Control-Relevant Discretization of Nonlinear State Delay Systems Using the Matrix Exponential Algorithm

2012 ◽  
Vol 224 ◽  
pp. 475-479
Author(s):  
Yuan Liang Zhang
Keyword(s):  
Data Representation ◽  
Matrix Exponential ◽  
Delay Systems ◽  
Continuous Systems ◽  
Discretization Method ◽  
Discretization Scheme ◽  
State Delay ◽  
Finite Dimensional Representation ◽  
The Matrix ◽  
Nonlinear State

In this paper, a new discretization method to obtain the sampled data representation of the nonlinear state delay control system is proposed. This discretization method is based on the matrix exponential computation. The mathematical structure of the new discretization scheme is explored. Then it is applied to obtain the discrete form of the nonlinear state delay continuous systems. The resulting time discretization method provides a finite dimensional representation for nonlinear control systems with state delay, thereby enabling the application of existing controller design techniques to such systems. The performance of the proposed discretization procedure is evaluated by means of the simulation study. In the simulation various sampling rates and time delay values are considered. The results demonstrate that the proposed discretization scheme can assure the system’s accuracy requirements.

Control-Relevant Discretization of Nonlinear Systems With Time-Delay Using Taylor-Lie Series

2004 ◽  
Vol 127 (1) ◽  
pp. 153-159 ◽  
Author(s):  
Nikolaos Kazantzis ◽  
K. T. Chong ◽  
J. H. Park ◽  
Alexander G. Parlos
Keyword(s):  
Time Delay ◽  
Nonlinear Control ◽  
Control Systems ◽  
Time Discretization ◽  
Data Representation ◽  
Nonlinear Control Systems ◽  
Discretization Method ◽  
Sampled Data ◽  
Lie Series ◽  
Finite Dimensional Representation

A new time-discretization method for the development of a sampled-data representation of a nonlinear continuous-time input-driven system with time delay is proposed. It is based on the Taylor-Lie series expansion method and zero-order hold assumption. The mathematical structure of the new discretization scheme is explored and characterized as useful for establishing concrete connections between numerical and system-theoretic properties. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. The resulting time-discretization provides a finite-dimensional representation for nonlinear control systems with time-delay enabling the application of existing controller design techniques. The performance of the proposed discretization procedure is evaluated using the case study of a two-degree-of-freedom mechanical system that exhibits nonlinear behavior. Various sampling rates and time-delay values are considered.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

scholarly journals Optimal parameter selection in Weeks’ method for numerical Laplace transform inversion based on machine learning

2021 ◽  
Vol 15 ◽  
pp. 174830262199962
Author(s):  
Patrick O Kano ◽  
Moysey Brio ◽  
Jacob Bailey
Keyword(s):  
Laplace Transform ◽  
Optimal Parameter ◽  
Large Data ◽  
Matrix Exponential ◽  
Resolvent Matrix ◽  
Data Set ◽  
Network Training ◽  
The Matrix ◽  
The Laplace Transform ◽  
Space Function

The Weeks method for the numerical inversion of the Laplace transform utilizes a Möbius transformation which is parameterized by two real quantities, σ and b. Proper selection of these parameters depends highly on the Laplace space function F( s) and is generally a nontrivial task. In this paper, a convolutional neural network is trained to determine optimal values for these parameters for the specific case of the matrix exponential. The matrix exponential eA is estimated by numerically inverting the corresponding resolvent matrix [Formula: see text] via the Weeks method at [Formula: see text] pairs provided by the network. For illustration, classes of square real matrices of size three to six are studied. For these small matrices, the Cayley-Hamilton theorem and rational approximations can be utilized to obtain values to compare with the results from the network derived estimates. The network learned by minimizing the error of the matrix exponentials from the Weeks method over a large data set spanning [Formula: see text] pairs. Network training using the Jacobi identity as a metric was found to yield a self-contained approach that does not require a truth matrix exponential for comparison.

A Splitting method for the calculation of the matrix exponential

Analysis ◽  
1994 ◽  
Vol 14 (2-3) ◽  
pp. 103-112 ◽  
Author(s):  
Eberhard U. Stichel
Keyword(s):  
Splitting Method ◽  
Matrix Exponential ◽  
The Matrix
Sign in / Sign up

Export Citation Format

Share Document