Order of Legendre-LMI conditions to assess stability of time-delay systems

IFAC PapersOnLine 55-36 (2022) 175–180 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2022 The Authors. This is an open access article under the CC BY-NC-ND license . Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2022.11.353 10.1016/j.ifacol.2022.11.353 2405-8963 Copyright © 2022 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) Order of Legendre-LMI conditions to assess stability of time-delay systems Mathieu Bajodek ∗,Alexandre Seuret ∗∗, Fr´ed´eric Gouaisbaut ∗. ∗LAAS-CNRS, Universit´e de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France ∗∗ Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters. Keywords: Time-delay systems; Stability conditions; LMI; Converse Lyapunov theorem; Polynomial approximation 1. INTRODUCTION An extension of the Lyapunov necessary and sufficient theorem has been introduced in Datko (1970) to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in Kharitonov (2013). Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Pad´e approximations in the frequency domain (see Golub and Van Loan (1989)) or pseudo-spectral methods in the time domain (see Gottlieb and Orszag (1977)). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part. In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see Gu et al. (2003)), linear (see Gu (2013); Medvedeva and Zhabko (2015)), polynomial (see spline methods) or exponential (see Egorov and Mondi´e (2014)). Applying an interpolation leads to necessary conditions of stability as in Egorov and Mondi´e (2014); Medvedeva and Zhabko (2019) or to sufficient conditions of stability based on discretized Lyapunov functional as in Gu et al. (2003). Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively Gu (2013) and Gomez et al. (2021)). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in Fridman (2014), which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the nfirst Legendre coefficients have been proposed in Seuret and Gouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of Bajodek et al. (2022) haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in Bajodek et al. (2022), and improves the estimation of the necessary order n∗. After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n∗expressed with respect to system parameters. Finally, the LMI test as well as the order n∗are computed on an academic example. Notations: Throughout the paper, N,Rm×p,Smand Sm + denote the sets of natural numbers, real matrices of size m×p, symmetric matrices of size mand positive symmet- Order of Legendre-LMI conditions to assess stability of time-delay systems Mathieu Bajodek ∗,Alexandre Seuret ∗∗, Fr´ed´eric Gouaisbaut ∗. ∗LAAS-CNRS, Universit´e de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France ∗∗ Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters. Keywords: Time-delay systems; Stability conditions; LMI; Converse Lyapunov theorem; Polynomial approximation 1. INTRODUCTION An extension of the Lyapunov necessary and sufficient theorem has been introduced in Datko (1970) to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in Kharitonov (2013). Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Pad´e approximations in the frequency domain (see Golub and Van Loan (1989)) or pseudo-spectral methods in the time domain (see Gottlieb and Orszag (1977)). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part. In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see Gu et al. (2003)), linear (see Gu (2013); Medvedeva and Zhabko (2015)), polynomial (see spline methods) or exponential (see Egorov and Mondi´e (2014)). Applying an interpolation leads to necessary conditions of stability as in Egorov and Mondi´e (2014); Medvedeva and Zhabko (2019) or to sufficient conditions of stability based on discretized Lyapunov functional as in Gu et al. (2003). Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively Gu (2013) and Gomez et al. (2021)). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in Fridman (2014), which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the nfirst Legendre coefficients have been proposed in Seuret and Gouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of Bajodek et al. (2022) haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in Bajodek et al. (2022), and improves the estimation of the necessary order n∗. After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n∗expressed with respect to system parameters. Finally, the LMI test as well as the order n∗are computed on an academic example. Notations: Throughout the paper, N,Rm×p,Smand Sm + denote the sets of natural numbers, real matrices of size m×p, symmetric matrices of size mand positive symmet- Order of Legendre-LMI conditions to assess stability of time-delay systems Mathieu Bajodek ∗,Alexandre Seuret ∗∗, Fr´ed´eric Gouaisbaut ∗. ∗LAAS-CNRS, Universit´e de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France ∗∗ Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters. Keywords: Time-delay systems; Stability conditions; LMI; Converse Lyapunov theorem; Polynomial approximation 1. INTRODUCTION An extension of the Lyapunov necessary and sufficient theorem has been introduced in Datko (1970) to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in Kharitonov (2013). Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Pad´e approximations in the frequency domain (see Golub and Van Loan (1989)) or pseudo-spectral methods in the time domain (see Gottlieb and Orszag (1977)). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part. In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see Gu et al. (2003)), linear (see Gu (2013); Medvedeva and Zhabko (2015)), polynomial (see spline methods) or exponential (see Egorov and Mondi´e (2014)). Applying an interpolation leads to necessary conditions of stability as in Egorov and Mondi´e (2014); Medvedeva and Zhabko (2019) or to sufficient conditions of stability based on discretized Lyapunov functional as in Gu et al. (2003). Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively Gu (2013) and Gomez et al. (2021)). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in Fridman (2014), which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the nfirst Legendre coefficients have been proposed in Seuret and Gouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of Bajodek et al. (2022) haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in Bajodek et al. (2022), and improves the estimation of the necessary order n∗. After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n∗expressed with respect to system parameters. Finally, the LMI test as well as the order n∗are computed on an academic example. Notations: Throughout the paper, N,Rm×p,Smand Sm + denote the sets of natural numbers, real matrices of size m×p, symmetric matrices of size mand positive symmet- Order of Legendre-LMI conditions to assess stability of time-delay systems Mathieu Bajodek ∗,Alexandre Seuret ∗∗, Fr´ed´eric Gouaisbaut ∗. ∗ LAAS-CNRS, Universit´e de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France ∗∗ Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters. Keywords: Time-delay systems; Stability conditions; LMI; Converse Lyapunov theorem; Polynomial approximation 1. INTRODUCTION An extension of the Lyapunov necessary and sufficient theorem has been introduced in Datko (1970) to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in Kharitonov (2013). Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Pad´e approximations in the frequency domain (see Golub and Van Loan (1989)) or pseudo-spectral methods in the time domain (see Gottlieb and Orszag (1977)). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part. In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see Gu et al. (2003)), linear (see Gu (2013); Medvedeva and Zhabko (2015)), polynomial (see spline methods) or exponential (see Egorov and Mondi´e (2014)). Applying an interpolation leads to necessary conditions of stability as in Egorov and Mondi´e (2014); Medvedeva and Zhabko (2019) or to sufficient conditions of stability based on discretized Lyapunov functional as in Gu et al. (2003). Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively Gu (2013) and Gomez et al. (2021)). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in Fridman (2014), which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the nfirst Legendre coefficients have been proposed in Seuret and Gouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of Bajodek et al. (2022) haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in Bajodek et al. (2022), and improves the estimation of the necessary order n∗. After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n∗expressed with respect to system parameters. Finally, the LMI test as well as the order n∗are computed on an academic example. Notations: Throughout the paper, N,Rm×p,Smand Sm + denote the sets of natural numbers, real matrices of size m×p, symmetric matrices of size mand positive symmet- Order of Legendre-LMI conditions to assess stability of time-delay systems Mathieu Bajodek ∗ ,Alexandre Seuret ∗∗ , Fr´ed´eric Gouaisbaut ∗. ∗LAAS-CNRS, Universit´e de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France ∗∗ Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters. Keywords: Time-delay systems; Stability conditions; LMI; Converse Lyapunov theorem; Polynomial approximation 1. INTRODUCTION An extension of the Lyapunov necessary and sufficient theorem has been introduced in Datko (1970) to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in Kharitonov (2013). Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Pad´e approximations in the frequency domain (see Golub and Van Loan (1989)) or pseudo-spectral methods in the time domain (see Gottlieb and Orszag (1977)). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part. In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see Gu et al. (2003)), linear (see Gu (2013); Medvedeva and Zhabko (2015)), polynomial (see spline methods) or exponential (see Egorov and Mondi´e (2014)). Applying an interpolation leads to necessary conditions of stability as in Egorov and Mondi´e (2014); Medvedeva and Zhabko (2019) or to sufficient conditions of stability based on discretized Lyapunov functional as in Gu et al. (2003). Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively Gu (2013) and Gomez et al. (2021)). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in Fridman (2014), which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the nfirst Legendre coefficients have been proposed in Seuret and Gouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of Bajodek et al. (2022) haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in Bajodek et al. (2022), and improves the estimation of the necessary order n∗. After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n∗expressed with respect to system parameters. Finally, the LMI test as well as the order n∗are computed on an academic example. Notations: Throughout the paper, N,Rm×p,Smand Sm + denote the sets of natural numbers, real matrices of size m×p, symmetric matrices of size mand positive symmet- Order of Legendre-LMI conditions to assess stability of time-delay systems Mathieu Bajodek ∗,Alexandre Seuret ∗∗, Fr´ed´eric Gouaisbaut ∗. ∗LAAS-CNRS, Universit´e de Toulouse, UPS, 7 avenue du Colonel Roche, 31077 Toulouse, France ∗∗ Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain Abstract: This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters. Keywords: Time-delay systems; Stability conditions; LMI; Converse Lyapunov theorem; Polynomial approximation 1. INTRODUCTION An extension of the Lyapunov necessary and sufficient theorem has been introduced in Datko (1970) to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in Kharitonov (2013). Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Pad´e approximations in the frequency domain (see Golub and Van Loan (1989)) or pseudo-spectral methods in the time domain (see Gottlieb and Orszag (1977)). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part. In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see Gu et al. (2003)), linear (see Gu (2013); Medvedeva and Zhabko (2015)), polynomial (see spline methods) or exponential (see Egorov and Mondi´e (2014)). Applying an interpolation leads to necessary conditions of stability as in Egorov and Mondi´e (2014); Medvedeva and Zhabko (2019) or to sufficient conditions of stability based on discretized Lyapunov functional as in Gu et al. (2003). Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively Gu (2013) and Gomez et al. (2021)). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in Fridman (2014), which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the nfirst Legendre coefficients have been proposed in Seuret and Gouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of Bajodek et al. (2022) haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in Bajodek et al. (2022), and improves the estimation of the necessary order n∗. After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n∗expressed with respect to system parameters. Finally, the LMI test as well as the order n∗are computed on an academic example. Notations: Throughout the paper, N,Rm×p,Smand Sm + denote the sets of natural numbers, real matrices of size m×p, symmetric matrices of size mand positive symmet- 176 Mathieu Bajodek et al. / IFAC PapersOnLine 55-36 (2022) 175–180 ric matrices of size m, respectively. For any square real matrix M,M⊤denotes the transpose of M,H(M) stands for M+M⊤and tril(M) stands for the triangular lower part of M. Furthermore, for any M∈Sm, its minimal and maximal eigenvalues are denoted σ(M) and ¯σ(M). The 2-norm of matrix Min Rm×pis |M|=¯σ(M⊤M). The vector u= vec(M) in Rmp×1collocates the columns of matrix Mand ‘vec−1’ denotes the inverse operator. Notation ⊗refers to the matrix Kronecker product. The set of square-integrable functions from (−h, 0) to Rm×pis noted L2(−h, 0; Rm×p). Let also H1(−h, 0; Rm×p) be the set of functions F, such that Fand F′are in L2(−h, 0; Rm×p). For any φin Rnx×L2(−h, 0; Rnz), the norm is given by ∥φ∥=|φ(0)|2+0 −h|φ(θ)|2dθ. Lastly, denote xtsuch that xt(θ)=x(t+θ), for any θin [−h, 0]. 