Tunable Rational Concave Factor

• The factor is a strictly decreasing, infinitely differentiable scalar gain that constructs concave Lyapunov comparison functions for stability analysis and actuator-constrained control.
• It provides explicit slope control to mitigate chattering and accelerates Lyapunov decay rates when integrated with control Lyapunov function quadratic programs (CLF–QPs).
• A systematic parameter tuning procedure enables balancing between accelerated decay and actuator saturation, as demonstrated in inverted pendulum and quadrotor examples.

A tunable rational concave factor is a strictly decreasing, infinitely differentiable scalar gain used to construct strictly concave comparison functions for Lyapunov-based stability analysis and control, particularly under actuation constraints. Its principal application is the acceleration of constrained Lyapunov decay rates in control-affine systems, notably when integrated with control Lyapunov function quadratic programs (CLF–QPs). This factor enables a strict increase in guaranteed windowed nominal exponential decay rates while respecting actuator saturation, offering explicit slope control to prevent chattering and full compatibility with standard and “flexible” CLF–QP design (Fan et al., 18 Nov 2025).

1. Mathematical Definition and Properties

The baseline linear comparison function is $\alpha_\ell(v) \coloneqq \sigma v$, $\sigma>0$. The rational concave factor introduces a strictly decreasing gain $s(v)>0$:

$$\alpha(v) \coloneqq s(v)\cdot\sigma v,\quad s(v) = \frac{k_\mathrm{min}\,v + k_\mathrm{max}\,\ell}{v + \ell}$$

where $k_\mathrm{min}, k_\mathrm{max}, \ell>0$, $k_\mathrm{min}<k_\mathrm{max}$.

Key properties:

• $s(0)=k_\mathrm{max}$ (maximal virtual gain at the origin),
• $\lim_{v\to\infty} s(v) = k_\mathrm{min}$ (minimal gain as Lyapunov value grows),
• $$s'(v)= -\frac{(k_\mathrm{max}-k_\mathrm{min})\ell}{(v+\ell)^2} < 0$$ (strictly decreasing),
• $$s''(v)= \frac{2(k_\mathrm{max}-k_\mathrm{min})\ell}{(v+\ell)^3} > 0$$ (convexity in the gain profile).

Defining $$F(v) \coloneqq v\,s(v) = \frac{k_\mathrm{min}\,v^2 + k_\mathrm{max}\,\ell v}{v+\ell}$$, direct differentiation gives $F'(v)>0$, $F''(v)<0$: $\alpha(v)=\sigma F(v)$ is strictly increasing and strictly concave, i.e., $\alpha\in\mathcal{K}_\mathrm{cave}$.

2. Slope Control and Lipschitz Bound

The parameter $k_\mathrm{max}$ determines the maximal slope of $\alpha$:

$$\frac{d\alpha}{dv} = \sigma\,F'(v) \leq \sigma\,k_\mathrm{max}$$

for all $v$. This explicit bound on the slope is instrumental for implementation in sampled systems, mitigating chattering by limiting rapid changes in the control action.

3. Achievable Windowed Nominal Rate and Endpoint Cap

Given evaluation window $[\epsilon, c]$, the crossing time for the comparison-ODE is:

$$T_\alpha(\epsilon, c) = \int_\epsilon^c \frac{1}{\alpha(v)}\,dv = \frac{1}{\sigma}\int_\epsilon^c \frac{v+\ell}{v(k_\mathrm{min}\,v+k_\mathrm{max}\,\ell)}\,dv$$

resulting in the closed-form

$$T_\alpha(\epsilon, c) = \frac{1}{\sigma}\left[ \frac{1}{k_\mathrm{max}}\ln\frac{c}{\epsilon} + \frac{k_\mathrm{max}-k_\mathrm{min}}{k_\mathrm{max}k_\mathrm{min}} \ln\frac{k_\mathrm{min}c+k_\mathrm{max}\ell}{k_\mathrm{min}\epsilon+k_\mathrm{max}\ell} \right].$$

