Sum-of-Squares Applications in Nonlinear Controller Synthesis

3.3 Modifications for Disturbance Attenuation Sontag Formula Feedback3.3 Modifications for Disturbance AttenuationFor linear systems, H ∞ design has become a widely accepted design paradigm that systematicallyaddresses performance and allows for conclusions in terms of uncertainties anddisturbance attenuation. For nonlinear systems, the corresponding property is the L 2 gain.Considering a disturbed systemẋ = f(x) + g d (x) dwith x(t) ∈ R n , f(t) ∈ R n , g d (t) ∈ R n × R n d, d(t) ∈ R n d,(49)we seek to guarantee a bound from the disturbance d to a certain performance index h interms of the L 2 norm√ ∫ ∞‖h‖ 2 = h(t) T h(t) dt . (50)0Note, that any closed loop systems using state feedback can be written in the form (49).The results in this section therefore implicitly apply to systems that use the Sontag formulafeedback developed in the previous sections.As an important conceptual difference to the approach of section 3.2, where we wereinterested in the stability of an equilibrium point, we now use the framework of boundedinput/bounded output stability. We are therefore interested in results of the form‖h‖ 2 < γ ‖d‖ 2 , ∀d with γ ∈ R + , (51)which for linear systems is equivalent to the H ∞ norm (see, e.g., Khalil [12]). From thetheory of dissipative systems [23], it is known that this inequality can be guaranteed to holdby finding a storage function with the supply rate γ 2 d T d − h T h. We formalize this asLemma 3:The system (49) has an L 2 gain from d to h that is less then γ if ∀d and ∀x ≠ 0there exists a positive definite, radially unbounded function V such that˙V < γ 2 d T d − h T h.Lemma 3 can be proved by integrating V along the system’s trajectories, which yieldsV (x ∞ ) − V (x 0 ) < γ 2 ‖d‖ 2 2 − ‖h‖ 2 2 ∀d and ∀x 0 (52)18