Viability-based computation of spatially constrained minimum time trajectories for an autonomous underwater vehicle: implementation and experiments.

A viability algorithm is developed to compute the constrained minimum time function for general dynamical systems. The algorithm is instantiated for a specific dynamics(Dubin’s vehicle forced by a flow field) in order to numerically solve the minimum time problem. With the specific dynamics considered, the framework of hybrid systems enables us to solve the problem efficiently. The algorithm is implemented in C using epigraphical techniques to reduce the dimension of the problem. The feasibility of this optimal trajectory algorithm is tested in an experiment with a Light Autonomous Underwater Vehicle (LAUV) system. The hydrodynamics of the LAUV are analyzed in order to develop a low-dimension vehicle model. Deployment results from experiments performed in the Sacramento River in California are presented, which show good performance of the algorithm.trajectories; underwater vehicle; viability algorithm; hybrid systems; implementation;

An Optimal Control Theory for the Traveling Salesman Problem and Its Variants

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.Comment: 24 pages, 8 figure

A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Comprehensive Energy Footprint Benchmarking of Strong Parallel Electrified Powertrain

In this paper we present a benchmark solution with higher number of continuous and discrete states and control levers using validated powertrain component models, where DP fails due to exponential rise in the computation time. The problem involves 13 states and 4 control levers, with complex interactions between multiple subsystems. Some of these variables are discrete while some are continuous. Some have slow dynamics while some have fast dynamics. A novel three step PS3 algorithm [1] which is presented in our prequel paper is used to obtain a near-optimal solution. PS3 algorithm makes use of pseudo spectral method for accurate state estimations. We present three scenarios where only fuel is minimized, only emissions are minimized and, lastly a combination of both fuel and emissions are minimized. All three cases are analyzed for their performance and computation time. The optimal compromise between fuel consumption and emissions are analyzed using a Pareto-front study. This large-scale powertrain optimization problem is solved for a P2 parallel hybrid architecture on a class 6 pick-up & delivery truck.Comment: Fixed typos, added discussio

On Distributed Implementation of Switch-Based Adaptive Dynamic Programming

Switch-based adaptive dynamic programming (ADP) is an optimal control problem in which a cost must be minimized by switching among a family of dynamical modes. When the system dimension increases, the solution to switch-based ADP is made prohibitive by the exponentially increasing structure of the value function approximator and by the exponentially increasing modes. This technical correspondence proposes a distributed computational method for solving switch-based ADP. The method relies on partitioning the system into agents, each one dealing with a lower dimensional state and a few local modes. Each agent aims to minimize a local version of the global cost while avoiding that its local switching strategy has conflicts with the switching strategies of the neighboring agents. A heuristic algorithm based on the consensus dynamics and Nash equilibrium is proposed to avoid such conflicts. The effectiveness of the proposed method is verified via traffic and building test cases

Approximate dynamic programming based solutions for fixed-final-time optimal control and optimal switching

Optimal solutions with neural networks (NN) based on an approximate dynamic programming (ADP) framework for new classes of engineering and non-engineering problems and associated difficulties and challenges are investigated in this dissertation. In the enclosed eight papers, the ADP framework is utilized for solving fixed-final-time problems (also called terminal control problems) and problems with switching nature. An ADP based algorithm is proposed in Paper 1 for solving fixed-final-time problems with soft terminal constraint, in which, a single neural network with a single set of weights is utilized. Paper 2 investigates fixed-final-time problems with hard terminal constraints. The optimality analysis of the ADP based algorithm for fixed-final-time problems is the subject of Paper 3, in which, it is shown that the proposed algorithm leads to the global optimal solution providing certain conditions hold. Afterwards, the developments in Papers 1 to 3 are used to tackle a more challenging class of problems, namely, optimal control of switching systems. This class of problems is divided into problems with fixed mode sequence (Papers 4 and 5) and problems with free mode sequence (Papers 6 and 7). Each of these two classes is further divided into problems with autonomous subsystems (Papers 4 and 6) and problems with controlled subsystems (Papers 5 and 7). Different ADP-based algorithms are developed and proofs of convergence of the proposed iterative algorithms are presented. Moreover, an extension to the developments is provided for online learning of the optimal switching solution for problems with modeling uncertainty in Paper 8. Each of the theoretical developments is numerically analyzed using different real-world or benchmark problems --Abstract, page v

State elimination for mixed-integer optimal control of partial differential equations by semigroup theory

Mixed-integer optimal control problems governed by partial differential equations (MIPDECOs) are powerful modeling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by partial differential equations that have the structure of an abstract ordinary differential equation in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments

Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

Mixed integer optimal compensation deals with optimization problems with integer- and real-valued control variables to compensate disturbances in dynamic systems. The mixed integer nature of controls could lead to intractability in problems of large dimensions. To address this challenge, we introduce a decomposition method which turns the original n-dimensional optimization problem into n independent scalar problems of lot sizing form. Each of these problems can be viewed as a two-player zero-sum game, which introduces some element of conservatism. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon, a step that mirrors a standard procedure in mixed integer programming. We apply the decomposition method to a mean-field coupled multi-agent system problem, where each agent seeks to compensate a combination of an exogenous signal and the local state average. We discuss a large population mean-field type of approximation and extend our study to opinion dynamics in social networks as a special case of interest