Unified Performance Control for Non-Square Nonlinear Systems with Relaxed Controllability

Bing Zhou; Kai Zhao; Yongduan Song; Zhen Chen

REVIEW 1 major objections 5 minor 34 references

Unified Performance Control for Non-Square Nonlinear Systems with Relaxed Controllability

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.13252 v2 pith:5HAY7KTI submitted 2024-11-20 eess.SY cs.SY

Bing Zhou , Kai Zhao , Yongduan Song , Zhen Chen This is my paper · ORCID

classification eess.SYcs.SY MSC 93C4093C1093C3593B52

keywords non-squareMIMOnonlinearsystemsprescribedperformancecontrolrelaxedcontrollabilityconditionactuatorfaultsadaptivebacksteppingover-actuatedunknowntime-varyinggain

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $P_i$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $P_i$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

But 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.

Lower-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.

Credit 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.

My 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.

Referee Report

1 major / 5 minor

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).

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 (1)

[§IV (Theorem 1 proof), Eq. (30)] 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).

minor comments (5)

[Assumption 2] 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.
[Step N, Eqs. (43), (50)-(51)] 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.
[Theorem 1 statement] 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.
[Eq. (52)] 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.
[Remark 6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1's result is conditional on explicitly stated Assumptions 1–4; the auxiliary matrices P_i are neither fitted nor used in the control law, and the cited transformation lemma is external support.

full rationale

The paper's main theorem is a conditional statement: under Assumptions 1–4, the control laws (21), (32), (38), and (46) keep all closed-loop signals bounded and keep the tracking error within the prescribed bounds H(−δ_j φ_j(t)) < e_j(t) < H(δ̄_j φ_j(t)). The relaxed controllability condition is Assumption 3, an existence assumption on auxiliary matrices P_i such that P_i g_i + g_i^T P_i and the final G_N are uniformly sign-definite. The paper does not fit the P_i from data, and it explicitly states in Remark 1 that 'the auxiliary matrix Pi is only required for stability analysis, as opposed to needing to be present in the control design' and in Remark 6 that 'the auxiliary matrix Pi appears in the Lyapunov function Vi, it is not directly used in the control laws (21), (32), (38) and (46).' The adaptive estimators estimate unknown constants θ_i assembled from the core functions in Assumption 4; no quantity that is called a prediction is produced from a fitted input. The prescribed-performance conclusion uses Lemma 1, whose proof is cited to [15]. Although [15] shares an author with the present paper (K. Zhao), the lemma is a published, externally available transformation result, and it is not used to define the controllability condition or to exclude alternative designs; it is the standard fact that boundedness of the transformed error s_j implies the original error stays within the performance bounds. That self-citation is therefore not load-bearing in a circular sense. The skeptic's objection—that Assumption 4 bounds only P_k and ∂P_k/∂t, while ∂P_k/∂X_{k−1} also contributes to Pdot_k in the Lyapunov derivatives—identifies a possible missing hypothesis or proof gap in the boundedness analysis, but it is not a circular reduction of the theorem's conclusion to its own inputs. No derivation step in the paper is, by construction, equivalent to an input or fitted parameter renamed as a prediction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The paper's central claim rests on four stated assumptions and one cited lemma. The key burden is Assumption 3, which postulates the existence of unknown auxiliary matrices P_i; this is a structural condition on the unknown gain and fault matrices that cannot be verified from the data. The design parameters (l, gamma, phi_0, phi_f, delta, delta_bar) are user-chosen performance specifications, not fitted parameters. No physical entities are introduced beyond the analysis-only matrices P_i.

free parameters (4)

l = 0.9936 in simulations
Shape parameter of the performance function H(phi)=l phi/sqrt(1-phi^2) and of the error transformation; user-selected, affects the size of the performance bound.
gamma = 0.9 in simulations
Decay rate in beta(t)=exp(-gamma t)cos^2(t); user-selected.
phi_0 and phi_f = 1 and 0.1 in simulations
Initial and final values of the scaling function phi(t); user-selected performance envelope.
delta_j and delta_bar_j = 1, or 0.8/1 and 1/0.6 in simulations
Asymmetry parameters in the performance bounds; user-selected.

