Linear Time-Varying Dynamic Systems Optimization via Higher-Order Method Using Shifted Chebyshev’s Polynomials

1999 ◽  
Vol 121 (2) ◽  
pp. 258-261 ◽  
Author(s):  
Xiaochun Xu ◽  
Sunil K. Agrawal
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Linear Time ◽  
Optimal Solution ◽  
Higher Order ◽  
Time Varying ◽  
Varying Coefficients ◽  
Higher Order Method ◽  
Time Varying Coefficients ◽  
Higher Order Differential Equations

For optimization of classes of linear time-varying dynamic systems with n states and m control inputs, a new higher-order procedure was presented by the authors that does not use Lagrange multipliers. In this new procedure, the optimal solution was shown to satisfy m 2p-order differential equations with time-varying coefficients. These differential equations were solved using weighted residual methods. Even though solution of the optimization problem using this procedure was demonstrated to be computation efficient, shifted Chebyshev’s polynomials are used in the paper to solve the higher-order differential equations. This further reduces the computations and makes this algorithm more appropriate for real-time implementation.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

2021 ◽  
Author(s):  
Edward Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z^(-1) which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z^(- i) by time functions which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of z^(i) at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

Adaptive Stabilization of Linear Second-Order Systems with Bounded Time-Varying Coefficients

2004 ◽  
Vol 10 (7) ◽  
pp. 963-978 ◽  
Author(s):  
Alexander V. Roup ◽  
Dennis S. Bernstein
Keyword(s):  
Linear Time ◽  
Adaptive Controller ◽  
Second Order ◽  
Lyapunov Methods ◽  
Time Varying ◽  
Adaptive Stabilization ◽  
Varying Coefficients ◽  
System States ◽  
Time Varying Coefficients ◽  
Second Order Systems

We consider adaptive stabilization for a class of linear time-varying second-order systems. Interpreting the system states as position and velocity, the system is assumed to have unknown, non-paranetric, bounded time-varying damping and stiffness coefficients. The coefficient bounds need not be known to implement the adaptive controller. Lyapunov methods are used to prove global convergence of the system states. For illustration, the controller is used to stabilize several example systems.

scholarly journals Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Moreno Concezzi ◽  
Roberto Garra ◽  
Renato Spigler
Keyword(s):  
Differential Equations ◽  
Numerical Approach ◽  
Numerical Results ◽  
Cauchy Type ◽  
Time Varying ◽  
Varying Coefficients ◽  
Fractional Relaxation ◽  
Time Varying Coefficients ◽  
Damped Oscillator

AbstractWe consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of the Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

Computing Numerical Solutions of Delayed Fractional Differential Equations With Time Varying Coefficients

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Venkatesh Suresh Deshmukh
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Metal Cutting ◽  
Numerical Solutions ◽  
Viscoelastic Damping ◽  
Time Varying ◽  
Varying Coefficients ◽  
Fractional Differential ◽  
Time Varying Coefficients ◽  
Cutting Processes

Fractional differential equations with time varying coefficients and delay are encountered in the analysis of models of metal cutting processes such as milling and drilling with viscoelastic damping elements. Viscoelastic damping is modeled as a fractional derivative. In the present paper, delayed fractional differential equations with bounded time varying coefficients in four different forms are analyzed using series solution and Chebyshev spectral collocation. A fractional differential equation with a known exact solution is then solved by the methodology presented in the paper. The agreement between the two is found to be excellent in terms of point-wise error in the trajectories. Solutions to the described fractional differential equations are computed next in state space and second order forms.

Solutions of Delayed Partial Differential Equations With Space-Time Varying Coefficients

2011 ◽  
Vol 7 (2) ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Initial Boundary ◽  
Time Varying ◽  
Initial Boundary Value Problems ◽  
Weakly Nonlinear ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
Initial Boundary Value

A constructive algorithm using Chebyshev spectral collocation is proposed for computing trustworthy approximate solutions of linear and weakly nonlinear delayed partial differential equations or initial boundary value problems, with continuous and bounded coefficients. The boundary conditions are assumed to be Dirichlet. The solution of linear problems is obtained at Chebyshev grid points in space and a given interval of time. The algorithm is then extended to systems with weak nonlinearities using perturbation series, which yields nonhomogeneous initial boundary value problems without delay. The proposed methodology is illustrated using examples of linear and weakly nonlinear heat and wave equations with bounded continuous space-time varying coefficients.

Sign in / Sign up

Export Citation Format

Share Document