law never estimates or inverts $g_i$ or $P_i$.","pith_inferences":["The paper leaves open how to verify Assumption 3 for a new plant; a systematic construction or semidefinite-programming test for $P_i$ would turn the relaxed condition from an illustration into a design tool.","Restricting $P_k$ to depend only on $X_{k-1}$ avoids algebraic loops but excludes plants whose controllability is restored only by a $P_k$ depending on the current state; a loop-free recursive treatment of $P_k(X_k)$ would broaden the class.","The non-monotonic rate $\\beta(t)=\\exp(-\\gamma t)\\cos^2(t)$ temporarily loosens the funnel before tightening it; this may help with infeasible initial conditions, but it also delays the tightest guarantee and is worth an explicit transient-performance trade-off analysis.","The core-function bounding of $P_i$ and the nonlinearities is modular enough that the same relaxed-condition framework could plausibly be ported to output-feedback or event-triggered implementations, though the paper does not address either."],"forward_implications":["The same controller structure covers both square ($A=I_n$) and non-square ($A=[I_n,\\Lambda]$, $n<m$) plants by choosing the allocation matrix.","Symmetric and asymmetric performance envelopes are selected through $\\delta_j$, $\\bar\\delta_j$, and $\\varphi(0)$ without changing the control law or redoing the stability proof.","Partial-loss-of-effectiveness faults that make $g\\rho+\\rho g^T$ indefinite are tolerated as long as the relaxed $P_i$ condition holds, with no fault detection or diagnosis module.","Because the gain matrix is never estimated or inverted, the design avoids the singularity problems of matrix-inverse adaptive control and does not require $\\rho(t)$ to be continuous or differentiable.","All closed-loop signals are bounded for all time, and the ultimate bound is computable from the design parameters, so the method gives a tunable trade-off between transient accuracy and control effort."],"supporting_citations":[{"why":"It supplies the classical controllability condition that Assumption 3 generalizes.","marker":"[6]"},{"why":"It establishes the prescribed-performance framework for MIMO systems that the unified performance function extends.","marker":"[8]"},{"why":"It embeds the actuator effectiveness matrix in the Lyapunov function and requires differentiability, a restriction the paper removes.","marker":"[12]"},{"why":"It proposes the sector-condition relaxation $Kg+g^TK^T$ with a known design parameter, a special case of the auxiliary-matrix condition.","marker":"[13]"},{"why":"It provides the error transformation and Lemma 1 that convert bounded transformed errors into the prescribed tracking-error bounds.","marker":"[15]"},{"why":"It introduces the controllability relaxation for square systems that this paper extends to non-square systems with faults.","marker":"[17]"},{"why":"It supplies the reaction-wheel spacecraft model and constrained-attitude fault-tolerant design used as the non-square example.","marker":"[23]"},{"why":"It provides the actuator allocation matrix for spacecraft with redundant reaction wheels that motivates the decomposition $A=[I_n,\\Lambda]$.","marker":"[26]"}],"fun_headline_variants":["Relaxed controllability unifies performance for non-square nonlinear systems","One backstepping law handles faults and unknown gains under relaxed controllability","Non-square systems get unified prescribed performance via relaxed controllability","Relaxed condition enables unified fault-tolerant control for non-square systems","Unified control for non-square systems under relaxed controllability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 3, that unknown symmetric positive definite auxiliary matrices $P_i$ exist (depending only on earlier states) making $P_i g_i+g_i^T P_i$ and the fault-embedded final term uniformly sign-definite with known sign, a condition the paper illustrates on two examples but gives no general way to verify or construct.","fun_headline_variants_meta":{"raw":{"variants":["Relaxed controllability unifies performance for non-square nonlinear systems","One backstepping law handles faults and unknown gains under relaxed controllability","Non-square systems get unified prescribed performance via relaxed controllability","Relaxed condition enables unified fault-tolerant control for non-square systems","Unified control for non-square systems under relaxed controllability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000427,"raw_usage":{"total_tokens":2192,"prompt_tokens":959,"completion_tokens":1233,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":1145}},"tokens_in":575,"tokens_out":1233,"duration_ms":8672,"temperature":1.0,"reasoning_tokens":1145,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:40:56.073026+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the reaction-wheel spacecraft with $A=D$, choose $\\rho(t)$ varying within $(0,1]^4$ so that for every positive definite $P(t)$ the matrix $P g_N\\rho A^T+A\\rho g_N^T P$ is indefinite at some time, then run the controller (46); if the tracking error leaves the funnel while Assumptions 1–4 are satisfied, the central claim is false.","supporting_citations":[{"cited_title":"Robust adaptive control for a class of mimo nonlinear systems with guaranteed error bounds,","cited_arxiv_id":null,"evidence_quote":"It supplies the classical controllability condition that Assumption 3 generalizes."},{"cited_title":"Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed perfor- mance,","cited_arxiv_id":null,"evidence_quote":"It establishes the prescribed-performance framework for MIMO systems that the unified performance function extends."},{"cited_title":"Fault-tolerant output-constrained control of unknown euler-lagrange systems with prescribed tracking accuracy,","cited_arxiv_id":null,"evidence_quote":"It embeds the actuator effectiveness matrix in the Lyapunov function and requires differentiability, a restriction the paper removes."},{"cited_title":"Output feedback performance recovery in the presence of uncertainties,","cited_arxiv_id":null,"evidence_quote":"It proposes the sector-condition relaxation $Kg+g^TK^T$ with a known design parameter, a special case of the auxiliary-matrix condition."},{"cited_title":"Unifying performance specifications in tracking control of mimo nonlinear systems with actuation faults,","cited_arxiv_id":null,"evidence_quote":"It provides the error transformation and Lemma 1 that convert bounded transformed errors into the prescribed tracking-error bounds."},{"cited_title":"Asymptotic tracking con- trol for uncertain mimo nonlinear systems with guaranteed performance and enhanced controllability,","cited_arxiv_id":null,"evidence_quote":"It introduces the controllability relaxation for square systems that this paper extends to non-square systems with faults."},{"cited_title":"Fault-tolerant control for full-state error constrained attitude tracking of uncertain spacecraft,","cited_arxiv_id":null,"evidence_quote":"It supplies the reaction-wheel spacecraft model and constrained-attitude fault-tolerant design used as the non-square example."},{"cited_title":"Fault-tolerant reduced- attitude control for spacecraft constrained boresight reorientation,","cited_arxiv_id":null,"evidence_quote":"It provides the actuator allocation matrix for spacecraft with redundant reaction wheels that motivates the decomposition $A=[I_n,\\Lambda]$."}],"review_version":1}