2. CONVERSE LYAPUNOV THEOREM FOR TIME-DELAY SYSTEMS 2.1 Presentation of the system Consider a linear time-delay system given by ˙x(t)=Ax(t)+Adx(t−h),∀t≥0, x(θ)=φ(θ),∀θ∈[−h, 0],(1) where the delay h>0 and matrices A, Adin Rnx×nxare constant and known. The initial condition φbelongs to D=       φ(θ)=φ0if θ=0, φ1(θ) if θ∈[−h, 0],such that φ0 φ1∈Rnx×H1(−h, 0; Rnx),φ 1(0) = φ0        . Definition 1. (GES). The trivial solution to (1) is said globally exponentially stable (GES), if there exist κ≥1 and µ>0 such that xt, solution to (1) generated by any initial condition φin D, verifies ∥xt∥≤κe−µt ∥φ∥,∀t≥0.(2) Remark 1. It is worth noticing that the regularity properties of the solutions to time-delay systems ensure that such a definition in the sense of ∥·∥-norm also implies that there exist ¯κ≥1 and ¯µ>0 such that |x(t)|≤¯κe−¯µt sup [−h,0]|φ(θ)|,∀t≥0.(3) 2.2 Existence of a complete Lyapunov functional Since Kharitonov and Zhabko (2003); Kharitonov (2013), it is well known that a necessary and sufficient stability condition of system (1) is the existence of a complete Lyapunov functional, which is recalled below. For any W1,W2and W3in Snx +and for any xtin D, define the following Lyapunov functional V(xt)=x⊤ t(0)U(0)xt(0) +2x⊤ t(0) 0 −h U(θ+h)Adxt(θ)dθ +0 −h0 −h x⊤ t(θ1)A⊤ dU(θ2−θ1)Adxt(θ2)dθ1dθ2 +0 −h x⊤ t(θ)θ+h hW2+W3xt(θ)dθ, (4) where Uis a matrix function from [−h, h] to Rnx×nx, called the Lyapunov matrix. Interestingly, referring to (Kharitonov, 2013, Section 2.10), a method is provided to build analytically this matrix. Its expression is given by U(θ)=vec−1�[In2 x0]eθM N−1−vec(W) 0if θ≤0, U⊤(−θ) if θ>0, (5) where matrices W=W1+W2+W3,M=−A⊤⊗Inx−A⊤ d⊗Inx Inx⊗A⊤ dInx⊗A⊤, N=A⊤⊗Inx+Inx⊗A⊤A⊤ d⊗Inx In2 x0+Inx⊗A⊤ d0 0−In2 xe−hM . (6) Such a complete Lyapunov functional Vhas been built so that its time-derivative along the trajectories of the system yields ˙ V(xt)=−1 h0 −hxt(0) hxt(θ) xt(−h)⊤W100 ∗1 hW20 ∗∗W3 xt(0) hxt(θ) xt(−h)dθ. (7) A necessary and sufficient condition of stability for system (1) based on this functional is formulated below. 2.3 Necessary and sufficient condition of stability The following theorem is obtained by application of the Lyapunov theorem. The proposed formulation is issued from (Gu et al., 2003, Theorem 5.19). Theorem 1. System (1) is GES if and only if there exist positive scalars α1,α 2,α 3such that Vdefined by (4) satisfies, for any xtin D, inequalities α1∥xt∥2+0 −h x⊤ t(θ)θ+h hW2xt(θ)dθ≤V(xt), V(xt)≤α2∥xt∥2, ˙ V(xt)≤−α3∥xt∥2, (8a) (8b) (8c) where ˙ Vdenotes the time-derivative of Valong (1). Remark 2. It is worth noticing that in the sense of the stability definition given by (3), the Lyapunov-Krasovskii version of this theorem also holds replacing (8) by α1|xt(0)|2≤V(xt)≤α2sup [−h,0]|xt(θ)|2, ˙ V(xt)≤−α3|xt(0)|2. (9a) (9b) Refer to Chaillet et al. (2019) for further details. Proof : To begin with, the sufficiency refers to the application of the Lyapunov theorem. The proof of necessity starts by noting that (8c) already holds by construction of the Lyapunov matrix (7). Indeed, expression (7) leads to α3= max(σ(W1), hσ(W2)). Equation (8b) is also easy to prove since Vis quadratic with respect to xt. Let focus now on (8a) and introduce a functional Was follows W(xt)=V(xt)−α1∥xt∥2−0 −h x⊤ t(θ)θ+h hW2xt(θ)dθ. According to (7), differentiating Walong the trajectories of (1) leads to ˙ W(xt)=−xt(0) xt(−h)⊤ W10 0W3xt(0) xt(−h) −α1xt(0) xt(−h)⊤ H(A)+InxAd A⊤ d−Inx xt(0) xt(−h). Mathieu Bajodek et al. / IFAC PapersOnLine 55-36 (2022) 175–180 177 ric matrices of size m, respectively. For any square real matrix M,M⊤denotes the transpose of M,H(M) stands for M+M⊤and tril(M) stands for the triangular lower part of M. Furthermore, for any M∈Sm, its minimal and maximal eigenvalues are denoted σ(M) and ¯σ(M). The 2-norm of matrix Min Rm×pis |M|=¯σ(M⊤M). The vector u= vec(M) in Rmp×1collocates the columns of matrix Mand ‘vec−1’ denotes the inverse operator. Notation ⊗refers to the matrix Kronecker product. The set of square-integrable functions from (−h, 0) to Rm×pis noted L2(−h, 0; Rm×p). Let also H1(−h, 0; Rm×p) be the set of functions F, such that Fand F′are in L2(−h, 0; Rm×p). For any φin Rnx×L2(−h, 0; Rnz), the norm is given by ∥φ∥=|φ(0)|2+0 −h|φ(θ)|2dθ. Lastly, denote xtsuch that xt(θ)=x(t+θ), for any θin [−h, 0]. 2. CONVERSE LYAPUNOV THEOREM FOR TIME-DELAY SYSTEMS 2.1 Presentation of the system Consider a linear time-delay system given by ˙x(t)=Ax(t)+Adx(t−h),∀t≥0, x(θ)=φ(θ),∀θ∈[−h, 0],(1) where the delay h>0 and matrices A, Adin Rnx×nxare constant and known. The initial condition φbelongs to D=       φ(θ)=φ0if θ=0, φ1(θ) if θ∈[−h, 0],such that φ0 φ1∈Rnx×H1(−h, 0; Rnx),φ 1(0) = φ0        . Definition 1. (GES). The trivial solution to (1) is said globally exponentially stable (GES), if there exist κ≥1 and µ>0 such that xt, solution to (1) generated by any initial condition φin D, verifies ∥xt∥≤κe−µt ∥φ∥,∀t≥0.