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Safe Stabilising Full-Order Affine Control Barrier Functions for Linear Systems (Extended)

Faisal Lawan, Joaquin Carrasco, Lanlan Su

arXiv:2609.12990v1eess.SYcs.SY

TL;DR

Safety filters can destabilise a closed loop through switching even when the nominal and filtered modes are individually stable, motivating a stability analysis for full-relative-degree affine constraints. The paper characterises common quadratic stability through a positive-real test and derives designs and synthesis conditions; the filter spectrum is set by barrier tuning, and the framework includes an explicit inactive-filter controller.

  • Problem

    A minimally invasive safety filter can create state-dependent switching, and existing results do not directly characterise common quadratic stability for full-relative-degree constraints.

    Source: §I · p. 1

  • Method

    The paper uses full-relative-degree structure to analyse the filtered mode, characterize common quadratic Lyapunov functions through a strongly strictly positive real polynomial ratio, and derive a storage function and reduced LMI condition.

    Source: §I · p. 2, §VI · p. 8

  • Results

    The filtered-mode spectrum is fixed by barrier tuning, and common quadratic certification is equivalent to the polynomial ratio being SSPR, yielding a scalar frequency-domain test and an explicit interval of admissible gains.

    Source: §VI · p. 8

  • Takeaways & Limitations

    For a prescribed nominal controller, the frequency-domain condition gives a scalar test for whether the nominal and filtered modes admit a common quadratic certificate.

    Source: §VI · p. 8

    Failure of the SSPR condition means no common quadratic certificate exists, but global exponential stability may still hold under state-dependent switching.

    Source: B. The Equivalence · p. 6

Abstract

from arXiv · show

Control barrier function safety filters enforce constraints by modifying a nominal input, but the resulting switching can destabilise the closed loop even when the nominal and filtered modes are individually stable. This paper presents a design framework for safe and globally exponentially stabilising controllers for linear systems with a single full-relative-degree affine constraint. We show that the filtered-mode spectrum is fixed by the barrier tuning and is independent of the plant, nominal controller, and quadratic-program weighting. This structure yields an explicit nominal controller for which the safety filter remains inactive everywhere. For a prescribed nominal controller, we prove that the nominal and filtered modes admit a strong common quadratic Lyapunov function if and only if the ratio of the nominal characteristic polynomial to the barrier polynomial is strongly strictly positive real. This equivalence characterises the existence of a common quadratic Lyapunov and provides a scalar frequency-domain test, along with an explicit interval of admissible gains. Building on these results, the extended analysis derives an explicit common storage function and reduces an existing LMI synthesis condition to a feasibility test in a single matrix variable. A flexible two-mass example explains a known instability mechanism and demonstrates how the proposed design restores safety and global exponential stability.

I. INTRODUCTION

Safety filters can preserve constraints while altering nominal control, and switching between individually stable modes can still destabilise the closed loop. This paper addresses whether stability is preserved when a minimally invasive filter wraps a nominal controller, focusing on the full-relative-degree case and common quadratic certificates.

I. INTRODUCTION

  • Stable nominal and filtered modes can still produce a diverging closed loop through repeated switching, so individual-mode stability is insufficient.The paper studies stability preservation when a minimally invasive safety filter is wrapped around a nominal controller.Source: §I · p. 1
  • The existing LMI and stability analysis for general relative degree does not directly characterise which full-relative-degree nominal controllers admit a common quadratic certificate.In this case, the invariant-zero requirement is automatically satisfied, leaving the interaction between nominal and filtered modes as the stability question.Source: §I · p. 1
  • For full relative degree, the constrained output and its derivatives describe the complete state, leaving no internal dynamics.This setting includes integrator chains constrained on the terminal output and the flexible two-mass system.Source: §I · p. 1
  • The paper characterises common quadratic certification through a strongly strictly positive real polynomial ratio and derives an explicit interval of admissible nominal gains.It also gives an inactive-filter design and reduces the existing LMI synthesis condition to one matrix variable.Source: §I · p. 2

C. Positive Realness and Passivity

The section connects positive-real conditions to passivity and sets out the linear-system safety-filter construction. The filter yields nominal and filtered modes while preserving forward invariance of the safe set.

