162 research outputs found
On the Hybrid Minimum Principle On Lie Groups and the Exponential Gradient HMP Algorithm
This paper provides a geometrical derivation of the Hybrid Minimum Principle
(HMP) for autonomous hybrid systems whose state manifolds constitute Lie groups
which are left invariant under the controlled dynamics of the
system, and whose switching manifolds are defined as smooth embedded time
invariant submanifolds of . The analysis is expressed in terms of extremal
(i.e. optimal) trajectories on the cotangent bundle of the state manifold .
The Hybrid Maximum Principle (HMP) algorithm introduced in \cite{Shaikh} is
extended to the so-called Exponential Gradient algorithm. The convergence
analysis for the algorithm is based upon the LaSalle Invariance Principle and
simulation results illustrate their efficacy
Graphon Control of Large-scale Networks of Linear Systems
To achieve control objectives for extremely large-scale complex networks
using standard methods is essentially intractable. In this work a theory of the
approximate control of complex network systems is proposed and developed by the
use of graphon theory and the theory of infinite dimensional systems. First,
graphon dynamical system models are formulated in an appropriate infinite
dimensional space in order to represent arbitrary-size networks of linear
dynamical systems, and to define the convergence of sequences of network
systems with limits in the space. Exact controllability and approximate
controllability of graphon dynamical systems are then investigated. Second, the
minimum energy state-to-state control problem and the linear quadratic
regulator problem for systems on complex networks are considered. The control
problem for graphon limit systems is solved in each case and approximations are
defined which yield control laws for the original control problems.
Furthermore, convergence properties of the approximation schemes are
established. A systematic control design methodology is developed within this
framework. Finally, numerical examples of networks with randomly sampled
weightings are presented to illustrate the effectiveness of the graphon control
methodology.Comment: 16 pages. To appear in the IEEE Transactions on Automatic Control,
2020. Originally announced July 201
Spectral Representations of Graphons in Very Large Network Systems Control
Graphon-based control has recently been proposed and developed to solve
control problems for dynamical systems on networks which are very large or
growing without bound (see Gao and Caines, CDC 2017, CDC 2018). In this paper,
spectral representations, eigenfunctions and approximations of graphons, and
their applications to graphon-based control are studied. First, spectral
properties of graphons are presented and then approximations based on Fourier
approximated eigenfunctions are analyzed. Within this framework, two classes of
graphons with simple spectral representations are given. Applications to
graphon-based control analysis are next presented; in particular, the
controllability of systems distributed over very large networks is expressed in
terms of the properties of the corresponding graphon dynamical systems.
Moreover, spectral analysis based upon real-world network data is presented,
which demonstrates that low-dimensional spectral approximations of networks are
possible. Finally, an initial, exploratory investigation of the utility of the
spectral analysis methodology in graphon systems control to study the control
of epidemic spread is presented.Comment: 8 pages, 58th IEEE Conference on Decision and Control (CDC 2019
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