A Hybrid Optimal Control Approach to Fuel-Efficient Aircraft Conflict Avoidance

Many interesting problems in numerical optimization are hybrid optimal control problems. Hybrid optimal control (HOC) problems include both continuous-valued variables and categorical variables in the problem formulation. For the types of problems envisioned here the categorical variables will specify the structure or sequence of events that qualitatively describes the trajectory or mission. Continuous variables are used in the modeling of the continuous dynamics. In this work, the mission planning problem is of interest. The problem is qualitatively different from the typical interception of multiple targets as the discrete variables now represent events like impulses, coast and thrust arcs that change the structure of the problem. In addition, the number of events in the categorical sequence is not fixed. For the dynamical assembly of events required for the solution of the mission planning problem, a scheme that defines events as modules consisting of parameters and constraints is presented. The method assembles the respective events one next to the other in time according to the given mission structure. For the generation of the initial guess, two new methods were developed that approximate optimal low-thrust trajectories. The first method, based on genetic algorithms (GA), handles the rendezvous constraints explicitly using a conditional penalty function. The second method, feasible region analysis (FRA), is based on GA and nonlinear programming (NLP), which allows taking advantage of the GA capabilities in finding a global optimum and NLP ability in handling constraints. A rendezvous problem with known solution is solved.

A Class of Linear Quadratic Gaussian Hybrid Optimal Control Problems with Realization–Independent Riccati Equations

Control Optimization of Discontinuous Hybrid Systems Using Embedding

Optimal control of constrained switched systems and application to electrical vehicle energy management

Loss control regions in optimal control problems

In the paper we present a new method for solving optimal control problems of a class of hybrid systems. We describe the new effective algorithm based on memetic algorithm (MA) for optimal control of switched systems. We concentrate on a class of problems in which a pre-specified sequence of active subsystems is known. Our aim is to find both the optimal switching instants and the optimal continuous inputs. The new approach, which we propose, decomposes the cost functional of the basic optimal control problem in Bolza form in two terms. The first term depends explicitly on a value of state variables at the final time. The second term depends on state and control trajectories. In oreder to solve those two tasks we used MA as the multi-objective optimization algorithm. In this paper we considered a fundamental bi-criteria case with two mentioned before functions: the value of state variables at the final time and the state and control trajectories. In order to find an approximation of Pareto frontier, we proposed new effective method based on genetic algorithm (GA) and local search (LS). Problems properties were taken into consideration in the design of our new approach of solving it. They were used to construct new algorithm inspired by the LS NSGA-II, which performed rather well in multi-criteria scheduling problem. Since simple genetic algorithms are efficient heuristics in searching for optimal solutions, but lack the accuracy of some more computational complex algorithms, a hybrid algorithm was constructed. It uses fast non-dominated sorting, in order to evaluate child population and allocate solutions to corresponding Pareto frontiers. I addition a local search method was used, in order to find more differentiated and better solutions. Clustering solutions from Pareto frontiers also improved diversity of solutions in child population. This approach can be used as a start point for searching for algorithm for solving optimal control problems of switched systems without pre-specified sequence of active subsystems. The performance of the algorithm is illustrated by the examples: (a) a hybrid optimal control problem with nonlinear dynamics and (b) a switched linear quadratic optimal control problem with long switching time intervals.

This chapter is devoted upon optimality, a topic in which the central result is the Pontryagin’s minimum principle. This important result is briefly approached from the classical calculus of variation. We show that the classical calculus of variation has some major limitations to modern control problems and motivate Pontryagin’s minimum principle. The Hamilton–Jacobi–Bellman method is discussed as an alternative approach to gain first-order necessary conditions for optimality. It is shown that both approaches correspond to each other under restrictive assumptions. The original Pontryagin’s minimum principle for continuous optimal control problems is not suitable for hybrid optimal control problems. However, a quite natural reformulation of the hybrid optimal control problem admits the classical theory for deduction of first-order necessary conditions in the sense of Pontryagin. The charm of this methodology is its comprehensible derivation.

This paper addresses an optimal control problem for a robot that has to find and collect a finite number of objects and move them to a depot in minimum time. The objects are modeled by point masses with a priori unknown locations in a bounded two-dimensional space. The robot has forth-order dynamics that change instantaneously at any pick-up or drop-off of an object. The corresponding hybrid Optimal Control Problem (OCP) is solved by a receding horizon scheme, where the derived lower bound for the cost-to-go is evaluated for the worst- and a probabilistic case, assuming a uniform distribution of the objects. We first present a time-driven approximate solution based on time and position space discretization. Due to the high computational cost of this solution, we alternatively propose an event-driven approximate approach based on a suitable motion parameterization. The solutions are compared in a numerical example, suggesting that the latter approach offers a significant computational advantage while yielding similar qualitative results compared to the former.

Chapter 1 - Introduction and Motivation

We develop an accurate model for the arrival traffic dynamics in terminal airspace that takes into account the weather forecast and runway configuration changes. The planning of runway configuration switching subject to weather constraints is formulated as a hybrid optimal control problem and a hierarchical approach is proposed to solve the problem. At the upper level of the hierarchy, a runway configuration sequence is determined based on weather forecast data and air traffic demand. At the lower level, the configuration switching times and the corresponding aircraft speeds are determined by solving a Mixed Integer Linear Program. We illustrate the utility of the approach by a case study inspired by operations in John F. Kennedy airport.

We consider a fairly general class of state-constrained nonlinear hybrid optimal control problems that are based on coordinatizing Sussmann's model. An event set generalizes the notion of a guard set, reset map, endpoint set as well as the switching set. We present a pseudospectral (PS) knotting method that discretizes the continuous-time variables of the problem. The discrete event conditions are imposed over the PS knots leading to a large, sparse, mixed-variable programming (MVP) problem. The Karush-Kuhn-Tucker conditions for the MVP are transformed in a manner that makes them closely resemble the discretized necessary conditions obtained from the hybrid minimum principle. A set of closure conditions are introduced to facilitate commuting the operations of dualization and discretization. An immediate consequence of this is a hybrid covector mapping theorem that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discretized covectors associated with the hybrid optimal control problem.

The 48-V diesel mild-hybrid electric vehicle equipped with an electrically heated catalyst is a cost-effective solution to reduce tailpipe emissions, while its supervisory control design is more challenging, due to cross-coupled energy and thermal subsystems. This article presents a distributed control design for the energy and thermal management system, aiming to optimize fuel economy and reduce catalyst warm-up time, by cooperation between electrical heating and engine load upshift. A control-oriented model of HEV and catalyst is developed, which captures the dynamics of the battery state of charge and catalyst temperature. The energy and thermal management problem is cast as a hybrid optimal control problem, to minimize entire trip fuel consumption under different switching (warm-up) times. An off-line solution method is designed to exploit the tradeoff between fuel and warm-up time based upon hybrid minimum principle and a nested shooting algorithm. For online implementation, a distributed supervisory control is proposed, which covers both cold-start and normal operations, and is compatible with the equivalent consumption minimization strategy. Both off-line and online methods are validated in simulation, using two driving cycles and five warm-up speeds.

Hybrid optimal control problems are studied for a general class of hybrid systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to terminal and running costs switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework and it is shown that under certain assumptions the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results and, in particular, a Riccati formalism for linear quadratic hybrid tracking problems is presented.

On the Minimum Principle and Dynamic Programming for Hybrid Systems

Many space mission planning problems may be formulated as hybrid optimal control problems (HOCP), i.e. problems that include both real-valued variables and categorical variables. In interplanetary trajectory design problems the categorical variables will typically specify the sequence of planets at which to perform ybys, and the real-valued