On Pole Placement and Spectral Abscissa Characterization for Time-delay Systems

15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Time Delay Systems Sinaia, Romania, September 2019 15th IFAC Workshop on Time9-11, Delay Systems Sinaia, Romania, September 2019 15th IFAC Workshop on Time9-11, Delay Systems Available online at www.sciencedirect.com Sinaia, Romania, September 9-11, 2019 15th IFAC Workshop on Time9-11, Delay Systems Sinaia, Romania, September 2019 Sinaia, Romania, September 9-11, 2019

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IFAC PapersOnLine 52-18 (2019) 55–60

On Pole Placement and Spectral Abscissa On Pole Placement and Spectral Abscissa On Pole Placement and Spectral Abscissa On Pole Placement and Spectral Abscissa Characterization for Time-delay Systems Characterization for Time-delay Systems On Pole Placement and Spectral Abscissa Characterization for Time-delay Systems Characterization for∗ Time-delay Systems ∗∗ Characterization for Time-delay Systems Fazia Fazia Bedouhene Bedouhene ∗∗ Islam Islam Boussaada Boussaada ∗∗ ∗∗

∗∗∗ Fazia Islam Silviu-Iulian ∗ Niculescu ∗∗ ∗∗∗ Fazia Bedouhene Bedouhene Islam Boussaada Boussaada Silviu-Iulian Niculescu ∗ Niculescu ∗∗∗ ∗∗ Silviu-Iulian ∗∗∗ Fazia Bedouhene Islam Boussaada Silviu-Iulian Niculescu ∗ ∗∗∗ Mouloud Mammeri Pures et Appliques, ∗ Laboratoire de Mathmatiques Silviu-Iulian Niculescu ∗ Laboratoire de Mathmatiques Pures et Appliques, Mouloud Mammeri Pures et Mouloud Mammeri University Tizi-Ouzou, Algeria. ∗ Laboratoire de Mathmatiques Laboratoire de Mathmatiques et Appliques, Appliques, University of of Pures Tizi-Ouzou, Algeria.Mouloud Mammeri ∗ University of Tizi-Ouzou, Algeria. E-mail: [email protected] Laboratoire de Mathmatiques Pures et Appliques, University of Tizi-Ouzou, Algeria.Mouloud Mammeri E-mail: [email protected] ∗∗ ˆ E-mail: [email protected] Ile-de-France, Equipe DISCO ∗∗ Inria SaclayUniversity of Tizi-Ouzou, ˆ E-mail: [email protected] EquipeAlgeria. DISCO & & L2S, L2S, ∗∗ Inria Saclay-Ile-de-France, ˆ Ile-de-France, Equipe DISCO CNRS-CentraleSup´ e lec-Universit´ e Paris Sud, Universit´ Paris ∗∗ Inria SaclayE-mail: [email protected] ˆ Inria SaclayIle-de-France, Equipe DISCO & &ee L2S, L2S, CNRS-CentraleSup´ elec-Universit´ e Paris Sud, Universit´ Paris Saclay, Saclay, ∗∗ CNRS-CentraleSup´ e lec-Universit´ e Paris Sud, Universit´ e Paris Saclay, 91192 Gif-sur-Yvette cedex, France & IPSA Ivry-sur-Seine, ˆ Inria SaclayIle-de-France, DISCO &e L2S, CNRS-CentraleSup´ elec-Universit´ e Paris Sud, Universit´ ParisFrance Saclay, 91192 Gif-sur-Yvette cedex, France &Equipe IPSA Ivry-sur-Seine, France 91192 Gif-sur-Yvette cedex, France & IPSA Ivry-sur-Seine, France Email: [email protected] CNRS-CentraleSup´ e lec-Universit´ e Paris Sud, Universit´ e Paris Saclay, 91192 Gif-sur-Yvette cedex, France & IPSA Ivry-sur-Seine, France Email: [email protected] ∗∗∗ Email: L2S, Gif-sur-Yvette CNRS-CentraleSup´ eelec-Universit´ ee Paris Sud, ee Paris ∗∗∗91192 cedex, France & IPSA Ivry-sur-Seine, France Email: [email protected] [email protected] lec-Universit´ Paris Sud, Universit´ Universit´ Paris ∗∗∗ L2S, CNRS-CentraleSup´ eeGif-sur-Yvette lec-Universit´ ee Paris Sud, Universit´ ee Paris Saclay, 91192 cedex, France ∗∗∗ L2S, CNRS-CentraleSup´ Email: [email protected] L2S, CNRS-CentraleSup´ lec-Universit´ Paris Sud, Universit´ Paris Saclay, 91192 Gif-sur-Yvette cedex, France ∗∗∗ Saclay, 91192 cedex, France Email: L2S, CNRS-CentraleSup´ lec-Universit´e Paris Universit´e Paris Saclay,[email protected] 91192 eGif-sur-Yvette Gif-sur-Yvette cedex,Sud, France Email: [email protected] Email: Saclay,[email protected] 91192 Gif-sur-Yvette cedex, France Email: [email protected] Email: [email protected] Abstract: This paper presents a systematic Abstract: This paper presents a systematic frequency frequency domain domain approach approach to to analyse analyse the the stability stability Abstract: This paper presents a systematic frequency domain approach to analyse the stability of reduced-order linear systems with single delay. More precisely, we address the problem of Abstract: This paper systematic domain approach to analyse stability of reduced-order linearpresents systemsa with single frequency delay. More precisely, we address thethe problem of of reduced-order linear systems with single delay. More precisely, we address the problem of the spectral abscissa characterization and the coexistence of non oscillating modes for such Abstract: This paper presents a systematic frequency domain approach to analyse the stability of linearcharacterization systems with single delay. More precisely, address modes the problem of thereduced-order spectral abscissa and the coexistence of non we oscillating for such the spectral abscissa and coexistence of non oscillating modes for functional differential equations. The dominancy of such non modes is of reduced-order linearcharacterization systems with delay. address the problem of the spectral abscissa characterization and the the coexistence of oscillating non we oscillating for such such functional differential equations. The single dominancy ofMore such precisely, non oscillating modesmodes is analytically analytically functional differential equations. The dominancy of such modes is shown for considered reduced order Time-delay the spectral abscissa characterization and the coexistence of oscillating non oscillating for such functional differential equations. of systems. such non non oscillating modesmodes is analytically analytically shown for the the considered reduced The orderdominancy Time-delay systems. shown for the considered reduced order Time-delay functional differential equations. The of systems. such non reserved. oscillating modes is analytically shown for © the considered reduced orderdominancy Time-delay systems. Copyright 