Multichannel feedback system response

Research on MIMO LTI Systems with Delays

I am researching a method to compute the response of feedback MIMO LTI systems with delays without explicitly inverting the transfer function matrix. The approach I attribute to myself, G. A. Kupriyanov [1], involves:

1. Reformulating the system as a matrix-vector "Duhamel's integral" (Russian: "Интеграл Дюамеля"; see also: convolution and impulse response) problem.

2. Deriving a Volterra integral equation of the first kind for the derivative of the output response, avoiding inversion of $\hat{\mathrm{\Phi }}(s)=\hat{\mathbf{I}}-\hat{\mathbf{L}}(s)\hat{\mathbf{A}}(s)$.

Key Steps

• Equations (1.7)-(1.11) in [1] show the reduction to:

$\underset{0}{\overset{t}{\int }}\hat{\mathrm{\Phi }}(t-\tau )\cdot \frac{\mathrm{d}\overrightarrow{R}}{\mathrm{d}\tau }\mathrm{d}\tau =\overrightarrow{\mathrm{\Psi }}(t)$

Notation

• $\overrightarrow{R}(t)$: Output vector
• $\hat{\mathbf{L}}(s)$: Forward Transfer Matrix
• $\hat{\mathbf{A}}(s)$: Feedback Transfer Matrix
• $\hat{\mathrm{\Phi }}(s)=\hat{\mathbf{I}}-\hat{\mathbf{L}}(s)\hat{\mathbf{A}}(s)$: Closed-loop matrix
• $\overrightarrow{\mathrm{\Psi }}(t)=\hat{\mathbf{L}}(t)\cdot \overrightarrow{f}(0)+\underset{0}{\overset{t}{\int }}\hat{\mathbf{L}}(t-\tau )\cdot \frac{\mathrm{d}\overrightarrow{f}}{\mathrm{d}\tau }\mathrm{d}\tau -\hat{\mathrm{\Phi }}(t)\cdot \overrightarrow{R}(0)$: combines input and initial conditions
• $\overrightarrow{f}(t)$: Input vector

Prior Art Review

Soviet Literature

• Ratmirov's operator methods (1970s)
• Tsypkin's integral criteria (1977)
• Yakubovich's frequency-domain inequalities

These avoid matrix inversion but focus on stability/synthesis, not explicit response calculation via Volterra equations.

Western Literature

• Lancaster's spectral decompositions
• Rosenbrock's polynomial matrices

None explicitly derive this Volterra-Duhamel combination for MIMO delays.

Research Question

Was this specific approach (matrix inversion-free Volterra equations for MIMO delay systems via Duhamel's integral) previously published — especially in Soviet-era works — or is it likely novel?

Context

• I claim no prior sources.
• I've conducted extensive literature searches without success.

References

• [1] G. A. Kupriyanov, "One method for solving MIMO LTI systems with delays", 2021.
• [2] Discussion in russian
• [3] Corresponding question (math.stackexchange.com)