Sum-of-Squares Applications in Nonlinear Controller Synthesis

2.3 Sum-of-Squares Relaxation for CLFs Control Lyapunov Functions and Sum-of-Squares2.3 Sum-of-Squares Relaxation for Control Lyapunov FunctionsRecall that the conditions for a control Lyapunov function are the positive definiteness ofthe function itself and the negative definiteness of L f V wherever L g V = 0. For polynomialsystems and polynomial control Lyapunov functions, the question whether a given functionis a CLF thus evolves around the positiveness of polynomials at values of x where anotherpolynomial’s value is zero.The important result used to assess positivity of a polynomial wherever another polynomialhas certain properties is taken from real algebraic geometry [3] and isTheorem 3: Positivstellensatz.The empty set condition⎧ ∣ ∣∣∣∣∣∣∣⎪⎨ q 1 ≥ 0, . . . , q nq ≥ 0x ∈ R n r 1 ≠ 0, . . . , r nr ≠ 0⎪⎩t 1 = 0, . . . , t nt = 0⎫⎪⎬= ∅⎪⎭is equivalent to the existence of polynomialsq ∈ C ( q 1 , . . . , q nq)r ∈ M (r 1 , . . . , r nr )t ∈ I (t 1 , . . . , t nt )such that q + r 2 + t = 0.The sets used in the theorem are the Multiplicative Monoid, Cone and IdealM (r 1 , . . . , r nr ) ={r k 11 rk 22 · · · r knrn r∣ }∣∣∣k 1 , k 2 , . . . , k nr ∈ NC ( ∣)l∑ ∣∣∣q 1 , . . . , q nq ={s 0 + s i b i l ∈ N, s i ∈ S, b i ∈ M ( ) } q 1 , . . . , q nq ,i=1{∑ nt ∣ }∣∣∣I (t 1 , . . . , t nt ) = t k p k p k ∈ P .k=1Observe, that for single polynomial constraints, i.e., n q = n r = n t = 1, these sets become,M (r) =C (q) =I (t) ={ ∣ } ∣∣∣r k k ∈ N{s 0 + s 1 q,{t p,}∣ s 0, s 1 ∈ S}∣ p ∈ P .,8