2019. The Authors. Published by Elsevier Ltd. All rights Keywords: Time-delay systems, Stability, Spectral abscissa, Control shown for the considered reduced order Time-delay systems. Keywords: Time-delay systems, Stability, Spectral abscissa, Control design, design, Pole Pole assignment, assignment, Keywords: Time-delay systems, Stability, Spectral abscissa, Control design, Pole Non oscillation Keywords: Time-delay systems, Stability, Spectral abscissa, Control design, Pole assignment, assignment, Non oscillation Non Keywords: Time-delay systems, Stability, Spectral abscissa, Control design, Pole assignment, Non oscillation oscillation 1. INTRODUCTION INTRODUCTION instance Non oscillation 1. instance P´ P´oolya lya and and Szeg˝ Szeg˝oo (1972). (1972). The The multiplicity multiplicity of of a a root root 1. instance P´ o lya and Szeg˝ o (1972). The multiplicity of itself is not important as such but its connection with the 1. INTRODUCTION INTRODUCTION instance P´olya and Szeg˝ of a a root root itself is not important aso (1972). such butThe its multiplicity connection with the Investigation of dynamical systems with time-delay is an itself is not important as such but its connection with the dominancy of this root is a meaningful tool for control 1. INTRODUCTION P´olya (1972). ofcontrol a root Investigation of dynamical systems with time-delay is an instance itself is not important aso is such butThe its multiplicity connection the dominancy of and this Szeg˝ root a meaningful tool forwith Investigation of dynamical systems with time-delay is an active research area that connects a wide range of sciendominancy of this root is a meaningful tool for control synthesis. To the best of our knowledge, the first time an is notTo important as such but its connection the Investigation dynamical systems awith is an itself active researchof area that connects widetime-delay range of sciendominancy ofthe thisbest root a knowledge, meaningful toolfirst forwith control synthesis. of isour the time an active research area that connects a wide range of scientific disciplines including mathematics, physics, engineersynthesis. To the best of our knowledge, the first time an analytical proof of the dominancy of a spectral value for Investigation of dynamical systems with time-delay is an dominancy of this root is a meaningful tool for control active research including area that mathematics, connects a wide range engineerof scien- synthesis. tific disciplines physics, theofbest our knowledge, the firstvalue time for an analytical To proof theofdominancy of a spectral tific disciplines including mathematics, physics, engineering, biology, economics etc. The present paper focuses analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented in active research area that connects a wide range of sciensynthesis. To the best of our knowledge, the first time an tific disciplines including mathematics, physics, engineering, biology, economics etc. The present paper focuses analytical of the dominancy of a spectral value for the scalar proof equation with a single delay was presented in ing, biology, economics etc. The present paper focuses on stability and stabilizing-controllers design for linear the scalar equation with a single delay was presented in Hayes (1950). The dominancy property is further explored tific disciplines including mathematics, physics, engineerproof of dominancy the dominancy of a isspectral value for ing,stability biology, and economics etc. The present paper focuses on stabilizing-controllers design for linear analytical the scalar equation with a single delay was presented in Hayes (1950). The property further explored on stability and stabilizing-controllers design for linear time-invariant retarded time-delay systems. The study of Hayes (1950). The dominancy property is further explored and analytically shown in the case of second-order sysing, biology, economics etc. The present paper focuses equation withina the single delay was presented in on stability and stabilizing-controllers design linear time-invariant retarded time-delay systems. Thefor study of the Hayes (1950). Theshown dominancy property further explored and scalar analytically case of issecond-order systime-invariant retarded time-delay systems. The study of conditions onand the equation parameters that guarantees the and analytically in the case systems and aa rightmost root design using on stabilityon stabilizing-controllers design linear (1950). Theshown dominancy issecond-order further explored time-invariant retarded time-delay systems. Thefor study of Hayes conditions the equation parameters that guarantees the and shown in assignment theproperty case of ofbased second-order systems analytically and rightmost root assignment based design using conditions the equation parameters that the exponential stability of solutions solutions is systems. question of study ongoing tems and aa rightmost root assignment design using is Boussaada al. time-invariant retarded time-delay The of delayed and analytically shown the case in ofbased second-order sysconditions on onstability the equation parameters that guarantees guarantees the exponential of is aa question of ongoing tems andstate-feedback rightmost root assignment based design et using delayed state-feedback isin proposed proposed in Boussaada et al. exponential stability of solutions is a question of ongoing interest and remains an open problem especially when delayed state-feedback is proposed in Boussaada et al. (2017, 2018b) where its applicability in damping active conditions onstability the equation parameters thatespecially guarantees the tems a rightmost root assignmentinbased design active using exponential ofansolutions is a question of ongoing interest and remains open problem when delayed state-feedback isapplicability