NOTATION AND SYMBOLS A(a) eigenvalue of A P(A) spectral radius of A PR(A) real spectrum radius of A o(A)and g(a) the largest and the smallest singular values of A ith singular value of A (A) condition number of a ofA:‖A Im(A),R(A) image(or range) space of A Ker(A), N(A) kernel (or null) space of A x(4) stable invariant subspace of A the stabilizing solution of an arE g*f convolution of g and f Inner p roduct orthogonal, ( a, 9=0 D rthogonal complement of D orthogonal complement of subspace S, e.g., 72 C2(-∞,∞) time domain square integrable functions C2+:=C2[0,∞) subspace of L2(-∞,∞) with functions zero for t<0 C2-:=C2(-∞,0] subspace of L2(-∞,∞) with functions zero for t>0 C2 gR) square integrable functions on Co including at oo H subspace of C2GR)with functions analytic in Re(s>0 subspace of C2GR)with functions analytic in Re(s)<o C∞(jR) functions bounded on Re(s)=0 including at oo the set of Coo gR) functions analytic in Re(s)>0 the set of Coo gR) functions analytic in Re(s)<o refix B and Bo closed and open unit ball,e.g.B△andB°△ prefix R real rational, e.g., RHoo and RH2, etc. Rp(s) rational proper transfer matrices shorthand for G(s) shorthand for state space realization C(sI-A)B+D n(G(s) number of right-half plane poles 7(G(s) number of imaginary axis poles winding number Fe(m, Q) lower lft Fu(M, Q) upper LFTxvi NOTATION AND SYMBOLS λ(A) eigenvalue of A ρ(A) spectral radius of A ρR(A) real spectrum radius of A σ(A) and σ(A) the largest and the smallest singular values of A σi(A) ith singular value of A κ(A) condition number of A kAk spectral norm of A: kAk = σ(A) Im(A), R(A) image (or range) space of A Ker(A), N(A) kernel (or null) space of A X−(A) stable invariant subspace of A Ric(H) the stabilizing solution of an ARE g ∗ f convolution of g and f ∠ angle h,i inner product x ⊥ y orthogonal, hx,yi = 0 D⊥ orthogonal complement of D S⊥ orthogonal complement of subspace S, e.g., H⊥ 2 L2(−∞,∞) time domain square integrable functions L2+ := L2[0,∞) subspace of L2(−∞,∞) with functions zero for t < 0 L2− := L2(−∞, 0] subspace of L2(−∞,∞) with functions zero for t > 0 L2(jR) square integrable functions on C0 including at ∞ H2 subspace of L2(jR) with functions analytic in Re(s) > 0 H⊥ 2 subspace of L2(jR) with functions analytic in Re(s) < 0 L∞(jR) functions bounded on Re(s) = 0 including at ∞ H∞ the set of L∞(jR) functions analytic in Re(s) > 0 H− ∞ the set of L∞(jR) functions analytic in Re(s) < 0 prefix B and Bo closed and open unit ball, e.g. B∆ and Bo∆ prefix R real rational, e.g., RH∞ and RH2, etc. Rp(s) rational proper transfer matrices G∼(s) shorthand for GT (−s)  A B C D  shorthand for state space realization C(sI − A)−1B + D η(G(s)) number of right-half plane poles η0(G(s)) number of imaginary axis poles wno(G) winding number F`(M,Q) lower LFT Fu(M,Q) upper LFT M?N star product