C. Positive Realness and Passivity

  • A transfer function is SSPR when it is analytic in the closed right half-plane and its real part is uniformly bounded below by a positive constant on the imaginary axis.For a minimal realization, SSPR is equivalent to the associated system being very strictly passive.Source: C. Positive Realness and Passivity · p. 2
  • The SCQLF condition in the auxiliary feedback analysis is interpreted as strict positive realness of the return difference 1 + λg.The condition applies to the return difference, not simply to the loop transfer function λg.Source: C. Positive Realness and Passivity · p. 2

D. Safety Filters for Linear Systems and Affine Constraints

  • The quadratic-program filter modifies a nominal linear controller to enforce the affine constraint, producing nominal and filtered modes.The filter is locally Lipschitz, and the safe set is forward invariant.Source: D. Safety Filters for Linear Systems… · p. 3
  • The analysis assumes the origin lies in the interior of the safe set and that the affine constraint has relative degree equal to the system order.Under full relative degree, the input pair is stabilisable.Source: D. Safety Filters for Linear Systems… · p. 3

III. SAFETY FILTERS OF FULL RELATIVE DEGREE

Full relative degree gives a complete-state coordinate system from the constrained output and its derivatives. This structure supplies input-output identities used in the subsequent analysis.

III. SAFETY FILTERS OF FULL RELATIVE DEGREE

  • Full relative degree lets the constrained output and its derivatives serve as coordinates for the full state, removing internal zero dynamics from the stability conditions.The resulting identities support the later analysis of the filtered mode.Source: §III · p. 3, A. Minimality and Rank of the Input… · p. 3

A. Minimality and Rank of the Input Matrix

  • The observability matrix is invertible, the pair (A, B) is controllable, and c⊤(sI − A0)−1B = Λ/pA0(s).The transfer-function identity is among the input-output relations established under full relative degree.Source: A. Minimality and Rank of the Input… · p. 3
  • In the proof, the observability and controllability matrices form an invertible product, establishing observability and controllability for a selected input direction.The argument then establishes controllability of (A, B) because that direction lies in the range of B.Source: A. Minimality and Rank of the Input… · p. 3
  • Full relative degree makes the invariant-zero condition of the earlier stability result automatic, leaving its negative-real-eigenvalue condition.The auxiliary system has transfer function θ/pA0(s) and no finite invariant zeros.Source: A. Minimality and Rank of the Input… · p. 3

B. The Filtered Mode

The filtered mode is similar to a companion matrix whose spectrum is set by the barrier polynomial. Under full relative degree, the barrier slopes determine the entire active-filter vector field, independently of the nominal gain and QP weight.

B. The Filtered Mode

  • The filtered-mode matrix is similar to the companion matrix Γ, so its spectrum is the barrier roots {−α_i}, and it depends on neither K nor G.The matrix identity also expresses the filtered mode through the observability coordinates.Source: B. The Filtered Mode · p. 4
  • When relative degree equals system order, the barrier slopes account for all filtered-mode eigenvalues and activating the filter fixes the entire vector field.For lower relative degree, the remaining eigenvalues can depend on the plant and nominal gain.Source: B. The Filtered Mode · p. 4
  • The absence of a positive real eigenvalue rules out a sufficient condition for unbounded trajectories, but other mechanisms may still produce them.The paper notes that the condition is sufficient, not necessary, and points to a later example of another mechanism.Source: B. The Filtered Mode · p. 4
  • The filtered mode is Hurwitz for every nominal gain and QP weight, but this alone does not ensure stability of the complete closed loop.The flexible two-mass example has both modes Hurwitz while instability is attributed to their interaction under persistent switching.Source: B. The Filtered Mode · p. 4

C. Safety by Design

Choosing the input-output-linearising gain makes the nominal and filtered modes coincide, so the filter stays inactive while the safe set remains forward invariant and the origin is globally exponentially stable.

