Merge pull request #101 from JuliaControl/freqseparation

add frequency-separation decomposition

src/RobustAndOptimalControl.jl

@@ -37,7 +37,7 @@ include("descriptor.jl")
 export ExtendedStateSpace, system_mapping, performance_mapping, noise_mapping, ssdata_e, partition, ss
 include("ExtendedStateSpace.jl")
 
-export modal_form, schur_form, hess_form
+export modal_form, schur_form, hess_form, frequency_separation
 include("canonical.jl")
 
 export δ, δr, δc, δss, nominal, UncertainSS, uss, blocksort, sys_from_particles, ss2particles

src/canonical.jl

@@ -156,4 +156,32 @@ function hess_form(sys)
     C = sys.C*Q
     D = sys.D
     ss(A,B,C,D, sys.timeevol), Q, F
+end
+
+"""
+    stateinds(sys, inds; zeroD = false)
+
+Extract a subsystem of the statevariables indicated in `inds`.
+If `zeroD = true`, the `D` matrix is set to zero.
+"""
+function stateinds(sys, inds; zeroD=false)
+    A = sys.A[inds, inds]
+    B = sys.B[inds, :]
+    C = sys.C[:, inds]
+    D = zeroD ? zero(sys.D) : sys.D
+    ss(A,B,C,D, sys.timeevol)
+end
+
+"""
+    frequency_separation(sys, ω)
+
+Decomponse `sys` into `sys = sys_slow + sys_fast`, where `sys_slow` contain all modes with eigenvalues with absolute value less than `ω` and `sys_fast` contain all modes with eigenvalues with absolute value greater than or equal to `ω`.
+"""
+function frequency_separation(sys, ω)
+    sysm, T, E = modal_form(sys)
+    slow_inds = findall(e->abs(e) < ω, E.values)
+    fast_inds = findall(e->abs(e) ≥ ω, E.values)
+    slow_sys = stateinds(sysm, slow_inds, zeroD=true)
+    fast_sys = stateinds(sysm, fast_inds, zeroD=false)
+    (; slow_sys, fast_sys, T, E, sysm)
 end

test/test_canonical_forms.jl

@@ -109,3 +109,12 @@ end
 end
 
 
+@testset "frequency_separation" begin
+    @info "Testing frequency_separation"
+
+    sys = ssrand(2,2,5)
+    res = frequency_separation(sys, 1)
+    slow_sys, fast_sys = res
+    # bodeplot([sys, slow_sys, fast_sys, slow_sys+fast_sys], legend=false, plotphase=false)
+    @test hinfnorm(fast_sys+slow_sys - sys)[1] < 1e-7
+end