Computing the \(H_\infty\)-norm of transfer functions that do not depend on parameters

We denote the algebra of real rational functions on the imaginary axis \(i \mathbb{R}\) which are proper and have no poles on \(i \mathbb{R}\) by \(RL_\infty := \Bigg\{ \frac{n(i \omega)}{d(i \omega)} \mid n,d \in \mathbb{R}[i \omega], \ \gcd(n,d) = 1, \\ \deg_\omega(n) \leq \deg_\omega(d), V_{i \mathbb{R}}(\langle d \rangle) = \emptyset \Bigg\}.\)

For a transfer function \(G \in \mathbb{R}(s)^{u \times v}\) such that \(G_{i \mathbb{R}} \in RL^u_{\infty}\), let \(\gamma > 0\) and \(\Phi_\gamma(s) := \gamma^2 I_v - G^T(-s)G(s),\) where \(G^T\) is the transpose of \(G\).

We then can write \(\det(\Phi_\gamma(i \omega)) = \frac{n(\omega, \gamma)}{d(\omega)},\) where \(n \in \mathbb{R}[\omega, \gamma]\) and \(d \in \mathbb{R}[\omega]\) are coprime. Since \(G\) has no poles on the imaginary axis, \(d(\omega)\) does not vanish on \(\mathbb{R}\).

We denote by \(\tilde{n} \in \mathbb{R}[\omega, y]\) the square-free part of \(n\). The \(H_\infty\)-norm of \(G\) is then given by:

$$\|G\|_\infty = \max \left\{ \pi_y \left( V_{\mathbb{R}} \left( \tilde{n}, \frac{\partial \tilde{n}}{\partial \omega} \right) \right) \cup V_{\mathbb{R}} \left( \langle L_\omega(\tilde{n}) \rangle \right) \right\},$$

where \(\pi_y\) gives the \(y\)-projection.

Below, we outline the, so-called, Sturm-Habicht method to compute \(\|G\|_\infty\) as presented in Yacine Bouzidi, Alban Quadrat, Fabrice Rouillier, and Grace Younes, "Computation of the \(\mathcal{L}_{\infty}\)-norm of Finite-Dimensional Linear Systems," in Mathematical Software – ICMS 2021, Springer, 2021.

Algorithm: Sturm-Habicht Method

Input: The polynomial \(P \in (\mathbb{Z}[y])[x]\) such that the curve \(\{(x, y) \in \mathbb{R}^2 \mid P(x, y) = 0\}\) is bounded in the \(y\)-direction.

Output: Isolating interval of \(\max \left\{ \pi_y \big( V_\mathbb{R} \big( \langle P, \frac{\partial P}{\partial x} \rangle \big) \big) \cup V_\mathbb{R}(\langle L_c(P) \rangle) \right\}.\)

Steps:

  1. Compute \(\{\mathrm{Sres}_j(P, \frac{\partial P}{\partial x}, x)\}_{j=0,\ldots,\deg_x(P)}\).

  2. Compute \(y_1 < \dots < y_m\), the real roots of \(\mathrm{Sres}_0\).

  3. For \(i\) from \(1\) to \(m\):

    • If \(y_{1-i+m} \in V_\mathbb{R}(\langle L_c(P) \rangle)\), return the isolating interval of \(y_{1-i+m}\).

    • Else if \(\mathrm{Sign\_Variations}(\{\mathrm{sign}(\mathrm{stha}_d(x)(y_{1-i+m})), \dots, \mathrm{sign}(\mathrm{stha}_1(y_{1-i+m}))\}) > 0\) return the isolating interval of \(y_{1-i+m}\).

Adding and loading the necessary packages

Info

This page is generated from a Pluto notebook. The notebook can be download from here, and to run it you need to have installed PaceControl.jl and the rest of the required packages locally and to activate your default Julia environement with 'using Pkg; Pkg.activate()'. 

begin
    using Pkg
    Pkg.activate();
    using  Nemo, AbstractAlgebra, PaceControl
end

Two ways to call the Hinf_StHa function:

1. With the transfer matrix \(G(s)\)

begin
    S1, (y, s) = rational_function_field(AbstractAlgebra.QQ, [:y, :s])
    G = [s//(s+1) -s//(s+1); -s//(s+1) 1//(s+1)]
end
2×2 Matrix{AbstractAlgebra.Generic.RationalFunctionFieldElem{Rational{BigInt}, AbstractAlgebra.Generic.MPoly{Rational{BigInt}}}}:
 s//(s + 1)   -s//(s + 1)
 -s//(s + 1)  1//(s + 1)
Hinf_StHa(G)
[1.59, 1.62]
Hinf_StHa(G,starting_precision=Int32(10))
[1.61792, 1.61804]

2. With the polynomial \(n(\gamma,\omega)\)

begin
    S, γ = polynomial_ring(Nemo.QQ, "γ");
    S2, ω = polynomial_ring(S, "ω");
    P = (4*γ − 1)*(4*γ + 1)*ω^4 − 4*(2*γ − 1)*(2*γ + 1) *ω^2 + 16*(γ − 1)*(γ + 1)
end
(16*γ^2 - 1)*ω^4 + (-16*γ^2 + 4)*ω^2 + 16*γ^2 - 16 
Hinf_StHa(P)
[1.09, 1.12]
Hinf_StHa(P,starting_precision=Int32(10))
[1.09436, 1.09448]