scholarly journals Balanced truncation of linear time-invariant systems over finite-frequency ranges

2020 ◽  
Vol 46 (6) ◽  
Author(s):  
Peter Benner ◽  
Xin Du ◽  
Guanghong Yang ◽  
Dan Ye
Keyword(s):  
Error Bounds ◽  
Linear Time ◽  
Upper And Lower Bounds ◽  
Single Frequency ◽  
Frequency Interval ◽  
Balanced Truncation ◽  
Frequency Dependent ◽  
Linear Time Invariant ◽  
Type Frequency ◽  
Time Invariant

AbstractThis paper discusses model order reduction of linear time-invariant (LTI) systems over limited frequency intervals within the framework of balanced truncation. Two new frequency-dependent balanced truncation methods are developed, one is single-frequency (SF)-type frequency-dependent balanced truncation to cope with the cases that only a single dominating point of the operating frequency interval is pre-known, and the other is interval-type frequency-dependent balanced truncation to deal with the case that both the upper and lower bounds of the relevant frequency interval are known a priori. Error bounds for both approaches are derived to estimate the approximation error over a pre-specified frequency interval. In contrast to other error bounds for frequency-weighted or frequency-limited balanced truncation, these bounds are given specifically for the interval under consideration and are thus often sharper than the global bounds for previous methods. We show that the new methods generally lead to good in-band approximation performance, and at the same time provide accurate error bounds under certain conditions. Examples are included for illustration.

Robust Controller Design for Descriptor-type Time-delay Systems

2021 ◽  
Vol 20 ◽  
pp. 289-294
Author(s):  
Altug Iftar
Keyword(s):  
Time Delay ◽  
Controller Design ◽  
Linear Time ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Robust Controller ◽  
Frequency Dependent ◽  
Stabilizing Controller ◽  
Linear Time Invariant ◽  
Time Invariant

Linear time-invariant descriptor-type time-delay systems are considered. A robust stabilizing controller design approach for such systems is introduced. Uncertainties both in the time-delays and in other system parameters are considered. A frequency-dependent scalar bound on such uncertainties is first derived. Once this bound is found, the controller design is completely based on the nominal model. However, satisfying a scalar frequency-dependent condition, which uses the derived bound, guarantees robust stability. An example is also presented to illustrate the proposed approach

Robust Stability Analysis of Distributed-Order Linear Time-Invariant Systems With Uncertain Order Weight Functions and Uncertain Dynamic Matrices

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Hamed Taghavian ◽  
Mohammad Saleh Tavazoei
Keyword(s):  
Weight Function ◽  
Robust Stability ◽  
Linear Time ◽  
Upper And Lower Bounds ◽  
Weight Functions ◽  
Order Linear ◽  
Linear Time Invariant ◽  
Robust Stability Analysis ◽  
Time Invariant ◽  
Distributed Order

Bounded-input bounded-output (BIBO) stability of distributed-order linear time-invariant (LTI) systems with uncertain order weight functions and uncertain dynamic matrices is investigated in this paper. The order weight function in these uncertain systems is assumed to be totally unknown lying between two known positive bounds. First, some properties of stability boundaries of fractional distributed-order systems with respect to location of eigenvalues of dynamic matrix are proved. Then, on the basis of these properties, it is shown that the stability boundary of distributed-order systems with the aforementioned uncertain order weight functions is located in a certain region on the complex plane defined by the upper and lower bounds of the order weight function. Thereby, sufficient conditions are obtained to ensure robust stability in distributed-order LTI systems with uncertain order weight functions and uncertain dynamic matrices. Numerical examples are presented to verify the obtained results.

Evaluation of fractional order of the discrete integrator. Part II

2020 ◽  
Vol 23 (2) ◽  
pp. 408-426
Author(s):  
Piotr Ostalczyk ◽  
Marcin Bąkała ◽  
Jacek Nowakowski ◽  
Dominik Sankowski
Keyword(s):  
Fractional Order ◽  
Output Signal ◽  
Linear Time ◽  
Mathematical Problem ◽  
Simple Method ◽  
Linear Time Invariant ◽  
Input And Output ◽  
Order Difference Equation ◽  
Time Invariant ◽  
Discrete Integrator

AbstractThis is a continuation (Part II) of our previous paper [19]. In this paper we present a simple method of the fractional-order value calculation of the fractional-order discrete integration element. We assume that the input and output signals are known. The linear time-invariant fractional-order difference equation is reduced to the polynomial in a variable ν with coefficients depending on the measured input and output signal values. One should solve linear algebraic equation or find roots of a polynomial. This simple mathematical problem complicates when the measured output signal contains a noise. Then, the polynomial roots are unsettled because they are very sensitive to coefficients variability. In the paper we show that the discrete integrator fractional-order is very stiff due to the degree of the polynomial. The minimal number of samples guaranteeing the correct order is evaluated. The investigations are supported by a numerical example.

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