In previous works of the author and others, max-plus methods have been explored for the solution of first-order, nonlinear Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. These methods exploit the max-plus linearity of the associated semigroups. In particular, although the problems are nonlinear, the semigroups are linear in the max-plus sense. These methods have been used successfully to compute solutions. Although they provide certain computational-speed advantages, they still generally suffer from the curse of dimensionality. Here we consider HJB PDEs in which the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. The approach to the solution will be rather general, but in order to ground the work, we consider only constituent Hamiltonians corresponding to long-run average-cost-per-unit-time optimal control problems for the development. We obtain a numerical method not subject to the curse of dimensionality. The method is based on construction of the dual-space semigroup corresponding to the HJB PDE. This dual-space semigroup is constructed from the dual-space semigroups corresponding to the constituent linear/quadratic Hamiltonians. The dual-space semigroup is particularly useful due to its form as a max-plus integral operator with a kernel obtained from the originating semigroup. One considers repeated application of the dual-space semigroup to obtain the solution.

A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs

The automotive hydro-pneumatic integrated suspen- sion model is nonlinear with large dimension. As a consequence, the nonlinear control methodology based on the traditional Hamilton-Jacoby-Isaacs equation is impractical in this appli- cation. An alternative so-called parameterized linear matrix inequality (PLMI) approach is proposed for solving this hard nonlinear control problem. The validity of the proposed approach is confirmed not only by detailed and realistic simula- tions but also by extensive experiments. Specifically, the proposed nonlinear control method outperforms the more classical feedback linearization control technique.

Control for an Integrated Suspension System via Parameterized Linear Matrix Inequality Characterizations

Max-plus summation of Fenchel -transformed semigroups for solution of nonlinear Bellman equations

In this paper we propose the use of max-plus approximation methods in nonlinear H/sub /spl infin// control. Using the max-plus linearity of the solution operator for the information state, we show how the information state can be represented as a max-plus expansion relative to what we call a monotone approximate basis. The forward time propagation of the information state is expressed as max-plus matrix multiplication. This leads to a natural approximation of the optimal value function and the optimal feedback function.

Max-plus methods in partially observed H/sub /spl infin// control

Max-plus methods have been explored for solution of first-order, nonlinear Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. These methods exploit the max-plus linearity of the associated semigroups. Although these methods provide advantages, they still suffer from the curse-of-dimensionality. Here we consider HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of quadratic forms. We obtain a numerical method not subject to the curse-of-dimensionality. The method is based on construction of the dual-space semigroup corresponding to the HJB PDE. This dual-space semigroup is constructed from the dual-space semigroups corresponding to the constituent quadratic Hamiltonians. The actual computations in the algorithm involve repeatedly computing coefficients of quadratics which are obtained as the maxima of two other quadratics.

/pdf/curse-of-dimensionality-free-method-for-bellman-pdes-with-3rrs17hqkd.pdf

Curse-of-Dimensionality Free Method for Bellman PDEs with Hamiltonian Written as Maximum of Quadratic Forms

The H, problem for nonlinear systems is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, partial differential equation. Interestingly, if one switches from the normal definition of addition and multiplication to the maxplus algebra (which is no more complex), the solution operator becomes a linear operator. The sciution can be expanded using a max-plus basis. The coefficients in this expansion satisfy a max-plus eigenvector equation for a matrix associated with this solution operator - thus transforming the nonlinear problem into a linear one. In fact there is a parameterized family of matrices for which this holds. Expressions and approximations for the coefficients in these matrices are given.

https://ieeexplore.ieee.org/iel5/6103/16405/00758250.pdf

William M. McEneaney

Papers

Max-Plus Eigenvector Representat ions for Nonlinear H, Value Functions