On the External Positivity of SISO Linear Dynamic Systems under a Class of Nonzero and Possibly Negative Initial Conditions Eventually Subject to Incommensurate Point Internal and External Delays

The property of external positivity of dynamic systems is commonly defined as the non-negativity of the output for all time under zero initial conditions and any given non-negative input for all time. This paper investigates the extension of that property for a structured class of initial conditions of a single-input single-output (SISO) linear dynamic system which can include, in general, certain negative initial conditions. The above class of initial conditions is characterized analytically based on the structure of the transfer function. The basic study is performed in the delay-free case, but extensions are then given for systems subject to a finite number of internal and external, in general incommensurate, point delays and for the closed-loop dynamic systems which incorporate a feedback compensator. The formulation relies on calculating the output based on the impulse responses by considering the relation of the mentioned sets of structured initial conditions with the zero-state response which allows to keep the non-negativity of the zero-input response and that of the total response provided the non-negativity for all time of the zero-state response.This research was funded by Spanish Government and European Commission, grant number RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and by the Basque Government, grant numberIT1207-19-The APC was funded by Spanish Government and European Commission grant number RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE)

On the Existence of Block-Diagonal Solutions to Lyapunov and H∞\mathcal{H}_{\infty} Riccati Inequalities

In this paper, we describe sufficient conditions when block-diagonal solutions to Lyapunov and $\mathcal{H}_{\infty}$ Riccati inequalities exist. In order to derive our results, we define a new type of comparison systems, which are positive and are computed using the state-space matrices of the original (possibly nonpositive) systems. Computing the comparison system involves only the calculation of $\mathcal{H}_{\infty}$ norms of its subsystems. We show that the stability of this comparison system implies the existence of block-diagonal solutions to Lyapunov and Riccati inequalities. Furthermore, our proof is constructive and the overall framework allows the computation of block-diagonal solutions to these matrix inequalities with linear algebra and linear programming. Numerical examples illustrate our theoretical results.Comment: This is an extended technical report. The main results have been accepted for publication as a technical note in the IEEE Transactions on Automatic Contro