Max-plus eigenvector representations for solution of nonlinear H/sub ∞/ problems: Basic concepts

Systems & Control Letters

2007

Max-plus methods have been explored for solution of first-order, nonlinear HamiltonJacobi-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 advantages, they still generally suffer from the curse-of-dimensionality. The natural analog to the Laplace transform in ordinary spaces is the Fenchel transform over max-plus spaces, the range space being referred to as the dual space. One can transform the semigroup operators into operators on the dual space. There are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems. Natural building blocks correspond to transforms of operators for linear/quadratic problems. In this paper, a method for exploiting operations in the Fenchel transform space is used to develop a method for certain problems such that the computational growth in the most time-consuming portion of the computations can be hugely reduced. Although the curse-of-dimensionality is not entirely avoided, the computational cost reductions are very high for some classes of problems.

Section: Problem Set-up and Results Reviewmentioning

confidence: 99%

Section: Problem Set-up and Results Reviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Systems & Control Letters

2007

“…(Nonetheless, we prefer to write the quadratics in the form of the previous section as it has proved efficient for max-plus methods for deterministic control, c.f. [11], [16], [18].) Since one cannot immediately obtain the affine-constituents case from the quadratic-constituents case of the previous section, we sketch the affine case here.…”

Section: Affine Formsmentioning

confidence: 99%

“…Here, max-plus methods are appropriate for problems with maximizing controllers and vice-versa. These methods include maxplus basis-expansion approaches [1], [2], [6], [7], [11], [15], [18], as well as the more recently developed curse-of-dimensionality-free methods [11], [16], [17]. However, stochastic control problems have eluded idempotent methods.…”

Section: Introductionmentioning

confidence: 99%

Distributed dynamic programming for discrete-time stochastic control, and idempotent algorithms

2011

Automatica

Previously, idempotent methods have been found to be extremely fast for solution of dynamic programming equations associated with deterministic control problems. The original methods exploited the idempotent (e.g., max-plus) linearity of the associated semigroup operator. However, it is now known that the curse-of-dimensionality-free idempotent methods do not require this linearity. Instead, it is sufficient that certain solution forms are retained through application of the associated semigroup operator. Here, we see that idempotent methods may be used to solve some classes of stochastic control problems. The key is the use of the idempotent distributive property. This allows one to apply the curse-of-dimensionality-free idempotent approach. We demonstrate this approach for a class of nonlinear, discrete-time stochastic control problems.

Min–Plus Eigenvector Methods for Nonlinear $\rm H_\infty$ Problems with Active Control

Collins

Optimal Control, Stabilization and Nonsmooth Analysis

2004

We consider the H ∞ problem for an onlinear s y stem. The corresponding d ynamic programming equation (DPE) is af ully nonlinear, first-order, steadystate partial differential equation (PDE), possessing aterm whichisquadratic in the gradient(for background, see [2], [4], [13], [25] among many notable others). The solutions are t y pically nonsmooth, and further, there are multiple viscosity solutions -t hat is, one does not even have uniqueness among the class of viscosity solutions (cf. [17]). Note that the nonuniqueness is not only due to an additivec onstant, but is much more difficult. If one removest he additiveconstantissue b y requiring the solution to be z ero at the origin, then, for example, the linear-quadratic case t y pically has twoclassical solutions and an infinite number of viscosity solutions [17]. Since the H ∞ problem is ad ifferential game, the PDE is, in general, aH amilton-Jacobi-Isaacs (HJI) PDE. The computation of the solution of an onlinear, steady -state, first-order PDE is t y pically quite difficult, and possibly even more so in the presence of the non-uniqueness mentioned above.S ome previous works in the general area of numerical methods for these problems are [3], [6], [7], [14], and the references therein.In recent y ears, an ew set of methods based on max-plus linearity have been developed for the H ∞ problem in the design case. In that problem class, afeedbackcontrol and disturbance attenuation parameter are chosen, and the associated PDE is solved to determine whether this is indeed an H ∞ controller with that disturbance attenuation level [18], [19], [21], [23]. Note that in that problem class, the PDE is aH amilton-Jacobi-Bellman (HJB) PDE. Here, we address ac lass of activec ontrol H ∞ problems where the controller is able