(2) Remark 1. It is worth noticing that the regularity properties of the solutions to time-delay systems ensure that such a definition in the sense of ∥·∥-norm also implies that there exist ¯κ≥1 and ¯µ>0 such that |x(t)|≤¯κe−¯µt sup [−h,0]|φ(θ)|,∀t≥0.(3) 2.2 Existence of a complete Lyapunov functional Since Kharitonov and Zhabko (2003); Kharitonov (2013), it is well known that a necessary and sufficient stability condition of system (1) is the existence of a complete Lyapunov functional, which is recalled below. For any W1,W2and W3in Snx +and for any xtin D, define the following Lyapunov functional V(xt)=x⊤ t(0)U(0)xt(0) +2x⊤ t(0) 0 −h U(θ+h)Adxt(θ)dθ +0 −h0 −h x⊤ t(θ1)A⊤ dU(θ2−θ1)Adxt(θ2)dθ1dθ2 +0 −h x⊤ t(θ)θ+h hW2+W3xt(θ)dθ, (4) where Uis a matrix function from [−h, h] to Rnx×nx, called the Lyapunov matrix. Interestingly, referring to (Kharitonov, 2013, Section 2.10), a method is provided to build analytically this matrix. Its expression is given by U(θ)=vec−1�[In2 x0]eθM N−1−vec(W) 0if θ≤0, U⊤(−θ) if θ>0, (5) where matrices W=W1+W2+W3,M=−A⊤⊗Inx−A⊤ d⊗Inx Inx⊗A⊤ dInx⊗A⊤, N=A⊤⊗Inx+Inx⊗A⊤A⊤ d⊗Inx In2 x0+Inx⊗A⊤ d0 0−In2 xe−hM . (6) Such a complete Lyapunov functional Vhas been built so that its time-derivative along the trajectories of the system yields ˙ V(xt)=−1 h0 −hxt(0) hxt(θ) xt(−h)⊤W100 ∗1 hW20 ∗∗W3 xt(0) hxt(θ) xt(−h)dθ. (7) A necessary and sufficient condition of stability for system (1) based on this functional is formulated below. 2.3 Necessary and sufficient condition of stability The following theorem is obtained by application of the Lyapunov theorem. The proposed formulation is issued from (Gu et al., 2003, Theorem 5.19). Theorem 1. System (1) is GES if and only if there exist positive scalars α1,α 2,α 3such that Vdefined by (4) satisfies, for any xtin D, inequalities α1∥xt∥2+0 −h x⊤ t(θ)θ+h hW2xt(θ)dθ≤V(xt), V(xt)≤α2∥xt∥2, ˙ V(xt)≤−α3∥xt∥2, (8a) (8b) (8c) where ˙ Vdenotes the time-derivative of Valong (1). Remark 2. It is worth noticing that in the sense of the stability definition given by (3), the Lyapunov-Krasovskii version of this theorem also holds replacing (8) by α1|xt(0)|2≤V(xt)≤α2sup [−h,0]|xt(θ)|2, ˙ V(xt)≤−α3|xt(0)|2. (9a) (9b) Refer to Chaillet et al. (2019) for further details. Proof : To begin with, the sufficiency refers to the application of the Lyapunov theorem. The proof of necessity starts by noting that (8c) already holds by construction of the Lyapunov matrix (7). Indeed, expression (7) leads to α3= max(σ(W1), hσ(W2)). Equation (8b) is also easy to prove since Vis quadratic with respect to xt. Let focus now on (8a) and introduce a functional Was follows W(xt)=V(xt)−α1∥xt∥2−0 −h x⊤ t(θ)θ+h hW2xt(θ)dθ. According to (7), differentiating Walong the trajectories of (1) leads to ˙ W(xt)=−xt(0) xt(−h)⊤ W10 0W3xt(0) xt(−h) −α1xt(0) xt(−h)⊤ H(A)+InxAd A⊤ d−Inx xt(0) xt(−h). Then, there exists a sufficiently small α1>0 such that W is negative. Integrating the above inequality from tto ∞, we obtain lim T→∞W(xT)−W(xt)≤0,∀t≥0. As xT−→ T→∞ 0 because system (1) is assumed to be GES, inequality W(xt)≥0 holds, which concludes the proof. □ A modified version of Theorem 1 will be exposed in the next section. Indeed, in practice, the closed-form of the Lyapunov matrix Ucannot be calculated precisely enough and the necessary and sufficient LMI conditions (8) need to be simplified to be tested. Then, it will rely on an approximation of the complete Lyapunov functional. 3. MODIFIED CONVERSE LYAPUNOV THEOREM 3.1 Legendre polynomials approximation The Lyapunov matrix Uis composed of a complex combination of exponential functions. In order to derive tractable numerical tests, the idea is to introduce approximations of U, which satisfy convergence properties to ease this purpose. Define the Legendre polynomials approximations at any order nin N∗as U(θ+h)=U1,nℓnθ+h h   U1,n(θ+h) +˜ U1,n(θ),∀θ∈[−h, 0], U(θ)=U2,nℓnθ+h 2h   U2,n(θ) +˜ U2,n(θ),∀θ∈[−h, h], (10) where matrix ℓnin Rnnx×nxcollocates the nfirst Legendre polynomials scaled on [0,1] ℓn(θ)=[ l0(θ)... l n−1(θ)]⊤⊗Inx,∀θ∈[0,1].