Sum-of-Squares Applications in Nonlinear Controller Synthesis
Sum-of-Squares Applications in Nonlinear Controller Synthesis
Sum-of-Squares Applications in Nonlinear Controller Synthesis
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3.3 Modifications for Disturbance Attenuation Sontag Formula Feedback) ( ) TReplac<strong>in</strong>g L f V <strong>in</strong> 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 <strong>in</strong>Lemma 4:There exists a state feedback law which guarantees the L 2 ga<strong>in</strong> from d to h isless than γ if there exists a positive def<strong>in</strong>ite, 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 aga<strong>in</strong> use sum-<strong>of</strong>-squares programm<strong>in</strong>g to search for a suitable functionV. As the condition <strong>in</strong> lemma 4 is quadratic <strong>in</strong> V , we use the Schur complement[ ]M LT≺ 0 ⇔ M − L T N −1 L ≺ 0 and N ≺ 0 (59)L Nto transform it <strong>in</strong>to the equivalent form⎡(L gdV⎣ L f V + hT 1h2 γ( ) TL gdV −I12 γ) ⎤To formulate this as a sum-<strong>of</strong>-squares problem, we then useLemma 5.⎦ ≺ 0 ∀x ≠ 0 such that L guV = 0 . (60)A polynomial matrix M(x) is positive semidef<strong>in</strong>ite for all x if the scalar polynomialy T M(x) y is a sum-<strong>of</strong>-squares <strong>in</strong> x and y.Us<strong>in</strong>g theorem 3 and the procedure described <strong>in</strong> section 2.3 then allows a sum-<strong>of</strong>-squaresrelaxation <strong>of</strong> condition (60) that leads toProgram 4.F<strong>in</strong>d 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