A unified framework for hybrid control (original) (raw)

A unified framework for hybrid control: b background, model, and theory

1994

We propose a very general framework for hybrid control problems which encompasses several types of hybrid phenomena considered in the literature. A specific control problem is studied in this framework, leading to an existence result for optimal controls. The "value function" associated with this problem is expected to satisfy a set of "generalized quasi-variational inequalities" which are formally derived.

A unified framework for hybrid control: model and optimal control theory

IEEE Transactions on Automatic Control, 1998

Complex natural and engineered systems typically possess a hierarchical structure, characterized by continuousvariable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today-from flight control to the factory floor-perform computer-coded checks and issue logical as well as continuous-variable control commands. The interaction of these different types of dynamics and information leads to a challenging set of "hybrid" control problems. We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure on which to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in an optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and near-optimal (precise) controls and derive "generalized quasi-variational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.

Optimal Control of Hybrid Systems

2005

This paper presents a method for optimal control of hybrid systems. An inequality of Bellman type is considered and every solution to this inequality gives a lower bound on the optimal value function. A discretization of this "hybrid Bellman inequality" leads to a convex optimization problem in terms of finitedimensional linear programming. From the solution of the discretized problem, a value function that preserves the lower bound property can be constructed. An approximation of the optimal feedback control law is given and tried on some examples.

On the Hybrid Optimal Control Problem: Theory and Algorithms

IEEE Transactions on Automatic Control, 2007

A class of hybrid optimal control problems (HOCP) for systems with controlled and autonomous location transitions is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented which together constitute generalizations of the standard Maximum Principle; these are given for the cases of open bounded control value sets and compact control value sets. The derivations in the paper employ: (i) classical variational and needle variation techniques; and (ii) a local controllability condition which is used to establish the adjoint and Hamiltonian jump conditions in the autonomous switching case. Employing the hybrid minimum principle (HMP) necessary conditions, a class of general HMP based algorithms for hybrid systems optimization are presented and analyzed for the autonomous switchings case and the controlled switchings case. Using results from the theory of penalty function methods and Ekeland's variational principle the convergence of these algorithms is established under reasonable assumptions. The efficacy of the proposed algorithms is illustrated via computational examples.

An Optimal Control Approach for Hybrid Systems

European Journal of Control, 2003

In this paper optimal control for hybrid systems will be discussed. While defining hybrid systems as causal and consistent dynamical systems, a general formulation for an optimal hybrid control problem is proposed. The main contribution of this paper shows how necessary conditions can be derived from the maximum principle and the Bellman principle. An illustrative example shows how optimal hybrid control via a set of Hamiltonian systems and using dynamic programming can be achieved. However, as in the classical case, difficulties related to numerical solutions exist and are increased by the discontinuous aspect of the problem. Looking for efficient algorithms remains a difficult and open problem which is not the purpose of this contribution.

Modeling, Analysis, and Optimal Control of a Class of Hybrid Systems

Discrete Event Dynamic Systems, 1998

We propose a modeling framework for a class of hybrid systems which arise in many manufacturing environments and study related optimal control problems. In this framework, discrete entities have a state characterized by a temporal component whose evolution is described by event-driven dynamics, and a physical component whose evolution is described by time-driven dynamics. As a first step towards developing

Optimal control of hybrid systems with an infinite set of discrete states

Journal of dynamical and control systems, 2003

Hybrid control systems are described by a family of continuous subsystems and a set of logic rules for switching between them. This paper concerns a broad class of optimization problems for hybrid systems, in which the continuous subsystems are modelled as differential inclusions. The formulation allows endpoint constraints and a general objective function that includes "transaction costs" associated with abrupt changes of discrete and continuous states, and terms associated with continuous control action as well as the terminal value of the continuous state. In consequence of the endpoint constraints, the value function may be discontinuous. It is shown that the collection of value functions (associated with all discrete states) is the unique lower semicontinuous solution of a system of generalized Bensoussan-Lions type quasi-variational inequalities, suitably interpreted for nondifferentiable, extended valued functions. It is also shown how optimal strategies and value functions are related. The proof techniques are system theoretic, i.e., based on the construction of state trajectories with suitable properties. A distinctive feature of the analysis is that it permits an infinite set of discrete states.

Optimal Control of Hybrid Systems Using a Feedback Relaxed Control Formulation

arXiv: Optimization and Control, 2015

We present a numerically tractable formulation for computing the optimal control of the class of hybrid dynamical systems whose trajectories are continuous. Our formulation, an extension of existing relaxed-control techniques for switched dynamical systems, incorporates the domain information of each discrete mode as part of the constraints in the optimization problem. Moreover, our numerical results are consistent with phenomena that are particular to hybrid systems, such as the creation of sliding trajectories between discrete modes.

Efficient solution of optimal control problems using hybrid systems

2005

We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem: synthesize optimal enabling conditions for switching between locations in which the control is constant. An algorithmic solution is obtained by translating the hybrid automaton to a finite automaton using a bisimulation and formulating a dynamic programming problem with extra conditions to ensure non-Zenoness of trajectories.

Necessary Optimality Conditions for a Class of Hybrid Optimal Control Problems

Lecture Notes in Computer Science, 2007

In this paper we study a class of Mayer-type hybrid optimal control problems. Using Lagrange techniques, we formulate a version of the Hybrid Maximum Principle for optimal control problems governed by hybrid systems with autonomous location transitions in the presence of additional target constraints.