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Wim Michiels - One of the best experts on this subject based on the ideXlab platform.

  • on the strong H2 Norm of differential algebraic systems with multiple delays finiteness criteria regularization and computation
    IEEE Transactions on Automatic Control, 2020
    Co-Authors: Marco A Gomez, Raphael M Jungers, Wim Michiels
    Abstract:

    The H2 Norm of an exponentially stable system described by Delay Differential Algebraic Equations (DDAEs) might be infinite due to the existence of hidden feedthrough terms and, as shown in this paper, it might become infinite as a result of infinitesimal changes to the delay parameters. We first introduce the notion of strong H2 Norm of semi-explicit DDAEs, a robustified measure that takes into account delay perturbations, and we analyze its properties. Next, we derive necessary and sufficient finiteness criteria for the strong H2 Norm, in terms of a frequency sweeping test over a hypercube, and in terms of a finite number of equalities involving multi-dimensional powers of a finite set of matrices. As the main contribution, we present a strengthened, sufficient, condition for finiteness of the strong H2 Norm, along with an algorithm for checking it, which has significantly better scalability properties in terms of both the dimension of the system and the number of delays. We show that the satisfaction of the novel condition is equivalent to the existence of a simultaneous block triangularization of the matrices of the delay difference equation associated to the DDAE.

  • computing the H2 Norm of large scale time delay systems
    IFAC Proceedings Volumes, 2013
    Co-Authors: Jelle Peeters, Wim Michiels
    Abstract:

    Abstract The H 2 Norm of an appropriately defined transfer function plays an important role in the field of systems and control, as a robustness measure with respect to noise or external disturbances. For a time-delay system the H 2 Norm of a transfer function can be approximated by rewriting the time-delay system as an infinite-dimensional linear system and applying a spectral discretization to become a system without delay. The H 2 Norm of this (larger) standard LTI system can be calculated by solving a Lyapunov equation. Downside of this method is the fact that the discretized model is of dimension nN , with n the size of the original time-delay system and N the number of discretization points. Therefore, a Krylov based model order reduction technique will be applied, which allows to reduce the order of the discretized model further down to a chosen value k nN and where the transfer function of the reduced order model matches several moments (function value and derivatives) at zero and infinity with the transfer function of the original time-delay system. In this paper, we describe in detail the procedure allowing to transform a large-scale time-delay system into a standard linear system of small dimension and show how the H 2 Norm of the resulting system is an accurate approximation for the H 2 Norm of the original time-delay system. We predict the error behavior of the approximation and demonstrate this on an application.

  • TDS - Computing the H2 Norm of large-scale time-delay systems
    IFAC Proceedings Volumes, 2013
    Co-Authors: Jelle Peeters, Wim Michiels
    Abstract:

    Abstract The H 2 Norm of an appropriately defined transfer function plays an important role in the field of systems and control, as a robustness measure with respect to noise or external disturbances. For a time-delay system the H 2 Norm of a transfer function can be approximated by rewriting the time-delay system as an infinite-dimensional linear system and applying a spectral discretization to become a system without delay. The H 2 Norm of this (larger) standard LTI system can be calculated by solving a Lyapunov equation. Downside of this method is the fact that the discretized model is of dimension nN , with n the size of the original time-delay system and N the number of discretization points. Therefore, a Krylov based model order reduction technique will be applied, which allows to reduce the order of the discretized model further down to a chosen value k nN and where the transfer function of the reduced order model matches several moments (function value and derivatives) at zero and infinity with the transfer function of the original time-delay system. In this paper, we describe in detail the procedure allowing to transform a large-scale time-delay system into a standard linear system of small dimension and show how the H 2 Norm of the resulting system is an accurate approximation for the H 2 Norm of the original time-delay system. We predict the error behavior of the approximation and demonstrate this on an application.

