Systems Engineering, **Machine** CM

Research School of Physical Sciences and Engineering

**Co-Investigator** Sonny Yuliar, Peter Dowell

Systems Engineering,

Research School of Physical Sciences and Engineering

**H-infinity control**

Our research activities have been in the field of robust H-infinity control for nonlinear dynamical systems. This field addresses the problem of achieving high performance systems in the presence of uncertainties. This problem arises in many engineering problems ranging from flight control, chemical process regulation to automated washing machines. The term "H-infinity" here is a conventional symbol to indicate that in the design process we minimize the maximum or worst effects of the uncertainties on the systems.

Our general goal is to develop a control synthesis theory as well as efficient numerical methods for computing the required control policies. The fundamental equations that we deal with are of the nature of dynamic programming ones. In particular, they are in the form of first order nonlinear partial differential equations (PDE's). We employ a finite difference technique to discretize these PDE's and use viscosity solution methods to analyse the convergence properties of the discretized equations. The development of the basic numerical methods was initiated in 1993. Our work since the beginning of 1994 has been focussing on obtaining accelerated versions of the basic methods.

**What are the basic questions addressed?**

The basic questions are: how to discretize the PDE

inf*u *in* U *sup*w* in *W * (
*DS(x)* . *f(x,u,w) + L(x,u,w)* ) = 0 in *X*,

in which *DS(x)* denotes the gradient of *S(x)* with respect to
*x*, *f,* *L* are determined from the problem data, *U*,
*W*, *X* are spaces of *u*, *w*, *x* respectively,
and how to compute a solution to the discretized equation efficiently.

**What are the results to date and the future of the work?**

We employ a finite difference method to discretize the PDE and a viscosity solution technique to obtain convergence properties. The discretized PDE is computed using value space and policy space iteration methods. These constitute our basic numerical schemes. We have also obtained accelerated versions of the basic schemes. The future of the work is to utilize the numerical method in a real control design problem.

**What computational techniques are used and why is a supercomputer
required?**

The computational techniques involve iteratively computing
*S ^{k}(x), k = 1,2,*...., for all

* S ^{k}(x) = *inf

in which *d* is the grid size, *p* and *c* are scalar functions,
and the notation *U ^{d}* indicates the discretized

**Publications**

*Numerical Approximation of the H[[infinity]] Norm for Nonlinear
Systems*, M R James, and S Yuliar, to appear in Automatica, June, 1995.

*Robust Output Feedback Control for Bilinear Systems*, C A Teolis, S
Yuliar, M R James, and J Baras, in Proc. 33rd IEEE Conference on Decision and
Control (CDC), USA, December 1994.