The effective windowed nominal rate, i.e., the exponential decay rate over $[\epsilon, c]$, is then

$$\sigma_\alpha(\epsilon, c) = \frac{\ln(c/\epsilon)}{T_\alpha(\epsilon,c)} = \frac{\sigma k_\mathrm{max}\ln(c/\epsilon)}{k_\mathrm{max}\ln(c/\epsilon)+ \left(\frac{k_\mathrm{max}-k_\mathrm{min}}{k_\mathrm{min}}\right)\ln\frac{k_\mathrm{min}c+k_\mathrm{max}\ell}{k_\mathrm{min}\epsilon+k_\mathrm{max}\ell}}.$$

To enforce endpoint consistency with input constraints (the endpoint cap), set $s(c)=r\leq1$ so $\alpha(c)=r\sigma c\leq\sigma c$. This yields:

$$r = \frac{k_\mathrm{min}\,c + k_\mathrm{max}\,\ell}{c+\ell},\qquad \ell = \frac{(r-k_\mathrm{min})c}{k_\mathrm{max}-r}$$

allowing explicit tuning between acceleration and peak actuation requirements.

4. Parameter Selection Recipe

A systematic four-step procedure is established for application-specific tuning within a desired window under actuator constraints:

1. Select $k_\mathrm{min}$: Small enough to ensure the shaped endpoint does not violate any known level-wise bound on actuation throughout $[\epsilon, c]$.
2. Choose $k_\mathrm{max}$: Sufficiently large so that, after setting $\ell$ via the endpoint normalization, the achieved nominal rate $\sigma_\alpha(\epsilon,c) \geq$ the desired value. If unsatisfactory, increase $k_\mathrm{max}$ or $r$ or decrease $k_\mathrm{min}$.
3. Compute $\ell$: Using $\ell = (r-k_\mathrm{min})c/(k_\mathrm{max}-r)$, ensuring endpoint normalization as above.
4. Verification and Adjustment: Recalculate $\sigma_\alpha(\epsilon,c)$; if actuator limits are exceeded, reduce $r$ or $k_\mathrm{max}$. Optionally, use $s(v^p)$ for $0

5. Integration into CLF–QP Frameworks

The factor modifies the CLF–QP by replacing the linear decay constraint:

$L_fV(x)+L_gV(x)u \leq -\sigma V(x)$

with the concave constraint:

$$L_fV(x)+L_gV(x)u \leq -\alpha(V(x)) = -\sigma s(V(x)) V(x)$$

Hard-constraint CLF–QP:

$$\min \|u\|^2 \ \text{subject to} \ L_fV + L_gV u + \sigma s(V)V \leq 0, \quad \|u\|_\infty\leq\theta$$

Soft-constraint version:

$$\min u^T H u + q\delta^2, \text{subject to:} \quad L_fV + L_gV u + \sigma s(V)V \leq \delta, \quad \|u\|_\infty\leq \theta, \quad \delta\geq0$$

All feasibility and acceleration guarantees for concave comparison functions apply directly ((Fan et al., 18 Nov 2025), Sec. IV).

6. Empirical Performance: Illustrative Examples

Demonstrated on two canonical systems with actuator saturation:

• Inverted pendulum: Baseline $\sigma=3$, $\theta=10$, $[\epsilon=10^{-4}c, c]$, $(k_\mathrm{min},k_\mathrm{max})=(0.1,2.3)$, $r=1\to0.6$, $\ell$ from endpoint condition. Achieved $\sigma_\alpha\approx6.0>3.0$ with the same endpoint and reduced peak torque and energy.
• Quadrotor attitude control: Baseline $\sigma=2$, $\theta=11$, $[\epsilon=10^{-3}c, c]$, $(k_\mathrm{min},k_\mathrm{max})=(0.8,2.5)$, $r=0.95$ or $0.85$. Achieved $\sigma_\alpha\approx4.3>2$, with lower peak and average torque than the “flexible” CLF–QP.

In each case, the tunable rational concave factor provides strictly higher windowed decay rates under the same or reduced endpoint cap, enhanced feasibility under actuator limits, and maximal slope capping to inhibit chattering.

7. Context and Theoretical Significance

The tunable rational concave factor establishes that strict concavity in Lyapunov comparison functions is both sufficient and necessary for achievable improvement over linear and convex counterparts under input constraints. Only concave shaping permits a decrease in the required actuation for a target nominal rate, and feasibility-preserving acceleration becomes possible whenever the linear comparison is feasible with margin on a sublevel set. All acceleration and feasibility properties follow directly from the functional form and can be systematized in control design via the provided recipe (Fan et al., 18 Nov 2025).