Barrier Cascade Construction

• Barrier cascade construction is a methodology that composes local barrier functions into a global safety certificate for ensuring invariant safe sets in complex dynamical systems.
• It unifies sum-of-squares and synthesis techniques to enable scalable safety verification and controller design in applications from quadrotor flight to power converters.
• The approach leverages SOS programming, CEGIS, and automata-based synthesis to manage compositional and hierarchical safety in nonlinear, stochastic, and hybrid systems.

Barrier cascade construction is a methodology for synthesizing safety certificates and associated controllers for high-dimensional and interconnected dynamical systems by composing local barrier functions or certificates into a global structure, typically under small-gain-type conditions. This approach enables formal safety guarantees for cascades of systems—ranging from nonlinear control-affine systems with polynomial dynamics, to large-scale stochastic switched or impulsive hybrid networks—by decomposing global invariance or “reach-avoid” specifications into modular sub-barriers that are hierarchically or compositionally merged via max, sum, or weighted-combination rules. The paradigm unifies sum-of-squares and synthesis-based techniques and is foundational for scalable safety verification and controller design in complex multi-domain applications (Schneeberger et al., 2023, Anand et al., 2021, Bieker et al., 2024, Nejati et al., 2020, Jagtap et al., 2020, Nejati et al., 2020).

1. Foundations of Barrier Cascade Construction

At its core, barrier cascade construction leverages local control barrier functions (CBFs), control Lyapunov functions (CLFs), or broader “pseudo-barrier” certificates for subsystems, and organizes them hierarchically or compositionally into a certificate for the global, interconnected, or switched system. A “cascade” in this context denotes either:

• A sequence of nested safe sets enforced by multiple barrier functions on the same system, forming a sequence $$\{h_i \le 0\} \subset \{h_{i-1} \le 0\} \subset \cdots$$ (“barrier cascading” in control-naïve settings) (Schneeberger et al., 2023), or
• The compositional, often max- or sum-type, combination of subsystem barriers for interconnected networks (“compositional barrier cascade”) (Anand et al., 2021, Bieker et al., 2024, Nejati et al., 2020, Jagtap et al., 2020, Nejati et al., 2020).

Barrier functions $h(x)$ define forward-invariant sets $\{x: h(x) \le 0\}$, with suitable system- and control-dependent differential or difference inequalities ensuring invariance under closed-loop trajectories. The cascade principle is invoked to coordinate multiple such certificates, either for more nuanced invariance/safety requirements or to scale barrier synthesis in multi-component systems.

2. Barrier Cascades in Control-Affine Polynomial Systems

For control-affine polynomial systems with dynamics $\dot{x} = f(x) + G(x)u$, the approach involves constructing a CLF $V(x)$ for stabilization and a cascade of CBFs $h_1(x), ..., h_t(x)$ whose zero-sublevel sets define nested safe sets $X_{\mathrm{s},i} = \{h_i(x) \le 0\}$ (Schneeberger et al., 2023). The central invariance and stability constraints are:

• $\nabla V \cdot [f + G(u)] \le -\alpha(V)$ (ensuring asymptotic stability),
• $\nabla h_i \cdot [f + G(u)] + \gamma_i h_i \ge 0$ on $\{h_i=0\}$ (ensuring forward invariance of each $h(x)$0),
• Subset inclusions $h(x)$1 (cascade structure) achieved via Sum-of-Squares (SOS) constraints $h(x)$2.

All safety and stability certificates are enforced by a single rational controller $h(x)$3. The constraints—including subset inclusion within allowable sets and strict positivity of denominators—are encoded as SOS certificates derived via Positivstellensatz. The resulting algorithm alternates between CLF/CBF synthesis (fixing $h(x)$4 and searching for $h(x)$5) and controller fitting, typically converging in a small number of steps; canonical semidefinite solver tools (SOSTOOLS, YALMIP) are employed (Schneeberger et al., 2023).

3. Cascading Barriers in Multi-Level and Hierarchical Architectures

Cascaded barrier architectures extend to hierarchical controllers in nonlinear, underactuated, or hybrid systems. In quadrotor flight control, for instance, layered CBFs enforce altitude and lateral domain constraints independently by embedding barrier conditions into separate quadratic programs (QPs) for high-level (altitude/thrust) and low-level (lateral/torques) loops. Each QP minimally perturbs the nominal reference, guaranteeing forward invariance of their respective domains; the intersection of the resulting super-level sets is shown to remain invariant under the composite input (Khan et al., 2019).

This physically motivated layered approach is a special case of barrier cascade construction, where the safety filter is recursively applied at multiple levels. The principle is to ensure that invariance at each level composes so that the intersection defines the global safe region, provided controller updates are sufficiently synchronized.

