Control Design and Robustness Analysis of Linear Time-Periodic Systems

2008 ◽  
Vol 3 (4) ◽  
Author(s):  
S. Kalender ◽  
H. Flashner
Keyword(s):  
Discrete Time ◽  
Linear Time ◽  
Robustness Analysis ◽  
Control Design ◽  
Mapping Technique ◽  
Parametric Uncertainties ◽  
Time Varying ◽  
Sampled Data ◽  
Time Dynamics ◽  
Robust Stability Analysis

This paper proposes a new design approach for control of periodically time-varying systems. The approach is based on the point-mapping technique to obtain an equivalent linear time-invariant sampled-data system for the linear periodically time-varying system with a piecewise parametrization of the control vector. This allows the known control design techniques for sampled-data systems to be applied. The proposed approach is then extended for analysis of robustness of the control design with respect to plant parametric uncertainties. This is achieved by computation of approximate discrete-time dynamics of the perturbed system by truncated point-mappings. By computing an upper norm bound on the error due to the truncated approximations, the robustness analysis of the system with respect to the parametric uncertainties is then formulated as a discrete-time structured singular value problem. Two numerical examples are considered to illustrate the approach and the extension of the approach for robust stability analysis.

A New Approach for Control of Linear Periodically Time Varying Systems

2006 ◽  
Author(s):  
S. Kalender ◽  
H. Flashner
Keyword(s):  
Inverted Pendulum ◽  
Linear Time ◽  
Control Design ◽  
Mapping Technique ◽  
Time Varying ◽  
Sampled Data ◽  
Periodic Forcing ◽  
Point Mapping ◽  
Robust Stability Analysis ◽  
Time Varying Systems

This paper proposes a new design approach for periodically time-varying systems. The approach is based on the application of point-mapping technique to obtain an equivalent linear time invariant sampled data system for the linear periodically time varying system with parameterized control vector, thus allowing the known control design techniques for sampled data systems to be applied. An extension of the approach for analysis of robustness of the control design with respect to parametric uncertainties is also presented. The extension is based on retaining linear as well as higher order terms in the uncertain parameters in the point-mapping algorithm. To illustrate the approach, stabilization of the trivial equilibrium point for a mathematical model representing the dynamics for a double inverted pendulum with periodic forcing is considered. The effectiveness of the extension of the approach for robust stability analysis is demonstrated also by considering uncertain periodic forcing parameters for the double inverted pendulum example.

Discrete-Time Robust Control of Linear Periodically Time-Varying Systems

2007 ◽  
Author(s):  
S. Kalender ◽  
H. Flashner
Keyword(s):  
Robust Control ◽  
Discrete Time ◽  
Robustness Analysis ◽  
Value Theory ◽  
Time Varying ◽  
Sampled Data ◽  
Point Mapping ◽  
Time Dynamic ◽  
Time Representation ◽  
Time Varying Systems

An approach for robust control of periodically time-varying systems is proposed. The approach combines the point-mapping formulation and a parameterization of the control vector to formulate an equivalent time-invariant discrete-time representation of the system. The discrete-time representation of the dynamic system allows for the application of known sampled-data control design methodologies. A perturbed, discrete-time dynamic model is formulated and plant parametric uncertainty are obtained using a truncated point-mapping algorithm. The error bounds due to point-mapping approximation are computed and a robustness analysis problem of the system due to parametric uncertainties is formulated using structured singular value theory. The proposed approach is illustrated by two design examples. Simulation studies show good performance robustness of the control system to parameter perturbations and system nonlinearities.

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.

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