Nonlinear Systems and Numerical Methods for HJB Equations

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9971546McEneaneyThe research is proceeding along two directions within the general area of control and estimation for nonlinear systems. The first direction involves a new class of numerical methods for such problems. Specifically, note that one of the most persistent difficulties with control and estimation of significantly nonlinear systems has been computational cost. Dynamic programming methods for H-infinity control and risk sensitive stochastic control lead to nonlinear PDEs (typically Hamilton-Jacobi-Bellman equations) which must be solved over some region of the state-space. Since the number of state variables can be quite high, the dimension of the regions over which one needs to solve these PDEs increases. This restricts the set of problems for which one can compute such controls to those with very low state-space dimension. By changes in the underlying definitions of addition and multiplication certain classes of nonlinear Hamilton-Jacobi-Bellman equations are transformed into problems with linear semi-groups. This property has led us to new methods that exploit this linearity. For instance, certain steady-state problems lead to spectral methods for linear problems (over the new algebras). It is hoped that the possible improvements in speed with these methods will lead to an ability to solve a wider range of problems. Some simple examples are already being tested. The second direction involves risk averse limits in nonlinear control systems. It is now well known that risk averse limits of risk sensitive controllers lead to robust/H-infinity controllers. However, the classes of systems for which this has actually been proven is rather small, and does not include most typical problems. We have recently obtained some uniqueness results that should allow one to extend the risk sensitive limit results for infinite time-horizon problems to a reasonably wide class. Analogous work on risk sensitive limits in the estimation problem is beginning as well.Many modern systems, from active automotive hydro-suspension systems to fighter aircraft, behave in a significantly nonlinear manner. These systems require active control in order to perform at increasingly competitive levels. There is a serious technical challenge in that the design of nonlinear controllers requires extremely heavy computational loads. In fact, the computation of controllers where the number of system state variables is more than two or three has been unattainable in general. The difficulty is that these computations require the solution of a nonlinear partial differential equation over a space whose dimension is that of the number of state variables. A recent advance is the observation that, despite the nonlinearity, the propagation of these solutions is often still linear IF one switches to the max-plus algebra in place of the traditional definitions of addition and subtraction. In the max-plus algebra, traditional addition is replaced by maximization, and traditional multiplication is replaced by addition. One of the goals of this project is to exploit this max-plus linearity to extend the class of nonlinear systems for which we can design active controllers.