proposed et al. (2017,and 2018b) where its inBoussaada damping interest and remains an open problem especially when the systems are of high order or having multiple and/or (2017, 2018b) where its applicability in damping active vibrations for a piezo-actuated beam is proved. See also exponential stability of solutions is a question of ongoing state-feedback proposed etalso al. interest and are remains anorder openorproblem especiallyand/or when delayed the systems of high having multiple (2017, 2018b) itsisapplicability inBoussaada damping vibrations for awhere piezo-actuated beaminis proved. Seeactive the are of high multiple distributed delays. In particular, inhaving frequency-domain, the (2017, vibrations for aawhere piezo-actuated beam is proved. See also Boussaada et (2018a); Boussaada (2018) interest anddelays. remains anorder openor especiallyand/or when 2018b) its applicability in Niculescu damping active the systems systems are of In high order orproblem having multiple and/or distributed particular, in frequency-domain, the vibrations for piezo-actuated beamand is proved. See also Boussaada et al. al. (2018a); Boussaada and Niculescu (2018) distributed delays. particular, in frequency-domain, problem reduces toIn the analysis of the distribution of the the which Boussaada et Boussaada and Niculescu (2018) an analytical proof for of the the systems are of high order orof multiple and/or vibrations for a (2018a); piezo-actuated is dominancy proved. See distributed delays. In particular, inhaving frequency-domain, problem reduces to the analysis the distribution of Boussaada et al. al. (2018a); Boussaada and Niculescu (2018) which exhibit exhibit an analytical proofbeam for the the dominancy ofalso the problem reduces to the analysis of the distribution of the roots of the corresponding characteristic equation, see for which exhibit an analytical proof for the dominancy of spectral value with maximal multiplicity for second-order distributed delays. particular, in the frequency-domain, et al. (2018a); Boussaada anddominancy Niculescu (2018) problem reduces toInthe analysis of distribution of the roots of the corresponding characteristic equation, see for Boussaada which exhibit an analytical proof for the of the the spectral value with maximal multiplicity for second-order roots corresponding characteristic equation, see for instance Bellman and Cooke (1963); Cooke and van van den spectral value with maximal multiplicity for controlled via proportional-derivative problem reduces toand theCooke analysis of theCooke distribution of den the which exhibit an analytical proof for the dominancy of the roots of of the the corresponding characteristic equation, see for systems instance Bellman (1963); and spectral value with maximal multiplicity for second-order second-order systems controlled via aa delayed delayed proportional-derivative instance Bellman and Cooke (1963); Cooke and den Driessche (1986); Walton and Marshall (1987); St´ p´ n systems controlled via proportional-derivative controller. roots of the corresponding characteristic equation, see for spectral value with maximal multiplicity for second-order instance Bellman and Cooke (1963); Cooke and van van den Driessche (1986); Walton and Marshall (1987); St´ eep´ aan systems controlled via aa delayed delayed proportional-derivative controller. Driessche (1986); Walton and Marshall (1987); St´ e p´ a n (1989); Hale and Lunel (1993); Michiels and Niculescu controller. instance Bellman Cooke (1963); Cooke andNiculescu van den controlled via a delayed proportional-derivative DriesscheHale (1986); Walton and Marshall (1987); St´ep´ an systems (1989); and and Lunel (1993); Michiels and controller. By this (1989); Hale and (1993); Michiels and (2007); Sipahi et al. al.Lunel (2011). this paper, paper, we we would would like like to to extend extend such such an an analytical analytical Driessche (1986); Walton and Marshall St´ep´an By controller. (1989); Sipahi Hale and Lunel (1993); Michiels (1987); and Niculescu Niculescu (2007); et (2011). By this paper, we would like to extend such an analytical characterization of the spectral abscissa for retarded time(2007); Sipahi et al. (2011). By this paper, we would like to extend such an analytical characterization of the spectral abscissa for retarded time(1989); Hale and Lunel (1993); Michiels and Niculescu (2007); Sipahipoint et al. of (2011). The starting the present work is an interesting characterization of the spectral abscissa for retarded timedelay system with real spectral values which are not necBy this paper, we would like to extend such an analytical The starting point of the present work is an interesting characterization of the spectral abscissa for retarded delay system with real spectral values which are nottimenec(2007); Sipahi et al. (2011). The starting point of the present work is an interesting property, discussed in recent studies, called Multiplicitydelay system with real spectral values which are not necessarily multiple. The effect of the coexistence of such non ofThe theeffect spectral abscissa for retarded The starting point of present work is anMultiplicityinteresting characterization property, discussed in the recent studies, called delay system with real spectral values which are nottimenecessarily multiple. of the coexistence of such non property, discussed in recent studies, called MultiplicityInduced-Dominancy denoted in the sequel (MID). As a essarily multiple. The effect of the coexistence of such non oscillatory modes on the asymptotic stability of the trivial The starting point of the present work is an interesting delay system with real spectral values which are not necproperty, discussed indenoted recent in studies, called (MID). MultiplicityInduced-Dominancy the sequel As a essarily multiple. effect of the coexistence