Small gain versus positive real modeling of real parameter uncertainty

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76326/1/AIAA-20872-692.pd

Singular Perturbations and Time-Scale Methods in Control Theory: Survey 1976-1982

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424U.S. Air Force / AFOSR 78-363

Applications of Singular Perturbation Techniques to Control Problems

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424National Science Foundation / NSF ECS 82-1763

The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed Ito differential equations

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of O(√ε) approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an O(ε) approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter ε is not precisely known. © 2012 Society for Industrial and Applied Mathematics.Vasile Dragan, Hiroaki Mukaidani and Peng Sh

The Statistical Dynamics of Nonequilibrium Control

Living systems, even at the scale of single molecules, are constantly adapting to changing environmental conditions. The physical response of a nanoscale system to external gradients or changing thermodynamic conditions can be chaotic, nonlinear, and hence difficult to control or predict. Nevertheless, biology has evolved systems that reliably carry out the cell’s vital functions efficiently enough to ensure survival. Moreover, the development of new experimental techniques to monitor and manipulate single biological molecules has provided a natural testbed for theoretical investigations of nonequilibrium dynamics. This work focuses on developing paradigms for both understanding the principles of nonequilibrium dynamics and also for controlling such systems in the presence of thermal fluctuations. Throughout this work, I rely on a perspective based on two central ideas in nonequilibrium statistical mechanics: large deviation theory, which provides a formalism akin to thermodynamics for nonequilibrium systems, and the fluctuation theorems which identify time symmetry breaking with entropy production. I use the tools of large deviation theory to explore concepts like efficiency and optimal coarse-graining in microscopic dynamical systems. The results point to the extreme importance of rare events in nonequilibrium dynamics. In the context of rare dynamical events, I outline a formal approach to predict efficient control protocols for nonequilibrium systems and develop computational tools to solve the resulting high dimensional optimization problems. The final chapters of this work focus on applications to self-assembly dynamics. I show that the yield of desired structures can be enhanced by driving a system away from equilibrium, using analysis inspired by the theory of the hydrophobic effect. Finally, I demonstrate that nanoscale, protein shells can be modeled and controlled to robustly produce monodisperse, nonequilibrium structures strikingly similar to the microcompartments observed in a variety of bacteria

Modelling protein localisation and positional information in subcellular systems

Cells and their component structures are highly organised. The correct function of many biological systems relies upon not only temporal control of protein levels but also spatial control of protein localisation within cells. Mathematical modelling allows us to quantitatively test potential mechanisms for protein localisation and spatial organisation. Here we present models of three examples of spatial organisation within individual cells. In the bacterium E. coli, the site of cell division is partly determined by the Min proteins. The Min proteins oscillate between the cell poles and suppress formation of the division ring here, thereby restricting division to midcell. We present a stochastic model of the Min protein dynamics, and use this model to investigate partitioning of the Min proteins between the daughter cells during cell division. The Min proteins determine the correct position for cell division by forming a timeaveraged concentration gradient which is minimal at midcell. Concentration gradients are involved in a range of subcellular processes, and are particularly important for obtaining positional information. By analysing the low copy number spatiotemporal uctuations in protein concentrations for a single polar gradient and two oppositelydirected gradients, we estimate the positional precision that can be achieved in vivo. We nd that time-averaging is vital for high precision. The embryo of the nematode C. elegans has become a model system for the study of cell polarity. At the one-cell stage, the PAR proteins form anterior and posterior domains in a dynamic process driven by contraction of cortical actomyosin. We present a continuum model for this system, including a highly simpli ed model of the actomyosin dynamics. Our model suggests that the known PAR protein interactions 5 are insu cient to explain the experimentally observed cytoplasmic polarity. We discuss a number of modi cations to the model which reproduce the correct cytoplasmic distributions