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Title:  Optimal nonlinear control and estimation using global domain linearization 
Author(s):  Wendt, Luke Adam 
Director of Research:  Levinson, Stephen E 
Doctoral Committee Chair(s):  Levinson, Stephen E 
Doctoral Committee Member(s):  HasegawaJohnson, Mark A; Makela, Jonathan J; Dell, Gary S; Rothganger, Fred H 
Department / Program:  Electrical & Computer Eng 
Discipline:  Electrical & Computer Engr 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  approximate
domain optimal nonlinear control estimation linearization generalized radial basis continuous dynamic differential feedback 
Abstract:  Alan Turing teaches that cognition is symbol processing. Norbert Wiener teaches that intelligence rests on feedback control. Thus, there are discrete symbols and continuous sensorymotor signals. Sensorimotor dynamics are wellrepresented by nonlinear differential equations. A possible construction of symbols could be based on equilibria. Language is a symbol system and is one of the highest expressions of cognition. Much of this comes from spatial reasoning, which requires embodied cognition. Spatial reasoning derives from motor function. This thesis introduces a novel generalized nonheuristic method of linearizing nonlinear differential equations over a finite domain. It is used to engineer optimal convergence to target sets, a general form of spatial reasoning. Ordinary differential equations are ubiquitous models in physics and engineering that describe a wide range of phenomena including electromechanical systems. This thesis considers ordinary differential equations expressed in statespace form. For a given initial state, these equations generate signals that are continuous in both time and state. The control engineering objective is to find input functions that steer these states to desired target sets using only the measured output of the system. The statespace domain containing the target set, along with its cost, can be thought of as a symbol for highlevel planning. Consider a basis state equal to a vector of nonlinear basis functions, computed from the state, where the state is generated from a multivariate nonlinear dynamic system. The basis state derivative can be expressed as a linear dynamic system with an additional error term. This thesis describes radial basis functions that minimize the error over the entire statespace domain, where the basis state equals zero if and only if the state is in a desired target set. This gives an approximate lineardynamic system, and if the basis state goes to zero, then the state goes to the target set. This form of linear approximation is global over the domain. Careful selection of the basis gives a fully generalizable relationship between linear stability and nonlinear stability. This form of linearization can be applied to optimal state feedback and state estimation problems. This thesis carefully introduces optimal state feedback control with an emphasis on optimal infinite horizon solutions to lineardynamic systems that have quadratic cost. A thorough introduction is also given to the optimal output feedback of lineardynamics systems. The detectable and stabilizable subspaces of a lineardynamic system are expressed in a generalized closed form. After introducing optimal control for linear systems, this thesis explores adaptive control from several different perspectives including: tuning, system identification, and reinforcement learning. Each of these approaches can be characterized as an optimal nonlinear output feedback problem. In each case, generalized representations can be found using a single layer of appropriately chosen nonlinear basis functions with linear parameterization. The primary focus of this thesis is to select these basis functions, in a fully generalized way, so that they have lineardynamics. When this can be achieved, infinite horizon state feedback and state estimation can be computed using wellknown closedform solutions. This thesis demonstrates how multivariate nonlinear dynamic systems defined on a finite domain can be approximated by computationally equivalent highdimensional lineardynamic systems using a generalized basis state. This basis state is computed with a single layer of biologically inspired radial basis functions. The method of linearization is described as "global domain linearization" because it holds over a specified domain, and therefore provides a global linear approximation with respect to that domain. Any optimal state estimation or state feedback is globally optimal over the domain of linearization. The tools of optimal linear control theory can be applied. In particular, control and estimation problems involving underactuated undermeasured nonlinear systems with generalized nonlinear reward can be solved with closedform infinite horizon linearquadratic control and estimation. The controllable, uncontrollable, stabilizable, observable, unobservable, and detectable subspaces can all be described in a meaningful generalized way. State estimation and state feedback can then be implemented in computationally efficient lowdimensional highly nonlinear form. Generalized optimal state estimation and state feedback for continuoustime continuousstate systems is necessary machinery for any highlevel symbolic planning that might involve unstable electromechanical systems. Symbols naturally form in the presence of more than one target state. This could provide a natural method of language acquisition. Given a state, all symbolic domains that intersect the state would have equilibrium and cost. These intersections define the legal grammar of symbol transition. An engineer or agent can design these symbols for highlevel planning. Generalized infinite horizon state feedback and state estimation can then be computed for the continuous system that each symbol represents using traditional linear tools with domain linearization. 
Issue Date:  20170709 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/98334 
Rights Information:  Copyright 2017 Luke A. Wendt 
Date Available in IDEALS:  20170929 
Date Deposited:  201708 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois