Thomas J. Böhme

Hybrid Systems and Hybrid Optimal Control

Whenever a dynamic system has continuous-valued control inputs and states, but can switch between multiple subsystems with different dynamical behaviors, different numbers of active inputs and states, etc., then the dynamic system can be modeled as a hybrid system. Hybrid systems occur naturally in many technical applications as well as in applications from natural sciences as biology or chemistry. A simple example of a discrete decision may be the on/off command of a heating system. More complex decisions are gear choices or different operation modes of the internal combustion engine in automotive applications. In this book, the focus is on switched systems, a subclass of hybrid systems that switch between subsystems only in response to a command. This subclass already covers a great range of technical problems. This chapter provides the basic definitions for switched systems necessary to formulate optimal control problems.

Introduction to Nonlinear Programming

This chapter provides a short introduction into nonlinear programming. It gives the reader a deeper insight into sequential quadratic programming methods and the sensitivity analysis of constrained nonlinear minimization problems, because these tools are fundamental to the optimal control algorithms proposed in the subsequent chapters.

Graph Theoretical Fundamentals for Sparse Matrices

We only state the graph theoretical concepts, which are in the scope of this book. A comprehensive introduction to graph theory can be found in the textbooks of Diestel [1], George et al. [2], Golumbic [3], and Wilson [4].

The Minimum Principle and Hamilton–Jacobi–Bellman Equation

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.

Predictive Real-Time Energy Management

This chapter discusses predictive control strategies for minimizing tank-to-wheel/tank-to-meters energy losses. An eco-driving management for battery electric vehicles known as predictive trip management is proposed and is implemented using a dynamic programming algorithm to calculate the recommended maximal vehicle speed to safely reach the target destination. This strategy is implemented on a rapid- prototyping hardware on PC level and is demonstrated on a subcompact BEV vehicle. Two different predictive real-time energy managements for (Plug-) HEVs are proposed and both use an indirect shooting method to solve a (switched) optimal control problem. The control strategies are implemented on a rapid prototype hardware on ECU level as (event-triggered) nonlinear model-predictive control. The PHEV strategy can be configured for the operation modes: charge-sustaining and charge-blending. The latter is used when the target destination provides a charging facility and the total driving distance exceeds the electrical range for the current state of charge. In this case the entire electrical energy can be depleted but the internal combustion engine has to be started several times, to prevent the high-voltage battery from falling below its minimum value before the target destination is reached.

Modeling Hybrid Vehicles as Switched Systems

This chapter describes the main layouts of hybrid powertrains including all relevant mechatronic subsystems. The focus of this chapter is to describe the hybrid powertrains as switched systems.

Discretization and Integration Schemes for Hybrid Optimal Control Problems

The practical problems of interest will seldom have an analytical solution and numerical integration is the only way to obtain information about the trajectory. In this chapter, the famous Runge–Kutta discretizations process is introduced. The determination of the Runge–Kutta order is briefly discussed and conditions up to the fourth order are given including the additional conditions for solving optimal control problems. Regarding optimal control problems only explicit and implicit Runge–Kutta discretizations which satisfy additional conditions for the adjoint differential equation are discussed.

Practical Implementation Aspects of Large-Scale Optimal Control Solvers

The direct transcription methods of optimal control problems lead to large-scale nonlinear programming problems. One suitable framework for the solution of this type of optimization problems is sequential quadratic programming, which is described in Chap. 2. But it is crucial for large-scale applications, that the SQP-algorithm takes into account the particular properties and structure of the objective and constraint functions. The Karush–Kuhn–Tucker matrices, which occur in the subproblems, must be sparse, so that the linear equation systems can be efficiently solved. To accomplish this task for general problems the structure of the matrix must be determined, the derivatives have to be calculated, and a sparse Quasi-Newton update has to be implemented.

Indirect Methods for Optimal Control

In this chapter, indirect methods to solve optimal control problems are discussed. Indirect methods rely on first-order necessary conditions, summarized in Pontryagin’s minimum principle, and attempt to locate control and state trajectories, which satisfy these conditions. An extension of the indirect shooting method for switched systems that yield a solution for systems of low complexity is presented in this chapter.

Advanced Vehicle Calibration

In this chapter, a set of optimal control problems are formulated and the appropriate algorithms described in this book will be applied to their solution. Results are then compared and the applicability of each algorithm is discussed. Yet, obtaining information for the calibration and functional design of energy management from the solution of an optimal control problem is a rather heuristic and cumbersome process and it is rather unlikely that a satisfying calibration will be obtained in a reasonable time span. Also, this process does not exploit the full potential of the underlying theory. With some further assumptions that have only minor effects on the quality of the solution, results from the optimal control problem solution can be used directly to obtain parameters and lookup tables for rule-based energy management, which dramatically facilitates the calibration process and improves the quality of the results obtained.