A consistent direct transcription method for optimal control problems in multibody dynamics

The present work deals with optimal control problems governed by differential-algebraic equations (DAEs). In particular, the control effort, which is necessary for moving a multibody system from one configuration to another, will be minimized. The orientation of the rigid bodies will be described using directors, which facilitates the integration of the equations of motion with an energy-momentum consistent time-stepping scheme [1]. This type of structure-preserving integrators offer outstanding numerical stability and robustness properties in comparison to the often applied generalized coordinates formulation. In the context of optimal control, other kinds of consistent integrators have been applied previously in [2] and [3]. We will test the different formulations with two numerical examples, a 3-link manipulator and a satellite. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symbolic-numeric efficient solution of optimal control problems for multibody systems

This paper presents an efficient symbolic-numerical approach for generating and solving the boundary value problem-differential algebraic equation (BVP-DAE) originating from the variational form of the optimal control problem (OCP). This paper presents the method for the symbolic derivation, by means of symbolic manipulation software (Maple), of the equations of the OCP applied to a generic multibody system. The constrained problem is transformed into a nonconstrained problem, by means of the Lagrange multipliers and penalty functions. From the first variation of the nonconstrained problem a BVP-DAE is obtained, and the finite difference discretization yields a nonlinear systems. For the numerical solution of the nonlinear system a damped Newton scheme is used. The sparse and structured Jacobians is quickly inverted by exploiting the sparsity pattern in the solution strategy. The proposed method is implemented in an object oriented fashion, and coded in C++ language. Efficiency is ensured in core routines by using Lapack and Blas for linear algebra.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

A ritz method for an optimal control problem

We generalize and simplify the proofs of the basic papers of Bosarge and Johnson (Refs. 1-3) on a variational procedure for approximating the solution of thestate regular problem. We derive generala priori error bounds for this procedure and apply these results to obtain asymptotic error bounds for the special case of spline-type approximations.     

A method for optimal control in boundary-layer flow

A methodology for active flow control which couples unsteady flow fields and controls is described. Active-control methods are used to maintain laminar flow in a region in which the natural instabilities lead to turbulent flow. The simplest form of control which might achieve this objective is the wave-cancellation approach. The case of boundary layer instability suppression is considered as the initial validation and test case. Control is effected through the injection or suction of fluid through a single orifice on the boundary. The optimal control theory provides an approach which does not require a priori knowledge of the flow. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multigrid method for constrained optimal control problems

We consider the fast and efficient numerical solution of linear-quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal-dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner.We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.     

A nonlinear finite element framework for flexible multibody dynamics

We present a uniform treatment of rigid body dynamics and nonlinear structural dynamics. The advocated approach is based on a rotationless formulation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holonomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional Hamiltonian systems. To demonstrate the superior numerical performance of the proposed methodology, the numerical examples deals with a multibody system containing both rigid and flexible bodies undergoing large deformations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

Spatiotemporal dynamics analysis and optimal control method for an SI reaction-diffusion propagation model

In the new social media era, it is becoming increasingly important to explore the propagation rules for rumors in social networks. This article is concerned with investigating a diffusive susceptible-infected rumor propagation model with a nonlinear propagation function in a spatially heterogeneous environment. We establish the uniform persistence and analyze the asymptotic behavior of the rumor-spreading steady state for the spatially heterogeneous model when one of the diffusion coefficients tends to zero. Moreover, to better reflect the effect of a time delay on the process of rumor propagation, we establish a spatially homogeneous model with a time delay and prove the existence and local stability of the corresponding equilibrium point. Furthermore, the optimal control in the spatially homogeneous environment case is derived. Finally, several numerical simulations are performed to verify the theoretical results in both spatially heterogeneous and spatially homogeneous systems.     

Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems

In a recent series of papers, the class of energy-conserving Runge-Kutta methods named Hamiltonian BVMs (HBVMs) has been defined and studied. Such methods have been further generalized for the efficient solution of general conservative problems, thus providing the class of Line Integral Methods (LIMs). In this paper we derive a further extension, which we name Enhanced Line Integral Methods (ELIMs), more tailored for Hamiltonian problems, allowing for the conservation of multiple invariants of the continuous dynamical system. The analysis of the methods is fully carried out and some numerical tests are reported, in order to confirm the theoretical achievements.     

A method for solving a quadratic optimal control problem

Hestenes' method of multipliers is used to approximate a quadratic optimal control problem. The global existence of a family of unconstrained problems is established. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original differentially constrained problem.The preparation of this paper was sponsored in part by the U.S. Army Research Office under Grant No. DA-31-124-ARO(D)-355.     

A pointwise quasi-Newton method for unconstrained optimal control problems

Summary For a class of unconstrained optimal control problems we propose a quasi-Newton method that exploits the structure of the problem. We define a new type of superlinear convergence for sequences in function spaces and prove superlinear convergence of the iterates generated by the quasi-Newton method in this sense.This author supported by NSF grants # DMS-8300841 and # DMS-8500844     

A filled function method for optimal discrete-valued control problems

In this paper, we present a new approach to solve a class of optimal discrete-valued control problems. This type of problem is first transformed into an equivalent two-level optimization problem involving a combination of a discrete optimization problem and a standard optimal control problem. The standard optimal control problem can be solved by existing optimal control software packages such as MISER 3.2. For the discrete optimization problem, a discrete filled function method is developed to solve it. A numerical example is solved to illustrate the efficiency of our method.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

非光滑最优控制问题的一种数值解法

针对非光滑最优控制问题提出一种分段数值解法.首先对问题进行全局拟谱离散,然后选取分点,将时间区域进行剖分,在每段区域上对问题进行离散,离散过程采用Chebyshev-Legendre拟谱方法,可以有效借助快速Legendre变换提高算法的运算效率,比现有算法在很大程度上节省了计算时间.给出了相关的理论分析,数值结果表明方法的高精度和有效性.