High-Relative-Degree Nonovershooting Constraints

• High-relative-degree nonovershooting constraints are safety specifications for dynamical systems where the control input influences the output via multiple derivatives to prevent overshoot.
• The methodology lifts the original constraint into a cascade of high-order barrier functions using class K functions to monitor successive derivatives and ensure forward invariance.
• These techniques are applied in adaptive cruise control, robotics, and aerospace by integrating control barrier functions with quadratic programming to maintain stringent safety guarantees.

High-relative-degree nonovershooting constraints are a class of safety specifications imposed on controlled dynamical systems where the control input first influences the output variable through multiple derivatives—i.e., the system’s relative degree is greater than one—and the constraint enforces that the output never “overshoots” a prescribed safe boundary. Formally, for an output $h(x)$ with relative degree $m>1$ in a control-affine system $\dot x = f(x) + g(x)u$, the nonovershooting requirement demands $h(x(t)) \ge 0$ (or, alternatively, $h(x(t)) \le 0$ depending on convention) at all times, even as the system responds to initial conditions, disturbances, or reference signals. Such constraints present technical challenges since standard control barrier functions (CBFs) or zeroing CBFs (ZCBFs) are inadequate when the input does not enter the first derivative of the constraint function. The design and enforcement of high-relative-degree nonovershooting constraints is a developing area in nonlinear control, robotic safety, automotive systems, and safety-critical reinforcement learning.

1. Mathematical Foundations and the Barrier Cascade Construction

The central methodological insight for high-relative-degree constraints is to “lift” the original output constraint $h(x)\ge 0$ into a chain or cascade of auxiliary barrier functions (high order barrier functions, HOBFs) that monitor successive derivatives up to the system relative degree. For a $C^m$ function $h:\mathbb{R}^n \to \mathbb{R}$ with relative degree $m$, define:

$$\begin{aligned} \psi_0(x,t) &= h(x,t), \ \psi_1(x,t) &= \frac{\partial \psi_0}{\partial t} + L_f \psi_0 + \alpha_1(\psi_0), \ \psi_2(x,t) &= \frac{\partial \psi_1}{\partial t} + L_f \psi_1 + \alpha_2(\psi_1), \ &\ \vdots \ \psi_m(x,t) &= \frac{\partial \psi_{m-1}}{\partial t} + L_f \psi_{m-1} + \alpha_m(\psi_{m-1}), \end{aligned}$$

where each $\alpha_i$ is a strictly increasing class $\mathcal{K}$ function. The nested sets

$$C_1(t) = \{ x: \psi_0(x,t) \geq 0 \},\, C_2(t) = \{ x: \psi_1(x,t) \geq 0 \},\,\ldots,\, C_m(t) = \{ x: \psi_{m-1}(x,t) \geq 0 \}$$

form a forward-invariant “cascade”: if at $t=0$ all $\psi_i \geq 0$, invariance of the intersection $\cap_{i=1}^m C_i(t)$ is implied by maintaining $\psi_m \geq 0$ along all trajectories, ensuring that $h(x(t),t)$ never becomes negative—i.e., the system does not overshoot the barrier (Xiao et al., 2019).

2. High-Order Control Barrier Functions for Control-Affine Dynamics

For systems with control inputs entering via a control-affine structure, the barrier-cascade is extended to high-order control barrier functions (HOCBFs). Let $\dot x = f(x) + g(x) u$ and suppose $h(x)$ has relative degree $m$, i.e., $L_g L_f^i h(x) = 0$ for $i = 0, \ldots, m-2$ but $L_g L_f^{m-1} h(x) \not= 0$. The high-relative-degree nonovershooting constraint is enforced via a single inequality:

$$L_f^m h(x) + L_g L_f^{m-1} h(x) u + O(h,x) + \alpha_m(\psi_{m-1}(x)) \geq 0,$$

where $O(h, x)$ collects all lower-order derivatives. Any control input $u(x)$ satisfying this for $x$ in the cascade intersection $\cap_{i=1}^m C_i$ guarantees forward invariance and nonovershooting. The requirement can also be recast as a chain of inequalities:

$$\psi_0 = h(x) \geq 0,\; \psi_1 = \dot h + \alpha_1(h) \geq 0,\; \ldots,\; \psi_m = \dot \psi_{m-1} + \alpha_m(\psi_{m-1}) \geq 0.$$

If each $\alpha_i$ is a class $\mathcal{K}$ function, these coupled inequalities ensure that $h(x(t))$ remains nonnegative, forbidding barrier violation (Xiao et al., 2019).

3. Tuning Class $\mathcal{K}$ Functions and Feasibility Regions

The choice of class $\mathcal{K}$ functions $\alpha_i$ controls both the transient response (enforcing “no-overshoot”) and the admissible control set size. Typical designs include:

• Linear: $\alpha_i(s) = k_i s$ (exponential-type barrier),
• Power-law: $\alpha_i(s) = k_i s^p$, $p>1$ (slower near zero, aggressive far away),
• Root-law: $\alpha_i(s) = k_i s^{1/2}$ (aggressive near zero).

