RK-CBC: Robust Krasovskii Control Barrier Certificates

• The paper presents a novel data-driven framework that synthesizes robust safety controllers for time-delayed polynomial systems using sum-of-squares programming.
• The methodology leverages Krasovskii functionals to aggregate current and delayed state variables, reducing complexity and enabling explicit treatment of delays and disturbances.
• Empirical benchmarks on academic, jet-compressor, and spacecraft systems demonstrate that RK-CBC maintains safety over extended trajectories with minimal data.

Robust Krasovskii Control Barrier Certificates (RK-CBC) are a formalism for enforcing infinite-horizon safety in discrete-time, input-affine, polynomial control systems with time-invariant delays, norm-bounded disturbances, and unknown dynamics. By combining the concept of control barrier certificates (CBC) with Krasovskii-type functionals, RK-CBC addresses the synthesis of robust safety controllers from input-state trajectories, using a data-driven sum-of-squares (SOS) programming approach. This aggregation of current and delayed state variables enables explicit treatment of delays and disturbances without requiring an explicit system model, thus allowing for robust, model-free safety policy synthesis. The following exposition details the mathematical framework, synthesis methodology, computational workflow, and reported performance for RK-CBC, as established in "A Data-Driven Krasovskii-Based Approach for Safety Controller Design of Time-Delayed Uncertain Polynomial Systems" (Akbarzadeh et al., 28 Jan 2026).

1. System Model and Formal Safety Specification

The RK-CBC framework considers discrete-time, input-affine uncertain polynomial systems with time-invariant delays and norm-bounded disturbances, described as dt-IAUPS-td systems. The system trajectory is governed by

$$x(k+1)=A_1\mathcal{M}(x(k)) + A_2\mathcal{M}(x(k-h)) + B\mathcal{G}(x(k), x(k-h))u(k) + w(k),$$

where $h \in \mathbb{N}^+$ is the delay, $\mathcal{M}(x)$ and $\mathcal{G}(x,x_h)$ are (unknown) monomial dictionaries of known maximal degree, $u(k)$ is the control input in a compact set $\mathcal{U}$, and $w(k)$ is an unknown disturbance with $\|w\|^2\le\delta$. The state history at time $k$ is denoted $$\mathbf{x}(k) = (x(k), x(k-1), \dots, x(k-h)) \in \mathcal{X}^{h+1}$$.

The safety specification requires that, for any initial state-history $\mathbf{x}(0)\in\mathcal{X}_a^{h+1}$ and any admissible disturbance sequence, the trajectory never enters the Cartesian unsafe set $$\mathcal{X}_b\times (\mathcal{X}\setminus\mathcal{X}_b)^h$$, where $\mathcal{X}_a\subset\mathcal{X}$ (safe), $\mathcal{X}_b\subset\mathcal{X}$ (unsafe), and $\mathcal{X}_a\cap \mathcal{X}_b=\emptyset$.

2. Robust Krasovskii Control Barrier Certificate: Definition and Theoretical Properties

An RK-CBC is a functional $$\mathcal{B}:\mathcal{X}^{h+1}\rightarrow \mathbb{R}_{\ge0}$$, parameterized by scalars $\beta>\eta>0$, $\lambda\in(0,1)$, and $\gamma>0$ (with $\gamma\delta\le (1-\lambda)\beta$), satisfying:

• Initial set: $\mathcal{B}(\mathbf{x})\le\eta$ for all $\mathbf{x}\in\mathcal{X}_a^{h+1}$.
• Unsafe set separation: $\mathcal{B}(\mathbf{x})\ge\beta$ for all $$\mathbf{x}\in\mathcal{X}_b\times (\mathcal{X}\setminus\mathcal{X}_b)^h$$.
• Decrease condition: For any $\mathbf{x}$ with $\mathcal{B}(\mathbf{x})<\beta$, there exists $u\in\mathcal{U}$ such that for all $w\in\mathcal{W}(\delta)$,

$$\mathcal{B}(\mathbf{x}(k+1)) - \lambda\mathcal{B}(\mathbf{x}(k)) \le \gamma \|w\|^2.$$

A control policy maintaining this condition is termed a robust safety controller (R-SC). The associated infinite-horizon safety theorem guarantees that, starting from any initial safe state-history, the trajectory remains forever outside the unsafe set when governed by an R-SC.

3. Data-Driven Synthesis via Sum-of-Squares Programming

Because the system matrices and monomial dictionaries are unknown, RK-CBC and R-SC synthesis is achieved directly from collected trajectories using a sequence of SOS and semidefinite relaxations:

• Data Lifting: Collect a trajectory of input-state data of length $T$, constructing $X_-$, $X_+$, $X_h$, $U_-$, $M_-$, $M_h$, $G$ using maximal degree information. Here, $M_-=[\mathcal{M}(x(0)),\ldots,\mathcal{M}(x(T-1))]$, $$G=\left[\mathcal{G}(x(k),x(k-h))u(k)\right]_{k=0}^{T-1}$$, etc.
• Certificate Candidate: Restrict to quadratic-Krasovskii certificates:

$$\mathcal{B}(\mathbf{x}) = x^\top P x + \kappa\sum_{i=1}^h \lambda^i x_i^\top P x_i$$

for $P\succ0$, $\kappa,\lambda\in(0,1)$.