(11) In this decomposition, the constant matrices U1,n and U2,n have been selected as the orthogonal projection of U(θ+h) and U(θ) on the interval [−h, 0] and [−h, h] respectively, and their expressions are given by U1,n =0 −h U(θ+h)ℓ⊤ nθ+h hdθIn, U2,n =1 2h −h U(θ)ℓ⊤ nθ+h 2hdθIn, (12) where the normalization matrix of Legendre polynomials is equal to In=IInx nwhere, for all S∈Snx, we define IS n=1 h1 0 ℓn(θ)ℓ⊤ n(θ)dθ−1 ⊗S=diag1 h,...,2n−1 h⊗S. (13) For any W1,W2and W3in Snx +and xtin D, we introduce a new functional Vn(xt)=x⊤ t(0)U(0)xt(0) +2x⊤ t(0) 0 −h U1,n(θ+h)Adxt(θ)dθ +0 −h0 −h x⊤ t(θ1)A⊤ dU2,n(θ2−θ1)Adxt(θ2)dθ1dθ2 +0 −h x⊤ t(θ)θ+h hW2+W3xt(θ)dθ. (14) Such a functional will be considered as an approximation of the original V. Denoting the error ˜ Vn=V−V n, we obtain ˜ Vn(xt)=2x⊤ t(0) 0 −h ˜ U1,n(θ)Adxt(θ)dθ +0 −h0 −h x⊤ t(θ1)A⊤ d˜ U2,n(θ2−θ1)Adxt(θ2)dθ1dθ2, (15) where ˜ U1,n and ˜ U2,n refer to the errors expressed in (10). Nevertheless, the price to pay is that the functional Vn generated by this approximation at order ndoes not necessarily verify the condition (8) of the complete one. In this sequel, we provide a new theorem stating a necessary and sufficient condition of stability even though the complete Lyapunov functional is not built by Ubut by its approximation U1,n and U2,n. This theorem naturally relies on ˜ U1,n and ˜ U2,n, which are seen as an approximation error. 3.2 A new necessary and sufficient condition of stability The main result of the paper is presented below. Theorem 2. Let W1,W2and W3in Snx +and assume that the following inequality holds Ψn(θ):=W1+H(˜ U1,n(0)Ad)Ψ 1 n(θ)−˜ U1,n(−h)Ad ∗1 hW2Ψ2 n(θ) ∗∗W3≻0,(16) for a given nin N∗and for all θin [−h, 0] with Ψ1 n(θ)=A⊤˜ U1,n(θ)Ad−˜ U′ 1,n(θ)Ad+A⊤ d˜ U2,n(θ)Ad, Ψ2 n(θ)=A⊤ d˜ U⊤ 1,n(θ)Ad−A⊤ d˜ U⊤ 2,n(θ+h)Ad. System (1) is GES if and only if there exist positive scalars α1,α 2,α 3such that Vndefined by (14) satisfies, for any xt in D, inequalities α1∥xt∥2≤V n(xt)≤α2∥xt∥2, ˙ Vn(xt)≤−α3∥xt∥2. (17a) (17b) Proof : Trivially, the sufficiency of the statement is verified by application of the Lyapunov theorem. For the necessity of the proposed statement, let us first note that the right-hand part of (17a) is directly obtained because the Lyapunov functional is quadratic with respect to xt. Let now focus on the two remained inequalities. Concerning (17b), the time-derivative of Vn=V−˜ Vn along (1) given by (7) and (15) leads to ˙ Vn(xt)=−1 h0 −hxt(0) xt(θ) xt(−h)⊤W100 ∗1 hW20 ∗∗W3 xt(0) xt(θ) xt(−h)dθ −2(Axt(0) + Adxt(−h))⊤0 −h ˜ U1,n(θ)Adxt(θ)dθ +2x⊤ t(0)˜ U1,n(θ)Adxt(θ) 0 −h + 0 −h ˜ U′ 1,n(θ)Adxt(θ)dθ −20 −hx⊤ t(θ1)A⊤ d˜ U2,n(θ2−θ1)Adxt(θ2)θ1=0 θ1=−hdθ2 + 0 −h 0 −h x⊤ t(θ1)A⊤ d(∂θ1+∂θ2)˜ U2,n(θ2−θ1)Adxt(θ2)dθ1dθ2. The last term vanishes and the previous expression boils down to ˙ Vn(xt)=−1 h0 −hxt(0) hxt(θ) xt(−h)⊤ Ψn(θ) xt(0) hxt(θ) xt(−h)dθ, (18) 178 Mathieu Bajodek et al. / IFAC PapersOnLine 55-36 (2022) 175–180 where Ψnis given in (16). Therefore if matrix Ψnis positive definite there exists a sufficiently small α3>0 such that Ψn(θ)≻α3I3nzwhich yields to (17b). In order to ensure the left-hand part of (17a), let us introduce the following functional Wn(xt)=Vn(xt)−α1∥xt∥2.(19) According to the previous calculations (18), differentiating Wnalong the trajectories of (1) leads to ˙ Wn(xt)=−1 h0 −hxt(0) hxt(θ) xt(−h)⊤ Ψn(θ)xt(0) hxt(θ) xt(−h)dθ −α1xt(0) xt(−h)⊤ H(A)+InxAd A⊤ d−Inz xt(0) xt(−h). (20) Therefore, if Ψnis positive definite, then there exists a sufficiently small α1>0 such that ˙ Wn(xt) is negative definite. Integrating then ˙ Wn(xt) from tto infinity yields lim T→∞ Wn(xT)−Wn(xt)=−Wn(xt)≤0, where we have used the assumption that system (1) is GES meaning that Wn(xT)−→ T→∞ 0. Hence, (19) ensures that there exists a sufficiently small α1>0 such that the right-hand part of inequality (17a) holds. □ In the following, inequality (8a) in Theorem 1 and inequality (17b) in Theorem 2 are translated into a numerical condition. 3.3 Necessary and sufficient LMI condition of stability For any (Pn,R,S)∈S(n+1)nx×Snx +×Snx +, introduce the Lyapunov candidate functional considered in Seuret and Gouaisbaut (2014, 2015) Vn(xt)= xt(0) 0 −hℓn(θ+h h)xt(θ)dθ⊤ Pn xt(0) 0 −hℓn(θ+h h)xt(θ)dθ +0 −h x⊤ t(θ)θ+h hR+Sxt(θ)dθ. (21) Using an integration by parts, the time-derivative of Vn along the trajectories of system (1) is given by ˙ Vn(xt)=2 xt(0) 0 −hℓn(θ+h h)xt(θ)dθ⊤ Pn Ad −ℓn(0) xt(−h) +2 xt(0) 0 −hℓn(θ+h h)xt(θ)dθ⊤ Pn A0 ℓn(1) Lnxt(0) 0 −hℓn(θ+h h)xt(θ)dθ +xt(0) xt(−h)⊤R+S0 0−Sxt(0) xt(−h) −1 h0 −h x⊤ t(θ)Rxt(θ)dθ, (22) where the derivation matrix of Legendre polynomials is Ln=−tril(ℓn(1)ℓ⊤ n(1) −ℓn(0)ℓ⊤ n(0))In,(23) In this context, the following theorem can be established. Theorem 3. For a given nin N∗, assume that inequality (16) holds (i.e. Ψn(θ)≻0, for all θin [−h, 0]). System (1) is GES if and only if there exist matrices (Pn, S, R)∈S(n+1)nx×Snx +×Snx +the two following linear matrix inequalities hold Ξ+ n=Pn+IS n≻0, Ξ− n=H(PnAn)+C⊤ n(R+S)Cn−1 hIR nPnBn ∗−S≺0, (24) where matrices are defined by IS n=00 0IS n,An=A0 ℓn(1) Ln,Bn=Ad −ℓn(0) ,Cn=Inx 0⊤ . Proof : Firstly, the proof of the sufficiency is provided in Seuret and Gouaisbaut (2015) using for a given order nin N∗the Lyapunov candidate functional given by (21) and applying the Lyapunov theorem. Secondly, to