Henrik Axelsson

Transition-time optimization for switched-mode dynamical systems

This note considers the problem of determining optimal switching times at which mode transitions should occur in multimodal, hybrid systems. It derives a simple formula for the gradient of the cost functional with respect to the switching times, and uses it in a gradient-descent algorithm. Much of the analysis is carried out in the setting of optimization problems involving fixed switching-mode sequences, but a possible extension is pointed out for the case where the switching-mode sequence is a part of the variable. Numerical examples testify to the viability of the proposed approach.

TRANSITION-TIME OPTIMIZATION FOR SWITCHED SYSTEMS

Abstract This paper proposes an algorithmic framework for optimal mode switches in hybrid dynamical systems. The problem is cast in the setting of optimal control, whose variable parameter consists of the switching times, and whose associated cost criterion is a functional of the state trajectory. The number of switching times (and hence of switching modes) is also a variable which may be unbounded, and therefore the optimization problem is not defined on a single metric space. Rather, it is defined on a sequence of spaces of possibly increasing dimensions. The paper characterizes optimality in terms of sequences of optimality functions and proposes an algorithm that is demonstrably convergent in this context.

Autonomous Formation Switching for Multiple, Mobile Robots

Abstract In this paper we investigate the question concerning what multi-agent formations to use in a given situation. In particular, we show how it is possible to produce a control strategy for teams of mobile robots that switches between different formations as a reaction to environmental changes. The feasibility of the approach is verified in simulation, where a steepest descent algorithm is combined with standard reactive behaviors that ensure that the individual robots avoid neighboring obstacles and robots, while approaching a desired target location

A Provably Convergent Algorithm for Transition-Time Optimization in Switched Systems

This paper concerns a mode-sequencing and switching-time optimization problem defined on autonomous switched-mode hybrid dynamical systems. The design parameter consists of two elements: (i) the sequence of dynamic-response functions associated with the modes, and (ii) the duration of each mode. The sequencing element is a discrete parameter which may render the problem of computing the optimal schedule exponentially complex. Therefore we are not seeking a global minimum, but rather a local solution in a suitable sense. To this end we endow the parameter space with a local continuous structure which allows us to apply gradient-descent techniques. With this structure, the problem is cast in the form of a nonlinear-programming problem defined on a sequence of nested Euclidean spaces with increasing dimensions. We charcterize suboptimality in an appropriate sense, define a corresponding convergence criterion, and devise a provably-convergent optimization algorithm.

Algorithm for Switching-Time Optimization in Hybrid Dynamical Systems

We consider the problem of minimizing a cost functional defined on the state trajectory of a switched-mode dynamical system with respect to the switching times. Following the derivation, in recent years, of various results concerning the gradient of the cost functional, we present a suitable algorithm, based on gradient projection, for computing local minima. Utilizing the problems special structure, we prove a convenient formula for the direction of descent, and apply the Armijo procedure for computing the step size. A potential extension to the optimal mode-insertion problem is discussed, and numerical examples are provided

Reactive robot navigation using optimal timing control

In this paper a solution is presented for the problem of avoiding obstacles while progressing towards a goal for a single robot. In particular, an optimal solution is obtained by allowing the robot to switch between a fixed number of behaviors and optimizing over what behaviors to use and when to switch between them. We moreover show that the structure of the switch law only depends on the distance between the obstacle and the goal. Hence, once initial simulations are done, the structure of the guard is known to the robot and, given that the robot knows the distance between the obstacle and the goal, it knows when to switch to obtain the optimal solution. Therefore the solution lends itself to real-time implementations. The feasibility of the approach is verified in real robotics experiments.

Optimal mode-switching for hybrid systems with unknown initial state

Abstract This paper concerns an optimal control problem defined on a class of switched mode hybrid dynamical systems. Such systems change modes whenever the state intersects certain surfaces that are defined in the state space. These switching surfaces are parameterized by a finite dimensional vector called the switching parameter. The optimization problem we consider is to minimize a given cost-functional with respect to the switching parameter under the assumption that the initial state of the system is not completely known. Instead, we assume that the initial state can be anywhere in a given set. We will approach this problem by minimizing the worst possible cost over the given set of initial states using results from minimax optimization. The results are then applied in order to solve a navigation problem in mobile robotics.

Optimal Switching Surfaces in Behavior-Based Robotics

In this paper an optimal solution is presented for the problem of avoiding obstacles while progressing towards a goal for a single robot. In particular, the solution is obtained by allowing the robot to switch between a fixed number of behaviors and optimizing over what behaviors to use and when to switch between them. It is moreover shown that the structure of the switching law only depends on the distance between the obstacle and the goal. Hence, once initial simulations are done, a guard can be generated with a fixed structure, and, given that the robot knows the distance between the obstacle and the goal, it knows when to switch in order to execute the pre-computed (optimal) solution. Therefore the solution lends itself nicely to real-time implementations. Experiments moreover verify that the proposed methods transitions well onto a real robotic platform

Trade-offs between precision and computation horizon in real-time optimal control of switched systems

In this paper the problem of real-time optimal control is considered. In particular, we will study dynamical systems that switch between different modes at controlled switching times. Ideally, one would want the switching controller to provide sufficiently good values for the switching times at every instant of time. However, due to the limited computational resources available in many real-time applications, questions concerning trade-offs between the computation horizon and the precision of the solution arise naturally. These trade-offs constitute the main focus of this paper, and a solution will be provided based on the minimization of a conservative bound on the error between the true gradient and the gradient obtained in real-time