3,864 research outputs found
Reduction of Second-Order Network Systems with Structure Preservation
This paper proposes a general framework for structure-preserving model
reduction of a secondorder network system based on graph clustering. In this
approach, vertex dynamics are captured by the transfer functions from inputs to
individual states, and the dissimilarities of vertices are quantified by the
H2-norms of the transfer function discrepancies. A greedy hierarchical
clustering algorithm is proposed to place those vertices with similar dynamics
into same clusters. Then, the reduced-order model is generated by the
Petrov-Galerkin method, where the projection is formed by the characteristic
matrix of the resulting network clustering. It is shown that the simplified
system preserves an interconnection structure, i.e., it can be again
interpreted as a second-order system evolving over a reduced graph.
Furthermore, this paper generalizes the definition of network controllability
Gramian to second-order network systems. Based on it, we develop an efficient
method to compute H2-norms and derive the approximation error between the
full-order and reduced-order models. Finally, the approach is illustrated by
the example of a small-world network
Optimal Control of Two-Player Systems with Output Feedback
In this article, we consider a fundamental decentralized optimal control
problem, which we call the two-player problem. Two subsystems are
interconnected in a nested information pattern, and output feedback controllers
must be designed for each subsystem. Several special cases of this architecture
have previously been solved, such as the state-feedback case or the case where
the dynamics of both systems are decoupled. In this paper, we present a
detailed solution to the general case. The structure of the optimal
decentralized controller is reminiscent of that of the optimal centralized
controller; each player must estimate the state of the system given their
available information and apply static control policies to these estimates to
compute the optimal controller. The previously solved cases benefit from a
separation between estimation and control which allows one to compute the
control and estimation gains separately. This feature is not present in
general, and some of the gains must be solved for simultaneously. We show that
computing the required coupled estimation and control gains amounts to solving
a small system of linear equations
Distributed Control with Low-Rank Coordination
A common approach to distributed control design is to impose sparsity
constraints on the controller structure. Such constraints, however, may greatly
complicate the control design procedure. This paper puts forward an alternative
structure, which is not sparse yet might nevertheless be well suited for
distributed control purposes. The structure appears as the optimal solution to
a class of coordination problems arising in multi-agent applications. The
controller comprises a diagonal (decentralized) part, complemented by a
rank-one coordination term. Although this term relies on information about all
subsystems, its implementation only requires a simple averaging operation
On Submodularity and Controllability in Complex Dynamical Networks
Controllability and observability have long been recognized as fundamental
structural properties of dynamical systems, but have recently seen renewed
interest in the context of large, complex networks of dynamical systems. A
basic problem is sensor and actuator placement: choose a subset from a finite
set of possible placements to optimize some real-valued controllability and
observability metrics of the network. Surprisingly little is known about the
structure of such combinatorial optimization problems. In this paper, we show
that several important classes of metrics based on the controllability and
observability Gramians have a strong structural property that allows for either
efficient global optimization or an approximation guarantee by using a simple
greedy heuristic for their maximization. In particular, the mapping from
possible placements to several scalar functions of the associated Gramian is
either a modular or submodular set function. The results are illustrated on
randomly generated systems and on a problem of power electronic actuator
placement in a model of the European power grid.Comment: Original arXiv version of IEEE Transactions on Control of Network
Systems paper (Volume 3, Issue 1), with a addendum (located in the ancillary
documents) that explains an error in a proof of the original paper and
provides a counterexample to the corresponding resul
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
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