Conditions for stabilizability of time‐delay systems with real‐rooted plant

Abstract

In this article we consider the 𝛾-stabilization of nth-order linear time-invariant dynamical systems using multiplicity-induced-dominancy-based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least n + 1 and at least n using an integral factorization of the corresponding characteristic function. A necessary condition for 𝛾-stabilizability is analyzed utilizing the property that the derivative of a 𝛾-stable quasipolynomial is also 𝛾-stable under certain conditions. Sufficient and necessary conditions are given for systems with real-rooted open-loop characteristic function: the delay intervals are determined where the conditions for dominancy and 𝛾-stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multilink inverted pendulum. K E Y W O R D Scharacteristic equation, dominant roots, feedback system, stabilizability, time delay INTRODUCTIONStabilization of unstable equilibria and orbits in the presence of communication delay is an important and challenging task in engineering applications. [1][2][3] Finding the control parameters that allow stable operation for large feedback delay is not a trivial task. 4 Stability diagrams can be used to visualize stability properties in the space of control parameters. [5][6][7] When performance with respect to settling time has to be optimized in linear time-invariant systems then one has to deal Abbreviations: LTI, linear time-invariant; MID, multiplicity-induced-dominancyThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of Kummer functions

“…Notice that the MID property may hold even when it is about a spectral value with a strictly-intermediate admissible multiplicity, as shown in [16] for second-order plants and in [4] for n th -order retarded equations admitting a real spectral value with multiplicity n + 1 and a finite dimensional part admitting exclusively real modes. However, in all generality, the limits of the MID property remain an open question.…”

“…The fact that a spectral value achieves maximum multiplicity imposes algebraic constraints on each of the system's "entries" (polynomial coefficients as well as the delay parameter). An MID-based approach is proposed in [4,5] operating the intimate representation of the quasipolynomial to provide conditions for one spectral value with an eligible intermediate multiplicity. This makes it possible to split the system parameters into two categories, some of them considered as model parameters (assumed to be fixed and known) and the remaining ones considered as values to be adjusted.…”

Stability, Delays and Multiple Characteristic Roots in Dynamical Systems: A Guided Tour

“…The dominancy of λ + has been shown by using the argument principle, see, for instance (Boussaada et al, 2020). A constructive proof of the dominancy of λ + may be also shown using the corresponding quasipolynomial factorization introduced in Bedouhene et al ( 2020) and recently extended in Balogh et al (2022) to arbitrary order systems such that in open-loop they admit only realrooted modes. 2…”

“…To give a sufficient condition for γ-stabilizability 26 we utilize the MID-property: the control parameters b i are tuned such that the characteristic function ∆(λ) has a real root λ 0 with multiplicity n + 1. The result from Balogh et al (2022) emphasizes the way to factorize a quasipolynomial admitting a multiple root. Proposition 7.3.…”

New Features of P3$δ$ software: Partial Pole Placement via Delay Action

“…In all the above cases, the maximal multiplicity of a real root or, equivalently, the maximal number of coexisting simple real roots is the integer n + m + 1. Furthermore, the idea to exploit the nature of the open-loop roots (real or complex) in control design was proposed for second-order systems in Boussaada et al (2020b) and further extended for arbitrary order systems with real-rooted plants in Balogh et al (2020Balogh et al ( , 2021.…”