Discrete-time Control Barrier Functions

• Discrete-time Control Barrier Functions are Lyapunov-inspired constraints that ensure system trajectories remain within predefined safe sets.
• They enable real-time implementation by synthesizing tractable affine and mixed-integer constraints for applications like lane keeping and obstacle avoidance.
• Their composable Boolean architecture provides necessary and sufficient conditions for safety in cyber-physical systems without excessive conservatism.

Discrete-time Control Barrier Functions (DTCBFs) are Lyapunov-like functions that provide necessary and sufficient conditions for the controlled invariance of safe sets in discrete-time dynamical systems. They form a rigorous foundation for safety-critical digital control, synthesizing tractable constraints that guarantee the existence of admissible control inputs keeping system trajectories within predetermined safe sets at each time step. This framework is crucial for real-time implementation in robotics, automotive safety, and other cyber-physical systems governed by digital controllers.

1. Mathematical Definition and Controlled Invariance

In discrete time, consider a system of the form

$x_{k+1} = f(x_k) + \tilde{g}(x_k) u_k,$

where $x \in \mathbb{R}^n$, $u \in U$ (admissible control set), $f$ and $\tilde{g}$ are nonlinear functions, and the safe set is defined by

$S = \{ x \in \mathbb{R}^n : h(x) \geq 0 \}$

for some real-valued function $h$. The core DTCBF condition is

$$\sup_{u \in U} h(f(x) + \tilde{g}(x) u) \geq 0 \qquad \forall x \in S,$$

ensuring that from any point in the safe set, there exists at least one admissible control preserving invariance of $S$ at the next step. This condition is both necessary and sufficient for forward controlled invariance in discrete time, in contrast to more conservative continuous-time relaxations. The set-valued map

$$K_S(x) = \{ u \in U : h(f(x) + \tilde{g}(x) u) \geq 0 \}$$

is the "safe input set," and it is maximized by formulating DTCBF conditions directly via the above invariance statement (Cavorsi et al., 2020).

2. DTCBF Construction for Partially Control Affine Systems

For systems partitioned as $x = (x_1, x_2)$, where $x_1$ evolves autonomously and $x_2$ is influenced by control,

$x_{k+1} = [ f_1(x_k); \, f_2(x_k) + g(x_k) u_k ]$

the DTCBF can be parameterized as

$h_A(x) = \mu(x_1) x_2 + \nu(x_1),$

with $\mu$ and $\nu$ nonlinear functions. The controlled invariance condition becomes

$$\sup_{u \in U} [ \mu(f_1(x))(f_2(x) + g(x)u) + \nu(f_1(x)) ] \geq 0.$$

Critically, this invariance constraint is affine in the control input $u$, facilitating tractable real-time optimization (Cavorsi et al., 2020). Such affine constraints enable deployment in quadratic programs (QPs) or mixed-integer QPs even when the state-dependence is nonlinear.

3. Boolean and Piecewise Compositions: Mixed-Integer Formulations

To accommodate complex safety specifications (e.g., multi-agent collision avoidance, obstacles with nonconvex safe sets), DTCBFs are composed using Boolean logic. The central encodings are:

• Conjunction ($\wedge$): Enforce $h_1(x) \geq 0$ and $h_2(x) \geq 0$, i.e., intersection of safe sets.
• Disjunction ($\vee$): Require at least one $h_i(x) \geq 0$; this is encoded via slack variables with SOS-1 (special ordered set of degree one) mixed-integer constraints.
• Negation ($\neg$): $-h_i(x) \geq 0$ encodes "not safe."
• Implication, XOR, equivalence are built via secondary compositions of the base logic atoms.

Piecewise DTCBFs arise for switched or mode-dependent dynamics. For a collection of partitions $(P_j)$, assign

$$\text{if } p_j(x) \leq 0 \text{ then } h_j(x) \geq 0,$$

which is encoded as an if-then via auxiliary mixed-integer constraints. The entire safety specification becomes a tractable mixed-integer linear or quadratic constraint system, suitable for real-time execution and allowing nonconvex safety regions (Cavorsi et al., 2020).

4. Real-Time Implementation and Affinity

The primary tractability result is that, for partially control affine systems and Boolean compositions, the DTCBF safety constraints are affine in $u$ (or piecewise-affine). This is essential for embedding into QPs or MIQPs solved reliably in real time. The safe input set

$$K_S(x) = \{ u \in U : h(f(x) + \tilde{g}(x) u) \geq 0 \}$$

is less conservative than alternative discrete-time CBF relaxations employing extra terms such as $(\gamma-1)h(x)$ or monotone comparison functions $\alpha(h(x)) - h(x)$. The direct controlled invariance approach maximizes the set of admissible controls and avoids sacrificing performance for safety (Cavorsi et al., 2020).

5. Applications: Lane Keeping and Obstacle Avoidance

DTCBFs have been applied to automotive safety-critical scenarios, focusing on digital implementations:

• Lane Keeping (LK): A practical DTCBF for LK is

$$h_{LK}(x) = \sqrt{2a_{max}(y_{max} - \operatorname{sgn}(v)y) + \tfrac{1}{4} a_{max}^2 t_s^2} - |v| - \tfrac{1}{2} a_{max} t_s$$

where $v$ is a lateral velocity expression. The affine safety constraint, coupled with acceleration constraints, is directly enforced in an MIQP, demonstrating enhanced lane centering compared to continuous-time approaches.

• Obstacle Avoidance (OA): Nonconvex safe sets due to obstacles are encoded via Boolean DTCBF compositions. Each sublane around an obstacle has its own DTCBF, and the disjunction

$$(h_{LK, \, left} \geq 0) \vee (h_{LK, \, right} \geq 0)$$

selects the safe sublane with the lowest control effort; the Boolean mixture is encoded as a mixed-integer constraint (Cavorsi et al., 2020).

6. Theoretical and Practical Implications

The DTCBF formalism provides:

• Necessity and sufficiency for controlled invariance: No unnecessary conservatism compared to relaxations.
• Composability: Support for complex, nonconvex, and piecewise safe sets, enabling broad applicability.
• Tractability: Affine and mixed-integer formulations with computational efficiency appropriate for digital controllers.
• Demonstrated effectiveness: In automotive simulations, DTCBF-based controllers provide better lane centering and more flexible obstacle avoidance than continuous-time CBF methods under digital (sampled-data) constraints.

This technical foundation paves the way for advanced safety-critical control synthesis in time-discretized frameworks, extending to robotics, autonomous vehicles, and general cyber-physical systems requiring hard safety guarantees at the digital implementation level. The formulation is amenable to future directions including learning-based synthesis (where invariance must be enforced on-the-fly), hybrid system safety, and distributed or multi-agent safety-critical control (Cavorsi et al., 2020).