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# Cost Density-Shaping for Stochastic Optimal Control

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posted on 2010-12-09, 00:00 authored by Matthew ZyskowskiIn stochastic optimal control theory, the complete specification of the probability density function of the random cost functional might be considered the most a designer can do when formulating an optimal control law. The field of cost cumulant controls has made considerable advances towards this capability in the past few decades, largely because of the advantage gained from controlling the cost cumulants instead of the cost moments. However, current cost cumulant control paradigms have left the deliberate specification of the probability density function for the random cost outside the designer's direct influence. The ability to design control laws upon the desired shape and location of the cost density would be highly valuable to control engineers, since there is evidence that the shape of the cost density under high-performance controllers directly corresponds to the resulting closed-loop system behavior.

This dissertation proposes a Multiple-Cumulant Cost Density-Shaping (MCCDS) optimization problem for the LQG framework. The control solution to the MCCDS optimization is derived using dynamic programming techniques; it is the finite-horizon, linear state-feedback control that minimizes a smooth, convex, scalar function of arbitrarily-many initial cost cumulants and target initial cost cumulants. The MCCDS theory is shown to generalize the Linear Quadratic Gaussian (LQG), 'k Cost Cumulant' (kCC), and Risk-Sensitive (RS) control paradigms for zero targets and linear performance indices. Additionally, the MCCDS framework enables the minimization of well-known distance functions between the cost density and a target cost density, such as the Kullback-Leibler Divergence, Bhattacharyya Distance, and the Hellinger Distance.

The finite-horizon MCCDS control is extended to the infinite-horizon in this dissertation. Other areas of investigation include MCCDS performance index construction, cost density-shaping minimax and Nash games, and a Statistical Target Selection (STS) iterative procedure for control design. Together MCCDS and STS enable the design of control laws with optimality among a family of target cost densities, which might be regarded as a new approach to robust LQG control design. For structures excited by seismic disturbances, numerical experiments show that the MCCDS controls identified thru STS can achieve greater vibration suppression than nominal kCC controllers, without any compromise to robust stability.

This dissertation proposes a Multiple-Cumulant Cost Density-Shaping (MCCDS) optimization problem for the LQG framework. The control solution to the MCCDS optimization is derived using dynamic programming techniques; it is the finite-horizon, linear state-feedback control that minimizes a smooth, convex, scalar function of arbitrarily-many initial cost cumulants and target initial cost cumulants. The MCCDS theory is shown to generalize the Linear Quadratic Gaussian (LQG), 'k Cost Cumulant' (kCC), and Risk-Sensitive (RS) control paradigms for zero targets and linear performance indices. Additionally, the MCCDS framework enables the minimization of well-known distance functions between the cost density and a target cost density, such as the Kullback-Leibler Divergence, Bhattacharyya Distance, and the Hellinger Distance.

The finite-horizon MCCDS control is extended to the infinite-horizon in this dissertation. Other areas of investigation include MCCDS performance index construction, cost density-shaping minimax and Nash games, and a Statistical Target Selection (STS) iterative procedure for control design. Together MCCDS and STS enable the design of control laws with optimality among a family of target cost densities, which might be regarded as a new approach to robust LQG control design. For structures excited by seismic disturbances, numerical experiments show that the MCCDS controls identified thru STS can achieve greater vibration suppression than nominal kCC controllers, without any compromise to robust stability.

## History

## Date Modified

2017-06-02## Defense Date

2010-12-03## Research Director(s)

Dr. Panos Antsaklis## Committee Members

Dr. Ron Diersing Dr. Peter Bauer Dr. Vijay Gupta Dr. Yih-Fang Huang## Degree

- Doctor of Philosophy

## Degree Level

- Doctoral Dissertation

## Language

- English

## Alternate Identifier

etd-12092010-104014## Publisher

University of Notre Dame## Program Name

- Electrical Engineering

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