Larger $p$ or $k_i$ expand the feasible set when the system state is far from the boundary ($\psi_{i-1} \gg 0$) but shrink the control region near the constraint ($\psi_{i-1} \approx 0$). Conversely, smaller gains soften constraints yet risk violating actuator bounds. To resolve control feasibility when input limitations clash with the barrier (i.e., hard bounds on $u$), penalty gains $p_i$ are introduced in $\alpha_i \to p_i \alpha_i$, tuning the tradeoff between constraint strength and input feasibility (Xiao et al., 2019).

4. Optimization-Based Synthesis via Quadratic Programming

High-relative-degree nonovershooting constraints are typically synthesized alongside control Lyapunov function (CLF) objectives in real-time quadratic programs (QPs). For a constraint $h(x)$ of relative degree $m$ and CLF $V(x)$, the QP at time $t$ is formulated as:

$$\begin{aligned} \min_{u,\delta} &\quad \tfrac{1}{2} [u^T H u + 2 F^T u + R \delta^2] \ \text{s.t.} &\quad L_f V + L_g V u + c_3 V \leq \delta \ &\quad L_f^m h + L_g L_f^{m-1} h u + O + \alpha_m(\psi_{m-1}) \geq 0 \ &\quad u_{\min} \leq u \leq u_{\max} \end{aligned}$$

with slack variable $\delta$ for CLF relaxation, and $H, F, R$ structuring the quadratic cost. When the HOCBF constraint conflicts with input bounds, penalty tuning of class $\mathcal{K}$ gains restores QP feasibility at the expense of slower enforcement near $\psi_{m-1} \approx 0$ (Xiao et al., 2019).

5. Practical Guidelines and Representative Applications

In adaptive cruise control (ACC), the safety constraint $h(x) = x_p - x_i - \delta \geq 0$ (with $x_p$ as lead vehicle position, $x_i$ as ego position, $\delta$ as minimum required distance) has relative degree two. Design proceeds by constructing $\psi_1 = \dot h + p h$, $\psi_2 = \ddot h + p \psi_1$ for linear $\mathcal{K}$, or using nonlinear alternatives for more aggressive enforcement. Tuning is performed so that the QP constraint is inactive away from the boundary but enforces nonovershooting as $h \to 0$. Simulations confirm that the methodology maintains $x_p - x_i \geq \delta$ for all time without overshoot, even under real actuator limits (Xiao et al., 2019).

Other domains utilizing these constructs include PDE systems with moving boundaries (stefan models) (Koga et al., 2021), reinforcement learning for black-box systems with hard constraints (Bouvier et al., 2024), interconnected safety-critical systems analyzed via small-gain techniques (Lyu et al., 2022), and robust control of spacecraft trajectories under input and disturbance bounds (Breeden et al., 2021).

6. Comparative Analysis and Current Limitations

HOCBF-based nonovershooting constraint enforcement generalizes and surpasses exponential (linear) barrier techniques and classical CBF backstepping. While effective in both single-agent and multi-agent (interconnected) systems, some challenges remain:

• Initialization: All cascade barrier functions $\psi_i$ must be positive at $t=0$.
• Feasibility: For systems with tight actuator bounds, aggressive class $\mathcal{K}$ choices may make QP infeasible near the constraint; penalty gains or alternative barrier function forms (e.g., rectified CBFs (Ong et al., 2024), robust predictive CBFs (Breeden et al., 2021)) can mitigate this.
• Complexity: Higher relative degree increases algebraic complexity in the control synthesis, particularly in systems with nonlinear, non-affine, or infinite-dimensional dynamics (Koga et al., 2021).

Adaptive schemes and observer-based frameworks provide extensions for cases where critical system parameters are uncertain, allowing the system to maintain practical nonovershooting safety with bounds on violation decaying as parameter estimates converge (Lyu et al., 23 Sep 2025).

7. Theoretical Guarantees and Extensions

The theoretical guarantee is formalized: for systems starting in the intersection of the cascade sets $C_1(0) \cap \ldots \cap C_m(0)$, enforcement of the HOCBF constraint $\psi_m \geq 0$ ensures that $h(x(t)) \geq 0$ for all $t \geq 0$—provable “no overshoot.” This extends naturally to interconnected systems with high-relative-degree constraints via input-to-state safety small-gain theorems (Lyu et al., 2022), robust and disturbance-observer-based schemes (Zinage et al., 2023), and fixed-time homogeneous feedbacks for rapid, nonovershooting convergence (Polyakov et al., 2022).

The methodology unifies safety-critical control around high-relative-degree nonovershooting via barrier-cascade constructions, careful class $\mathcal{K}$ tuning, input feasibility mechanisms, and optimization-based real-time implementation. This forms a robust and versatile foundation for enforcing rigorous safety constraints in complex nonlinear, multi-agent, and safety-critical environments (Xiao et al., 2019).