C. Safety by Design

  • Setting the nominal gain to K⋆ removes switching by making the nominal and filtered modes coincide.The filtered gain is the input-output-linearising gain; this construction is unavailable when relative degree r < n because the zero dynamics remain untouched.Source: C. Safety by Design · p. 4
  • The gain K⋆ makes the safety filter inactive everywhere, leaving a closed loop with poles at the negatives of the barrier class-K slopes.The resulting safe set is forward invariant and the origin is globally exponentially stable.Source: C. Safety by Design · p. 4
  • With α_i = ρ ᾱ_i, the required nominal gain grows as ∥K⋆(ρ)∥ = Θ(ρ^n), so aggressive barrier tuning can demand aggressive control.The next design analysis quantifies how far the nominal gain can move from this inactive-filter choice.Source: C. Safety by Design · p. 4

IV. A POSITIVE-REAL CHARACTERISATION

The analysis treats the filtered mode as a fixed reference and the nominal mode as a rank-one perturbation. An auxiliary feedback system expresses their relationship through a scalar return difference.

IV. A POSITIVE-REAL CHARACTERISATION

  • The nominal and filtered modes are rank-one perturbations of the reference matrix eA along the same input direction, differing only in output direction.This structure motivates testing whether the two modes share a quadratic Lyapunov function in the frequency domain.Source: §IV · p. 4, A. The Auxiliary Feedback System and… · p. 4

A. The Auxiliary Feedback System and its Return Difference

  • The auxiliary system uses w = −λy, with state matrix eA − λv1v3⊤; its return difference is 1 + λg, not merely the loop transfer function λg.The scalar return difference can be written as a ratio of characteristic polynomials; in the studied family, hK = 1 + λg.Source: A. The Auxiliary Feedback System and… · pp. 4–5
  • The transfer function g is fixed by the plant and barrier, while the plant, nominal gain, and QP weighting enter the related hK representation through pA0.The auxiliary realisation is minimal when none of the barrier poles −α_i is an eigenvalue of A0; this generic condition is assumed thereafter.Source: A. The Auxiliary Feedback System and… · pp. 4–5

B. The Equivalence

For a prescribed nominal controller, a common quadratic Lyapunov certificate for the nominal and filtered modes is equivalent to a positive-real condition on their return difference. The extended analysis supplies an explicit common storage function and applies the certificate to the switched closed loop.

B. The Equivalence

  • The nominal and filtered modes admit an SCQLF if and only if the return difference hK = pA0/φ is SSPR, equivalently satisfying the stated strict frequency-domain condition.When these equivalent conditions hold, A0 is Hurwitz and the auxiliary system is very strictly passive.Source: B. The Equivalence · p. 5
  • The storage function V(z) = z⊤Pz certifies both modes when P, L, W, and ε satisfy the stated equations.The same quadratic storage function certifies very strict passivity of the auxiliary system and serves as the common Lyapunov function.Source: B. The Equivalence · pp. 5–6
  • If pA0/φ is SSPR, the origin of the state-dependent safety-filtered system is globally exponentially stable.The proof applies the common Lyapunov inequalities to the continuous piecewise-affine closed loop.Source: B. The Equivalence · p. 6
  • Failure of the SSPR condition rules out an SCQLF, but does not by itself rule out global exponential stability of the state-dependent switched system.The switching is state-dependent and confined to R±, rather than arbitrary.Source: B. The Equivalence · p. 6

C. A Family of Nominal Gains

A one-parameter family interpolates from the inactive-filter gain to the open-loop plant, turning the positive-real condition into an explicit interval of admissible gains.

C. A Family of Nominal Gains

  • The family interpolates from the input-output-linearising gain at λ = 0 to zero nominal gain and the open-loop plant at λ = 1.Along this line, the return difference is hλ = 1 + λg.Source: C. A Family of Nominal Gains · p. 6
  • The extrema used to calculate the admissible interval come from ω = 0, real stationary frequencies, and the limiting value 0 as ω → ∞.Finite stationary frequencies solve N′D − ND′ = 0.Source: C. A Family of Nominal Gains · p. 6
  • The gain in the stated interval renders the origin globally exponentially stable; every nonzero gain outside it admits no SCQLF for the two modes.The interval is expressed using the minimum and maximum of Re{g(jω)} over frequency.Source: C. A Family of Nominal Gains · p. 6

D. Feasibility of the Design Inequalities

The section reduces the existing two-LMI controller synthesis to a feasibility test in Q alone. This determines whether that construction can produce a nominal controller without searching over its gain.

