Toward a Partial Pole Placement for Systems of Differential Equations With Commensurate Delays: The Planar System Case

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Abstract
Recent studies have introduced a stabilization framework entitled Partial Pole Placement (PPP) for linear time-invariant single-input/single-output (LTI SISO) systems with input delay. By exploiting the delay as a control parameter, this approach enables the precise prescription of the closed-loop solutions exponential decay rate by directly assigning the spectral abscissa of the associated quasipolynomial. The foundation of PPP lies in a spectral property called Multiplicity-induced-dominancy (MID). For quasipolynomials with a single delay, the generic MID (GMID) property serves as both a cornerstone and the most striking manifestation of the MID principle, asserting that a spectral value with maximalmultiplicity is necessarily dominant. However, a key limitation of the GMID is its restricted parametric flexibility, which constrains its applicability to control design. In this paper, we show how to harness the MID more effectively by highlighting the advantages of intermediate multiplicities for control applications in multiple-input/multiple-output (MIMO) systems with input delay. To this end, the validity of the intermediate (overorder) MID (IMID) property is investigated, then exploited in the design of a state-feedback achieving a prescribed exponential stabilization of the solution of planar dynamical systems with input delay.
Funding Information
  • SPECTRE-EDP Project
  • CNRS-AFRICA Joint Research Program

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