prove the necessity, we take support on Theorem 2. For all W1,W2and W3in Snx +, let matrices Pn=U(0) Qn ∗T n,R=W2,S=W3,(25) Qn=0 −h U(θ+h)Adℓ⊤ n(θ)Indθ, Tn= 0 −h 0 −h Inℓn(θ1)A⊤ dU(θ2−θ1)Adℓ⊤ n(θ2)Indθ1dθ2. Note that (Pn,R,S) belongs to S(n+1)nx×Snx +×Snx +using the symmetric property of U. Moreover, the Lyapunov candidate functional Vngiven by (21) is the Legendre approximation (14) of the Lyapunov converse functional V. Assume that inequality (16) holds at order nand that system (1) is GES. Applying (8a) in Theorem 1 and (17b) in Theorem 2, there exist positive scalars α1,α 3such for the following inequalities hold α1∥xt∥2+0 −h x⊤ t(θ)θ+h hRxt(θ)dθ≤V(xt), ˙ Vn(xt)≤−α3∥xt∥2. (26a) (26b) Then, these inequalities are applied for xtin Dn⊂Dgiven by Dn=           φ(θ)=       φ0if θ=0, ℓ⊤ nθ+h hζnif θ∈(−h, 0), φ−hif θ=−h, such that [φ0ζnφ−h]⊤∈R(n+2)nx.            . Thus, inequalities (26a),(26b) lead to Ξ+ n≻0 and Ξ− n≺0, respectively, which concludes the proof. □ Theorem 3 provides a necessary and sufficient LMI condition for the stability of system (1). Nonetheless, it does not guarantee the existence of an order nsuch that the LMI must be satisfied. The estimation of this order is reserved to the next section. 4. ESTIMATION OF THE NECESSARY ORDER 4.1 Convergence properties In the light of Dunkl and Xu (2001), we provide convergence properties satisfied by Legendre polynomial approximation in order to estimate an order from which the necessary and sufficient stability condition holds. Firstly, the Lyapunov matrix Uassociated to Wdefined in (5) is continuous on [−h, h] and satisfies Mathieu Bajodek et al. / IFAC PapersOnLine 55-36 (2022) 175–180 179 where Ψnis given in (16). Therefore if matrix Ψnis positive definite there exists a sufficiently small α3>0 such that Ψn(θ)≻α3I3nzwhich yields to (17b). In order to ensure the left-hand part of (17a), let us introduce the following functional Wn(xt)=Vn(xt)−α1∥xt∥2.(19) According to the previous calculations (18), differentiating Wnalong the trajectories of (1) leads to ˙ Wn(xt)=−1 h0 −hxt(0) hxt(θ) xt(−h)⊤ Ψn(θ)xt(0) hxt(θ) xt(−h)dθ −α1xt(0) xt(−h)⊤ H(A)+InxAd A⊤ d−Inz xt(0) xt(−h). (20) Therefore, if Ψnis positive definite, then there exists a sufficiently small α1>0 such that ˙ Wn(xt) is negative definite. Integrating then ˙ Wn(xt) from tto infinity yields lim T→∞ Wn(xT)−Wn(xt)=−Wn(xt)≤0, where we have used the assumption that system (1) is GES meaning that Wn(xT)−→ T→∞ 0. Hence, (19) ensures that there exists a sufficiently small α1>0 such that the right-hand part of inequality (17a) holds. □ In the following, inequality (8a) in Theorem 1 and inequality (17b) in Theorem 2 are translated into a numerical condition. 3.3 Necessary and sufficient LMI condition of stability For any (Pn,R,S)∈S(n+1)nx×Snx +×Snx +, introduce the Lyapunov candidate functional considered in Seuret and Gouaisbaut (2014, 2015) Vn(xt)= xt(0) 0 −hℓn(θ+h h)xt(θ)dθ⊤ Pn xt(0) 0 −hℓn(θ+h h)xt(θ)dθ +0 −h x⊤ t(θ)θ+h hR+Sxt(θ)dθ. (21) Using an integration by parts, the time-derivative of Vn along the trajectories of system (1) is given by ˙ Vn(xt)=2 xt(0) 0 −hℓn(θ+h h)xt(θ)dθ⊤ Pn Ad −ℓn(0) xt(−h) +2 xt(0) 0 −hℓn(θ+h h)xt(θ)dθ⊤ Pn A0 ℓn(1) Lnxt(0) 0 −hℓn(θ+h h)xt(θ)dθ +xt(0) xt(−h)⊤R+S0 0−Sxt(0) xt(−h) −1 h0 −h x⊤ t(θ)Rxt(θ)dθ, (22) where the derivation matrix of Legendre polynomials is Ln=−tril(ℓn(1)ℓ⊤ n(1) −ℓn(0)ℓ⊤ n(0))In,(23) In this context, the following theorem can be established. Theorem 3. For a given nin N∗, assume that inequality (16) holds (i.e. Ψn(θ)≻0, for all θin [−h, 0]). System (1) is GES if and only if there exist matrices (Pn, S, R)∈S(n+1)nx×Snx +×Snx +the two following linear matrix inequalities hold Ξ+ n=Pn+IS n≻0, Ξ− n=H(PnAn)+C⊤ n(R+S)Cn−1 hIR nPnBn ∗−S≺0, (24) where matrices are defined by IS n=00 0IS n,An=A0 ℓn(1) Ln,Bn=Ad −ℓn(0) ,Cn=Inx 0⊤ . Proof : Firstly, the proof of the sufficiency is provided in Seuret and Gouaisbaut (2015) using for a given order nin N∗the Lyapunov candidate functional given by (21) and applying the Lyapunov theorem. Secondly, to prove the necessity, we take support on Theorem 2. For all W1,W2and W3in Snx +, let matrices Pn=U(0) Qn ∗T n,R=W2,S=W3,(25) Qn=0 −h U(θ+h)Adℓ⊤ n(θ)Indθ, Tn= 0 −h 0 −h Inℓn(θ1)A⊤ dU(θ2−θ1)Adℓ⊤ n(θ2)Indθ1dθ2. Note that (Pn,R,S) belongs to S(n+1)nx×Snx +×Snx +using the symmetric property of U. Moreover, the Lyapunov candidate functional Vngiven by (21) is the Legendre approximation (14) of the Lyapunov converse functional V. Assume that inequality (16) holds at order nand that system (1) is GES. Applying (8a) in Theorem 1 and (17b) in Theorem 2, there exist positive scalars α1,α 3such for the following inequalities hold α1∥xt∥2+0 −h x⊤ t(θ)θ+h hRxt(θ)dθ≤V(xt), ˙ Vn(xt)≤−α3∥xt∥2. (26a) (26b) Then, these inequalities are applied for xtin Dn⊂Dgiven by Dn=           φ(θ)=       φ0if θ=0, ℓ⊤ nθ+h hζnif θ∈(−h, 0), φ−hif θ=−h, such that [φ0ζnφ−h]⊤∈R(n+2)nx.            . Thus, inequalities (26a),(26b) lead to Ξ+ n≻0 and Ξ− n≺0, respectively, which concludes the proof. □ Theorem 3 provides a necessary and sufficient LMI condition for the stability of system (1). Nonetheless, it does not guarantee the existence of an order nsuch that the LMI must be satisfied. The estimation of this order is reserved to the next section. 4. ESTIMATION OF THE NECESSARY ORDER 4.1 Convergence properties In the light of Dunkl and Xu (2001), we provide convergence properties satisfied by Legendre polynomial approximation in order to estimate an order from which the necessary and sufficient stability condition holds. Firstly, the Lyapunov matrix Uassociated to Wdefined in (5) is continuous on [−h, h] and satisfies Fig. 1. Estimation of the necessary order n∗for example (36). sup [−h,0) U(d)≤|M|dρ|W|,∀d∈N,(27) where U(d)is the dth derivative of U, the parameter ρ=√nxeh|M|N−1,(28) and the matrices M,Nand Ware given by (6). Thanks to this smoothness, the uniform convergence of ˜ U1,n towards zero is supergeometric as proposed in Property 1 below. Property 1. For all n≥6, the Legendre approximation error ˜ U1,n satisfies the inequalities sup [−h,0] ˜ U1,n≤µn−⌈µ⌉ρ1 (n−3 2)...(⌈µ⌉+3 2)|W|,(29a) hsup [−h,0] ˜ U′ 1,n≤µn−⌈µ⌉ρ1 (n−7 2)...(⌈µ⌉+3 2)|W|,(29b) where µ=h|M| 2,ρ 1=2π3 ⌈µ⌉ρ, (30) and matrices M,Nand Ware given by (6). Proof : The proof can be found in (Wang and Xiang, 2012, Theorem 2.5) using Legendre polynomials properties and d=n−1−⌈µ⌉successive integrations by parts. □ Property 2. For any n≥4, the Legendre approximation error ˜ U2,n satisfied the inequality sup θ∈[−h,h]˜ U2,n≤ρ2 √n−3|W|,(31) where ρ2=√2πh(1 + √nxπheh|M|M2N−1),(32) and matrices M,Nand Ware given by (6). Proof : The proof is provided in Bajodek (2022). □ The convergence properties of Legendre polynomial approximation ensures the existence of an order n∗from which Theorem 3 holds, as explained below. Fig. 2. Sufficient order nto satisfy LMI (24) for example (36). 4.2 Estimation of the orders The following theorem estimates the orders for which Theorem 3 can be applied. Theorem 4. Theorem 3 holds for orders n≥6 for which the following inequality holds 1−µn−⌈µ⌉ρ′ 1 (n−7 2)...(⌈µ⌉+3 2)−ρ′ 2 √n−3≥0.(33) where scalars ρ′ 1,ρ′ 2are equal to    ρ′ 1= (3 + |A|+1 h+|Ad|)(2 + h)|Ad|ρ1, ρ′ 2= 2(2 + h)|Ad|2ρ2, (34) Proof : Choosing matrices W1=Inx,W2=hInzand W3=Inz, it is worth noticing that Ψ1 nand Ψ2 ngiven by (16) are upper bounded by Ψ1 n≤(|A|+1 h)µn−⌈µ⌉ρ1 (n−7 2)...(⌈µ⌉+3 2)+|Ad|ρ2 √n−3(2 + h)|Ad|, Ψ2 n≤µn−⌈µ⌉ρ1 (n−7 2)...(⌈µ⌉+3 2)+ρ2 √n−3(2 + h)|Ad|2. From Schur complement, inequality (16) is satisfied under the conditions                      1−(3 + |A|+1 h)µn−⌈µ⌉ρ1 (n−7 2)...(⌈µ⌉+3 2)+|Ad|ρ2 √n−3(2+h)|Ad|≥0, 1−(|A|+1 h+|Ad|)µn−⌈µ⌉ρ1 (n−7 2)...(⌈µ⌉+3 2)+2|Ad|ρ2 √n−3(2+h)|Ad|≥0, 1−(1 + |Ad|)µn−⌈µ⌉ρ1 (n−7 2)...(⌈µ⌉+3 2)+|Ad|ρ2 √n−3(2+h)|Ad|≥0, (35) Consequently, the estimated order n∗satisfy (33), which concludes the proof. □ Thanks to the approximation of Uby Legendre polynomials and leaning on the supergeometric convergence rate of error ˜ U1,n, we greatly reduce the estimate of the order n∗ proposed in Bajodek et al. (2022). These estimated orders are shown in the last example section. 180 Mathieu Bajodek et al. / IFAC PapersOnLine 55-36 (2022) 175–180 5. NUMERICAL EXAMPLE Consider the system (1) with A=01 00 ,A d=00 −k1−k2,h=1,(36) for proportional gains k1>0 and k2≥0. Figure 1 reports the necessary orders n∗given by (33) that corresponds to system (36) for several gains (k1,k 2) in [0,1] ×[0,1.5]. These orders n∗still very large to test the necessary and sufficient LMI conditions (24). As a complement, Figure 2 reports the sufficient order n such that our sufficient condition of stability holds. Increasing the order nfrom 1 to n∗, we are looking for the first order ˇnfrom which the LMI condition (24) is true. One remarks that there are lower than the necessary order, which leaves a room for forthcoming improvements. 6. CONCLUSIONS In this work, three steps leading to necessary and sufficient LMI conditions of stability are highlighted. First, the complete Lyapunov functional is presented. Second, from Theorem 2, we are now able to quantify the degree of conservatism introduced by an approximation process. Finally, the necessary and sufficient Legendre-LMI test provided by Theorem 3 is established. The size of the LMI test can be calculated thanks to Theorem 4. Following the same methodology, this work paves the way to develop new LMI conditions for larger classes of systems. 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