We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approximate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with three examples of rigid body attitude control using momentum wheels.

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

A game theoretic algorithm to compute local stabilizing solutions to HJBI equations in nonlinear H∞ control

In this paper, we study nonlinear control of a spacecraft symmetric about its principal axis with two control torques. Using a computationally efficient H ∞ control design procedure, attitude stabilization and command tracking problems of the axisymmetric spacecraft are solved locally. The proposed nonlinear H ∞ control approach uses higher order Lyapunov functions and reformulates the difficult Hamilton―Jacobian―Isaacs inequalities as semidefinite optimization conditions. Sum-of-squares programming techniques are then applied to obtain computationally tractable solutions, from which nonlinear control laws will be constructed. The nonlinear H ∞ control designs for spacecraft are capable of exploiting the most suitable forms of Lyapunov functions for performance improvement.

Nonlinear H_infinity Control Designs with Axisymmetric Spacecraft Control

Solving Hamilton-Jacobi-Bellman (HJB) equations is essential in feedback optimal control. Using the solution of HJB equations, feedback optimal control laws can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, numerically solving HJB equations for general nonlinear systems is unfeasible due to the curse of dimensionality. In this paper, we develop a new computational method of solving HJB equations. The method is causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as 6-D HJB equations for the attitude control of rigid bodies. The method is applied to the optimal attitude control of a satellite system using momentum wheels. The accuracy of the numerical solution is verified at a set of randomly selected sample points.

A Causality Free Computational Method for HJB Equations with Application to Rigid Body Satellites

In this paper, we propose a computational scheme of solving the output feedback Hinfin control problem for a class of nonlinear systems with polynomial vector field. The output feedback control design problem will be decomposed into a state feedback and an output estimation problems. Resorting to higher order Lyapunov functions, two Hamilton-Jacobian-Isaacs (HJI) inequalities are first formulated as semi-definite optimization conditions. Sum-of-squares (SOS) programming techniques are then applied to obtain computationally tractable solutions, from which a nonlinear control law will be constructed. The closed-loop system is asymptotically stabilizable by the nonlinear output feedback control and achieves good Hinfin performance under the exogenous disturbances.

Nonlinear output feedback H ∞ control for polynomial nonlinear systems

The design of the control of an aircraft encountering wind shear after takeoff is treated as a problem of minimizing the H/sub infinity / gain between the wind shear and the vertical acceleration of the aircraft. The method of nonlinear H/sub infinity / control is used to design the control law. For the reason of robustness, no information or assumptions about wind shear structure or intensity are used in the design of the controller. The approximate solution of the Hamilton-Jacobi-Isaacs equation is used to design a nonlinear controller to improve the performance. Simulations under both linear and nonlinear controllers are carried out. The performance of the aircraft is good and the angle of attack is smooth. >

W. Kang

Papers

Flight control in a windshear via nonlinear H/sub infinity / methods