Citation
Horowitz, Matanya Benasher (2014) Efficient Methods for Stochastic Optimal Control. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/D40A9E03. http://resolver.caltech.edu/CaltechTHESIS:05312014011052261
Abstract
The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.
In the last decade researchers have found that under certain, fairly nonrestrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.
This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving suboptimality gap. The resulting approximate solutions are shown to be guaranteed over and underapproximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and loworder polynomials.
The optimizationbased SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a lowrank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.
The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multitask planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to precompute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  stochastic optimal control, hamilton jacobi bellman equation, polynomial optimization, sum of squares programming,  
Degree Grantor:  California Institute of Technology  
Division:  Engineering and Applied Science  
Major Option:  Control and Dynamical Systems  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  14 May 2014  
NonCaltech Author Email:  matanya.horowitz (AT) gmail.com  
Record Number:  CaltechTHESIS:05312014011052261  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:05312014011052261  
DOI:  10.7907/D40A9E03  
Related URLs: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  8453  
Collection:  CaltechTHESIS  
Deposited By:  Matanya Horowitz  
Deposited On:  03 Jun 2014 19:03  
Last Modified:  11 Apr 2019 17:50 
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