  • Characterizing and Computing the ${\cal H}_{2}$ Norm of Time-Delay Systems by Solving the Delay Lyapunov Equation
    IEEE Transactions on Automatic Control, 2011
    Co-Authors: Elias Jarlebring, Joris Vanbiervliet, Wim Michiels
    Abstract:

    It is widely known that the solutions of Lyapunov equations can be used to compute the H2 Norm of linear time-invariant (LTI) dynamical systems. In this paper, we show how this theory extends to dynamical systems with delays. The first result is that the H2 Norm can be computed from the solution of a generalization of the Lyapunov equation, which is known as the delay Lyapunov equation. From the relation with the delay Lyapunov equation we can prove an explicit formula for the H2 Norm if the system has commensurate delays, here meaning that the delays are all integer multiples of a basic delay. The formula is explicit and contains only elementary linear algebra operations applied to matrices of finite dimension. The delay Lyapunov equations are matrix boundary value problems. We show how to apply a spectral discretization scheme to these equations for the general, not necessarily commensurate, case. The convergence of spectral methods typically depends on the smoothness of the solution. To this end we describe the smoothness of the solution to the delay Lyapunov equations, for the commensurate as well as for the non-commensurate case. The smoothness properties allow us to completely predict the convergence order of the spectral method.

  • Computing H2 Norms and their derivatives for time-delay systems, using Krylov based model order reduction
    2011
    Co-Authors: Jelle Peeters, Wim Michiels
    Abstract:

    it was shown in [1], that the H2 Norm ||γ(s)||2 can be calculated by rewriting the time-delay system in a linear infinitedimensional form and applying a spectral discretization to become a system without delay. The H2 Norm of this (larger) standard LTI system, ||γN(s)||2, can be easily calculated, resulting in an approximation for ||γ(s)||2. Downside of this method is the fact that the discretized model is of order nN, with n the size of the original time-delay system and N the number of discretization points. In [2], a Krylov based model order reduction technique for time-delay systems is suggested which allows to reduce the order of the discretized model further down to a chosen value k

Rachid Malti - One of the best experts on this subject based on the ideXlab platform.

Jocelyn Sabatier - One of the best experts on this subject based on the ideXlab platform.

  • H2-Norm for mesh optimization with application to electro-thermal modeling of an electric wire in automotive context
    Communications in Nonlinear Science and Numerical Simulation, 2017
    Co-Authors: Christophe Farges, Mathieu Chevrié, Jocelyn Sabatier, Franck Guillemard, Laetitia Pradere
    Abstract:

    In automotive application field, reducing electric conductors dimensions is significant to decrease the embedded mass and the manufacturing costs. It is thus essential to develop tools to optimize the wire diameter according to thermal constraints and protection algorithms to maintain a high level of safety. In order to develop such tools and algorithms, accurate electro-thermal models of electric wires are required. However, thermal equation solutions lead to implicit fractional transfer functions involving an exponential that cannot be embedded in a car calculator. This paper thus proposes an integer order transfer function approximation methodology based on a spatial discretization for this class of fractional transfer functions. Moreover, the H2-Norm is used to minimize approximation error. Accuracy of the proposed approach is confirmed with measured data on a 1.5 mm2 wire implemented in a dedicated test bench.

  • Finite length wire dynamical modeling for automotive applications using H2-Norm based approximation of a fractional model
    2016
    Co-Authors: Mathieu Chevrié, Christophe Farges, Jocelyn Sabatier, Franck Guillemard, Laetitia Pradere
    Abstract:

    In automotive application field, reducing electric conductors dimensions is significant to decrease the embedded mass and the manufacturing costs. It is thus essential to develop tools to optimize the wire diameter according to thermal constraints and protection algorithms to maintain a high level of safety. In order to develop such tools and algorithms, accurate electro-thermal models of electric wires are required. However, thermal equation solutions lead to implicit fractional transfer functions involving an exponential that cannot be embedded in a car calculator. This paper thus proposes an integer order approximation methodology based on a spatial discretization for this class of fractional transfer functions. Moreover, the H2-Norm is used to minimize approximation error. Accuracy of the proposed approach is confirmed with measured data on a 1.5 mm2 wire implemented in a dedicated test bench.

