Funnel control of nonlinear systems

• N denotes the positive integers, while N0 denotes the non-negative integers.Source: Nomenclature · p. 2
• L∞(I,Rn) contains measurable, essentially bounded functions from an interval I to Rn.Source: Nomenclature · p. 2
• loc(I,Rn) denotes measurable, locally essentially bounded functions from an interval I to Rn.Source: Nomenclature · p. 2
• W k,∞(I,Rn) contains functions whose values and weak derivatives through order k are essentially bounded.Source: Nomenclature · p. 2
• Ck(V,Rn) denotes k-times continuously differentiable functions, with C(V,Rn) defined as C0(V,Rn).Source: Nomenclature · p. 2

1 Introduction

• Theorem 1.9 develops feedback that makes the tracking error meet prescribed transient-behavior and asymptotic-accuracy objectives for the system class and admissible references.The class is modeled by r-th order functional differential equations, with feedback using output and reference information.Source: §1 · p. 2

1.1 Novelties and literature

• The proposed design is presented as a simpler alternative to backstepping and derivative-heavy controllers, whose complexity grows with relative degree.Earlier arbitrary-relative-degree backstepping approaches require increasing powers of a gain function, while a later design uses successive derivatives of auxiliary error variables.Source: §1.1 · p. 3
• The system class covers a larger class of dead-zone inputs, and the design allows continuous surjective switching functions that properly contain the Nussbaum class.Dead-zone nonlinearities can severely affect control performance in applications such as hydraulic servo valves and electronic motors.Source: §1.1 · p. 4
• Unbounded funnel functions permit exact asymptotic tracking but can make the controller’s product computation ill-conditioned, whereas bounded functions allow arbitrarily small nonzero prescribed asymptotic error.For example, an admissible exponentially growing function yields exponentially fast convergence to zero; bounded functions avoid the ill-conditioning associated with multiplying very large and very small terms.Source: §1.1 · p. 4

1.2 System class

• The system class combines a high-gain property with causal, locally Lipschitz, bounded-input bounded-output internal operators; for linear strict-relative-degree systems, the latter corresponds to a minimum-phase assumption.The class also encompasses systems with arbitrary state-space dimension, including infinite-dimensional internal dynamics.Source: §1.2 · pp. 6, 9

1.3 Control objectives

• The performance funnel shapes transient behavior and asymptotic accuracy through its boundary, and may widen later to accommodate periodic disturbances or strongly varying references.At each t > 0, the funnel cross-section is an open ball of radius 1/ϕ(t).Source: §1.3 · pp. 9–10

1.4 Funnel control structure

• Feedback uses the output and its first r−1 derivatives, plus the reference value and, for some ˆr ≤ r, accessible reference derivatives.The introduction notes that the strategy can cope with unavailable reference derivatives, and simulations support the expectation that more available derivatives may improve controller behavior.Source: §1.1 · pp. 4–5, §1.4 · p. 10
• The controller is proportional feedback of an information vector with scalar gain, and its recursive signals are directly implementable without successive auxiliary-error derivatives as in.The paper describes this structure as substantially less complex than previous approaches for larger r.Source: §1.4 · pp. 11–12
• With bounded ϕ, the controller guarantees approximate tracking to a prescribed accuracy; unbounded ϕ permits exact asymptotic tracking when ˆr = r but has limited practical utility.The paper notes that computing ϕ(t)e(t) with unbounded ϕ can be ill-conditioned.Source: §1.4 · pp. 12–13
• A simulation illustrates the probing capability: the input reaches an appropriate interval of control values independent of the sign choice for σ.For σ = +1 the input rises sharply near the funnel boundary, while for σ = −1 it remains in a bounded interval.Source: §1.4 · p. 13, Figure 3 · p. 14

1.5 Funnel control – main result

• Theorem 1.9 guarantees global solutions, bounded input and output, and tracking error uniformly inside the prescribed funnel for admissible references under its stated assumptions.If ˆr = r and ϕ is unbounded, the error and its derivatives through order r−1 converge to zero.Source: §1.5 · pp. 14–15
• Under additional conditions, Corollary 1.10 gives bounds on the tracking error and its first ˆr−2 derivatives, simpler than bounds in [8, Prop. 3.2].The conditions include ˆr ≥ 2, ϕ(0) > 0, and monotonically non-decreasing α′.Source: §1.5 · p. 16

2 Examples

• The class N^m,r encompasses linear MIMO systems with strict relative degree r and asymptotically stable zero dynamics.The section also treats control directions and input nonlinearities, including dead-zone inputs.Source: §2 · p. 16

2.1 The prototypical linear system class

• For the linear prototype, the class properties are organized around a causal, locally Lipschitz operator that maps bounded functions to bounded functions.The linear system is introduced to relate the operator properties and the high-gain property to the system representation.Source: §2.1 · p. 16

2.1.1 Strict relative degree

• Strict relative degree lets the linear system be transformed into Byrnes-Isidori form and represented as a functional differential system.The system has CA^kB = 0 for k = 0,...,r−2 and invertible Γ = CA^(r−1)B.Source: §2.1.1 · pp. 17–18

