940 research outputs found
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
Distributed Robust Control for Systems with Structured Uncertainties
We present D-Phi iteration: an algorithm for distributed, localized, and
scalable robust control of systems with structured uncertainties. This
algorithm combines the System Level Synthesis (SLS) parametrization for
distributed control with stability criteria from L1, L-infinity, and nu robust
control. We show in simulation that this algorithm achieves near-optimal
nominal performance (within 12% of the LQR controller) while doubling or
tripling the stability margin (depending on the stability criterion) compared
to the LQR controller. To the best of our knowledge, this is the first
distributed and localized algorithm for structured robust control; furthermore,
algorithm complexity depends only on the size of local neighborhoods and is
independent of global system size. We additionally characterize the suitability
of different robustness criteria for distributed and localized computation, and
discuss open questions on the topic of distributed robust control.Comment: Submitted to CDC 202
Multiscale differential Riccati equations for linear quadratic regulator problems
We consider approximations to the solutions of differential Riccati equations
in the context of linear quadratic regulator problems, where the state equation
is governed by a multiscale operator. Similarly to elliptic and parabolic
problems, standard finite element discretizations perform poorly in this
setting unless the grid resolves the fine-scale features of the problem. This
results in unfeasible amounts of computation and high memory requirements. In
this paper, we demonstrate how the localized orthogonal decomposition method
may be used to acquire accurate results also for coarse discretizations, at the
low cost of solving a series of small, localized elliptic problems. We prove
second-order convergence (except for a logarithmic factor) in the
operator norm, and first-order convergence in the corresponding energy norm.
These results are both independent of the multiscale variations in the state
equation. In addition, we provide a detailed derivation of the fully discrete
matrix-valued equations, and show how they can be handled in a low-rank setting
for large-scale computations. In connection to this, we also show how to
efficiently compute the relevant operator-norm errors. Finally, our theoretical
results are validated by several numerical experiments.Comment: Accepted for publication in SIAM J. Sci. Comput. This version differs
from the previous one only by the addition of Remark 7.2 and minor changes in
formatting. 21 pages, 12 figure
System Level Synthesis
This article surveys the System Level Synthesis framework, which presents a
novel perspective on constrained robust and optimal controller synthesis for
linear systems. We show how SLS shifts the controller synthesis task from the
design of a controller to the design of the entire closed loop system, and
highlight the benefits of this approach in terms of scalability and
transparency. We emphasize two particular applications of SLS, namely
large-scale distributed optimal control and robust control. In the case of
distributed control, we show how SLS allows for localized controllers to be
computed, extending robust and optimal control methods to large-scale systems
under practical and realistic assumptions. In the case of robust control, we
show how SLS allows for novel design methodologies that, for the first time,
quantify the degradation in performance of a robust controller due to model
uncertainty -- such transparency is key in allowing robust control methods to
interact, in a principled way, with modern techniques from machine learning and
statistical inference. Throughout, we emphasize practical and efficient
computational solutions, and demonstrate our methods on easy to understand case
studies.Comment: To appear in Annual Reviews in Contro
Distributed Design for Decentralized Control using Chordal Decomposition and ADMM
We propose a distributed design method for decentralized control by
exploiting the underlying sparsity properties of the problem. Our method is
based on chordal decomposition of sparse block matrices and the alternating
direction method of multipliers (ADMM). We first apply a classical
parameterization technique to restrict the optimal decentralized control into a
convex problem that inherits the sparsity pattern of the original problem. The
parameterization relies on a notion of strongly decentralized stabilization,
and sufficient conditions are discussed to guarantee this notion. Then, chordal
decomposition allows us to decompose the convex restriction into a problem with
partially coupled constraints, and the framework of ADMM enables us to solve
the decomposed problem in a distributed fashion. Consequently, the subsystems
only need to share their model data with their direct neighbours, not needing a
central computation. Numerical experiments demonstrate the effectiveness of the
proposed method.Comment: 11 pages, 8 figures. Accepted for publication in the IEEE
Transactions on Control of Network System
Nonnormality and the localized control of extended systems
The idea of controlling the dynamics of spatially extended systems using a
small number of localized perturbations is very appealing - such a setup is
easy to implement in practice. However, when the distance between controllers
generating the perturbations becomes large, control fails due to increasing
sensitivity of the system to noise and nonlinearities. We show that this
failure is due to the fact that the evolution operator for the controlled
system becomes increasingly nonnormal as the distance between controllers
grows. This nonnormality is the result of control and can arise even for
systems whose evolution operator is normal in the absence of control.Comment: 4 pages, 4 figure
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
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