such non oscillatory modes The on the asymptotic stability ofofthe trivial Induced-Dominancy denoted in the sequel (MID). As a matter of fact, it is shown that multiple spectral valoscillatory modes on the asymptotic stability of the trivial solution will be explored. In particular, the coexistence of property, discussed in recent studies, called Multiplicityessarily multiple. The effect of the coexistence of such non Induced-Dominancy in themultiple sequel (MID). a oscillatory matter of fact, it isdenoted shown that spectralAs valmodes on the asymptotic stability of the trivial solution will be explored. In particular, the coexistence of matter of fact, it is shown that multiple spectral values for Time-delay systems can be characterized using solution will be explored. In particular, the coexistence real spectral values makes them rightmost-roots of P S Induced-Dominancy denoted in the sequel (MID). As a oscillatory modes onvalues the asymptotic stability of the trivial matter fact, it issystems shown can that multiple spectralusing values for of Time-delay be characterized solution be explored. In particular, the coexistence spectral makes them rightmost-roots of P SB B realwill ues for Time-delay be characterized using Birkhoff/Vandermonde-based approach; seespectral for instance instance real spectral values makes them rightmost-roots of S the corresponding quasipolynomial. Furthermore, if fact, it issystems shown can that multiple val- P solution be explored. particular, the coexistence ues for of Time-delay systems can be characterized using aamatter Birkhoff/Vandermonde-based approach; see for realwill spectral values In makes them rightmost-roots of P SB the quasipolynomial. Furthermore, if they they Bcorresponding aBoussaada Birkhoff/Vandermonde-based approach; see for instance and Niculescu (2016b,a, 2014); Boussaada et al. the corresponding quasipolynomial. Furthermore, if they are negative, this guarantees the asymptotic stability of ues for Time-delay systems can be characterized using real spectral values makes them rightmost-roots of P S a Birkhoff/Vandermonde-based approach; see for instance Boussaada and Niculescu (2016b,a, 2014); Boussaada et al. the corresponding quasipolynomial. Furthermore, if they are Bnegative, this guarantees the asymptotic stability of Boussaada and Niculescu (2016b,a, 2014); Boussaada et al. (2016). More precisely, in the previous works, it is emphaare negative, this guarantees the asymptotic stability the trivial solution. a Birkhoff/Vandermonde-based approach; see for instance the corresponding quasipolynomial. Furthermore, if they Boussaada and Niculescu (2016b,a, 2014); Boussaada et al. (2016). More precisely, in the previous works, it is empha- are negative, this guarantees the asymptotic stability of of the trivial solution. (2016). More precisely, in the previous works, it emphasized that the admissible multiplicity of the zero spectral trivial solution. Boussaada and Niculescu 2014); Boussaada et al. the are negative, this guarantees the asymptotic stability of (2016). More precisely, in (2016b,a, the previous works, it is is spectral emphasized that the admissible multiplicity of the zero the trivial solution. The remaining paper is organized as follows; in Section sized the admissible multiplicity of the value isMore bounded by the the generic Polya and Szeg¨ ospectral bound The remaining paper is organized as follows; in Section 2 2 (2016). in the previous works, it iso emphathe trivial solution. sized that that theprecisely, admissible multiplicity of the zero zero spectral value is bounded by generic Polya and Szeg¨ bound The remaining paper is organized as follows; in Section 2 we summarize some important facts on the coexistence non value is bounded by the generic Polya and Szeg¨ o bound denoted P S , which is nothing but the degree of the The remaining paper is organized as follows; in Section 2 we summarize some important facts on the coexistence non B sized that the admissible multiplicity of the zero spectral value is bounded by the generic Polya anddegree Szeg¨ o of bound denoted P SB , which is nothing but the the oscillatory we summarize some important facts on the coexistence non modes for the first and second order time-delay Thesummarize remaining paper is organized asonfollows; intime-delay Section 2 denoted P S which is nothing but the degree of the corresponding (i.e the number of the inwe some important facts the coexistence non oscillatory modes for the first and second order B ,, quasipolynomial value is bounded by the generic Polya and Szeg¨ o bound denoted P S which is nothing but the degree of the corresponding B quasipolynomial (i.e the number of the inoscillatory modes for the first and second order time-delay differential equation with aa single delay. Section 33 contains we summarize some important facts on the coexistence non corresponding quasipolynomial (i.e the number of the involved polynomials plus their degree minus one), see for oscillatory modes for the first and second order time-delay differential equation with single delay. Section contains denotedpolynomials P SB , quasipolynomial which is their nothing the degree of the corresponding (i.ebut theminus number of the in- oscillatory volved plus degree one), see for differential equation aa single 33 contains forwith the first and delay. secondSection order time-delay volved plus one), see equation with single delay. Section contains corresponding quasipolynomial (i.e theminus number of the in- differential modes volved polynomials polynomials plus their their degree degree minus one), see for for differential equation with a single delay. Section 3 contains volved polynomials plus their degree minus one), see for Copyright © 2019 IFAC 126 2405-8963 Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved.

