Sum-of-Squares Applications in Nonlinear Controller Synthesis

2.1 Artstein-Sontag Theorem Control Lyapunov Functions and Sum-of-SquaresV becomes a Lyapunov function for the closed loop system:˙V = L f V + L g V u⎧⎪⎨ L f V < 0 if L g V = 0 , x ≠ 0= ( ) 2 ( ) 2 ⎪⎩ −√L f V + L g V < 0 if L g V ≠ 0 , x ≠ 0 .(4)It is easy to verify that the condition of theorem 1 reduces to the question of whether thereexists a suitable V withL f V < 0 ∀x ≠ 0 such that L g V = 0 , (5)as for any L g V ≠ 0 the inequality can be satisfied by some choice of u. This can looselybe interpreted as “the Lyapunov function being accessible from the input, wherever itsdynamics are unstable”. Once a control Lyapunov function is found, the Sontag feedbackformula (3) can be used to guarantee global stability via state feedback. The design taskthus reduces mainly to finding a CLF. 1As an introductory example of CLFs, the special case of single input linear systemsẋ = A x + B u (6)is presented. To show global stability, we consider a quadratic Lyapunov functionV = x T P x , P ≻ 0 . (7)The time derivative along the closed loop system’s trajectories is˙V = (A x + B u) T P x + x T P (A x + B u)( )= x T A T P + P A x +}2 x T {{P B}u . (8)} {{ } L g VL f VThe first term (L f V ) describes the system dynamics without input and is in fact the classicalLyapunov inequality, as the autonomous system is stable if and only if A T P + P A ≺ 0.The control term relaxes this requirement, as the autonomous system has to be stable onlyin those regions of the state space where the input does not affect ˙V .1 It is noteworthy that for any control Lyapunov function V , the scaled function α V with α ∈ R + is alsoa CLF and that the Sontag formula yields the same feedback law for all these scaled CLFs.4