scholarly journals Optimal Preview Control for Linear Discrete-Time Periodic Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Fucheng Liao ◽  
Mengyuan Sun ◽  
Usman
Keyword(s):  
Discrete Time ◽  
Original Problem ◽  
Control Method ◽  
Original System ◽  
Periodic Systems ◽  
Invariant System ◽  
Preview Control ◽  
Time Invariant ◽  
Time Periodic ◽  
Time Invariant System

In this paper, the optimal preview tracking control problem for a class of linear discrete-time periodic systems is investigated and the method to design the optimal preview controller for such systems is given. Initially, by fully considering the characteristic that the coefficient matrices are periodic functions, the system can be converted into a time-invariant system through lifting method. Then, the original problem is also transformed into the scenario of time-invariant system. Later on, the augmented system is constructed and the preview controller of the original system is obtained with the help of existing preview control method. The controller comprises integrator, state feedback, and preview feedforward. Finally, the simulation example shows the effectiveness of the proposed preview controller in improving the tracking performance of the close-loop system.

On The Response of Linear Time-Periodic Systems Subjected to Deterministic and Stochastic Excitations

1996 ◽  
Vol 2 (2) ◽  
pp. 219-249 ◽  
Author(s):  
J.M. Spires ◽  
S.C. Sinha
Keyword(s):  
Chebyshev Polynomial ◽  
Linear Time ◽  
Original System ◽  
Invariant Form ◽  
Periodic Systems ◽  
Solution Technique ◽  
Stiffness Matrices ◽  
Original Time ◽  
Time Invariant ◽  
Time Periodic

In many situations, engineering systems modeled by a set of linear, second-order differential equations, with periodic damping and stiffness matrices, are subjected to external excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it is shown that the Liapunov-Floquet transformation matrix associated with the system can be computed, and the original time-periodic system can be put into a time- invariant form. In this paper, these techniques are applied in finding the transient response of periodic systems subjected to deterministic and stochastic forces. Two formulations are presented. In the first formulation, the response of the original system is computed directly. In the second formulation, first the original system is transformed to a time-invariant form, and then the response is found by determining the response of the time-invariant system. Both formulations use the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through a Chebyshev polynomial expansion technique. Results for some time-invariant and periodic systems are included, as illustrative examples.

Lyapunov Perron Transformation for Quasi-Periodic Systems and its Applications

2021 ◽  
pp. 1-26
Author(s):  
Susheelkumar Cherangara Subramanian ◽  
Sangram Redkar
Keyword(s):  
Periodic System ◽  
Normal Forms ◽  
Identity Transformation ◽  
Periodic Systems ◽  
Invariant System ◽  
External Excitation ◽  
P Transformation ◽  
State Augmentation ◽  
Time Invariant ◽  
Time Invariant System

Abstract This paper depicts the application of symbolically computed Lyapunov Perron (L-P) Transformation to solve linear and nonlinear quasi-periodic systems. The L-P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of Normal Forms are used to compute the L-P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time invariant system as the quasi-periodic parametric excitation terms are replaced by ‘fictitious’ states. This nonlinear system can be reduced to a linear system via Normal Forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L-P transformation. The L-P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu-Hill type quasi-periodic system. The results obtained via the L-P transformation approach match very well with the numerical integration and analytical results.

Response of Linear Time-Periodic Systems Subjected to Stochastic Excitations: A Chebyshev Polynomial Approach

1995 ◽  
Author(s):  
J. M. Spires ◽  
S. C. Sinha
Keyword(s):  
Chebyshev Polynomial ◽  
Original System ◽  
Invariant Form ◽  
Periodic Systems ◽  
Solution Technique ◽  
Mean Square ◽  
Stochastic Excitations ◽  
Mean Square Response ◽  
Time Invariant ◽  
Time Periodic

Abstract In many situations engineering systems modeled by a system of linear second order differential equations with periodic damping and stiffness matrices are subjected to stochastic excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it has been shown that the Liapunov-Floquet transformation matrix can be computed, and the original time-periodic system can be put into a time invariant form. In this paper, these techniques are applied in finding the transient mean square response and transient autocorrelation response of periodic systems subjected to stochastic forcing vectors. Two formulations are presented. In the first formulation, the mean square response of the original system is computed directly. In the second formulation, the original system is transformed to a time-invariant form. The autocorrelation response is found by determining the response of the time-invariant system. Both formulations utilize the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through Chebyshev polynomial expansion. Results for some time-invariant and periodic systems are included, as illustrative examples.

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