Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations

Kandala, Shanti Swaroop and Vyasarayani, C P (2019) Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations. PhD thesis, Indian Institute of Technology Hyderabad.


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The dynamics of time-delayed systems (TDS) are governed by delay differential equa- tions (DDEs), which are infinite dimensional and pose computational challenges. The Galerkin approximation method is one of several techniques to obtain the spectrum of DDEs for stability and stabilization studies. In the literature, Galerkin approximations for DDEs have primarily dealt with second-order TDS (second-order Galerkin method), and the for- mulations have resulted in spurious roots, i.e., roots that are not among the characteristic roots of the DDE. Although these spurious roots do not affect stability studies, they never- theless add to the complexity and computation time for control and reduced-order modelling studies of DDEs. A refined mathematical model, called the first-order Galerkin method, is proposed to avoid spurious roots, and the subtle differences between the two formulations (second-order and first-order Galerkin methods) are highlighted with examples. For embedding the boundary conditions in the first-order Galerkin method, a new pseudoinverse-based technique is developed. This method not only gives the exact location of the rightmost root but also, on average, has a higher number of converged roots when compared to the existing pseudospectral differencing method. The proposed method is combined with an optimization framework to develop a pole-placement technique for DDEs to design closed-loop feedback gains that stabilize TDS. A rotary inverted pendulum system apparatus with inherent sensing delays as well as deliberately introduced time delays is used to experimentally validate the Galerkin approximation-based optimization framework for the pole placement of DDEs. Optimization-based techniques cannot always place the rightmost root at the desired location; also, one has no control over the placement of the next set of rightmost roots. However, one has the precise location of the rightmost root. To overcome this, a pole- placement technique for second-order TDS is proposed, which combines the strengths of the method of receptances and an optimization-based strategy. When the method of receptances provides an unsatisfactory solution, particle swarm optimization is used to improve the location of the rightmost pole. The proposed approach is demonstrated with numerical studies and is validated experimentally using a 3D hovercraft apparatus. The Galerkin approximation method contains both converged and unconverged roots of the DDE. By using only the information about the converged roots and applying the eigenvalue decomposition, one obtains an r-dimensional reduced-order model (ROM) of the DDE. To analyze the dynamics of DDEs, we first choose an appropriate value for r; we then select the minimum value of the order of the Galerkin approximation method system at which at least r roots converge. By judiciously selecting r, solutions of the ROM and the original DDE are found to match closely. Finally, an r-dimensional ROM of a 3D hovercraft apparatus in the presence of delay is validated experimentally.

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IITH Creators:
IITH CreatorsORCiD
Vyasarayani, Chandrika Prakashhttp://orcid.org/0000-0002-3396-0484
Item Type: Thesis (PhD)
Uncontrolled Keywords: Role placement, Galterkin approximations, Reduced order modelling, Method of receptances, TD1585
Subjects: Physics > Mechanical and aerospace
Divisions: Department of Mechanical & Aerospace Engineering
Depositing User: Team Library
Date Deposited: 31 Dec 2019 09:07
Last Modified: 04 Mar 2022 07:03
URI: http://raiith.iith.ac.in/id/eprint/7284
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