D. Feasibility of the Design Inequalities

  • For any feasible pair (Q, Y), the resulting gain is K = −YQ⁻¹ and P = Q⁻¹ defines an SCQLF for the nominal and filtered modes.The filtered-mode inequality is independent of the gain, so for a fixed Q it holds either for every Y or none.Source: D. Feasibility of the Design… · pp. 6–7
  • The coupled design inequalities are feasible exactly when some positive-definite Q satisfies both inequalities in (37).If such a Q exists, the second inequality holds for every Y, and the elimination lemma guarantees a Y satisfying the first.Source: D. Feasibility of the Design… · pp. 6–7
  • The Q-only test settles whether the synthesis of [6, Lem. 3] can produce a nominal controller at all.The positive-definite Q satisfying the first inequality form a nonempty open convex cone; a change of variables makes that inequality depend only on the barrier parameters and reusable across plants sharing O.Source: D. Feasibility of the Design… · pp. 6–7

V. THE FLEXIBLE TWO-MASS EXAMPLE

The flexible two-mass example applies the stability analysis to a system where the filtered mode is stable but the nominal-filter interaction still depends on barrier tuning. Its setup has full relative degree four, and the nominal mode is also Hurwitz for every positive tuning parameter.

V. THE FLEXIBLE TWO-MASS EXAMPLE

  • The results are applied to [8, Sec. III.B], which [6, Sec. V] identifies as a case where eA is Hurwitz while A0eA has negative real eigenvalues, so [6, Thm. 2] does not apply.Source: §V · p. 7

A. System Data

  • The two unit masses are coupled by a spring of stiffness 0.1 N m−1, with damping d = 0.01 m and the first mass position constrained.The barrier is h0(x) = p1 + d; the constraint has relative degree r = n = 4, and φ(s) = (s + α)^4.Source: A. System Data · p. 7

B. The Two Modes

  • For every α > 0, the filtered mode has spectrum {−α}, while the nominal mode has roots −0.4399±0.2418j and −0.1758±0.6049j.Thus both modes are Hurwitz for all α > 0, leaving the SSPR test of pA0/φ as the remaining common-quadratic-stability question.Source: B. The Two Modes · p. 7

C. The Positive-Real Test

For a prescribed nominal controller, the positive-real test determines whether the two modes admit an SCQLF, but failure of that certificate need not mean instability under state-dependent switching. In the two-mass example, redesign at α = 4 restores a safe, globally exponentially stable closed loop.

C. The Positive-Real Test

  • The positive-real test (43) holds below the threshold, with the violation first appearing at ω = 0.960; independently, a negative real eigenvalue appears at the same threshold.The value (44) agrees with the numerical finding in [8, Sec. IV] that an SCQLF exists for 0 < α < 1.Source: C. The Positive-Real Test · p. 7

D. What the Threshold Predicts

  • At α = 0.8 the trajectory converges after one filter activation; at α = 2 it still converges with persistent switching, and at α = 4 it diverges.The observed divergence threshold near α ≈ 3.25 matches [8, Sec. IV].Source: D. What the Threshold Predicts · p. 7, Figure 3 · p. 8
  • Loss of the SCQLF between α⋆ and α ≈ 3.25 does not itself imply instability, because switching is state-dependent and confined to R±.The conclusion identifies the gap between certificate failure and instability as a subject for future work.Source: D. What the Threshold Predicts · p. 8, §VI · p. 8

E. Redesigning the Nominal Gain

VI. CONCLUSIONS

  • The extended analysis derives an explicit common storage function and reduces existing LMI synthesis to a feasibility condition in one matrix variable.The storage function from the KYP factorisation is itself an SCQLF for the two modes.Source: §VI · p. 8
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