2.1.2 Minimum phase

• For strict-relative-degree systems, the operator’s BIBO property is closely related to asymptotic stability of the zero dynamics.For stabilizable and detectable systems, the cited equivalences connect this property to the transfer function having no zeros in C≥0 and to external stability of the associated operator.Source: §2.1.2 · p. 18

2.1.3 Sign-definite high-frequency gain matrix

• The system has the high-gain property if and only if its high-frequency gain matrix Γ is sign definite.The equivalent condition is v⊤Γv ≠ 0 for every nonzero v.Source: §2.1.3 · p. 18

2.2 Known and unknown control directions

• For unknown control direction, the design permits any continuous surjective probing function, a larger class than Nussbaum functions.The passage gives a continuous function satisfying the paper’s condition that does not have the Nussbaum properties.Source: §2.2 · p. 20

2.3 Input nonlinearities

• The generic formulation’s high-gain property accommodates a variety of input nonlinearities, including scalar systems with a continuous nonzero-valued control-direction function.The scalar example imposes conditions on β that make it unbounded in opposite directions at the two ends of its domain.Source: §2.3 · pp. 20–21

2.4 Dead-zone input

• The dead-zone model allows its left and right branch functions to be neither differentiable nor monotone.The deadband parameters and branch functions may be unknown, subject to the stated endpoint and limiting conditions.Source: §2.4 · p. 21

3.1 Mass-on-car system

• For system (26) under controllers (9) and (27), performance is comparable, but controller (9) is feasible in some situations outside controller (27)’s scope; Fig. 5 uses ϑ = π.These include asymptotic tracking, unknown control direction, and operation when instantaneous ˙yref(t) is unavailable.Source: §3.1 · p. 23, Figure 5 · p. 23
• In simulation of system (26) with ϑ = 0, controller (9) requires more input action than controller (28), which is significantly more complex and harder to implement.Their performance is broadly similar; the implementation-complexity issue becomes more severe for relative degrees higher than three.Source: §3.1 · p. 24, Figure 6 · p. 24

3.2 Nonlinear MIMO system

• In the nonlinear multi-input, multi-output manipulator example, controllers (9) and (30), applied to (29), exhibit nearly identical performance in simulation over the interval [0,10].The manipulator is planar, rigid, has revolute joints, and has two degrees of freedom.Source: §3.2 · pp. 24–25
• The paper suggests that controller behavior differs with nontrivial internal dynamics, while it is practically identical for the manipulator’s trivial internal dynamics.The authors describe internal dynamics as the presumable cause, not a demonstrated explanation.Source: §3.2 · p. 26

3.3 A nonlinear system with dead-zone input

• The dead-zone example belongs to the system class for which funnel-control feasibility follows from the high-gain property and the internal-dynamics conditions.The internal dynamics are shown to be input-to-state stable and to have globally defined solutions.Source: §3.3 · pp. 27–28
• With dead-zone input, the simulation achieves asymptotic tracking when reference-derivative information is available; using less information degrades performance.The input shows jumps attributed to the dead-zone, and the reduced-information case uses a bounded funnel.Source: §3.3 · p. 28

4 Conclusion

• Funnel tracking objectives are achieved for a large class of higher-relative-degree nonlinear systems described by functional differential equations satisfying a high-gain property.The class assumes BIBO internal dynamics; the feedback strategy is described as simple for higher relative degree.Source: §4 · p. 28
• PDE systems are not treated, although systems with a well-defined relative degree and infinite-dimensional internal dynamics may be susceptible to these control laws.The passage gives a linearized moving water tank with sloshing effects as an example.Source: §4 · p. 28
• For infinite-dimensional systems without a well-defined relative degree, these results do not apply, and funnel feasibility must be investigated directly for the nonlinear closed-loop system.The passage points to studies of boundary-controlled heat equations [48], boundary control systems, monodomain equations [6], and the Fokker-Planck equation [4].Source: §4 · p. 29
• Non-derivative funnel control remains unresolved above relative degree three, even for SISO linear systems with positive high-frequency gain and asymptotically stable zero dynamics.The cited pre-compensator approaches cover relative degree at most three.Source: §4 · p. 29

Appendix A Proofs

• After recasting the feedback-controlled system as an initial-value problem, a variant of [28, Thm. B.1] yields a maximal solution x:[−h,ω)→R^n, 0<ω≤∞; the proof shows ω=∞, so this solution is global.Source: Appendix A · pp. 30, 31, 37
• The first transformed error stays inside the unit ball, which ensures the tracking error evolves in the performance funnel.For k = 1, the proof states that ∥e1(t)∥ = ϕ(t)∥e(t)∥ < 1.Source: Appendix A · p. 31
• An induction bounds the intermediate transformed errors and their associated functions, with essential boundedness of their derivatives established along the way.The proof obtains these bounds first for k = 1 and then for k ∈ {2,...,r −1}.Source: Appendix A · pp. 34–35
• The proof establishes boundedness of the final gain function and concludes that the maximal solution extends globally, with ω = ∞.For negative- or positive-definite high-gain properties, the adapted argument also concludes boundedness of αr and global existence.Source: Appendix A · pp. 37–38