2,153 research outputs found
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
Optimal control of multiscale systems using reduced-order models
We study optimal control of diffusions with slow and fast variables and
address a question raised by practitioners: is it possible to first eliminate
the fast variables before solving the optimal control problem and then use the
optimal control computed from the reduced-order model to control the original,
high-dimensional system? The strategy "first reduce, then optimize"--rather
than "first optimize, then reduce"--is motivated by the fact that solving
optimal control problems for high-dimensional multiscale systems is numerically
challenging and often computationally prohibitive. We state sufficient and
necessary conditions, under which the "first reduce, then control" strategy can
be employed and discuss when it should be avoided. We further give numerical
examples that illustrate the "first reduce, then optmize" approach and discuss
possible pitfalls
Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors
Landscape is one of the key notions in literature on biological processes and
physics of complex systems with both deterministic and stochastic dynamics. The
large deviation theory (LDT) provides a possible mathematical basis for the
scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss
explicitly two issues in singularly perturbed stationary diffusion processes
arisen from nonlinear differential equations: (1) For a process whose
corresponding ordinary differential equation has a stable limit cycle, the
stationary solution exhibits a clear separation of time scales: an exponential
terms and an algebraic prefactor. The large deviation rate function attains its
minimum zero on the entire stable limit cycle, while the leading term of the
prefactor is inversely proportional to the velocity of the non-uniform periodic
oscillation on the cycle. (2) For dynamics with multiple stable fixed points
and saddles, there is in general a breakdown of detailed balance among the
corresponding attractors. Two landscapes, a local and a global, arise in LDT,
and a Markov jumping process with cycle flux emerges in the low-noise limit. A
local landscape is pertinent to the transition rates between neighboring stable
fixed points; and the global landscape defines a nonequilibrium steady state.
There would be nondifferentiable points in the latter for a stationary dynamics
with cycle flux. LDT serving as the mathematical foundation for emergent
landscapes deserves further investigations.Comment: 4 figur
Rapid near-optimal trajectory generation and guidance law development for single-stage-to-orbit airbreathing vehicles
General problems associated with on-board trajectory optimization, propulsion system cycle selection, and with the synthesis of guidance laws were addressed for an ascent to low-earth-orbit of an air-breathing single-stage-to-orbit vehicle. The NASA Generic Hypersonic Aerodynamic Model Example and the Langley Accelerator aerodynamic sets were acquired and implemented. Work related to the development of purely analytic aerodynamic models was also performed at a low level. A generic model of a multi-mode propulsion system was developed that includes turbojet, ramjet, scramjet, and rocket engine cycles. Provisions were made in the dynamic model for a component of thrust normal to the flight path. Computational results, which characterize the nonlinear sensitivity of scramjet performance to changes in vehicle angle of attack, were obtained and incorporated into the engine model. Additional trajectory constraints were introduced: maximum dynamic pressure; maximum aerodynamic heating rate per unit area; angle of attack and lift limits; and limits on acceleration both along and normal to the flight path. The remainder of the effort focused on required modifications to a previously derived algorithm when the model complexity cited above was added. In particular, analytic switching conditions were derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another for two cases: the case in which engine cycle operations can overlap, and the case in which engine cycle operations are mutually exclusive. The resulting guidance algorithm was implemented in software and exercised extensively. It was found that the approximations associated with the assumed time scale separation employed in this work are reasonable except over the Mach range from roughly 5 to 8. This phenomenon is due to the very large thrust capability of scramjets in this Mach regime when sized to meet the requirement for ascent to orbit. By accounting for flight path angle and flight path angle rate in construction of the flight path over this Mach range, the resulting algorithm provides the means for rapid near-optimal trajectory generation and propulsion cycle selection over the entire Mach range from take-off to orbit
Rapid near-optimal aerospace plane trajectory generation and guidance
Effort was directed toward the problems of the real time trajectory optimization and guidance law development for the National Aerospace Plane (NASP) applications. In particular, singular perturbation methods were used to develop guidance algorithms suitable for onboard, real time implementation. The progress made in this research effort is reported
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