19,865 research outputs found
On Weak Topology for Optimal Control of Switched Nonlinear Systems
Optimal control of switched systems is challenging due to the discrete nature
of the switching control input. The embedding-based approach addresses this
challenge by solving a corresponding relaxed optimal control problem with only
continuous inputs, and then projecting the relaxed solution back to obtain the
optimal switching solution of the original problem. This paper presents a novel
idea that views the embedding-based approach as a change of topology over the
optimization space, resulting in a general procedure to construct a switched
optimal control algorithm with guaranteed convergence to a local optimizer. Our
result provides a unified topology based framework for the analysis and design
of various embedding-based algorithms in solving the switched optimal control
problem and includes many existing methods as special cases
Strong Stationarity Conditions for Optimal Control of Hybrid Systems
We present necessary and sufficient optimality conditions for finite time
optimal control problems for a class of hybrid systems described by linear
complementarity models. Although these optimal control problems are difficult
in general due to the presence of complementarity constraints, we provide a set
of structural assumptions ensuring that the tangent cone of the constraints
possesses geometric regularity properties. These imply that the classical
Karush-Kuhn-Tucker conditions of nonlinear programming theory are both
necessary and sufficient for local optimality, which is not the case for
general mathematical programs with complementarity constraints. We also present
sufficient conditions for global optimality.
We proceed to show that the dynamics of every continuous piecewise affine
system can be written as the optimizer of a mathematical program which results
in a linear complementarity model satisfying our structural assumptions. Hence,
our stationarity results apply to a large class of hybrid systems with
piecewise affine dynamics. We present simulation results showing the
substantial benefits possible from using a nonlinear programming approach to
the optimal control problem with complementarity constraints instead of a more
traditional mixed-integer formulation.Comment: 30 pages, 4 figure
Discrete Mechanics and Optimal Control Applied to the Compass Gait Biped
This paper presents a methodology for generating locally optimal control policies for simple hybrid mechanical systems, and illustrates the method on the compass gait biped. Principles from discrete mechanics are utilized to generate optimal control policies as solutions of constrained nonlinear optimization problems. In the context of bipedal walking, this procedure provides a comparative measure of the suboptimality of existing control policies. Furthermore, our methodology can be used as a control design tool; to demonstrate this, we minimize the specific cost of transport of periodic orbits for the compass gait biped, both in the fully actuated and underactuated case
Consistent Approximations for the Optimal Control of Constrained Switched Systems
Though switched dynamical systems have shown great utility in modeling a
variety of physical phenomena, the construction of an optimal control of such
systems has proven difficult since it demands some type of optimal mode
scheduling. In this paper, we devise an algorithm for the computation of an
optimal control of constrained nonlinear switched dynamical systems. The
control parameter for such systems include a continuous-valued input and
discrete-valued input, where the latter corresponds to the mode of the switched
system that is active at a particular instance in time. Our approach, which we
prove converges to local minimizers of the constrained optimal control problem,
first relaxes the discrete-valued input, then performs traditional optimal
control, and then projects the constructed relaxed discrete-valued input back
to a pure discrete-valued input by employing an extension to the classical
Chattering Lemma that we prove. We extend this algorithm by formulating a
computationally implementable algorithm which works by discretizing the time
interval over which the switched dynamical system is defined. Importantly, we
prove that this implementable algorithm constructs a sequence of points by
recursive application that converge to the local minimizers of the original
constrained optimal control problem. Four simulation experiments are included
to validate the theoretical developments
Determination of the adjoint state evolution for the efficient operation of a hybrid electric vehicle
To minimize the fuel consumption in hybrid electric vehicles, it is necessary to define a strategy for the management of the power flows within the vehicle. Under the assumption that the velocity to be developed by the vehicle is known a priori, this problem may be posed as a nonlinear optimal control problem with control and state constraints. We find the solution to this problem using the optimality conditions given by the Pontryagin Maximum Principle. This leads to boundary value problems that we solve using a software tool named PASVA4. On real time operation, the velocity to be developed by the vehicle is not known in advance. We show how the adjoint state obtained from the former problem may be used as a weighing factor, called ‘‘equivalent consumption’’. This weighing factor may be used to design suboptimal real time algorithms for power management.Fil: Perez, Laura Virginia. Universidad Nacional de Rio Cuarto. Facultad de Ingeniería. Grupo de Electronica Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: de Angelo, Cristian Hernan. Universidad Nacional de Rio Cuarto. Facultad de Ingeniería. Grupo de Electronica Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pereyra, Víctor. San Diego State University; Estados Unido
L∞ estimates on trajectories confined to a closed subset, for control systems with bounded time variation
The term ‘distance estimate’ for state constrained control systems refers to an estimate on the distance of an arbitrary state trajectory from the subset of state trajectories that satisfy a given state constraint. Distance estimates have found widespread application in state constrained optimal control. They have been used to establish regularity properties of the value function, to establish the non-degeneracy of first order conditions of optimality, and to validate the characterization of the value function as a unique solution of the HJB equation. The most extensively applied estimates of this nature are so-called linear L∞L∞ distance estimates. The earliest estimates of this nature were derived under hypotheses that required the multifunctions, or controlled differential equations, describing the dynamic constraint, to be locally Lipschitz continuous w.r.t. the time variable. Recently, it has been shown that the Lipschitz continuity hypothesis can be weakened to a one-sided absolute continuity hypothesis. This paper provides new, less restrictive, hypotheses on the time-dependence of the dynamic constraint, under which linear L∞L∞ estimates are valid. Here, one-sided absolute continuity is replaced by the requirement of one-sided bounded variation. This refinement of hypotheses is significant because it makes possible the application of analytical techniques based on distance estimates to important, new classes of discontinuous systems including some hybrid control systems. A number of examples are investigated showing that, for control systems that do not have bounded variation w.r.t. time, the desired estimates are not in general valid, and thereby illustrating the important role of the bounded variation hypothesis in distance estimate analysis
An hybrid system approach to nonlinear optimal control problems
We consider a nonlinear ordinary differential equation and want to control
its behavior so that it reaches a target by minimizing a cost function. Our
approach is to use hybrid systems to solve this problem: the complex dynamic is
replaced by piecewise affine approximations which allow an analytical
resolution. The sequence of affine models then forms a sequence of states of a
hybrid automaton. Given a sequence of states, we introduce an hybrid
approximation of the nonlinear controllable domain and propose a new algorithm
computing a controllable, piecewise convex approximation. The same way the
nonlinear optimal control problem is replaced by an hybrid piecewise affine
one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce
the global structure of the hybrid optimal control steering the system to the
target
- …