assumptions (5)

domain assumption Assumption 1: The desired trajectory y* and its derivatives up to nth order are known and bounded; state vector x_i is available for control design.
Stated in Section II; standard for output-feedback-free backstepping.
domain assumption Assumption 2: The control gain matrix g_i can be decomposed as g_i = A b_i with known full-row-rank allocation matrix A = [I_n, Lambda_1,...,Lambda_{m-n}] and unknown b_i.
Stated in Section II; restricts to over-actuated systems n<m and assumes a known allocation structure.
domain assumption Assumption 3: Existence of unknown symmetric positive definite matrices P_i such that P_i g_i + g_i^T P_i (i<N) and P_N g_N rho A^T + A rho g_N^T P_N are uniformly positive definite with known sign.
Stated in Section II; the key relaxed controllability condition that the whole stability proof relies on.
domain assumption Assumption 4: Existence of known nonnegative core functions phi_fi, phi_k1, phi_k2 and unknown constants bounding f_i+d_i, P_k, and partial P_k/partial t.
Stated in Section II; used to define the computable functions Phi_i in the adaptive laws.
standard math Lemma 1 of [15]: boundedness of the transformed variable s_j implies the prescribed performance bounds (3) for the tracking error.
Used in Section III-B and in the proof of Theorem 1; proof omitted and credited to [15].

invented entities (1)

Auxiliary Lyapunov matrices P_i(t)
purpose: To guarantee the relaxed controllability condition through the positive definiteness of P_i g_i + g_i^T P_i and P_N g_N rho A^T + A rho g_N^T P_N; they appear in the Lyapunov functions but not in the controller.
Postulated in Assumption 3; not measured or estimated, so their existence is an unverified structural assumption.

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Cite this review

Pith. "Pith review of Unified Performance Control for Non-Square Nonlinear Systems with Relaxed Controllability." pith.science (2026). https://pith.science/paper/5HAY7KTI

@misc{pith2026241113252,
  author       = {Pith},
  title        = {Pith review of: Unified Performance Control for Non-Square Nonlinear Systems with Relaxed Controllability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5HAY7KTI}},
  note         = {Machine review of arXiv:2411.13252}
}

read the original abstract

In this paper, we investigate the problem of unified prescribed performance tracking for a class of non-square strict-feedback nonlinear systems under relaxed controllability conditions. By using a skillful matrix decomposition and introducing some feasible auxiliary matrices, a more generalized controllability condition than the current state of the art is constructed, which can be applied to both square and non-square nonlinear systems subject to actuator faults and unknown yet time-varying control gain. Incorporating the relaxed controllability conditions and the uniform performance specifications into the backstepping design procedure, a prescribed performance fault-tolerant controller is developed that can achieve different performance demands without modifying the controller structure, which is more flexible and practical.In addition, the destruction of the system stability by unknown controllability auxiliary matrices and unknown nonlinearities is circumvented by embedding the available core information of the state-dependent uncertainties into the design procedure. Both theoretical analysis and numerical simulation demonstrate the effectiveness and benefits of the proposed method.

Figures

Figures reproduced from arXiv: 2411.13252 by the authors.

**Figure 1.** The evolution of ej with different initial conditions, i = 1, 2, 3. 0 2 4 6 8 10 Time(s) -2 0 2 0 2 4 6 8 10 Time(s) -1 0 1 0 2 4 6 8 10 Time(s) -2 -1 0 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Control input signals under x1(0) = [0.5, 0.5, 0.5]T [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 4.** Control input signals under x1(0) = [1, 0.5, 0.5]T . The key design parameters are selected as δj = ¯δj = 1, j = 1, 2, 3. The control parameters are given as: κ1 = 1, σ1 = 0.01 and µ1 = 0.1. Since configuration matrix D can also be interpreted as the actuator allocation matrix, we set A = D. Similarly, it can be verified that gρDT + DρgT is not uniformly positive definite or negative definite for all t ∈ (0, ∞], but… view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: The evolution of ej under δj = 0.8, ¯δj = 1, i = 1, 2, 3. 0 2 4 6 8 10 Time(s) -1 -0.5 0 0.5 0 2 4 6 8 10 Time(s) -1 -0.5 0 0.5 0 2 4 6 8 10 Time(s) -1 -0.5 0 0.5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: The evolution of ej under δj = 1, ¯δj = 0.6, i = 1, 2, 3. VI. CONCLUSION In this paper, a unified prescribed performance tracking control method with generalized controllability condition is proposed for a class of non-square nonlinear strict-feedback systems with actu…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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