5,294 research outputs found
Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function
Eigenvalue assignment problem of a linear scalar system with a single
discrete delay is analytically and exactly solved. The existence condition of
the desired eigenvalue is established when the current and delay states are
present in the feedback loop. Design of the feedback controller is then
followed. Furthermore, eigenvalue assignment for the input-delay system is also
obtained as well. Numerical examples illustrate the procedure of assigning the
desired eigenvalue
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations
Several recent methods used to analyze asymptotic stability of
delay-differential equations (DDEs) involve determining the eigenvalues of a
matrix, a matrix pencil or a matrix polynomial constructed by Kronecker
products. Despite some similarities between the different types of these
so-called matrix pencil methods, the general ideas used as well as the proofs
differ considerably. Moreover, the available theory hardly reveals the
relations between the different methods.
In this work, a different derivation of various matrix pencil methods is
presented using a unifying framework of a new type of eigenvalue problem: the
polynomial two-parameter eigenvalue problem, of which the quadratic
two-parameter eigenvalue problem is a special case. This framework makes it
possible to establish relations between various seemingly different methods and
provides further insight in the theory of matrix pencil methods.
We also recognize a few new matrix pencil variants to determine DDE
stability.
Finally, the recognition of the new types of eigenvalue problem opens a door
to efficient computation of DDE stability
Dynamic Multi-Vehicle Routing with Multiple Classes of Demands
In this paper we study a dynamic vehicle routing problem in which there are
multiple vehicles and multiple classes of demands. Demands of each class arrive
in the environment randomly over time and require a random amount of on-site
service that is characteristic of the class. To service a demand, one of the
vehicles must travel to the demand location and remain there for the required
on-site service time. The quality of service provided to each class is given by
the expected delay between the arrival of a demand in the class, and that
demand's service completion. The goal is to design a routing policy for the
service vehicles which minimizes a convex combination of the delays for each
class. First, we provide a lower bound on the achievable values of the convex
combination of delays. Then, we propose a novel routing policy and analyze its
performance under heavy load conditions (i.e., when the fraction of time the
service vehicles spend performing on-site service approaches one). The policy
performs within a constant factor of the lower bound (and thus the optimal),
where the constant depends only on the number of classes, and is independent of
the number of vehicles, the arrival rates of demands, the on-site service
times, and the convex combination coefficients.Comment: Extended version of paper presented in American Control Conference
200
Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations
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
Design of a Wide Area Controller Using Eigenstructure Assignment in Power Systems
Small signal stability has become a major concern for power system operators around the world. This has resulted from constantly evolving changes in the power system ranging from increased number of interconnections to ever increasing demand of power. In highly stressed operating conditions, even a small disturbance such as a load change can make the system unstable resulting in small signal instability. The main reason for small signal instability in power systems is an inter-area mode/s becoming unstable. Inter-area modes involve a group of generators oscillating against each other. Traditionally, power system stabilizers installed on the synchrous machines were used to damp the inter-area modes. However, they may not be very suitable to perform the job since they use local I/O signals which do not have a good controllability/observability of the inter-area modes. Recent advancements in phasor measurement technology has resulted in fast acquisition of time-synchronized measurements throughout the system. Thus, instead of using local controllers, an idea of a wide area controller (WAC) was proposed by the power systems community that would use global signals. This dissertation demonstrates the design of a WAC using eigenstructure assignment technique. This technique provides the freedom to assign a few eigenvalues and corresponding left or right eigenvectors for Multi-Input-Multi-Output (MIMO) systems. This technique forms a good match for designing a WAC since a WAC usually uses multiple I/O signals and a power system only has a few inter-area modes that might lead to instability. The last chapter of this dissertation addresses an important aspect of controller design, i.e., robustness of the controller to uncertainties in operating point and time delay of feedback signals. The operating point of a power system is highly variable in nature and thus the designed WAC should be able to damp the inter-area modes under these variations. Also, a transmission delay is associated due to routing of remote signals. This time delay is known to deteriorate the performance of the controller. A single controller will be shown to achieve robustness against both these uncertainties
Information flow and cooperative control of vehicle formations
We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability
Towards Optimal Distributed Node Scheduling in a Multihop Wireless Network through Local Voting
In a multihop wireless network, it is crucial but challenging to schedule
transmissions in an efficient and fair manner. In this paper, a novel
distributed node scheduling algorithm, called Local Voting, is proposed. This
algorithm tries to semi-equalize the load (defined as the ratio of the queue
length over the number of allocated slots) through slot reallocation based on
local information exchange. The algorithm stems from the finding that the
shortest delivery time or delay is obtained when the load is semi-equalized
throughout the network. In addition, we prove that, with Local Voting, the
network system converges asymptotically towards the optimal scheduling.
Moreover, through extensive simulations, the performance of Local Voting is
further investigated in comparison with several representative scheduling
algorithms from the literature. Simulation results show that the proposed
algorithm achieves better performance than the other distributed algorithms in
terms of average delay, maximum delay, and fairness. Despite being distributed,
the performance of Local Voting is also found to be very close to a centralized
algorithm that is deemed to have the optimal performance
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