Sum-of-Squares Applications in Nonlinear Controller Synthesis

3.1 Modifications for Performance Sontag Formula Feedbackdetermines the performance of the control law. A desirable shape might be obtained fromthe solution of the linear quadratic regulator (LQR) problem.We motivate this idea by introducing the modified Sontag feedback law⎧0 if L g V = 0(⎪⎨√u = −R−1 (L g V ) T( ) 2 ( ) ( )) TL(L g V ) R −1 (L g V ) T f V + L f V + x T Q x L g V R −1 L g V⎪⎩if L g V ≠ 0 .which for any Q ≻ 0 , R ≻ 0 still yields global stability of the closed loop system because⎧⎪⎨ L f V < 0 if L g V = 0 , x ≠ 0˙V = ( ) 2 ( )⎪⎩ −√ ( ) TL f V + x T Q x L g V R −1 L g V < 0 if L g V ≠ 0 , x ≠ 0 .It still is continuous everywhere except possibly at x = 0. Furthermore for linear systemsẋ = A x + B u (34)the control law (32) is equivalent to the linear quadratic regulator(32)(33)u = −R −1 B T P x (35)with 0 = A T P + P A + Q − P B R −1 B T P , P ≻ 0 , P = P T , (36)that is known to minimize the cost functional∫ ∞0x T Q x + u T R u dt . (37)To see this, note that for a quadratic control Lyapunov function V = x T P x, the feedbacklaw (32) can be written in explicit form asu = −R−1 g T P xx T P g R −1 g T P xFor linear systems this becomes( √)x T P f + (x T P f) 2 + x T Q x x T P g R −1 g T P x . (38)√u = −R −1 B T P x xT P A x + (x T P A x) 2 + x T Q x x T P B R −1 B T P xx T P B R −1 B T . (39)P x14