Copyright © 2019 IFAC 126 Copyright © under 2019 IFAC 126 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 126 10.1016/j.ifacol.2019.12.206 Copyright © 2019 IFAC 126

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Fazia Bedouhene et al. / IFAC PapersOnLine 52-18 (2019) 55–60

the main result of this paper. We address the problem of the spectral abscissa characterization and the coexistence of non oscillating modes for the third-order differential equation with a single delay. Section 4 is dedicated to illustrate the main result of the paper on Mach number regulation in a wind tunnel by the assignment of four dominant equidistributed real roots. In Section 5, some concluding remarks end the paper. 2. PRELIMINARIES Let consider the generic n-order system with a single time delay: x(t) ˙ = A0 x(t) + A1 x(t − τ ). (1) Here τ is a positive constant delay and the matrices Aj ∈ Mn (R) for j = 0 . . . 1. It is well known that the asymptotic behavior of the solutions of (1) is determined from the spectrum designating the set of the roots of the associated characteristic function (denoted in the sequel ∆(s, τ )). Namely, the characteristic function corresponding to system (1) is a quasipolynomial ∆ : C × R+ → C of the form: ∆(s, τ ) = det(sI − A0 − A1 e−τ s ). (2) Asymptotic stability of the trivial solution and oscillatory behavior of (1) are known. In particular, the zero solution of this equation is asymptotically stable if and only if all roots of (3) lie in the left half plane, and a given solution of (1) is said to be non oscillatory if it corresponds to a real root of (3). This section summarizes the main findings reported in Amrane et al. (2018), concerning the question of coexistence of P SB negative real roots for (3), when in particular ∆(s, τ ) = P0 (s) + P1 (s)e−τ s . (3) where deg(P0 ) = n, deg(P1 ) = 0 for n = 1 and n = 2. Let’s consider the following systems x(t) ˙ + ax(t) + bx(t − τ ) = 0,

(4)

x ¨(t) + ax(t) ˙ + bx(t) + αx(t − τ ) = 0. (5) The characteristic equation associated to (4) and (5) are respectively as follows: ∆1 (s, τ ) := s + a + b exp(−sτ ) = 0,

(6)

(7) ∆2 (s, τ ) = s2 + as + b + α exp(−sτ ) = 0. Theorem 1. For a given delay τ > 0, the system (4) admits two distinct real spectral values at s = s1 and s = s2 , with s2 < s1 , if and only if  s exp(−s1 τ ) − s1 exp(−s2 τ )   a = a(s1 , s2 , τ ) := 2 ; exp(−s2 τ ) − exp(−s1 τ ) (8) s − s 1 2   b = b(s1 , s2 , τ ) := . exp(−s2 τ ) − exp(−s1 τ )

• Moreover, both spectral values s1 and s2 of (4) are negative, if and only if equation a(s1 , s2 , τ ) = 0 admits a positive solution in τ . Furthermore, in such a case the zero solution of (4) is asymptotically stable. • The spectral value s1 is nothing but the spectral abscissa corresponding to (4). 127

Theorem 2. The system (5) admits three distinct real spectral values s3 , s2 and s1 with s3 < s2 < s1 if and only if the parameters a, b and α satisfy   1 i+j  a(τ ) := (−1) (s2i − s2j ) exp(−sk τ )    Q(τ )  i,j,k∈Λ    i
(9)

where

Q(τ ) =



i,j,k∈Λ i
(−1)

i+j

(si − sj ) exp(−sk τ ).

