Funnel-Based Control Method

• Funnel-based control is a framework that confines system errors within time-varying bounds (funnels), ensuring robust tracking and constraint satisfaction.
• It employs high-gain, barrier-like feedback combined with explicit normalization to maintain errors within prescribed limits despite model uncertainties.
• Experimental and simulation validations on robotic platforms demonstrate its efficacy in enforcing actuator limits through a closed-form, approximation-free control strategy.

A funnel-based control method is a mathematically rigorous framework designed to confine system errors, output trajectories, or other quantities of interest within time-varying bounds—referred to as "funnels"—specified by designer-chosen performance functions. Funnels are employed both for tracking and constraint satisfaction in dynamical systems. The core concept combines high-gain, barrier-like feedbacks with explicit normalization, ensuring robust evolution within prescribed boundaries independently of model uncertainties or input constraints.

1. Mathematical Foundations of Funnel Constraints

Funnel-based control operates by defining time-varying, component-wise performance envelopes for tracking errors and enforcing their invariance under the closed-loop dynamics. For a system with state $x(t) \in \mathbb{R}^n$ and velocity $v(t)$, position-level and velocity-level error variables are

• Position error: $$e_x(t) := x(t) - x_\mathrm{ref}(t) \in \mathbb{R}^n$$
• Velocity error: $e_v(t) := v(t) - v_r(t) \in \mathbb{R}^n$, with reference velocity $v_r(t)$ formally designed.

Funnels are constructed using vector-valued boundaries, typically exponential decays: $$\rho_{x,i}(t) = e^{-\mu_{x,i} t}(p_{x,i} - q_{x,i}) + q_{x,i}, \quad 0<q_{x,i}<p_{x,i}$$ and similarly for velocity: $$\rho_{v,i}(t) = e^{-\mu_{v,i} t}(p_{v,i} - q_{v,i}) + q_{v,i}$$ Constraints are imposed as $-\rho_x(t) < e_x(t) < \rho_x(t)$ and $-\rho_v(t) < e_v(t) < \rho_v(t)$ at all times.

Normalization is performed via

$$\epsilon_x(t) := \operatorname{diag}(\rho_x(t))^{-1} e_x(t) \in (-1,1)^n, \qquad \epsilon_v(t) := \operatorname{diag}(\rho_v(t))^{-1} e_v(t) \in (-1,1)^n$$

This mapping transforms the original error coordinates into a normalized space suitable for bounded-feedback synthesis.

2. Input Constraints and Feasibility Conditions

Funnel-based control achieves input constraint satisfaction by design rather than post hoc saturation. Explicit feasibility conditions guarantee that feedback laws interacting with the normalized errors do not drive either the errors or control inputs outside the allowable regions. These are two-stage:

1. Velocity bounds (keeping up with funnel contraction and reference motion):

$\bar v \succeq \mu_x (p_x - q_x) + \bar v_r$

where $\bar v$ is the actuator velocity limit and $\bar v_r$ the reference speed bound.

2. Torque bounds (compensating for worst-case inertial/Coriolis/gravity and disturbance effects):

$$\bar \tau \succeq \frac{1}{\overline{m}} \left[ \max(-\underline V_M, \overline V_M) + \underline{m}_i \overline d + \mu_v(p_v - q_v) + \overline a_r \right]$$

$$\overline{m}, \underline{m}_i, \overline{V}_M, \underline{V}_M, \overline d, \overline a_r$$ are global bounds for inertia, velocity nonlinearities, disturbance forces, and reference acceleration.

If these conditions are satisfied, state trajectories and control inputs will both remain within prescribed constraints by virtue of the feedback architecture.

3. Closed-Form Controller Synthesis: Approximation-Free Strategies

Funnel-based control employs component-wise, non-adaptive, closed-form control laws, using only normalization and bounded transformations:

• Active correction law (when all feasibility conditions are met):
  • Reference velocity feedback:

  $v_r(t) = -\bar v \cdot \Psi(\epsilon_x(t))$ - Final torque command:

  $\tau(t) = -\bar \tau \cdot \Psi(\epsilon_v(t))$

Here, $\Psi: [-1,1] \rightarrow [-1,1]$ is a smooth, non-decreasing "bounding" function ensuring boundedness of commands.

• Stop-deviation strategy (if feasibility is violated):
  • Switch to a "zeroing" function $\Psi^0$ so that $$\Psi^0_i(\pm 1) = \pm 1, \lim_{s \to \pm \infty} \Psi^0(s) = 0$$. For out-of-funnel normalized error, control torque fades to zero, halting further deviation and preventing actuator saturation or unsafe operation.

Unlike adaptation-based or neural methods, the approach is strictly approximation-free—the unknown EL dynamics enter only via the feasibility conditions, not the online feedback laws.

4. Stability, Robustness, and Barrier Arguments

The stability analysis relies on a barrier/contraction Lyapunov-type method, forgoing classical quadratic Lyapunov functions. The invariance proof hinges on contradiction: if at any time the normalized error were to reach the boundary ($\pm 1$), the closed-loop dynamics would necessarily force the error back into the interior, provided finite bounds hold. Global boundedness is ensured as long as the initial state is within the normalized funnel and the feasibility conditions are satisfied.

Robustness to uncertainties and disturbances is structurally built-in. All properties of the unknown EL system—model uncertainties, bounded external disturbances—are incorporated explicitly via the feasibility inequalities. No parameter adaptation or system inversion is required. If the system is driven outside the prescribed funnel due to severe disturbances, safety is preserved via the stop-deviation law.

5. Experimental and Simulation Validation

The efficacy and generality of funnel-based control under explicit input constraints are demonstrated across several platforms:

• 2R SCARA manipulator (simulated):
  • $\bar \tau = [10,10]$ Nm, $\bar v = [6,6]$ rad/s; funnel parameters $p_x=[0.2,0.2]$, $q_x=[0.02,0.02]$, $\mu_x=0.1I$; under a $2$ Nm disturbance, max $|e_x| < \rho_x$, $|\tau| < 10$ Nm.
• 7-DOF Franka Emika Panda (hardware):
  • $\bar \tau = 8$ Nm, $$|\theta_i - \theta_{\mathrm{ref},i}| < \rho_{x,i}(t)$$, $|\tau_i| < 8$ Nm in various operating regimes: nominal, payload, and abrupt jerks.
• Omnidirectional mobile robot (hardware):
  • $\bar v_x=\bar v_y=0.1$ m/s, $\bar \omega=0.1$ rad/s. Under severe jerks, the error is recovered into the funnel via saturation; zeroing control successfully halts unsafe motion.

These results corroborate robust funnel invariance and actuator constraint enforcement in the presence of nominal and adversarial conditions.

6. Comparative Perspective and Conservatism

Relative to conventional funnel control, this method is fundamentally approximation-free, relying on no model adaptation, state or parameter estimation, or neural/concurrent learning architectures. Actuator limits are encoded directly via bounded transformation and feasibility checks, not handled post hoc.

Strengths:

• Closed-form, smooth, any-time implementable laws.
• Explicit, designer-chosen input constraints fully enforced.

Limitations:

• The feasibility conditions may be conservative and require access to global bounds of inertia, velocity nonlinearities, and disturbances.
• Excessively tight funnels or limited actuators necessitate switching to the stop-deviation law, which results in loss of tracking performance.

This framework extends the applicability of funnel control to real robotic systems constrained by actuation, with rigorous barrier-based stability and invariance guarantees, and it is validated in both simulation and hardware using conservative but explicit bounds (Das et al., 2 Jul 2025).