7. 2 Storage functions with quadratic supply rates As described in the previous section, one can search for storage functions by considering linearly parameterized sets of storage function candidates. It turns out that storage functions derived for subsystems of a given system can serve as convenient building blocks (i.e. the components Vg of V). Indeed, assume that Va =va(a(t) are storage functions with supply rates aa=a(z(t)). Typically, a(t)includes x(t) as its component, and has some additional elements, such as inputs, outputs, and othe nonlinear combinations of system states and inputs. If the objective is to find a storage function V, with a given supply rate o, one can search for V. in the form V(x(t)=∑v(x(),τ≥0 where Ta are the search parameters. Note that in this case it is known a-priori that every V in(7.7)is a storage function with supply rate (2(t) Therefore, in order to find a storage function with supply rate o.=o.(2()), it is sufficient to find T≥0 such that ∑n(2)≤(2) (7.9) When o, aa are generic functions, even this simplified task can be difficult. However, in the important special case when o, and g are quadratic functionals, the search for Tg in (7.9)becomes a semidefinite program In this section, the use of storage functions with quadratic supply rates is discussed 7.2.1 Storage functions for LTI systems A quadratic form V(r)=I'PI is a storage function for LTI system Ar+ Bw (7.10 with quadratic supply rate (,⑦) a if and only if matrix inequality(7.5) is satisfied The well-known Kalman-Popou-Yakubouich Lemma, or positive real lemma gives use requency domain condition for eristence of such P= Pl for given A, B, 25 7.2 Storage functions with quadratic supply rates As described in the previous section, one can search for storage functions by considering linearly parameterized sets of storage function candidates. It turns out that storage functions derived for subsystems of a given system can serve as convenient building blocks (i.e. the components Vq of Vˆ ). Indeed, assume that Vq = Vq(x(t)) are storage functions with supply rates ψq = ψq(z(t)). Typically, z(t) includes x(t) as its component, and has some additional elements, such as inputs, outputs, and othe nonlinear combinations of system states and inputs. If the objective is to find a storage function V� with a given supply rate ψ�, one can search for V� in the form N V (x(t)) = Vq(x(t)), φq ∗ 0, (7.7) q=1 where φq are the search parameters. Note that in this case it is known a-priori that every V� in (7.7) is a storage function with supply rate N ψ(z(t)) = φqψq(z(t)). (7.8) q=1 Therefore, in order to find a storage function with supply rate ψ� = ψ�(z(t)), it is sufficient to find φq ∗ 0 such that N φ1ψq(¯z) → ψ�(¯z) � z. ¯ (7.9) q=1 When ψ�, ψq are generic functions, even this simplified task can be difficult. However, in the important special case when ψ� and ψq are quadratic functionals, the search for φq in (7.9) becomes a semidefinite program. In this section, the use of storage functions with quadratic supply rates is discussed. 7.2.1 Storage functions for LTI systems A quadratic form V (¯x) = ¯x∗ Px¯ is a storage function for LTI system x˙ = Ax + Bw (7.10) with quadratic supply rate � �∗ � � x x ¯ ¯ ψ(¯ w) = ¯ x, ¯ � w w¯ if and only if matrix inequality (7.5) is satisfied. The well-known Kalman-Popov-Yakubovich Lemma, or positive real lemma gives useful frequency domain condition for existence of such P = P∗ for given A, B, �