In this case, α is necessarily negative. • The spectral value s1 is negative if and only if there exists τ0 > 0 such that a(τ0 ) + s2 = 0. This guarantees the asymptotic stability of the system. • The root s1 is the spectral abscissa of (5). 2.1 Application to a control problem Let us focus on the second order system x ¨(t) + ax(t) ˙ + bx(t) = u(t), (10) where u is the unknown control and a and b are known parameters. Assume that the system (10) is unstable in the uncontrolled case, namely when u(t) = 0. This arises for instance if a < 0. Our aim is to design a control u under the form: u(t) = −αx(t) − βx(t − τ ), (11) that stabilizes the closed loop system: x ¨(t) + ax(t) ˙ + (b + α)x(t) + βx(t − τ ) = 0, (12) by pole assignment method. It is therefore a question to assign three negative spectral values, s1 , s2 and s3 , to the characteristic equation associated to (12). This pole assignment is then interpreted by the conditions (9), in which case b := b + α, allowing the computation of the parameters α, β and τ > 0 of the control (11). By choosing s1 , s2 and s3 equi-distributed (s1 − s2 = s2 − s3 = d > 0), we obtain, in view of (12), the following values of the parameters τ , α and β:  b = 1/2 a2 + as1 − ad − 2 s1 d + s1 2 + 1/2 d2 ,     β = −1/2 −8 s d + 3 d2 − 4 ad + 4 s 2 + 4 as + a2  ×   1 1 1     − −s1d+d −3 d + 2 s1 + a ,   2 s1 − d + a       −3 d + 2 s1 + a    τ = ln d−1 2 s1 − d + a

The asymptotic stability of the nontrivial solution of the closed-loop system (12) is guaranteed by the dominance property of s1 established in Theorem 2, and the positivity of the delay τ .

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2.2 About the geometric structure of the envelope curve associated to (6) and (7) (1) It is well known that the classical envelope curve has a connected geometric structure (see for instance Niculescu et al. (2010)). Interestingly, when considering two real distinct spectral values, the connected structure of the envelope may be lost, see Fig. 1. τ ∈ [0.1, τ ∗ ]

10

τ ∈ [τ ∗ , τ ∗∗ ]

4

5

2

0

0

−5

−2

−10

0

−5 −4

−4 −5

0

5

10

τ ≥ τ ∗∗

5

−4

−2

0

2

−2

0

2

Fig. 1. Envelope curve of the characteristic equation (6). Case of co-existence of two simple real roots s1 = −1, s2 = −2. Here τ ∗ ≈ 0.67288 and τ ∗∗ = ln(2). (2) Such a geometry is encountered in Qiao and Sipahi (2013), where an analytical study and synthesis of rightmost eigenvalues of x(t) ˙ = Ax(t − τ ) is considered. (3) Likewise, the geometric structure of the envelope curve of the quasipolynomial (7), defined by  x2 + y 2 − A0 2 − A1 2 e−τ x = 0,     −a 1 0 0 with A0 = and A1 = , may loose −b 0 −α 0 its connection as observed in the first order equation, depending on the distribution of the roots s1 , s2 , s3 and the delay τ . More precisely, three cases can be observed according to the distance of the root s3 with respect to the centered circle, of radius R = A0 2 . τ ∈ [0.1, τ˜], with τ˜ = 1.12665

15

10 ≤ τ ≤ 90

10 > τ > τ˜ 15

15

10

10

10 5 y

5

5 s2 s1

s3

s2 s1

0 s3

0 −5 −10

s2 s1

s3 0

−5

−5

−10

−10

−15

−15

x2 + y2 = A0 22

−15 −10

−5

0 x

5

10

15 −15

−10

−5

x

0

5

10 −15

−10

−5

0 x

5

10

15

Fig. 2. Envelope curve of the characteristic equation (7). Case s1 = −2, s2 = −3, s3 = −13.

Case s1 = −2, s2 = −3, s3 = −6 15

15

x2 + y2 = A0 22

10

y

y

0

−5

−5

−10

−10

−15 −10

Let consider the generic 3-order equation with a single time delay x(3) (t)+a2 x(2) (t)+a1 x(1) (t)+a0 x(t)+αx(t−τ ) = 0. (13) The characteristic equation associated to (13) is given by: ∆3 (s, τ ) = s3 + a2 s2 + a1 s + a0 + α exp(−sτ ) = 0. (14) To simplify some formulas, let us introduce the notation [s1 , s2 ]t := ts1 + (1 − t) s2 . Theorem 3. i) System (13) admits four distinct real spectral values s4 , s3 , s2 and s1 with s4 < s3 < s2 < s1 if and only if 4 4   (si − sj ) = 0. (−1)k exp(−sk τ ) Q(τ ) := In this case, the coefficients ai , i = 1, . . . , 3, and α are uniquely determined as a continuous function with respect to the delay τ > 0. The parameter variable α(τ ) is necessarily positive for every τ > 0, and satisfies 4  1 α (τ ) = (si − sj ) = Q (τ ) i
τ3

