Conversion of Linear Time-Invariant Delay-Differential Equations with External Input and Output into Representation as Time-Delay Feedback Systems

2012 ◽  
Vol 5 (4) ◽  
pp. 200-209 ◽  
Author(s):  
Tatsuya YAMAZAKI ◽  
Tomomichi HAGIWARA
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Linear Time ◽  
External Input ◽  
Feedback Systems ◽  
Linear Time Invariant ◽  
Input And Output ◽  
Delay Differential ◽  
Time Invariant

Solution and Parameter Estimation in Linear Time-Invariant Delayed Systems Using Laguerre Polynomial Expansion

1983 ◽  
Vol 105 (4) ◽  
pp. 297-301 ◽  
Author(s):  
Fan-Chu Kung ◽  
Hua Lee
Keyword(s):  
Time Delay ◽  
Large Scale ◽  
Laguerre Polynomial ◽  
Linear Time ◽  
Small Time ◽  
Polynomial Expansion ◽  
Feedback Systems ◽  
Linear Time Invariant ◽  
Small Time Delay ◽  
Time Invariant

Finite dimensional Laguerre polynomial expansion can be applied to approximate the solution of linear time-invariant systems. Here, we extend this approach to systems with small time delay by introducing a delay matrix operator to the system equations. In addition, parameters of linear unity-feedback systems with small time delay can also be estimated by using the Laguerre expansion. In this paper we provide algorithms for large scale systems to avoid direct matrix inversion and give examples to demonstrate the approximation.

scholarly journals Robust stability of positive linear time delay systems under time-varying perturbations

2015 ◽  
Vol 63 (4) ◽  
pp. 947-954
Author(s):  
P.H.A. Ngoc ◽  
C.T. Tinh
Keyword(s):  
Time Delay ◽  
Robust Stability ◽  
Linear Time ◽  
Delay Systems ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Novel Approach ◽  
Delay Differential Systems ◽  
Delay Differential ◽  
Time Invariant

Abstract By a novel approach, we get explicit robust stability bounds for positive linear time-invariant time delay differential systems subject to time-varying structured perturbations or non-linear time-varying perturbations. Some examples are given to illustrate the obtained results. To the best of our knowledge, the results of this paper are new.

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Robust Control ◽  
Differential Equations ◽  
Time Domain ◽  
Delay Differential Equations ◽  
Linear Time ◽  
Lambert W Function ◽  
Linear Time Invariant ◽  
Robust Control Design ◽  
Delay Differential ◽  
Eigenvalue Assignment

An approach to eigenvalue assignment for systems of linear time-invariant (LTI) delay differential equations (DDEs), based upon the solution in terms of the matrix Lambert W function, is applied to the problem of robust control design for perturbed LTI systems of DDEs, and to the problem of time-domain response specifications. Robust stability of the closed-loop system can be achieved through eigenvalue assignment combined with the real stability radius concept. For a LTI system of DDEs with a single delay, which has an infinite number of eigenvalues, the recently developed Lambert W function-based approach is used to assign a dominant subset of them, which has not been previously feasible. Also, an approach to time-domain specifications for the transient response of systems of DDEs is developed in a way similar to systems of ordinary differential equations using the Lambert W function-based approach.

Decentralized time-delay controller design for systems described by delay differential-algebraic equations

2021 ◽  
pp. 014233122110159
Author(s):  
H. Ersin Erol ◽  
Altuğ İftar
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Controller Design ◽  
Linear Time ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Design Algorithm ◽  
Linear Time Invariant ◽  
Delay Differential ◽  
Time Invariant

This paper presents a complete approach for designing stabilizing linear time-invariant decentralized finite-dimensional or retarded time-delay output feedback controllers for linear time-invariant systems of delay differential-algebraic equations. The proposed approach is based on the sequential design of the local controllers by using a centralized controller design algorithm. In this sequential design approach, the local controller to be designed at each step is determined depending on the mobility of the rightmost modes with respect to the controllers that have not yet been designed and closed with the system. Since no predefined sequence is followed, a sequence that can target the least effort and dimension for each agent can be aimed. Also, in the proposed approach, a base controller effort can be targeted for each control agent, so that the effort required to stabilize the system can be distributed among the local controllers. In the centralized controller design algorithm used for the design of each local controller, the parameters of the controllers are changed stepwise in a quasi-continuous way to shift the targeted rightmost modes towards the stable area. For a time-delay controller, the desired mode placement can be achieved by applying small changes stepwise to the elements of the matrices and the time-delays of the controller while time-delays remain always non-negative. The effect of small perturbations on the time-delays in the open-loop system or to be added by the controller to be designed is taken into account to ensure some degree of robustness against all possible perturbations on the delays. The effectiveness of the proposed design approach is demonstrated by a numerical example.

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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