F: R"HR which admits efficient check of non-negativity of its elements is the set of quadratic for F() for which nonnegativity is equivalent to positive semidefiniteness of the coefficient matrix Q This observation is exploited in the linear-quadratic case, when f, g are linear functions ∫(,)=A+B,9(z,⑦)=Cz+ and o is a quadratic form (,⑦) Then it is natural to consider quadratic storage function candidates V(a)=iPD only, and(.3)transforms into the(symmetric)matri inequality PA+AP PB BP (7.5) Since this inequality is linear with respect to its parameters P and 2, it can be solve elatively efficiently even when additional linear constraints are imposed on P and 2 Note that a quadratic functional is non-negative if and only if it can be represented as a sum of squares of linear functionals. The idea of checking non-negativity of a functional by trying to represent it as a sum of squares of functions from a given linear set can be used in searching for storage functions of general nonlinear systems as well. Indeed, let H: RXR"R and V: R"HR be arbitrary vector-valued functions. For every TER, condition(7.3)with V(z)=7V() is implied by the identity rV(f(G,))-TV()+H(x,)SH(,)=0(正,)V∈X,∈W,(7.6) S=S>0 is a positive semidefinite symmetric matrix. Note that both the storage function candidate parameter T and the"sum of squares"parameter S=s20 enter constraint(7.6)linearly. This, the search for a valid storage function is reduced to semidefinite program. In practice, the scalar components of vector H should include enough elements so that identity(7.6)can be achieved for every T E R by choosing an appropriate S=S'(not necessarily positivie semidefinite). For example, if f, g, o are polynomials, it may be a good idea to use a polynomial V and to define h as the vector of monomials up to a given degree� � 4 F : Rn ∈� R which admits efficient check of non-negativity of its elements is the set of quadratic forms � �∗ � � x x ¯ ¯ ∗ F(¯x) = Q , (Q = Q ) 1 1 for which nonnegativity is equivalent to positive semidefiniteness of the coefficient matrix Q. This observation is exploited in the linear-quadratic case, when f, g are linear functions f(¯x, w¯) = Ax¯ + Bw, ¯ g(¯x, w¯) = Cx¯ + Dw, ¯ and ψ is a quadratic form � �∗ � � x x ¯ ¯ ψ(¯ w) = ¯ x, ¯ � . w w¯ Then it is natural to consider quadratic storage function candidates V (¯x) = ¯x x ∗ P ¯ only, and (7.3) transforms into the (symmetric) matrix inequality PA + A∗ P PB → �. (7.5) B∗ P 0 Since this inequality is linear with respect to its parameters P and �, it can be solved relatively efficiently even when additional linear constraints are imposed on P and �. Note that a quadratic functional is non-negative if and only if it can be represented as a sum of squares of linear functionals. The idea of checking non-negativity of a functional by trying to represent it as a sum of squares of functions from a given linear set can be used in searching for storage functions of general nonlinear systems as well. Indeed, let Hˆ : Rn × Rm ∈� RM and Vˆ : Rn ∈� RN be arbitrary vector-valued functions. For every φ ≤ RN , condition (7.3) with x) = φ ∗ Vˆ V (¯ (¯x) is implied by the identity x, ¯ φ ∗ ˆ x) + ˆ x, ¯ H(¯ w) = ψ(¯ w) � ¯ ¯ ∗ Vˆ (f(¯ w)) − φ V (¯ H(¯ w) ∗ S ˆ x, ¯ x, ¯ x ≤ X, w ≤ W, (7.6) as long as S = S∗ ∗ 0 is a positive semidefinite symmetric matrix. Note that both the storage function candidate parameter φ and the “sum of squares” parameter S = S∗ ∗ 0 enter constraint (7.6) linearly. This, the search for a valid storage function is reduced to semidefinite program. In practice, the scalar components of vector Hˆ should include enough elements so that identity (7.6) can be achieved for every φ ≤ RN by choosing an appropriate S = S∗ (not necessarily positivie semidefinite). For example, if f, g, ψ are polynomials, it may be a good idea to use a polynomial Vˆ and to define Hˆ as the vector of monomials up to a given degree