Randomized algorithms and global optimization for optimal and robust control

This dissertation explores the capabilities of randomized global optimization algorithms on a selected set of difficult control problems for which analytic solutions do not exist. Two control problems are considered. The first is to synthesize the open-loop control of a clutch-to-clutch shift in an automotive automatic transmission. The second is a robust stability analysis problem with real parametric uncertainty.In the first part of the dissertation, the global optimization problem is presented. Finding the global minimum of a generic cost function can very difficult since the function may have many local minima or be nondifferentiable. These and other challenges present in global optimization are addressed. Following this is a description of the features that a general global optimization algorithm should incorporate as well as examples of such algorithms. Our focus is on randomized algorithms.Next, we introduce a simplified model of an automotive powertrain containing a clutch-to-clutch shifting automatic transmission and motivate the clutch-to-clutch shifting problem. The problem is to find the optimal clutch pressure control input trajectories which produce a comfortable shift and that will not wear down the transmission components prematurely. While the automotive powertrain model is a simplified one, it is complex enough that computational means must be used to solve the optimal control problem. In the approach of this study, a cost function is formulated which captures the requirements of a good shift. Each evaluation of the cost function requires a dynamic simulation of the powertrain model to be run. Secondly, the clutch pressure controls are parameterized and are used as the input parameters to the cost function. In this way, the minimizer of the cost function is the optimal control input. The randomized algorithms approach to finding the global minimum is shown to be an effective technique for finding the optimal control. In addition, a local optimization algorithm is also tested for comparison purposes and shown that it consistently finds inferior solutions.Even for finite dimensional linear time-invariant control systems, the problem of robust stability analysis is very difficult to handle analytically. It has been shown that in the case of real parametric uncertainty these problems are NP-hard.Due to these results on computational complexity, we propose an alternative approach to the robust stability problem which may be useful. Rather than concentrating on analytic results, the strategy is to pose the problem as a global optimization problem and use randomized algorithms to solve it. Two different problem settings of the real...