### Funnel Control With Saturation: Nonlinear SISO Systems \$ \$

Norman Hopfe, Achim Ilchmann, Eugene P. Ryan
2010 IEEE Transactions on Automatic Control
Tracking -by the system output -of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of nonlinear, single-input, single-output systems modelled by functional differential equations and subject to input saturation. Prespecified is a parameterized performance funnel within which the tracking error is required to evolve; transient and asymptotic behaviour of the tracking error is influenced through choice of parameter values which
more » ... e the funnel. The control structure is a saturating error feedback with time-varying non-monotone gain designed to evolve in such a way as to preclude contact with the funnel boundary. A feasibility condition -formulated in bounds of the plant data, the saturation bound, the funnel data, the reference signal and the initial data -is presented under which the tracking objective is achieved, whilst maintaining boundedness of the state and gain function. Nomenclature: R + := [0, ∞); C(I, R ℓ ), I ⊂ R, is the space of continuous functions I → R ℓ ; L ∞ (I, R ℓ ) is the space of measurable, essentially bounded functions f : I → R ℓ , with norm f ∞ := ess sup t∈I y(t) ; the space of measurable, locally essentially bounded functions f : I → R ℓ is denoted by L ∞ loc (I, R ℓ ); if ℓ = 1, we simply write L ∞ (I) and L ∞ loc (I); W 1,∞ (R + ) is the space of absolutely continuous functions r : R + → R with r,ṙ ∈ L ∞ (R + ). A function β : R + → R + is called a K function if it is continuous and strictly increasing, with β(0) = 0; the class of unbounded K functions is denoted by K ∞ . A continuous function α : R + × R + → R + is called a KL function if α(·, t) ∈ K for all t ∈ R + and, for all s ∈ R + , α(s, ·) is non-increasing with α(s, t) → 0 as t → ∞.