Funnel control with input filter for nonlinear systems with arbitrary relative degree

Abstract: This paper addresses output reference tracking with prescribed transient performance for unknown nonlinear multi-input multi-output systems with arbitrary relative degree. We propose a novel derivative-free extension of funnel control based on a collection of filter variables that estimate the output derivatives. The resulting controller ensures that the tracking error evolves within prescribed performance bounds, while avoiding differentiation of the output signal and maintaining a simple structure with only a small number of tuning parameters. The effectiveness of the proposed approach is illustrated by a numerical example.

• The paper introduces a derivative-free funnel controller for uncertain nonlinear MIMO systems of arbitrary relative degree, using only r−1 first-order filters instead of measuring output derivatives.
• The main theorem guarantees global solutions, uniform tracking-funnel satisfaction, and bounded inputs, filter states, and output derivatives when initial feasibility and a BIBO internal-dynamics condition hold.
• The controller requires no estimate of the input-gain matrix and reduces dynamic order and tuning complexity compared with funnel pre-compensators, although simulations indicate moderately higher control bandwidth.

Problem setting and motivation

Funnel control enforces output reference tracking within a prescribed performance funnel $F_\varphi = \{(t,e) : \varphi(t)\|e\| < 1\}$ for uncertain nonlinear MIMO systems, using high-gain feedback with mild structural assumptions. Its principal practical drawback is that classical designs require the output derivatives $y^{(1)},\dots,y^{(r-1)}$ to be measurable, which restricts applicability to systems of relative degree $r \geq 2$. Existing remedies each carry costs: the virtual-output approach of Chowdhury and Khalil depends on system dynamics and is not model-free; the filter-based design of Ilchmann et al. involves high powers of a large gain; the funnel pre-compensator cascade requires $r(r-1)$ auxiliary dynamic variables and an estimate of the input gain matrix $\Gamma$; and sample-and-hold schemes are limited to relative degree two and require a sampling time tied to system properties.

This paper proposes a derivative-free funnel controller for unknown nonlinear MIMO systems of arbitrary relative degree, built on $r-1$ first-order filter variables that act as surrogates for the unavailable output derivatives. It generalizes prior work on relative degree two (Dennstädt et al., 19 Dec 2025) while substantially reducing dynamic order and tuning effort relative to the pre-compensator approach.

System class

The plant has the form

$$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$

with $r \geq 2$, constant matrices $R_i$, and $\Gamma$ whose symmetric part $y^{(1)},\dots,y^{(r-1)}$0 is positive definite. The internal dynamics are captured by a causal, locally Lipschitz operator $y^{(1)},\dots,y^{(r-1)}$1 satisfying a BIBO property: boundedness of $y^{(1)},\dots,y^{(r-1)}$2 alone implies boundedness of $y^{(1)},\dots,y^{(r-1)}$3. The authors note this is a stronger assumption than in classical funnel control, where output derivatives are assumed available; here boundedness must be inferred from the output only, since derivatives are inaccessible. The explicit linear terms $y^{(1)},\dots,y^{(r-1)}$4 are retained because they violate the BIBO property of $y^{(1)},\dots,y^{(r-1)}$5.

The control objective is that for any reference $y^{(1)},\dots,y^{(r-1)}$6, the tracking error evolves within the funnel determined by a user-chosen $y^{(1)},\dots,y^{(r-1)}$7 with $y^{(1)},\dots,y^{(r-1)}$8.

Controller structure

The controller introduces $y^{(1)},\dots,y^{(r-1)}$9 filter variables satisfying a lower-triangular chain,

$r \geq 2$0

and defines nested error signals $r \geq 2$1 and $r \geq 2$2, with control law

$r \geq 2$3

The interpretation is that $r \geq 2$4 tracks the relative-degree-one funnel law applied to $r \geq 2$5, with residual errors $r \geq 2$6 confined to constant funnels of radii $r \geq 2$7 by construction. The controller uses no differentiation of either output or reference, requires only $r \geq 2$8 dynamic variables (versus $r \geq 2$9 for the pre-compensator cascade), and its parameters — $r(r-1)$0, $r(r-1)$1, and initial values $r(r-1)$2 — can be chosen without system knowledge except for the initial output value $r(r-1)$3.

