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== 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 <math>\mathbf{\hat{\Phi}}(s) = \mathbf{\hat{I}} - \mathbf{\hat{L}}(s)\mathbf{\hat{A}}(s)</math>.

=== Key Steps ===

* Equations (1.7)-(1.11) in [1] show the reduction to:
: <math>\int_0^t \mathbf{\hat{\Phi}}(t-\tau) \cdot \frac{\mathrm d\vec{R}}{\mathrm d\tau} \, \mathrm d\tau = \vec{\Psi}(t)</math>

=== Notation ===

* <math>\vec{R}(t)</math>: Output vector
* <math>\mathbf{\hat{L}}(s)</math>: Forward Transfer Matrix
* <math>\mathbf{\hat{A}}(s)</math>: Feedback Transfer Matrix
* <math>\mathbf{\hat{\Phi}}(s) = \mathbf{\hat{I}} - \mathbf{\hat{L}}(s)\mathbf{\hat{A}}(s)</math>: Closed-loop matrix
* <math>\vec{\Psi}(t) = \mathbf{\hat{L}}(t) \cdot \vec{f}(0) + \int_0^t \mathbf{\hat{L}}(t-\tau) \cdot \frac{\mathrm d\vec{f}}{\mathrm d\tau} \, \mathrm d\tau - \mathbf{\hat{\Phi}}(t) \cdot \vec{R}(0)</math>: combines input and initial conditions
* <math>\vec{f}(t)</math>: 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, "[https://github.com/1spb-org/recycling-CAD/blob/main/ideas/J001-One-Method-Of-Solving-MIMO-LTI-Systems.pdf One method for solving MIMO LTI systems with delays]", 2021.
* [2]  [https://alterlit.ru/post/76566 Discussion in russian]
* [3]  [https://math.stackexchange.com/questions/5057549/was-the-matrix-inversion-free-volterra-integral-approach-for-mimo-lti-systems-wi Corresponding question (math.stackexchange.com)]

[[Category:Control theory]]
[[Category:Linear time-invariant systems]]
[[Category:Multiple-input multiple-output systems]]