88 research outputs found
An Optimal Controller Architecture for Poset-Causal Systems
We propose a novel and natural architecture for decentralized control that is
applicable whenever the underlying system has the structure of a partially
ordered set (poset). This controller architecture is based on the concept of
Moebius inversion for posets, and enjoys simple and appealing separation
properties, since the closed-loop dynamics can be analyzed in terms of
decoupled subsystems. The controller structure provides rich and interesting
connections between concepts from order theory such as Moebius inversion and
control-theoretic concepts such as state prediction, correction, and
separability. In addition, using our earlier results on H_2-optimal
decentralized control for arbitrary posets, we prove that the H_2-optimal
controller in fact possesses the proposed structure, thereby establishing the
optimality of the new controller architecture.Comment: 32 pages, 9 figures, submitted to IEEE Transactions on Automatic
Contro
H_2-Optimal Decentralized Control over Posets: A State-Space Solution for State-Feedback
We develop a complete state-space solution to H_2-optimal decentralized
control of poset-causal systems with state-feedback. Our solution is based on
the exploitation of a key separability property of the problem, that enables an
efficient computation of the optimal controller by solving a small number of
uncoupled standard Riccati equations. Our approach gives important insight into
the structure of optimal controllers, such as controller degree bounds that
depend on the structure of the poset. A novel element in our state-space
characterization of the controller is a remarkable pair of transfer functions,
that belong to the incidence algebra of the poset, are inverses of each other,
and are intimately related to prediction of the state along the different paths
on the poset. The results are illustrated by a numerical example.Comment: 39 pages, 2 figures, submitted to IEEE Transactions on Automatic
Contro
A partial order approach to decentralized control
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 173-177).In this thesis we consider the problem of decentralized control of linear systems. We employ the theory of partially ordered sets (posets) to model and analyze a class of decentralized control problems. Posets have attractive combinatorial and algebraic properties; the combinatorial structure enables us to model a rich class of communication structures in systems, and the algebraic structure allows us to reparametrize optimal control problems to convex problems. Building on this approach, we develop a state-space solution to the problem of designing Hâ‚‚-optimal controllers. Our solution is based on the exploitation of a key separability property of the problem that enables an efficient computation of the optimal controller by solving a small number of uncoupled standard Riccati equations. Our approach gives important insight into the structure of optimal controllers, such as controller degree bounds that depend on the structure of the poset. A novel element in our state-space characterization of the controller is a pair of transfer functions, that belong to the incidence algebra of the poset, are inverses of each other, and are intimately related to estimation of the state along the different paths in the poset. We then view the control design problem from an architectural viewpoint. We propose a natural architecture for poset-causal controllers. In the process, we establish interesting connections between concepts from order theory such as Mobius inversion and control-theoretic concepts such as state estimation, innovation, and separability principles. Finally, we prove that the Hâ‚‚-optimal controller in fact posseses the proposed controller structure, thereby proving the optimality of the architecture.by Parikshit Shah.Ph.D
Stabilization of Networked Control Systems with Sparse Observer-Controller Networks
In this paper we provide a set of stability conditions for linear
time-invariant networked control systems with arbitrary topology, using a
Lyapunov direct approach. We then use these stability conditions to provide a
novel low-complexity algorithm for the design of a sparse observer-based
control network. We employ distributed observers by employing the output of
other nodes to improve the stability of each observer dynamics. To avoid
unbounded growth of controller and observer gains, we impose bounds on their
norms. The effects of relaxation of these bounds is discussed when trying to
find the complete decentralization conditions
Localized LQR Optimal Control
This paper introduces a receding horizon like control scheme for localizable
distributed systems, in which the effect of each local disturbance is limited
spatially and temporally. We characterize such systems by a set of linear
equality constraints, and show that the resulting feasibility test can be
solved in a localized and distributed way. We also show that the solution of
the local feasibility tests can be used to synthesize a receding horizon like
controller that achieves the desired closed loop response in a localized manner
as well. Finally, we formulate the Localized LQR (LLQR) optimal control problem
and derive an analytic solution for the optimal controller. Through a numerical
example, we show that the LLQR optimal controller, with its constraints on
locality, settling time, and communication delay, can achieve similar
performance as an unconstrained H2 optimal controller, but can be designed and
implemented in a localized and distributed way.Comment: Extended version for 2014 CDC submissio
Optimal Output Feedback Architecture for Triangular LQG Problems
Distributed control problems under some specific information constraints can
be formulated as (possibly infinite dimensional) convex optimization problems.
The underlying motivation of this work is to develop an understanding of the
optimal decision making architecture for such problems. In this paper, we
particularly focus on the N-player triangular LQG problems and show that the
optimal output feedback controllers have attractive state space realizations.
The optimal controller can be synthesized using a set of stabilizing solutions
to 2N linearly coupled algebraic Riccati equations, which turn out to be easily
solvable under reasonable assumptions.Comment: To be presented at 2014 American Control Conferenc
Fast M\"obius and Zeta Transforms
M\"obius inversion of functions on partially ordered sets (posets)
is a classical tool in combinatorics. For finite posets it
consists of two, mutually inverse, linear transformations called zeta and
M\"obius transform, respectively. In this paper we provide novel fast
algorithms for both that require time and space, where and is the width (length of longest antichain) of
, compared to for a direct computation. Our approach
assumes that is given as directed acyclic graph (DAG)
. The algorithms are then constructed using a chain
decomposition for a one time cost of , where is the number of
edges in the DAG's transitive reduction. We show benchmarks with
implementations of all algorithms including parallelized versions. The results
show that our algorithms enable M\"obius inversion on posets with millions of
nodes in seconds if the defining DAGs are sufficiently sparse.Comment: 16 pages, 7 figures, submitted for revie
Optimal Control for LQG Systems on Graphs---Part I: Structural Results
In this two-part paper, we identify a broad class of decentralized
output-feedback LQG systems for which the optimal control strategies have a
simple intuitive estimation structure and can be computed efficiently. Roughly,
we consider the class of systems for which the coupling of dynamics among
subsystems and the inter-controller communication is characterized by the same
directed graph. Furthermore, this graph is assumed to be a multitree, that is,
its transitive reduction can have at most one directed path connecting each
pair of nodes. In this first part, we derive sufficient statistics that may be
used to aggregate each controller's growing available information. Each
controller must estimate the states of the subsystems that it affects (its
descendants) as well as the subsystems that it observes (its ancestors). The
optimal control action for a controller is a linear function of the estimate it
computes as well as the estimates computed by all of its ancestors. Moreover,
these state estimates may be updated recursively, much like a Kalman filter
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