0

x

5

10

15

−10

1

2

[0,1]3

(1 − t) (1 − θ) e

a1 (τ ) = 5

10

Fig. 3. Envelope curve of the characteristic equation (7). Case s1 = −2, s2 = −3, and s3 = −R (right), and 0 > s3 > −R (left). 128

a0 (τ ) =

i
i=k

4 4 4   1  (−1)k e−sk τ (si − sj ) si sj ; Q(τ ) i,j=1 i,j=1 k=1

0 x

] dλdθdt

θ t

Sketch of the proof First, let us find conditions on the coefficients ai , for i = 0, · · · , 2, and α ensuring the coexistence of P SB real spectral values (here P SB = 4) for the quasipolynomial (14), recall that P SB is Polya and Szeg¨ o bound. For, assume that (14) admits four real spectral values s1 > s2 > s3 > s4 . This means that s3i +a2 s2i +a1 si +a0 +α exp(−si τ ) = 0, for all i = 1, · · · , 4. (15) From which we deduce the following values of the coefficients ai , for i = 0, · · · , 2, and α:

s3

−5

−τ [s2 ,[s3 ,[s4 ,s1 ]λ ]

ii) The spectral value s1 is negative if and only if there exists τ0 > 0 such that a2 (τ0 ) + s2 + s3 = 0. This guarantees the asymptotic stability of the system.

−15 −5

i,j=1 i
k=1

s2 s1

s2 s1

0

3.1 Four real poles assignment is possible

k=1

5 s3

3. ON POLE-PLACEMENT FOR THIRD-ORDER EQUATIONS WITH A SINGLE DELAY

4 4 4   1  k+1 −sk τ a2 (τ ) = (−1) e (si − sj ) si ; Q(τ ) i,j=1 i=1

 √ Case s1 = −2, s2 = −3, s3 = − 5 35 + 31

10

5

57

i
i
4 4 4   1  (−1)k+1 e−sk τ (si − sj ) si sj ; Q(τ ) i,j=1 i,j=1 k=1

i
i
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4 1  α(τ ) = (si − sj ); Q(τ ) i,j=1

3

≤τ α



[0,1]3

i
Clearly, ai , for i = 0, · · · , 2, and α are well-defined, for any value of the delay τ , if and only if the following condition Q (τ ) = 0, ∀τ > 0. (16) is fulfilled. So, let us check the condition (16) . By rearranging the terms in Q, and using the mean value theorem, we get Q (τ ) = τ 3

4 

i
with ˜ )= Q(τ



2

˜ )>0 (si − sj ) Q(τ

(1 − t) (1 − θ) e

−τ [s2 ,[s3 ,[s4 ,s1 ]λ ]

(17)

] dλdθdt.

θ t

The existence and uniqueness of the coefficients a2 (τ ), a1 (τ ), a0 (τ ), α (τ )) is then proved. The positiveness of ˜ α is provided by (17) and the sign of Q. Since the mapping τ → a2 (τ ) + s2 + s3 is continuous and increasing from −∞ to −s1 when τ varies in R+∗ , this means that the mapping τ → a2 (τ ) + s2 + s3 takes positive values if and only if s1 < 0. Also , if and only if, there exists (a unique) root τ0 > 0 to equation a2 (τ ) + s2 + s3 = 0. 3.2 The dominancy of s1 for (13) To study the stability of the system (13), we need to study the dominancy of s1 by using an adequate factorization of the quasi-polynomial ∆3 in (14). Theorem 4. The root s1 is the spectral abscissa of (13). Sketch of the proof Rewrite the quasipolynomial ∆3 as: ∆3 (s, τ ) = (s − s1 ) (s − s2 ) (s − s3 ) P (s, τ )

s3 + a2 s2 + a1 s + a0 + α exp (−τ s) (s − s1 ) (s − s2 ) (s − s3 ) . Define the quantities: b0 := a0 + s1 s2 s3 ; b1 := a1 − 3  3 si sj ; and b2 := a2 + i=1 si . Some tedious algebraic P (s, τ ) =

i,j=1 i=j

manipulations allows to write P (s, τ )

where P˜ (s, τ ) =

P (s, τ ) = 1 − τ 3 αP˜ (s, τ ) 

2

(1 − t) (1 − θ) e

−τ [s,[s2 ,[s1 ,s3 ]λ ]

<τ α



2

(1 − t) (1 − θ) e

−τ [s1 ,[[s2 ,s3 ]λ ,s4 ]

(18)

] dλdθdt

θ t

[0,1]3 3

= τ αQ (τ ) = 1 which is inconsistent. 4. MACH NUMBER REGULATION IN A WIND TUNNEL: EQUIDISTRIBUTED DOMINANT-ROOTS ASSIGNMENT Roughly speaking, the Mach number regulation in a wind tunnel is based on the Navier-Stokes equations for unsteady flow and contains control laws for temperature and pressure regulation.

[0,1]3

with

3

    −τ s, s ,[s ,s ] (1 − θ) e [ [ 1 2 3 λ ]θ ]t  dλdθdt

] dλdθdt.

