Sum-of-Squares Applications in Nonlinear Controller Synthesis

3.3 Modifications for Disturbance Attenuation Sontag Formula Feedback) ( ) TReplacing L f V in the Sontag formula (32) with β = L f V +(L 14 γ 2 gdV L gdV + h T hyields the desired result, as for the closed loop system condition (58) becomes−√β 2 + x T Q x( )L guV R −1( ) TL guV < 0 ∀x ≠ 0which trivially holds. We summarize these results inLemma 4:There exists a state feedback law which guarantees the L 2 gain from d to h isless than γ if there exists a positive definite, radially unbounded function V withL f V + 1 ( ) ( ) T4 γ 2 L gdV L gdV + h T h < 0 ∀x ≠ 0 such that L guV = 0 .It is now possible to again use sum-of-squares programming to search for a suitable functionV. As the condition in lemma 4 is quadratic in V , we use the Schur complement[ ]M LT≺ 0 ⇔ M − L T N −1 L ≺ 0 and N ≺ 0 (59)L Nto transform it into the equivalent form⎡(L gdV⎣ L f V + hT 1h2 γ( ) TL gdV −I12 γ) ⎤To formulate this as a sum-of-squares problem, we then useLemma 5.⎦ ≺ 0 ∀x ≠ 0 such that L guV = 0 . (60)A polynomial matrix M(x) is positive semidefinite for all x if the scalar polynomialy T M(x) y is a sum-of-squares in x and y.Using theorem 3 and the procedure described in section 2.3 then allows a sum-of-squaresrelaxation of condition (60) that leads toProgram 4.Find V such that:(L gdV⎛ ⎛ ⎡− ⎝s ⎝y T ⎣ L f V + hT 1h2 γ( ) TL gdV −I12 γ) ⎤⎞V − l⎞ 1⎦ y⎠ + l 2 + p L g V ⎠s∈ S∈ S∈ Sp, l 1 , l 2 ∈ Pl 1 , l 2 ≻ 0 .20