Main result

The main theorem establishes that, provided the initial data satisfy strict feasibility conditions ($r(r-1)$4 and $r(r-1)$5), every maximal solution of the closed loop:

• is global ($r(r-1)$6);
• keeps the tracking error uniformly inside the funnel, i.e., there exists $r(r-1)$7 with $r(r-1)$8 for all $r(r-1)$9;
• yields bounded input $\Gamma$0, bounded filter variables $\Gamma$1, and bounded output derivatives $\Gamma$2.

The proof proceeds in six steps. The key technical device is a family of auxiliary signals $\Gamma$3, defined as polynomial combinations of $\Gamma$4 and $\Gamma$5, shown via a Gronwall argument to remain bounded using the BIBO property of $\Gamma$6. A Vandermonde-based argument then constructs linear combinations $\Gamma$7 that isolate $\Gamma$8 plus finitely many bounded terms, yielding boundedness of successive output derivatives by induction. Funnel feasibility follows from a contradiction argument on the boundary $\Gamma$9: choosing $r-1$0 so that $r-1$1 exceeds a computable bound $r-1$2 involving $r-1$3, $r-1$4, $r-1$5, $r-1$6, $r-1$7, $r-1$8, and $r-1$9 forces the derivative of $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$0 negative at the boundary. A parallel induction shows each $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$1 stays strictly below $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$2, keeping the denominators $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$3 bounded away from zero and hence $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$4 bounded. Global existence follows because the closure of the solution graph remains compact in the domain where the vector field is well-defined.

Two structural consequences deserve emphasis. First, unlike the pre-compensator designs, no estimate of $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$5 is required — only positive definiteness of its symmetric part enters the analysis. Second, the closed-loop dimension grows linearly rather than quadratically in $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$6.

Numerical illustration

The controller is applied to a nonlinear relative-degree-three MIMO example from Lanza's pre-compensator study, featuring a time-varying disturbance $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$7 absorbed into the operator $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$8, nonlinearity $$y^{(r)}(t) = \sum_{i=1}^{r-1} R_i y^{(i)}(t) + f\big(T(y,\dots,y^{(r-1)})(t)\big) + \Gamma u(t),$$9, and coupling matrix $r \geq 2$0. With tuning parameters $r \geq 2$1, $r \geq 2$2, $r \geq 2$3, zero-initialized filters, and a funnel function matching the transient specification of the earlier study, the simulation (MATLAB ode15s, tolerances $r \geq 2$4/$r \geq 2$5) shows comparable transient performance but a moderately higher control bandwidth than the pre-compensator-based controller, while requiring fewer differential equations and fewer tuning parameters. No quantitative performance metrics beyond these qualitative comparisons are reported.

Limitations and open questions

Several caveats are stated explicitly by the authors. The BIBO assumption on $r \geq 2$6 is stronger than what classical funnel control requires, since internal-state boundedness must follow from output boundedness alone. Feasibility of small values of $r \geq 2$7 — which improve performance — demands precise knowledge of the initial output $r \geq 2$8, so aggressive tightening of the inner funnels trades against initialization requirements. Like all funnel controllers, the scheme cannot exclude transient violations of the funnel constraint under measurement noise or discretization effects; robustness to such effects is not analyzed here. Finally, the theoretical guarantee concerns continuous-time implementation; integration with other funnel-based schemes and extension to networked settings remain unaddressed.

Conclusion

The paper delivers a derivative-free funnel controller for nonlinear MIMO systems of arbitrary relative degree, replacing derivative measurement with a chain of $r \geq 2$9 filters and nested constant-funnel error bounds. The main theorem guarantees global solutions, uniform funnel satisfaction, and boundedness of all closed-loop signals under feasibility conditions on the initial data, with no knowledge of the gain matrix beyond positive definiteness of its symmetric part. Relative to the funnel pre-compensator, it reduces both dynamic order and tuning complexity, at the cost of a somewhat higher control bandwidth observed in simulation.