Perturbation methods in large scale systems

Perturbation Methods are physically motivated tools to model dynamic interactions in naturally decentralized hierarchical structures. Application of the singular perturbation approach to differential game problems is discussed. Perturbation methods are useful for dealing with systems that can be approximated by a system of simpler structure. The difference between the actual system structure and the simplified system structure is modelled as a set of parameters which when neglected result in the simplified system. Two classes of perturbation methods are of interest: regular perturbations and singular perturbations. In regular perturbations, neglecting the parameters simplifies the system but does not reduce its order, e.g., perturbation parameters representing weak coupling among interconnected systems. In singular perturbations neglecting the parameters reduces the order of the system, e.g., perturbation parameters representing fast modes in a two-time scale system. The parameterization of the model simplifying process has provided analytical tools to study the behavior of simplified models. One aspect of perturbation analysis which I would rather emphasize here is its use to show the well-posedness of control design problems. To make myself clear I will consider a specific example, i.e. the linear quadratic optimal control problem where the system equations contain perturbation parameters either regularly or singularly. If the optimization problem is solved for the actual model and for the simplified model where the perturbation parameters are neglected, then it is well-known that for sufficiently small parameters the two solutions are sufficiently close. This basic continuity property of the solution has a theoretical as well as a practical significance. Differential game problems do not necessarily have the well-posedness property of the optimal control problem. A recent investigation of singularly perturbed Nash and Stackelberg strategies reveals an important relation between well-posedness and feedback information. It has been also found that while a direct simplification of the system may lead to an ill-posed simplified problem, an alternative simplification which takes into consideration the particular problem in hand may be well-posed. These and other aspects of well-posedness of design problems are now under investigation.