Numerical Methods for Nonlinear Optimal Control Problems and Their Applications in Indoor Climate Control

Efficiency, comfort, and convenience are three major aspects in the design of control systems for residential Heating, Ventilation, and Air Conditioning (HVAC) units. In this dissertation, we study optimization-based algorithms for HVAC control that minimizes energy consumption while maintaining a desired temperature, or even human comfort in a room. Our algorithm uses a Computer Fluid Dynamics (CFD) model, mathematically formulated using Partial Differential Equations (PDEs), to describe the interactions between temperature, pressure, and air flow. Our model allows us to naturally formulate problems such as controlling the temperature of a small region of interest within a room, or to control the speed of the air flow at the vents, which are hard to describe using finite-dimensional Ordinary Partial Differential (ODE) models. Our results show that our HVAC control algorithms produce significant energy savings without a decrease in comfort. Also, we formulate a gradient-based estimation algorithm capable of reconstructing the states of doors in a building, as well as its temperature distribution, based on a floor plan and a set of thermostats. The estimation algorithm solves in real time a convection-diffusion CFD model for the air flow in the building as a function of its geometric configuration. We formulate the estimation algorithm as an optimization problem, and we solve it by computing the adjoint equations of our CFD model, which we then use to obtain the gradients of the cost function with respect to the flow’s temperature and door states. We evaluate the performance of our method using simulations of a real apartment in the St. Louis area. Our results show that the estimation method is both efficient and accurate, establishing its potential for the design of smarter control schemes in the operation of high-performance buildings. The optimization problems we generate for HVAC system\u27s control and estimation are large-scale optimal control problem. While some optimal control problems can be efficiently solved using algebraic or convex methods, most general forms of optimal control must be solved using memory-expensive numerical methods. In this dissertation we present theoretical formulations and corresponding numerical algorithms that can find optimal inputs for general dynamical systems by using direct methods. The results show these algorithms\u27 performance and potentials to be applied to solve large-scale nonlinear optimal control problem in real time

A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.Comment: 123 pages, revised versio

Mathematical control theory and Finance

Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio

Fast, Optimal, and Safe Motion Planning for Bipedal Robots

Bipedal robots have the potential to traverse a wide range of unstructured environments, which are otherwise inaccessible to wheeled vehicles. Though roboticists have successfully constructed controllers for bipedal robots to walk over uneven terrain such as snow, sand, or even stairs, it has remained challenging to synthesize such controllers in an online fashion while guaranteeing their satisfactory performance. This is primarily due to the lack of numerical method that can accommodate the non-smooth dynamics, high degrees of freedom, and underactuation that characterize bipedal robots. This dissertation proposes and implements a family of numerical methods that begin to address these three challenges along three dimensions: optimality, safety, and computational speed. First, this dissertation develops a convex relaxation-based approach to solve optimal control for hybrid systems without a priori knowledge of the optimal sequence of transition. This is accomplished by formulating the problem in the space of relaxed controls, which gives rise to a linear program whose solution is proven to compute the globally optimal controller. This conceptual program is solved using a sequence of semidefinite programs whose solutions are proven to converge from below to the true optimal solution of the original optimal control problem. Moreover, a method to synthesize the optimal controller is developed. Using an array of examples, the performance of this method is validated on problems with known solutions and also compared to a commercial solver. Second, this dissertation constructs a method to generate safety-preserving controllers for a planar bipedal robot walking on flat ground by performing reachability analysis on simplified models under the assumption that the difference between the two models can be bounded. Subsequently, this dissertation describes how this reachable set can be incorporated into a Model Predictive Control framework to select controllers that result in safe walking on the biped in an online fashion. This method is validated on a 5-link planar model. Third, this dissertation proposes a novel parallel algorithm capable of finding guaranteed optimal solutions to polynomial optimization problems up to pre-specified tolerances. Formal proofs of bounds on the time and memory usage of such method are also given. Such algorithm is implemented in parallel on GPUs and compared against state-of-the-art solvers on a group of benchmark examples. An application of such method on a real-time trajectory-planning task of a mobile robot is also demonstrated. Fourth, this dissertation constructs an online Model Predictive Control framework that guarantees safety of a 3D bipedal robot walking in a forest of randomly-placed obstacles. Using numerical integration and interval arithmetic techniques, approximations to trajectories of the robot are constructed along with guaranteed bounds on the approximation error. Safety constraints are derived using these error bounds and incorporated in a Model Predictive Control framework whose feasible solutions keep the robot from falling over and from running into obstacles. To ensure that the bipedal robot is able to avoid falling for all time, a finite-time terminal constraint is added to the Model Predictive Control algorithm. The performance of this method is implemented and compared against a naive Model Predictive Control method on a biped model with 20 degrees of freedom. In summary, this dissertation presents four methods for control synthesis of bipedal robots with improvements in either optimality, safety guarantee, or computational speed. Furthermore, the performance of all proposed methods are compared with existing methods in the field.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162880/1/pczhao_1.pd

A study of optimization and optimal control computation : exact penalty function approach

In this thesis, We propose new computational algorithms and methods for solving four classes of constrained optimization and optimal control problems. In Chapter 1, we present a brief review on optimization and optimal control. In Chapter 2, we consider a class of continuous inequality constrained optimization problems. The continuous inequality constraints are first approximated by smooth function in integral form. Then, we construct a new exact penalty function, where the summation of all these approximate smooth functions in integral form, called the constraint violation, is appended to the objective function. In this way, we obtain a sequence of approximate unconstrained optimization problems. It is shown that if the value of the penalty parameter is sufficiently large, then any local minimizer of the corresponding unconstrained optimization problem is a local minimizer of the original problem. For illustration, three examples are solved using the proposed method.From the solutions obtained, we observe that the values of their objective functions are amongst the smallest when compared with those obtained by other existing methods available in the literature. More importantly, our method finds solutions which satisfy the continuous inequality constraints.In Chapter 3, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. However, the existing gradient-based optimization techniques have difficulty to solve this equivalent nonlinear optimization problem effectively due to the new quadratic inequality constraint. Thus, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton types of methods.It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.In Chapter 4, we investigate the optimal design of allpass variable fractional delay (VFD) filters with coefficients expressed as sums of signed powers-of-two terms, where the weighted integral squared error is minimized. A new optimization procedure is proposed to generate a reduced discrete search region. Then, a new exact penalty function method is developed to solve the optimal design of allpass variable fractional delay filter with signed powers-of-two coefficients. Design examples show that the proposed method is highly effective. Compared with the conventional quantization method, the solutions obtained by our method are of much higher accuracy. Furthermore, the computational complexity is low.In Chapter 5, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent optimal control problem subject to original constraints and additional linear and quadratic constraints, where the decision variables are taking values from a feasible region, which is the union of some continuous sets. However, due to the new quadratic constraints, standard optimization techniques do not perform well when they are applied to solve the transformed problem directly.We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective function, forming a penalized objective function. This leads to a sequence of approximate optimal control problems, each of which can be solved by using optimal control techniques, and consequently, many optimal control software packages, such as MISER 3.4, can be used. Convergence results how that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude this chapter with some numerical results for two train control problems.In Chapter 6, some concluding remarks and suggestions for future research directions are made