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Direct Trajectory Optimization and Costate Estimation of FiniteHorizon and InfiniteHorizon Optimal Control Problems Using a Radau pseudospectral Method
 Computational Optimization and Applications
, 2011
"... A method is presented for direct trajectory optimization and costate estimation using global collocation at LegendreGaussRadau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the ..."
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Cited by 37 (24 self)
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A method is presented for direct trajectory optimization and costate estimation using global collocation at LegendreGaussRadau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the terminal point. The resulting integration matrix is nonsingular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finitehorizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem (NLP) to the costates of the optimal control problem. Finally,
Direct Trajectory Optimization Using a Variable LowOrder Adaptive Pseudospectral Method
 AIAA Journal of Spacecraft and Rockets
"... A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high cur ..."
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Cited by 15 (12 self)
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A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the mesh, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the polynomial. An efficient iterative method is then described for accurately solving a general nonlinear optimal control problem. Using four examples, the adaptive pseudospectral method described in this paper is shown to be more efficient than either a global pseudospectral method or a fixedorder method. Nomenclature C = path constraint function D = N N 1 Radau pseudospectral differentiation matrix E = maximum absolute solution error Fd = magnitude of drag force, N Fg = magnitude of gravity force, N Fl = magnitude of lift force, N
Practical stabilization through realtime optimal control
 Proc. of American Control Conference
, 2006
"... Abstract — Infinitehorizon, nonlinear, optimal, feedback control is one of the fundamental problems in control theory. In this paper we propose a solution for this problem based on recent progress in realtime optimal control. The basic idea is to perform feedback implementations through a domain t ..."
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Cited by 8 (5 self)
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Abstract — Infinitehorizon, nonlinear, optimal, feedback control is one of the fundamental problems in control theory. In this paper we propose a solution for this problem based on recent progress in realtime optimal control. The basic idea is to perform feedback implementations through a domain transformation technique and a Radau based pseudospectral method. Two algorithms are considered: free sampling frequency and fixed sampling frequency. For both algorithms, a theoretical analysis for the stability of the closedloop system is provided. Numerical simulations with random initial conditions demonstrate the techniques for a flexible robot arm and a benchmark inverted pendulum problem. I.
Optimal Feedback Control: Foundations, Examples, and Experimental Results for a New Approach
"... Typical optimal feedback controls are nonsmooth functions. Nonsmooth controls raise fundamental theoretical problems on the existence and uniqueness of state trajectories. Many of these problems are frequently addressed in control applications through the concept of a Filippov solution. In recent ye ..."
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Cited by 2 (1 self)
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Typical optimal feedback controls are nonsmooth functions. Nonsmooth controls raise fundamental theoretical problems on the existence and uniqueness of state trajectories. Many of these problems are frequently addressed in control applications through the concept of a Filippov solution. In recent years, the simpler concept of a solution has emerged as a practical and powerful means to address these theoretical issues. In this paper, we advance the notion of Carathéodory – solutions that stem from the equivalence between closedloop and feedback trajectories. In recognizing that feedback controls are not necessarily closedform expressions, we develop a sampling theorem that indicates that the Lipschitz constant of the dynamics is a fundamental sampling frequency. These ideas lead to a new set of foundations for achieving feedback wherein optimality principles are interwoven to achieve stability and system performance, whereas the computation of optimal controls is at the level of first principles. We demonstrate these principles by way of pseudospectral methods because these techniques can generate Carathéodory – solutions at a sufficiently fast sampling rate even when implemented in a MATLAB ® environment running on legacy computer hardware. To facilitate an exposition of the proposed ideas to a wide audience, we introduce the core principles only and relegate the intricate details to numerous recent references. These principles are then applied to generate pseudospectral feedback controls for the slew maneuvering of NPSAT1, a spacecraft conceived, designed, and built at the Naval Postgraduate School and scheduled to be launched in fall 2007. I.
Costate Approximation in Optimal Control Using Integral Gaussian Quadrature Orthogonal Collocation Methods
"... Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at LegendreGauss and LegendreGaussRadau points. It is shown that the derivative of the costate of the continuoustime optimal control problem is equal to the negative ..."
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Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at LegendreGauss and LegendreGaussRadau points. It is shown that the derivative of the costate of the continuoustime optimal control problem is equal to the negative of the costate of the integral form of the continuoustime optimal control problem. Using this continuoustime relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LegendreGauss and LegendreGaussRadau collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LegendreGauss and LegendreGaussRadau collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LegendreGauss or LegendreGaussRadau points. 1
Pseudospectral Optimal Control for Military and Industrial Applications
, 2006
"... Abstract—During the last decade, pseudospectral methods for optimal control, the focus of this tutorial session, have been rapidly developed as a powerful tool to enable new applications that were previously considered impossible due to the complicated nature of these problems. The purpose of this t ..."
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Abstract—During the last decade, pseudospectral methods for optimal control, the focus of this tutorial session, have been rapidly developed as a powerful tool to enable new applications that were previously considered impossible due to the complicated nature of these problems. The purpose of this tutorial section is to introduce this advanced technology to a wider community of control system engineering. We bring in experts of pseudospectral methods from academia, industry, and military and DoD to present topics covering a large spectrum of pseudospectral methods, including the theoretical foundation, numerical techniques of pseudospectral optimal control, and military/industry applications. Over the last few years, pseudospectral (PS) methods for solving optimal control problems have moved rapidly from mathematical theory to realworld applications. For example,
Numerical Solutions to Optimal PowerFlowConstrained Vibratory Energy Harvesting Problems
"... Abstract — This study addresses the formulation of optimal numerical controllers for stochasticallyexcited vibratory energy harvesters in which a singledirectional power electronic converter is used to regulate powerflow. Singledirectional converters have implementation advantages for smallsca ..."
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Abstract — This study addresses the formulation of optimal numerical controllers for stochasticallyexcited vibratory energy harvesters in which a singledirectional power electronic converter is used to regulate powerflow. Singledirectional converters have implementation advantages for smallscale applications, but restrict the domain of feasible controllers. Optimizing the average power generated in such systems can be accomplished by formulating the constrained control problem in terms of stochastic HamiltonJacobi theory. However, solving the stochastic HamiltonJacobi equation (HJE) is challenging because it is a nonlinear partial differential equation. As such, we investigate the capability of the pseudospectral (PS) method to solve the HJE with mixed statecontrol constraints. The performance of the PS controller is computed for a singledegreeoffreedom resonant oscillator with electromagnetic coupling. We compare the PS performance to the performance of the optimal static admittance controller as well as the optimal unconstrained linearquadraticGaussian controller. Index Terms — Energy harvesting, optimal control, constrained control systems, stochastic systems.
AAS 09405 PSEUDOSPECTRAL OPTIMAL CONTROL ON ARBITRARY GRIDS
"... In advancing our prior work on a unified theory for pseudospectral (PS) optimal control, we present new results for PS methods over arbitrary grids. These results provide a way to compare performances among different PS methods and suggest guidelines to choose the proper grids and discretization app ..."
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In advancing our prior work on a unified theory for pseudospectral (PS) optimal control, we present new results for PS methods over arbitrary grids. These results provide a way to compare performances among different PS methods and suggest guidelines to choose the proper grids and discretization approaches for solving optimal
Nomenclature
"... A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both themesh spacing and the degree of the polynomial on eachmesh interval until a specified error tolerance is satisfied. In regions of relatively high curva ..."
Abstract
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A variableorder adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both themesh spacing and the degree of the polynomial on eachmesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the mesh, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the polynomial. An efficient iterativemethod is then described for accurately solving a general nonlinear optimal control problem. Using four examples, the adaptive pseudospectral method described in this paper is shown to be more efficient than either a global pseudospectral method or a fixedorder method.