4. Compositional Cascade Methods for Large-Scale and Interconnected Systems

For interconnected, (possibly stochastic) networks, the barrier cascade paradigm is formalized via compositional construction. Each subsystem $h(x)$6 is assigned a local barrier or pseudo-barrier $h(x)$7 satisfying trajectory decay/containment properties with respect to its own dynamics and couplings. These sub-barriers are merged into a global barrier $h(x)$8 using max or weighted-sum compositions:

• Max-cascade: $h(x)$9,
• Sum-cascade: $\{x: h(x) \le 0\}$0.

Here, gains/scaling functions $\{x: h(x) \le 0\}$1 and weights $\{x: h(x) \le 0\}$2 are derived based on small-gain arguments, guaranteeing that cyclic gain compositions remain $\{x: h(x) \le 0\}$3—a direct application of nonlinear small-gain theory (Anand et al., 2021, Bieker et al., 2024, Nejati et al., 2020, Jagtap et al., 2020, Nejati et al., 2020). Under these compositional conditions, the constructed $\{x: h(x) \le 0\}$4 serves as a valid global barrier certificate, ensuring that if every local barrier prevents a local “unsafe transition,” then the interconnected cascade remains globally safe.

For switched, hybrid, or impulsive systems, extra care is taken to accommodate switching signals, jump maps, or dwell-time constraints—usually through augmented barrier certificates and adjusted small-gain conditions (Nejati et al., 2020, Bieker et al., 2024). Probabilistic systems require corresponding stochastic barrier functions and martingale-type inequalities to certify bounded exit probabilities over finite horizons (Nejati et al., 2020, Anand et al., 2021).

5. Automata-Based and Logic-Guided Cascade Construction

Complex specifications (e.g., temporal logic, co-Büchi or DFA-type acceptance) can be systematically decomposed into finite collections of reach-avoid subproblems by automata-theoretic analysis. For each segment $\{x: h(x) \le 0\}$5 in the run-tree of the complement automaton, a local reach-avoid task is identified and a corresponding barrier (certificate) is synthesized. These certificates are then composed with small-gain-based cascade rules as above.

A switching policy—typically governed by a small, product automaton (“switching DFA”)—selects, at runtime, the correct local controller as the system trace traverses the automata states. This methodology yields hybrid policies with provable global probabilistic safety guarantees; the overall safety probability is bounded by the product or sum-product of the local segment probabilities (Anand et al., 2021, Nejati et al., 2020, Jagtap et al., 2020).

6. Computational Synthesis and Heuristics

Barrier cascade construction is computationally realized via a combination of SOS programming and Counter-Example Guided Inductive Synthesis (CEGIS). The steps are:

• Choose a parametric ansatz (e.g., degree, template) for each local barrier/certificate and feedback policy,
• Formulate the certificate and invariance/decay/safety conditions as a system of polynomial or semi-algebraic (SOS) inequalities,
• For SOS, pose as a semidefinite program and solve numerically,
• For CEGIS, iteratively alternate between candidate parameter optimization and search for counterexamples (SMT-based), refining the candidate until all conditions are met on a sufficiently large finite set.

For cascade architectures, local synthesis is performed first; composite gains/matrix scalings are then computed and checked for small-gain admissibility. The approach is well-suited for scalability, supporting implementation on high-dimensional or large-scale multi-agent systems (Schneeberger et al., 2023, Anand et al., 2021, Nejati et al., 2020, Jagtap et al., 2020, Nejati et al., 2020, Bieker et al., 2024).

7. Applications and Case Studies

Barrier cascade constructions have been demonstrated across control theory and networked systems:

Application	Cascade Method	System Type
Nonlinear power converter	Nested SOS/Cascade	Polynomial control-affine
Quadrotor flight control	Cascaded QPs	SE(3), underactuated
Room temperature regulation (large)	Max/Sum cascade	Hybrid/discrete interconnect
SIR epidemiology (multi-region)	Max-cascade	Switched-impulsive
Kuramoto oscillator network	Sum-cascade	Stochastic hybrid

In these examples, barrier cascade construction proved critical to achieving tractable safety certificate and controller design where global direct approaches would be infeasible or overly conservative (Schneeberger et al., 2023, Khan et al., 2019, Jagtap et al., 2020, Nejati et al., 2020, Anand et al., 2021, Bieker et al., 2024).

• "SOS Construction of Compatible Control Lyapunov and Barrier Functions" (Schneeberger et al., 2023)
• "Barrier Functions in Cascaded Controller: Safe Quadrotor Control" (Khan et al., 2019)
• "From Small-Gain Theory to Compositional Construction of Barrier Certificates for Large-Scale Stochastic Systems" (Anand et al., 2021)
• "Compositional Construction of Barrier Functions for Switched Impulsive Systems" (Bieker et al., 2024)
• "Compositional Construction of Control Barrier Functions for Continuous-Time Stochastic Hybrid Systems" (Nejati et al., 2020)
• "Compositional Construction of Control Barrier Functions for Interconnected Control Systems" (Jagtap et al., 2020)
• "Compositional Construction of Control Barrier Certificates for Large-Scale Stochastic Switched Systems" (Nejati et al., 2020)