  • H2-Norm of a class of fractional transfer functions suited for modeling diffusive phenomena
    2015
    Co-Authors: Mathieu Chevrié, Christophe Farges, Jocelyn Sabatier, Rachid Malti
    Abstract:

    This paper focuses on the H2-Norm of a class of implicit fractional order transfer functions well suited to describe input-output behaviour of diffusive systems. First, analytical expression of the H2-Norm of this kind of transfer function is established. This result is then used to evaluate the quality of an integer order approximation of such an implicit fractional transfer function.

  • H2-Norm of fractional transfer functions of implicit type
    Communications in Nonlinear Science and Numerical Simulation, 2015
    Co-Authors: Rachid Malti, Mathieu Chevrié, Christophe Farges, Jocelyn Sabatier
    Abstract:

    Abstract This paper studies the H 2 -Norm (or impulse response energy) of fractional transfer functions of implicit type. Stability conditions are first shown to be identical as in rational systems with all poles located in the open left half complex plane. Then, analytical expressions of the H 2 -Norm are derived for elementary fractional transfer functions of the first and the second kind cascaded with a pure fractional integrator. Next, general boundedness conditions are established in terms of transfer function relative degree. Three illustrative examples are finally proposed. The first one evaluates the quality of a rational approximation of a fractional model of implicit type on the basis of the H 2 -Norm of the error signal. The second one evaluates the Integral Squared Error of a CRONE control loop and compares it to a classical proportional-derivative controller in a vehicle suspension. Finally, the third one allows to set up an implicit fractional preshaping filter for closed-loop control.

  • H2-Norm of fractional transfer functions of implicit type of the first kind
    IFAC Proceedings Volumes, 2014
    Co-Authors: Mathieu Chevrié, Christophe Farges, Rachid Malti, Jocelyn Sabatier
    Abstract:

    This paper studies the H2-Norm, or impulse response energy, of fractional transfer functions of implicit type. The analytical expression of the H2-Norm is first derived for an elementary fractional transfer function of the first kind with a single real pole. Series connection of such a transfer function with a pure fractional integrator and with another implicit transfer function of the first kind are then studied. Results developed in the paper are finally used to derive a criterion to evaluate the quality of an integer order approximation for an implicit type fractional order model of the first kind.

Makan Fardad - One of the best experts on this subject based on the ideXlab platform.

  • Frequency Analysis and Norms of Distributed Spatially Periodic Systems
    IEEE Transactions on Automatic Control, 2008
    Co-Authors: Makan Fardad, Mihailo R. Jovanovic, Bassam Bamieh
    Abstract:

    We investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients. We develop a spatial-frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems. Using this representation, we introduce the notion of the H2 Norm for this class of systems and provide algorithms for its computation. A stochastic interpretation of the H2 Norm is given in terms of spatially cyclostationary random fields and spectral-correlation density operators. When the periodic coefficients are viewed as feedback modifications of spatially invariant systems, we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems. Two examples from physics are provided to illustrate the main results.

  • Brief paper: H2 Norm of linear time-periodic systems: A perturbation analysis
    Automatica, 2008
    Co-Authors: Mihailo R. Jovanovic, Makan Fardad
    Abstract:

    We consider a class of linear time-periodic systems in which the dynamical generator A(t) represents the sum of a stable time-invariant operator A"0 and a small-amplitude zero-mean T-periodic operator @eA"p(t). We employ a perturbation analysis to develop a computationally efficient method for determination of the H"2 Norm. Up to second order in the perturbation parameter @e we show that: (a) the H"2 Norm can be obtained from a conveniently coupled system of Lyapunov and Sylvester equations that are of the same dimension as A"0; (b) there is no coupling between different harmonics of A"p(t) in the expression for the H"2 Norm. These two properties do not hold for arbitrary values of @e, and their derivation would not be possible if we tried to determine the H"2 Norm directly without resorting to perturbation analysis. Our method is well suited for identification of the values of period T that lead to the largest increase/reduction of the H"2 Norm. Two examples are provided to motivate the developments and illustrate the procedure.

Bernhard P. Lampe - One of the best experts on this subject based on the ideXlab platform.

