7.3I t storage functions A number of important results in nonlinear system analysis rely on storage functions for which no explicit formula is known. It is frequently sufficient to provide a lower bound for the storage function(for example, to know that it takes only non-negative values), and to have an analytical expression for the supply rate function o In order to work with such"implicit"storage functions, it is helpful to have theorems which guarantee existence of non-negative storage functions for a given supply rate. In this regard, Theorem 7. 1 can be considered as an example of such result, stating existence of a storage function for a linear and time invariant system as an implication of a frequenc dependent matrix inequality. In this section we present a number of such statements which can be applied to nonlinear systems 7.3.1 Implicit storage functions for abstract systems Consider a system defined by behavioral set B=2l of functions 2: 0,oo)+R9. As usually, the system can be autonomous, in which case z(t)is the output at time t, or with an input, in which case z(t)=u(t);w(t) combines vector input v(t)and vector output Theorem 7.2 Let o: RHR be a function and let b be a behavioral set, consisting of some functions z: 0, oo)+R. Assume that the composition o(a(t)) is integrable over every bounded interval (to, t1) in R+ for all z E B. For to, tE R+ define T(2,to,t)=/o(2(7)d The following conditions are equivalent for all z e B defining same state as zo at time to, is bounded from belo. (a) for every zo E B and to E R+ the set of values I(a, to, t), taken for all t> to and (b)there erists a non-megative storage function V: BxR++R+(such that V(21, t) V(a2, t)whenever z1 and z2 define same state of B at time t)with supply rate o Moreover, when condition(a)is satisfied, a storage function V from(b) can be defined by V(zo(), to)=-infI(a, to, t), 12) where the infimum is taken over all t> to and over all z e B defining same state as zo at time t Proof Implication(b)=(a)follows directly from the definition of a storage function which requires V(z0, t1)-V(zo, to)<I(a, to, t1)7 7.3 Implicit storage functions A number of important results in nonlinear system analysis rely on storage functions for which no explicit formula is known. It is frequently sufficient to provide a lower bound for the storage function (for example, to know that it takes only non-negative values), and to have an analytical expression for the supply rate function ψ. In order to work with such “implicit” storage functions, it is helpful to have theorems which guarantee existence of non-negative storage functions for a given supply rate. In this regard, Theorem 7.1 can be considered as an example of such result, stating existence of a storage function for a linear and time invariant system as an implication of a frequency￾dependent matrix inequality. In this section we present a number of such statements which can be applied to nonlinear systems. 7.3.1 Implicit storage functions for abstract systems Consider a system defined by behavioral set B = {z} of functions z : [0,⊂) ∈� Rq . As usually, the system can be autonomous, in which case z(t) is the output at time t, or with an input, in which case z(t) = [v(t); w(t)] combines vector input v(t) and vector output w(t). Theorem 7.2 Let ψ : Rq ∈� R be a function and let B be a behavioral set, consisting of some functions z : [0,⊂) ∈� Rq . Assume that the composition ψ(z(t)) is integrable over every bounded interval (t0, t1) in R+ for all z ≤ B. For t0, t ≤ R+ define � t I(z, t0, t) = ψ(z(φ ))dφ. t0 The following conditions are equivalent: (a) for every z0 ≤ B and t0 ≤ R+ the set of values I(z, t0, t), taken for all t ∗ t0 and for all z ≤ B defining same state as z0 at time t0, is bounded from below; (b) there exists a non-negative storage function V : B×R+ ∈� R+ (such that V (z1, t) = V (z2, t) whenever z1 and z2 define same state of B at time t) with supply rate ψ. Moreover, when condition (a) is satisfied, a storage function V from (b) can be defined by V (z0(·), t0) = − inf I(z, t0, t), (7.12) where the infimum is taken over all t ∗ t0 and over all z ≤ B defining same state as z0 at time t0. Proof Implication (b)≥(a) follows directly from the definition of a storage function, which requires V (z0, t1) − V (z0, t0) → I(z, t0, t1) (7.13)