In this paper, we take a control-theoretic approach to answering some standard questions in statistical mechanics, and use the results to derive limitations of classical measurements. A central problem is the relation between systems which appear macroscopically dissipative but are microscopically lossless. We show that a linear system is dissipative if, and only if, it can be approximated by a linear lossless system over arbitrarily long time intervals. Hence lossless systems are in this sense dense in dissipative systems. A linear active system can be approximated by a nonlinear lossless system that is charged with initial energy. As a by-product, we obtain mechanisms explaining the Onsager relations from time-reversible lossless approximations, and the fluctuation-dissipation theorem from uncertainty in the initial state of the lossless system. The results are applied to measurement devices and are used to quantify limits on the so-called observer effect, also called back action, which is the impact the measurement device has on the observed system. In particular, it is shown that deterministic back action can be compensated by using active elements, whereas stochastic back action is unavoidable and depends on the temperature of the measurement device.
Substantial work has already been done in the control community in formulating various results of classical thermodynamics in a more mathematical framework. In @cite_10 @cite_8 , the second law of thermodynamics is derived and a control-theoretic heat engine is obtained (in @cite_17 these results are generalized). In @cite_26 , a rigorous dynamical systems approach is taken to derive the laws of thermodynamics using the framework of dissipative systems @cite_28 @cite_23 . In @cite_37 , it is shown how the entropy flows in Kalman-Bucy filters, and in @cite_18 Linear-Quadratic-Gaussian control theory is used to construct heat engines. In @cite_40 @cite_2 @cite_42 , the problem of how lossless systems can appear dissipative (compare with @cite_13 @cite_35 @cite_33 above) is discussed using various perspectives. In @cite_38 , how the direction of time affects the difficulty of controlling a process is discussed.
@cite_13 @cite_35 @cite_37 @cite_26 @cite_18 @cite_33 @cite_38 @cite_8 @cite_28 @cite_42 @cite_40 @cite_23 @cite_2 @cite_10 @cite_17
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The second law of thermodynamics is studied from the point of view of stochastic control theory. We find that the feedback control laws which are of interest are those which depend only on average values, and not on sample path behavior. We are lead to a criterion which, when satisfied, permits one to assign a temperature to a stochastic system in such a way as to have Carnot cycles be the optimal trajectories of optimal control problems. Entropy is also defined and we are able to prove an equipartition of energy theorem using this definition of temperature. Our formulation allows one to treat irreversibility in a quite natural and completely precise way. In this paper, we focus on a particular class of nonlinear affine control systems of the form xdot=f(x)+Bu, where the drift f is a multi-affine vector field (i.e., affine in each state component), the control distribution B is constant, and the control u is constrained to a convex set. For such a system, we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant. We then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time. If the control constraints are polyhedral, we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle. This work is motivated by the need to construct discrete abstractions for continuous and hybrid systems, in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs. We show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators This paper develops new methods to model and control the aggregated power demand from a population of thermostatically controlled loads, with the goal of delivering services such as regulation and load following. Previous work on direct load control focuses primarily on peak load shaving by directly interrupting power to loads. In contrast, the emphasis of this paper is on controlling loads to produce relatively short time scale responses (hourly to sub-hourly), and the control signal is applied by manipulation of temperature set points, possibly via programmable communicating thermostats or advanced metering infrastructure. To this end, the methods developed here leverage the existence of system diversity and use physically-based load models to inform the development of a new theoretical model that accurately predicts – even when the system is not in equilibrium – changes in load resulting from changes in thermostat temperature set points. Insight into the transient dynamics that result from set point changes is developed by deriving a new exact solution to a well-known hybrid state aggregated load model. The eigenvalues of the solution, which depend only on the thermal time constant of the loads under control, are shown to have a strong effect on the accuracy of the model. The paper also shows that load heterogeneity – generally something that must be assumed away in direct load control models – actually has a positive effect on model accuracy. System identification techniques are brought to bear on the problem, and it is shown that identified models perform only marginally better than the theoretical model. The paper concludes by deriving a minimum variance control law, and demonstrates its effectiveness in simulations wherein a population of loads is made to follow the output of a wind plant with very small changes in the nominal thermostat temperature set points. In this thesis we present a technique for the composition of robot control laws in dynamical environments. We propose a challenging robotic task, called Dynamical Pick and Place, in which a robot equipped with merely a soft paddle must capture and contain a ball, safely negotiate it past obstacles, and bring it to rest at a desired location. We develop a composition technique for local controllers that provides a formal guarantee of the stability of the switching behavior required in this task, and provide descriptive statistics of a working implementation. Our robotic system displays unusually dexterous behavior in the face of significant system noise, and recovers gracefully from large unexpected perturbations caused by the experimenters. Our approach to controller composition makes use of the funnel as a metaphor for asymptotic stability, is motivated by the pre-image backchaining techniques developed by Lozano-Perez, Mason and Taylor, and extends their ideas from quasi-static environments to systems with full dynamics. We introduce the concepts of "dynamical obstacle avoidance" and "dynamical safety" for systems with only intermittent control of their environment, and show that it is important not only that the system avoid obstacles directly, but also that the system will never reach an obstacle before getting another chance to effect control. The Dynamical Pick and Place problem addressed by this thesis is a difficult control problem, but an easy planning problem. The system we develop provides a way to engage more powerful AI planning tools without sacrificing access to the stability arguments of dynamical systems theory.