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.
Abstract
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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