Silviu-Iulian Niculescu

This article describes the set of stabilizing parameters for a double integrator using two delayed feedback gains. The system is normalized so that the parameter space becomes 3-dimensional. A stability analysis for a given gain ratio is obtained using the stability crossing curves with two delays as the parameters. A global geometric description of the stable region is obtained by analyzing the evolution and bifurcation of stability crossing curves as the gain ratio changes.

A series of recent works have shown that, for a system of linear functional differential equations, a spectral value having a multiplicity exceeding the order of the system tends to correspond to the spectral abscissa of the system, a property called MID for multiplicity-induced-dominancy . In particular, when this multiplicity coincides with the degree of the characteristic quasipolynomial, this property is called generic MID (GMID), in opposition to the intermediate MID (IMID), which corresponds to a multiplicity strictly smaller than the degree. The GMID has been fully characterized for single-delay retarded as well as neutral delay-differential equations thanks to the representation of the corresponding quasipolynomial in terms of a Kummer hypergeometric function . However, apart from partial results, in full generality, no result of the literature enables the characterization of the dominance of a spectral value having an intermediate multiplicity, which is essentially due to the lack of existing results among the open literature pertaining to linear combinations of Kummer functions’ zeros distribution. In this work, we overcome this difficulty and we further investigate the MID to cover the so-called over-order MID , that is, the cases where the multiplicity is larger than the order of the corresponding differential equation.

This work addresses the design of adaptive observer, where both the parameters and state variables of the system are estimated simultaneously. For this purpose, we consider a Linear Time-Invariant system with delay in the output channel, and we address the problem of the semiautonomous adaptive cruise control system with the aim of estimating the cyberattack jointly with the system dynamics in the presence of transmission delay in the communication channel. The dynamics of the estimation errors are expressed by delay-differential equations, and its stability analysis makes use of the multiplicity-induced-dominancy property for spectral values of the error estimation equation Numerical simulations are provided to show the convergence of the proposed observer in presence of delays to estimate the states and the cyber-attack jointly.

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.

Motivated by recently obtained factorizations of characteristic functions of some classes of time-delay systems admitting a root with the largest possible multiplicity in terms of Kummer confluent hypergeometric functions, this paper provides a new representation of some Kummer functions with integer coefficients in terms of iterated integrals of an exponential kernel. As a consequence of the existing links between Kummer, Whittaker, and modified Bessel functions, the latter classes of special functions also take advantage of such an iterated integral representation. We also express characteristic functions of the aforementioned classes of time-delay systems in terms of iterated integrals, and illustrate how such an iterated integral representation allow us to obtain information on the location of the spectrum of the system, at least in some low-order cases.