Measures of input-output amplification (across frequency) Largest singular value Hilbert-Schmidt norm (power spectral density) Systems with one spatial variable Two point boundary value problems EE 8235: Lecture 2 Lecture 2: Input-output norms; Pseudospectra Singular Value Decomposition of the frequency response operator Input-output norms { worst-case amplification of deterministic disturbances H norm: measure of robustness H 2 norm: { energy of the impulse response variance amplification Pseudospectra of linear operators
EE 8235: Lecture 2 2 Example: cantilever beam µ ψ tt + αei ψ txxxx + EI ψ xxxx = ψ(, t) =, ψ x (, t) = αei ψ txxx (l, t) + EIψ xxx (l, t) = u(t), ψ xx (l, t) = input: output: u(t) y(t) ψ(l, t) = y(t) magnitude: phase:
EE 8235: Lecture 2 3 Example: diffusion equation on L 2, Distributed input and output fields Frequency response operator φ t (y, t) = φ yy (y, t) + d(y, t) φ(y, ) = φ(±, t) = φ(y, ω) = T (ω) d(, ω) (y) = (jωi yy ) d(, ω) (y) = T ker (y, η; ω) d(η, ω) dη
EE 8235: Lecture 2 4 Two point boundary value realizations of T (ω) Input-output differential equation T (ω) : Spatial state-space realization T (ω) : { φ (y, ω) jω φ(y, ω) = d(y, ω) φ(±, ω) = x (y, ω) x (y, ω) x = 2(y, ω) jω x 2 (y, ω) φ(y, ω) = x (y, ω) x 2 (y, ω) x (, ω) = x 2 (, ω) + + d(y, ω) x (, ω) x 2 (, ω)
EE 8235: Lecture 2 5 Frequency response operator Evolution equation Spatial differential operators Frequency response operator E φ t (y, t) = F φ(y, t) + G d(y, t) ϕ(y, t) = C φ(y, t) F = F ij = f ij,k (y) dk dy k nij k = T (ω) = C (jωe F) G
EE 8235: Lecture 2 6 Singular Value Decomposition of T (ω) compact operator T (ω): H in H out ϕ(y, ω) = T (ω) d(, ω) (y) = n = σ n (ω) u n (y, ω) v n, d T (ω) T (ω) u n (, ω) (y) = σ 2 n(ω) u n (y, ω) {u n } orthonormal basis of H out T (ω) T (ω) v n (, ω) (y) = σ 2 n(ω) v n (y, ω) {v n } orthonormal basis of H in σ (ω) σ 2 (ω) > d(y, ω) = v m (y, ω) ϕ(y, ω) = σ m (ω) u m (y, ω) σ (ω): the largest amplification at any frequency
EE 8235: Lecture 2 7 Input-output gains Determined by singular values of T (ω) H norm: an induced L 2 gain (of a system) T 2 worst case amplification: = sup output energy input energy = sup ω σ 2 (ω) σ 2 (ω)
d Nominal Dynamics Γ modeling uncertainty (can be nonlinear or time-varying) small-gain theorem: ϕ EE 8235: Lecture 2 8 Robustness interpretation stability for all Γ with Γ γ γ < T LARGE worst case amplification small stability margins
Both σ (ω) and T (ω) 2 HS power spectral density: T (ω) 2 HS = trace ( T (ω) T (ω) ) = Enabling tool: TPBVRs of T (ω) and T (ω) n = σ 2 n (ω) can be computed efficiently using Chebfun T (ω) 2 HS : Jovanović & Bamieh, Syst. Control Lett. 6 EE 8235: Lecture 2 9 Hilbert-Schmidt norm of T (ω) σ (ω): Lieu & Jovanović, J. Comput. Phys. (submitted; also: arxiv:2.579v) software: Frequency Responses of PDEs in Chebfun H 2 norm: variance amplification T 2 2 = 2 π T (ω) 2 HS dω
EE 8235: Lecture 2 ψ ψ 2 = d s + λ WORST CASE AMPLIFICATION sup energy of ψ 2 energy of d A toy example λ ψ R λ 2 ψ 2 ψ R = sup T (jω) 2 = R2 (λ λ 2 ) 2 2π ω + s + λ 2 ψ 2 d VARIANCE AMPLIFICATION T (jω) 2 dω = R 2 λ λ 2 (λ + λ 2 ) ROBUSTNESS d s + λ ψ R s + λ 2 ψ 2 small-gain theorem: stability for all Γ with Γ γ Γ modeling uncertainty (can be nonlinear or time-varying) γ < λ λ 2 /R
EE 8235: Lecture 2 A note on computation of H 2 and H norms H 2 norm Operator Lyapunov equation φ t (y, t) = A φ(y, t) + B d(y, t) ϕ(y, t) = C φ(y, t) T 2 2 = trace ( C X C ) A X + X A = B B H norm E-value decomposition of Hamiltonian in conjunction with bisection T γ A γ B B γ C C A has at least one imaginary e-value
Realization of T T : f T g = d x (y) = A (y) x(y) + B (y) d(y) ϕ(y) = C (y) x(y) = N a x(a) + N b x(b) T ϕ EE 8235: Lecture 2 2 Spatial state-space realization of T (ω) Cascade connection of T and T Realization of T T : z (y) = A (y) z(y) C (y) f(y) g(y) = B (y) z(y) = M a z(a) + M b z(b) Ma M b N a N b =
EE 8235: Lecture 2 3 Integral form of a differential equation D diffusion equation: differential form Auxiliary variable: Integrate twice ( E + ( ) D (2) jωi φ(y) = d(y) ) E φ(y) = ν(y) = D (2) φ (y) φ (y) = φ(y) = y y ν(η ) dη + k = ( η2 = J (2) ν (y) + K (2) k J () ν (y) + k ) ν(η ) dη dη 2 + (y + ) k 2 k
D diffusion equation: integral form ( I jωj (2)) ν(y) jω K (2) k = d(y) 2 k2 k = Eliminate k from the equations to obtain ( E + (I jωj (2) + 2 ) jω (y + ) E J (2) ν(y) = d(y) ) E J (2) ν(y) EE 8235: Lecture 2 4 More suitable for numerical computations than differential form integral operators and point evaluation functionals are well-conditioned
Book Trefethen and Embree: Spectra and Pseudospectra Online resources Talk by Nick Trefethen: Pseudospectra and EigTool Software: { Pseudospectra Gateway EigTool EE 8235: Lecture 2 5 Pseudospectra perturbed system: ψ t = (A + Γ) ψ ɛ-pseudospectrum: σ ɛ (A) = { s C; (si A) > /ɛ } = {s C; s σ(a + Γ), Γ < ɛ} can be converted to an input-output problem