Reducing the Prediction Horizon in NMPC: An Algorithm Based Approach

In order to guarantee stability, known results for MPC without additional terminal costs or endpoint constraints often require rather large prediction horizons. Still, stable behavior of closed loop solutions can often be observed even for shorter horizons. Here, we make use of the recent observation that stability can be guaranteed for smaller prediction horizons via Lyapunov arguments if more than only the first control is implemented. Since such a procedure may be harmful in terms of robustness, we derive conditions which allow to increase the rate at which state measurements are used for feedback while maintaining stability and desired performance specifications. Our main contribution consists in developing two algorithms based on the deduced conditions and a corresponding stability theorem which ensures asymptotic stability for the MPC closed loop for significantly shorter prediction horizons.Comment: 6 pages, 3 figure

Analysis of unconstrained nonlinear MPC schemes with time varying control horizon

For discrete time nonlinear systems satisfying an exponential or finite time controllability assumption, we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints. Based on our formula, we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop

Theory and Application of Autoproducts

Acoustics is a branch of physics largely governed by linear field equations. Linearity carries with it the implication that only the frequencies broadcast by acoustic sources can be measured in the surrounding acoustic medium. However, nonlinearities introduced not in the physical world, but in the mathematical and signal processing realm, have the potential to change frequency content. In this dissertation, nonlinear mathematical constructions termed ‘autoproducts’ are created which have the potential to shift frequencies from the measured, in-band frequencies to other higher or lower frequencies which may no longer be in-band. These out-of-band autoproduct fields did not physically propagate in the environment, and yet, this research has found that autoproducts can nonetheless mimic genuine out-of-band fields in a number of different acoustic environments. Approximately half of this dissertation addresses the theory of autoproducts. More specifically, mathematical analyses and simple acoustic models are used to uncover the reasons for how this frequency-shifting behavior works, and what its limitations are. It is found that there are no inherent limitations on the frequencies considered, and that in single-path environments, like plane or spherical waves, autoproducts mimic out-of-band fields in all or nearly all circumstances, respectively. However, in multipath environments, the mimicry of out-of-band fields by autoproducts is no longer so complete. Though, with bandwidth averaging techniques, it is found that the difference in time-of-arrivals of multiple paths is an important parameter: if it is larger than the inverse of the bandwidth available for averaging, then autoproducts can succeed in mimicking out-of-band fields. Other theoretical considerations include the effects of diffraction behind barriers and the effects of strong refraction. Strengths and limitations of autoproducts are assessed with a variety of simple acoustic models, and conclusions are drawn as to the predicted capabilities of autoproduct-based techniques. The other half of this dissertation covers applications of autoproducts. More specifically, it focuses on the use of autoproducts to perform physics-based source localization, especially for applications in the shallow ocean. Existing techniques are well-known to be very sensitive to uncertainties in the acoustic environment (e.g. the sound speed), especially at high frequencies (nominally greater than 1 kHz in the shallow ocean). Through the use of autoproducts, measured fields at high frequency can be shifted to much lower frequencies, where they can be processed with much more robustness to environmental uncertainties. In one of the main results of this dissertation, it is shown that a remote acoustic source broadcasting sound between 11 and 33 kHz in a 106-meter-deep, downward refracting sound channel could be localized using measurements from a sparse array located 3 km away. The data from the method suggest that autoproduct-based source localization can make physics-based array signal processing robust at arbitrarily high frequencies – a novel and important contribution to existing literature. Overall, by developing the theory for, and exploring applications of, these nonlinear mathematical constructions, the extent to which autoproducts are fundamentally limited is assessed, and new signal processing techniques are developed which have the potential to significantly improve the robustness of source localization algorithms for uncertain multipath environments. Through this study, significant portions of the necessary theoretical foundation have been laid, which will aid in the further development of robust, autoproduct-based signal processing techniques.PHDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145865/1/bworthma_1.pd

A Distributed NMPC Scheme without Stabilizing Terminal Constraints

International audienceWe consider a distributed NMPC scheme in which the individual systems are coupled via state constraints. In order to avoid violation of the constraints, the subsystems communicate their individual predictions to the other subsystems once in each sampling period. For this setting, Richards and How have proposed a sequential distributed MPC formulation with stabilizing terminal constraints. In this chapter we show how this scheme can be extended to MPC without stabilizing terminal constraints or costs.We show theoretically and by means of numerical simulations that under a suitable controllability condition stability and feasibility can be ensured even for rather short prediction horizons