• Feedback Parameterization: Assume $$u = \widetilde{F}_1(x,x_h) x + \widetilde{F}_2(x,x_h) x_h$$, leveraging a polynomial factorization $\mathcal{M}(x)=L(x)x$ and defining regressor matrices for state and delay.
• Decrease Condition as LMI: Substitute this structure into the one-step difference condition; obtain a matrix inequality (in variables $P^{-1}$, $\widetilde{F}_1$, $\widetilde{F}_2$) that is affine in the unknown constants but tractable via data-driven S-procedure and SOS relaxation.
• Imposing Set Conditions: Use semialgebraic representations for $\mathcal{X}_a$, $\mathcal{X}_b$, and $\mathcal{X}$, enforcing that $x^\top P x$ satisfies the initial and unsafe set bounds inside $\mathcal{X}_a$, $\mathcal{X}_b$ via SOS multipliers.
• SOS Program: Collect all decision variables and define the final SOS feasibility problem, including positivity, spectral, and structural constraints. The D-invariant $P$ and gain maps $F_1, F_2$ yield the certified controller $$u(k)=F_1(x(k),x(k-h))\,x(k)+F_2(x(k),x(k-h))\,x(k-h)$$.

4. Computational Workflow and Complexity Analysis

The synthesis procedure follows these steps:

1. Collect a state-input trajectory under arbitrary excitation and construct lifted monomial data.
2. Initialize scalar hyperparameters ($\kappa$, $\lambda$, $\mu_1$, $\mu_2$).
3. Formulate the SOS program as specified, using tools such as SOSTOOLS and an SDP solver (e.g., MOSEK).
4. Solve for the RK-CBC and the robust safety controller parameters.
5. Invert $P^{-1}$, compute controller gains, determine $\gamma=(1+1/\mu_1+1/\mu_2)\|P^{1/2}\|^2$, and select $(\eta,\beta)$ to satisfy the robust decrease-to-bound.

The computational bottleneck is the SOS-based SDP in $(x,x_h) \in \mathbb{R}^{2n}$ with polynomial degree dictated by monomials and system delay. The Krasovskii functional reduces the complexity from $(h+1)$-fold cross products to at most two-fold, improving tractability when $h>1$. Problems are typically tractable for $n\leq4$ and moderate polynomial degree ($\leq4$); for higher dimensions, sparsity or compositionality can be exploited.

5. Empirical Results and Benchmarks

The effectiveness of the RK-CBC framework was demonstrated on three benchmarks, with all systems unknown to the designer and affected by delay and disturbance. Table 1 summarizes the main parameters and results.

System	$T$	$n$	Degree	$h$	$\delta$	$\mu_1$	$\mu_2$	$\kappa$	$\lambda$	$\gamma$	$\eta$	$\beta$
Academic (Sec 5.1)	10	2	2	3	$1.8\times10^{-3}$	0.59	0.92	0.38	0.94	28.28	36.41	40.43
Jet-compressor (5.2)	10	2	3	4	$8.0\times10^{-4}$	0.63	0.92	0.23	0.91	7.17	36.70	38.13
Spacecraft (5.3)	13	3	2	3	$1.2\times10^{-3}$	0.68	0.96	0.41	0.93	171.38	$1.07\times10^3$	$1.10\times10^3$

Case studies included an academic polynomial system ($n=2$, $h=3$), a jet-engine compressor ($n=2$, $h=4$), and a spacecraft attitude control system ($n=3$, $h=3$). Closed-loop simulations over multiple disturbance trajectories showed that certified controllers kept trajectories safe from entering the unsafe region over 50–100 steps (depending on benchmark). SOS solutions returned positive definite $P$ and polynomial feedback gains, with all RK-CBC conditions deterministically satisfied.

6. Practical Considerations and Limitations

The Krasovskii aggregation in the RK-CBC framework substantially decreases the Cartesian-product complexity of delayed state histories, which would otherwise grow combinatorially with $h$. The guarantee of safety is deterministic (confidence 1), and the data requirement is minimal: only a single sufficiently informative trajectory is necessary, without any assumption of i.i.d. data or repeated experiments.

However, the methodology is currently limited to polynomial systems of moderate state dimension and degree. Extension to rational or general nonlinear systems, or to larger-scale models, would require further structural assumptions or decomposition. Numerical tractability is influenced by the choice of monomial basis, relaxation degree in the SOS program, and SDP solver tolerances. Proper scaling enhances feasibility and can speed up computations.

7. References and Context

RK-CBC builds upon prior work on control barrier certificates (Prajna & Rantzer 2007; Ames 2019; Akbarzadeh et al. 2024), Krasovskii-type functionals for delay systems (Fridman 2014), and practical sum-of-squares toolchains (SOSTOOLS 2004). The approach directly addresses the gap in safety analysis for uncertain, time-delayed, input-affine polynomial systems by providing tractable, data-driven, robust certification procedures (Akbarzadeh et al., 28 Jan 2026).