2,789 research outputs found
Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators
We present an estimator-based control design procedure for flow control,
using reduced-order models of the governing equations, linearized about a
possibly unstable steady state. The reduced models are obtained using an
approximate balanced truncation method that retains the most controllable and
observable modes of the system. The original method is valid only for stable
linear systems, and we present an extension to unstable linear systems. The
dynamics on the unstable subspace are represented by projecting the original
equations onto the global unstable eigenmodes, assumed to be small in number. A
snapshot-based algorithm is developed, using approximate balanced truncation,
for obtaining a reduced-order model of the dynamics on the stable subspace. The
proposed algorithm is used to study feedback control of 2-D flow over a flat
plate at a low Reynolds number and at large angles of attack, where the natural
flow is vortex shedding, though there also exists an unstable steady state. For
control design, we derive reduced-order models valid in the neighborhood of
this unstable steady state. The actuation is modeled as a localized body force
near the leading edge of the flat plate, and the sensors are two velocity
measurements in the near-wake of the plate. A reduced-order Kalman filter is
developed based on these models and is shown to accurately reconstruct the flow
field from the sensor measurements, and the resulting estimator-based control
is shown to stabilize the unstable steady state. For small perturbations of the
steady state, the model accurately predicts the response of the full
simulation. Furthermore, the resulting controller is even able to suppress the
stable periodic vortex shedding, where the nonlinear effects are strong, thus
implying a large domain of attraction of the stabilized steady state.Comment: 36 pages, 17 figure
Necessary conditions involving Lie brackets for impulsive optimal control problems
We obtain higher order necessary conditions for a minimum of a Mayer optimal
control problem connected with a nonlinear, control-affine system, where the
controls range on an m-dimensional Euclidean space. Since the allowed
velocities are unbounded and the absence of coercivity assumptions makes big
speeds quite likely, minimizing sequences happen to converge toward
"impulsive", namely discontinuous, trajectories. As is known, a distributional
approach does not make sense in such a nonlinear setting, where instead a
suitable embedding in the graph space is needed. We will illustrate how the
chance of using impulse perturbations makes it possible to derive a Higher
Order Maximum Principle which includes both the usual needle variations (in
space-time) and conditions involving iterated Lie brackets. An example, where a
third order necessary condition rules out the optimality of a given extremal,
concludes the paper.Comment: Conference pape
General existence and uniqueness of viscosity solutions for impulse control of jump-diffusions
General theorems for existence and uniqueness of viscosity solutions for
Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVI) with integral
term are established. Such nonlinear partial integro-differential equations
(PIDE) arise in the study of combined impulse and stochastic control for
jump-diffusion processes. The HJBQVI consists of an HJB part (for stochastic
control) combined with a nonlocal impulse intervention term.
Existence results are proved via stochastic means, whereas our uniqueness
(comparison) results adapt techniques from viscosity solution theory. This
paper is to our knowledge the first treating rigorously impulse control for
jump-diffusion processes in a general viscosity solution framework; the jump
part may have infinite activity. In the proofs, no prior continuity of the
value function is assumed, quadratic costs are allowed, and elliptic and
parabolic results are presented for solutions possibly unbounded at infinity
limit solutions for control systems
For a control Cauchy problem on an interval , we
propose a notion of limit solution verifying the following properties: i)
is defined for (impulsive) inputs and for standard,
bounded measurable, controls ; ii) in the commutative case (i.e. when
for all ),
coincides with the solution one can obtain via the change of coordinates that
makes the simultaneously constant; iii) subsumes former concepts
of solution valid for the generic, noncommutative case.
In particular, when has bounded variation, we investigate the relation
between limit solutions and (single-valued) graph completion solutions.
Furthermore, we prove consistency with the classical Carath\'eodory solution
when and are absolutely continuous.
Even though some specific problems are better addressed by means of special
representations of the solutions, we believe that various theoretical issues
call for a unified notion of trajectory. For instance, this is the case of
optimal control problems, possibly with state and endpoint constraints, for
which no extra assumptions (like e.g. coercivity, bounded variation,
commutativity) are made in advance
Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations
An advantageous feature of piecewise constant policy timestepping for
Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation
schemes, and indeed different meshes, can be used for the resulting linear
equations for different control parameters. Standard convergence analysis
suggests that monotone (i.e., linear) interpolation must be used to transfer
data between meshes. Using the equivalence to a switching system and an
adaptation of the usual arguments based on consistency, stability and
monotonicity, we show that if limited, potentially higher order interpolation
is used for the mesh transfer, convergence is guaranteed. We provide numerical
tests for the mean-variance optimal investment problem and the uncertain
volatility option pricing model, and compare the results to published test
cases
Data compression for estimation of the physical parameters of stable and unstable linear systems
A two-stage method for the identification of physical system parameters from experimental data is presented. The first stage compresses the data as an empirical model which encapsulates the data content at frequencies of interest. The second stage then uses data extracted from the empirical model of the first stage within a nonlinear estimation scheme to estimate the unknown physical parameters. Furthermore, the paper proposes use of exponential data weighting in the identification of partially unknown, unstable systems so that they can be treated in the same framework as stable systems. Experimental data are used to demonstrate the efficacy of the proposed approach
Numerical methods for an optimal order execution problem
This paper deals with numerical solutions to an impulse control problem
arising from optimal portfolio liquidation with bid-ask spread and market price
impact penalizing speedy execution trades. The corresponding dynamic
programming (DP) equation is a quasi-variational inequality (QVI) with solvency
constraint satisfied by the value function in the sense of constrained
viscosity solutions. By taking advantage of the lag variable tracking the time
interval between trades, we can provide an explicit backward numerical scheme
for the time discretization of the DPQVI. The convergence of this discrete-time
scheme is shown by viscosity solutions arguments. An optimal quantization
method is used for computing the (conditional) expectations arising in this
scheme. Numerical results are presented by examining the behaviour of optimal
liquidation strategies, and comparative performance analysis with respect to
some benchmark execution strategies. We also illustrate our optimal liquidation
algorithm on real data, and observe various interesting patterns of order
execution strategies. Finally, we provide some numerical tests of sensitivity
with respect to the bid/ask spread and market impact parameters
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