θ t

[0,1]3

To prove dominancy property for s1 , let us assume that there exists some s0 = ζ + jη a root of (14) such that ζ > s1 . This means that P (s0 , τ ) = 0. Combining this fact and the positiveness of α, we get 1 = τ 3 αP˜ (s, τ ) 129

As an illustrative example for the applicative potential of the proposed main result, let revisit the following simplified model of Mach number regulation proposed in Manitius (1984) and consists of a system of three state equations with a delay in one of the state variables. It is stressed that in steady-state operating conditions, the dynamic response of the Mach number perturbations ξ1 to small perturbations in the guide vane angle actuator ξ2 are governed by:   ξ˙1 (t) = −aξ1 (t) + k a ξ2 (t − τ )  (19) ξ˙2 (t) = ξ3 (t)  ˙ 2 2 ξ3 (t) = −ω ξ2 (t) − 2ζωξ3 (t) + ω u(t)

where a, ω, ζ, k and τ are parameters depending on the operating point and presumed constant when the perturbations ξi are small. Moreover, following the experimental parameter values of the wind tunnel developed at NASA Langley Research Center, the parameters a, ω, ζ, τ are positive. In Manitius (1984), a feedback consisting of a linear combination of state variables and weighted integrals of some of the state variables over a period equal to the time delay, where the spectrum of the closed-loop system is finite (consists of three eigenvalues). However, our method does not render the closed-loop system finite dimensional but only involves controlling its rightmost root. In Boussaada et al. (2018a) the control law u(t) = − ωα2 ξ2 (t) − ωβ02 ξ2 (t − τ ) − ωβ12 ξ3 (t − τ ) is proposed allowing to the closed-loop quasipolynomial function:   ∆(s, τ ) = (s+a) (sβ1 +β0)e−sτ +s2 +2 s ζ ω+ω 2 +α . (20) Thanks to such a factorization and since a is a positive parameter, the aim in Boussaada et al. (2018a) were to establish conditions on parameters such that the rightmost root of the second factor of (20) has a negative real part and the MID property for second order systems were exploited. Here, we propose the control law: u(t) = αξ1 (t) + βξ2 (t) + γξ3 (t) which gives in closed-loop the following quasipolynomial function ∆(s, τ ) =   s3 + (2 ζω − ω 2 γ + a)s2 + (1 − β − aΓ)ω 2 + 2aζω s + (1 − β) aω 2 − ω 2 α e−sτ ka

(21)

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Fazia Bedouhene et al. / IFAC PapersOnLine 52-18 (2019) 55–60

Using Theorem 3 and assuming that sl = s1 − (l − 1) d for l = 2, . . . , 4 one obtains the following values of the controller gains:  −a2 − 3 ad − 8 d2 d3 e−aτ   α = −3 , β = 1 + , ω 2 ka ω2 (22)   γ = 2 ζ + −2 a − 3 d , ω ω2 as well as the precise value of the spectral abscissa and the distance between two successive assigned roots: ln (2) aτ + ln (2) s1 = − , d= . τ τ

Fig. 4. The spectrum distribution of (21) satisfying (22) exhibiting the four-roots placement in two configurations. In blue, the spectrum corresponding to a = 5, τ = 2. In red, the spectrum corresponding to a = 3, τ = 1. In both cases, the dominancy of the four assigned equidistributed real rootsis underlined.

5. CONCLUSION In this note, we extended some recent results by the authors on pole-placement for Time-delay systems. This new result emphasizes a new delayed controller-design based on the trivial solution’s decay rate assignment. The potential applicability of the approach is illustrated through the regulation of Mach number in a wind tunnel. Further insights on the applicability of the presented method in damping active vibrations can be found in Tliba et al. (2019). REFERENCES Amrane, S., Bedouhene, F., Boussaada, I., and Niculescu, S.I. (2018). On qualitative properties of low-degree quasipolynomials: Further remarks on the spectral abscissa and rightmost-roots assignment. Bull. Math. Soc. Sci. Math. Roumanie, Tome 61. No. 4(109), 361–381. Bellman, R. and Cooke, K. (1963). Differential-difference equations. Academic Press, New York. Boussaada, I. and Niculescu, S.I. (2014). Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices. 21st International Symposium on Mathematical Theory of Networks and Systems, 1 – 8. Boussaada, I. and Niculescu, S.I. (2016a). Characterizing the codimension of zero singularities for time-delay systems. Acta Applicandae Mathematicae, 145(1), 47– 88. 130

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St´ep´ an, G. (1989). Retarded Dynamical Systems: Stability and Characteristic Functions. Pitman research notes in mathematics series. Longman Sci. and Tech. Tliba, S., Boussaada, I., Bedouhene, F., and Niculescu, S.I. (2019). Active vibration control through quasipolynomial based controller. In (Submitted), 1 – 6. Walton, K. and Marshall, J.E. (1987). Direct method for tds stability analysis. IEE Proceedings D - Control Theory and Applications, 134(2), 101–107.

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