Theorem 7.1 Assume that the pair(A, B)is controllable. A symmetric matric P=P satisfying(7.5) erists if and only if ≥0 whenever ju:=Az+ Bt for some u∈R Moreover, if there exists a matric K such that A+ BK is a Hurwitz matric, and <0 then all such matrices P= P are positive semidefinite Example 7.2 Let G(s)=C(sI-A)-B+D be a stable transfer function(i.e.matrix A is a Huewitz matrix) with a controllable pair(A, B). Then G u)l< 1 for all wE R if and only if there exists P=P>0 such that 2P(A+B)≤|l2-Cz+D2∈R",∈R This can be proven by applying Theorem 7.1 with (,)=|2-|Cz+D2 ndK=0 7.2.2 Storage functions for sector nonlinearities Whenever two components v= v(t)and w w(t) of the system trajectory z z(t) are related in such a way that the pair(u(t), w(t)) lies in the cone between the two lines w=k1v and v=k20, v=0 is a storage function for a(z(t)=(u(t)-k1(t)(k2v(t)-t() For example, if w(t)=v(t) then a(a(t))=v(tw(t). If w(t)= sin(t)sin(u(t)then a(2(t)=|(t)2-|(t)2 7.2.3 Storage for scalar memoryless nonlinearity Whenever two components v=v(t)and w= w(t) of the system trajectory z= 2(t)are related by w(t)=o(u(t)), where o: R + R is an integrable function, and v(t)is a component of system state, V(r(t))=v(u(t)is a storage function with supply rate a( 2(t)=u(t)w(t) where v(y)=/o(r)6 Theorem 7.1 Assume that the pair (A, B) is controllable. A symmetric matrix P = P ∗ satisfying (7.5) exists if and only if � �∗ � � x¯ w¯ � x¯ w¯ ∗ 0 whenever jσx¯ = Ax¯ + Bw¯ for some σ ≤ R. (7.11) Moreover, if there exists a matrix K such that A + BK is a Hurwitz matrix, and � �∗ � � I I � → 0, K K then all such matrices P = P∗ are positive semidefinite. Example 7.2 Let G(s) = C(sI − A)−1B + D be a stable transfer function (i.e. matrix A is a Huewitz matrix) with a controllable pair (A, B). Then |G(jσ)| → 1 for all σ ≤ R if and only if there exists P = P∗ ∗ 0 such that w| 2 w| 2 2¯x x + B ¯ x + D ¯ � ¯ ¯ ∗ P(A¯ w) → | ¯ − |C¯ x ≤ Rn, w ≤ Rm. This can be proven by applying Theorem 7.1 with ψ(¯x, w¯) = |w¯| x + D ¯ 2 − |C¯ w| 2 and K = 0. 7.2.2 Storage functions for sector nonlinearities Whenever two components v = v(t) and w = w(t) of the system trajectory z = z(t) are related in such a way that the pair (v(t), w(t)) lies in the cone between the two lines w = k1v and v = k2v, V ∞ 0 is a storage function for ψ(z(t)) = (w(t) − k1v(t))(k2v(t) − w(t)). For example, if w(t) = v(t)3 then ψ(z(t)) = v(t)w(t). If w(t) = sin(t)sin(v(t)) then 2 ψ(z(t)) = |v(t)| 2 − |w(t)| . 7.2.3 Storage for scalar memoryless nonlinearity Whenever two components v = v(t) and w = w(t) of the system trajectory z = z(t) are related by w(t) = �(v(t)), where � : R ∈� R is an integrable function, and v(t) is a component of system state, V (x(t)) = �(v(t)) is a storage function with supply rate ψ(z(t)) = v˙(t)w(t), where � y �(y) = �(φ )dφ. 0