failure to construct proper and stable approximate inverse

[baggepinnen]

The following error is surprising to me, I am trying to find a stable approximate inverse $R$ to the system $P$ (finding approximate solution $P R \approx T_r$) using the function grasol and I have ensured that $T_r$ has higher order than $P$ so that $R$ will be proper. I'm not finding a way to accomplish this, what am I missing?

P = let
    tempA = [-127.71761592895362 33.740472583436485 18.69926152087514 16.2653468435812 -60.8537683435043 -27.663561859737232; 33.599695985371504 -18.33957948184755 5.435950763181253 6.438453963791473 -50.57194688335815 -10.88056707117063; 0.0 -9.984850557442169 5.909777138354576 9.645266687309263 -28.204903435199924 39.84215207598821; 0.0 0.0 -3.200344495993629 -2.3302121892316534 21.190934019367457 53.843586702867505; 0.0 0.0 0.0 -0.09447508503844063 -5.45325278997154 -22.807939980462518; 0.0 0.0 0.0 0.0 0.21710913034924129 0.41541933413572085]
    tempB = [-6.380782719358301; 0.0; 0.0; 0.0; 0.0; 0.0;;]
    tempC = [0.0 2.1420348851324356 -0.7829492912082094 -2.8288802598425318 -1.5366043434257315 0.17412523537137548]
    tempD = [0.0;;]
    dss(tempA, tempB, tempC, tempD)
end

Tr = let
    tempA = [-388.88888888888886 -253.18287037037038 -183.14733099708505 -79.49102907859596 -41.401577645102066 -23.959246322397036 -11.884546786903291; 256.0 0.0 0.0 0.0 0.0 0.0 0.0; 0.0 128.0 0.0 0.0 0.0 0.0 0.0; 0.0 0.0 128.0 0.0 0.0 0.0 0.0; 0.0 0.0 0.0 64.0 0.0 0.0 0.0; 0.0 0.0 0.0 0.0 32.0 0.0 0.0; 0.0 0.0 0.0 0.0 0.0 16.0 0.0]
    tempB = [4.0; 0.0; 0.0; 0.0; 0.0; 0.0; 0.0;;]
    tempC = [0.0 0.0 0.0 0.0 0.0 0.0 2.971136696725823]
    tempD = [0.0;;]
    dss(tempA, tempB, tempC, tempD)
end

R, info = grasol(P, Tr)
dss2ss(R)

julia> dss2ss(R)
ERROR: The system is possibly improper: try with simple_infeigs = false

julia> dss2ss(R, simple_infeigs = false)
ERROR: controllable higher order infinite eigenvalues present

[andreasvarga]

The condition for the solvability of the model-matching problem, i.e., the lack of zeros on the boundary of stability domain, is not satisfied, because P has two infinite zeros

julia> gzero(P)
6-element Vector{Float64}:
  3.880280056208058
 -3.968112108105213
 -4.213613389538763
 -0.8064516128932318
 Inf
 Inf

Therefore, there exists no stable solution. The computed "solution" is improper (has two infinite poles), so dss2ss can not compute a proper transfer function.

julia> gpole(R)
12-element Vector{ComplexF64}:
 -56.052604966110174 + 0.0im
 -55.855985733623704 + 0.3910698295096717im
 -55.855985733623704 - 0.3910698295096717im
  -55.43640193417488 + 0.4684925698796138im
  -55.43640193417488 - 0.4684925698796138im
  -55.12575429359079 + 0.2007315642242232im
  -55.12575429359079 - 0.2007315642242232im
 -0.8064516128971927 + 0.0im
  -4.213613389532109 + 0.0im
  -3.968112108110124 + 0.0im
                 Inf + 0.0im
                 Inf + 0.0im

[baggepinnen]

Thanks for explaining, what is your recommended way to obtain the desired approximate inverse that is also stable?

[andreasvarga]

I am not quite sure, but you could try the following:

julia> Tr = dss(1.);

julia> R, info = grasol(P, Tr);

julia> gpole(R)
6-element Vector{Float64}:
 -4.213613389556274
 -3.968112108012967
 -3.8802800562817055
 -0.8064516128932349
 Inf
 Inf

julia> N, M = glcf(R);

julia> glinfnorm(P*R-Tr,atol = 1.e-4)
(1.0, 0.0)

julia> glinfnorm(P*N-Tr,atol = 1.e-4)
(1.0, 0.0)

R is the Hinf-optimal solution of min||P*R - 1||, but is improper. You can make it proper using a stable left (or right) coprime factorization and use the numerator N as a suboptimal solution. Depending on your problem, this could work ... or not. Both R and N have practically the same norm of the residual, namely 1.

[baggepinnen]

Thanks for the suggestion!