Concave Comparison Functions for Accelerating Constrained Lyapunov Decay

TL;DR

The paper addresses fundamental limits of Lyapunov decay under actuator bounds and shows that shaping the Lyapunov comparison function as a strictly concave function yields faster guaranteed decay on a window while reducing required actuation due to the endpoint cap. It develops a windowed performance framework, proves decay ordering and necessity of concavity, and introduces a constructive rational concave factor that is Lipschitz and compatible with CLF-QP implementations. The results demonstrate feasibility-preserving acceleration and provide practical tuning guidelines, backed by case studies on inverted pendulums and quadrotor attitude control under saturation. The approach offers a versatile, design-focused alternative to rate-scheduling in CLF-based controllers and has potential to improve robustness and performance in a range of Lyapunov-based control architectures.

Abstract

What limits how fast a Lyapunov function can decay under input bounds? We address this question by showing how the shape of Lyapunov comparison functions governs guaranteed decay for control affine systems. Using a windowed nominal exponential rate together with the endpoint cap induced by actuator limits, we establish a strict ordering: concave comparison functions strictly outperform linear and convex ones, and strict concavity is necessary to improve the best achievable global exponential rate under a fixed endpoint cap. We derive a computable lower bound on the required actuation level for a target nominal rate and show that only concave shaping can reduce this level under the endpoint cap. We then establish a feasibility-preserving acceleration result: whenever a margin exists on a sublevel set, a feasible linear comparison can be replaced by a concave one that preserves feasibility while strictly increasing the guaranteed windowed decay. Finally, we give a tunable rational concave factor with controlled slope that yields a constructive design and integrates with CLF QP, as illustrated by examples.

Figures (5)

• Figure 1: Sketch: decay cap, endpoint cap, evaluation window $[\epsilon,c]$. The concave curve lies above linear and convex comparison, hence the largest $\sigma_\alpha(\epsilon,c)$ is guaranteed.
• Figure 2: Assignable decay-rate cap $D_{\max}(x,\theta)$ along the trajectory for $\theta=10$ and the comparison functions. The endpoint value $\alpha(c)$ strictly restricts the assignable exponential rate.
• Figure 3: Inverted pendulum: Normalized Lyapunov response $V(x(t))/V(x_0)$, instantaneous exponential rate $-\dot V/V$, and control input $u$. For windows with $\epsilon\le 10^{-1}c$, all concave designs achieve a strictly larger nominal rate than the linear baseline (shorter crossing times). For very small $V\lesssim 10^{-4}c$, the instantaneous rate dip because $\delta/V$ grows as $V\to0$. In contrast, with the hard (mini-norm) CLF controller ($\delta\equiv0$), the instantaneous rate remains increasing toward equilibrium, consistent with the slack analysis as shown in Fig.\ref{['fig:inv-mininorm']}.
• Figure 4: Inverted pendulum: Mini-norm controllers with concave constraints where the instantaneous rate tracks the designed profiles.
• Figure 5: Quadrotor: normalized Lyapunov response $V(x(t))/V(x_0)$, instantaneous exponential rate $-\dot V/V$, and control input norm $\|u\|_2$. The concave CLF–QP yields faster decay than the flexible ES–CLF–QP after $t\approx0.5$ s; along most of the trajectory, the instantaneous rate exceeds the baseline $\sigma=2$, consistent with the feasibility-preserving acceleration theorem. With $r=0.85$, faster decay is achieved with a lower peak input $\|u\|_\infty$ and reduced energy. Decreasing $r$ further reduces the actuation peak, consistent with endpoint-dominated actuation level.

Theorems & Definitions (32)

• Definition 1: Control Lyapunov function
• Definition 2: Nominal rate
• Lemma 1: Composition of nominal rates
• proof
• Lemma 2: Dynamic-factor characterization
• proof
• Definition 3: Concave Factor
• Proposition 1: Ordering under equal endpoint cap
• proof
• Theorem 1: Acceleration under a relaxed endpoint cap