  • Stochastic analysis and H2—Norm of linear periodic operators
    Computer Controlled Systems, 2000
    Co-Authors: Efim N. Rossenwasser, Bernhard P. Lampe
    Abstract:

    In this chapter we investigate the response of a linear periodic operator to an input signal x(t), that is a centered, and in a loose sense, stationary stochastic process. The last property means that, Astrom (1970) $$E[x(t)] = 0$$ (8.1) $$E[x({{t}_{1}})x({{t}_{2}})] = {{K}_{x}}({{t}_{2}} - {{t}_{1}})$$ (8.2) where E denotes the operator of mathematical expectation and K x (t) is the autocorrelation function of the signal x(t). If we take t1 = t and t2 = t + τ, then $$E[x(t)x(t + \tau )] = {{K}_{x}}(\tau )$$ (8.3) and, as is well known $${{K}_{x}}(t) = {{K}_{x}}( - t).$$ (8.4)

  • stochastic analysis and H2 Norm of linear periodic operators
    2000
    Co-Authors: Efim N. Rossenwasser, Bernhard P. Lampe
    Abstract:

    In this chapter we investigate the response of a linear periodic operator to an input signal x(t), that is a centered, and in a loose sense, stationary stochastic process. The last property means that, Astrom (1970) $$E[x(t)] = 0$$ (8.1) $$E[x({{t}_{1}})x({{t}_{2}})] = {{K}_{x}}({{t}_{2}} - {{t}_{1}})$$ (8.2) where E denotes the operator of mathematical expectation and K x (t) is the autocorrelation function of the signal x(t). If we take t1 = t and t2 = t + τ, then $$E[x(t)x(t + \tau )] = {{K}_{x}}(\tau )$$ (8.3) and, as is well known $${{K}_{x}}(t) = {{K}_{x}}( - t).$$ (8.4)

  • Statistische Analyse und H2 — Norm linearer periodischer Operatoren
    Digitale Regelung in kontinuierlicher Zeit, 1997
    Co-Authors: Yephim N. Rosenwasser, Bernhard P. Lampe
    Abstract:

    In diesem Kapitel wird die Aufgabe gestellt, die Reaktion eines linearen periodischen Operators auf ein Eingangssignal x(t) zu untersuchen, das ein zentralisierter im weiteren Sinne stationarer stochastischer Prozes ist, Lange (1971); Astrom (1970); Karlin (1966). Weiterhin wird vereinbart, das $$E[x(t)]=0$$ (8.1) $$E[x({{t}_{1}})x({{t}_{2}})]={{K}_{x}}({{t}_{2}}-{{t}_{1}})$$ (8.2) sind, worin E der Operator fur die mathematische Erwartung und K x (t) die Autokorrelationsfunktion des Signals x(t) sind. Wenn man t1 = t und t2 = t + τ setzt, dann bekommt man $$E[x(t)x(t+T)]={{K}_{x}}(T)$$ (8.3) und bekanntlich ist $${{K}_{x}}(t)={{K}_{x}}(-t)$$ (8.4)

  • statistische analyse und H2 Norm linearer periodischer operatoren
    1997
    Co-Authors: Yephim N. Rosenwasser, Bernhard P. Lampe
    Abstract:

    In diesem Kapitel wird die Aufgabe gestellt, die Reaktion eines linearen periodischen Operators auf ein Eingangssignal x(t) zu untersuchen, das ein zentralisierter im weiteren Sinne stationarer stochastischer Prozes ist, Lange (1971); Astrom (1970); Karlin (1966). Weiterhin wird vereinbart, das $$E[x(t)]=0$$ (8.1) $$E[x({{t}_{1}})x({{t}_{2}})]={{K}_{x}}({{t}_{2}}-{{t}_{1}})$$ (8.2) sind, worin E der Operator fur die mathematische Erwartung und K x (t) die Autokorrelationsfunktion des Signals x(t) sind. Wenn man t1 = t und t2 = t + τ setzt, dann bekommt man $$E[x(t)x(t+T)]={{K}_{x}}(T)$$ (8.3) und bekanntlich ist $${{K}_{x}}(t)={{K}_{x}}(-t)$$ (8.4)