Summaries - Office of Research & Innovation
Back Further Development of the Patchy Method for Solving the Partial Difference and Differential Equations of Nonlinear Control
|Division||Graduate School of Engineering & Applied Science|
|Investigator(s)||Krener, Arthur J.|
|Sponsor||Air Force Office of Scientific Research (Air Force)|
There are great needs for effective algorithms to control highly nonlinear plants such as modem aircraft and spacecraft. Fortunately there has been great progress in nonlinear control theory over the last three decades and many of the needed algorithms have been developed. However their implementation has lagged behind. The principle reason for this is computational. There are few effective computational methods available to solve the nonlinear partial differential equations that are required by the nonlinear theory. This is in contrast to linear control where the theory has been complimented by excellent numerical methods such as those in the MATLAB Control Toolbox. The, goal of this research project is to continue to develop the numerical tools needed to solve Hamilton Jacobi Bellman partial differential equations and others that arise in nonlinear control theory. We have made substantial progress to date using patchy methods to solve Hamilton Jacobi Bellman partial differential equations, the dynamic programming equations and the Francis Byrnes Isidori equation for regulation of nonlinear plants. These equations admit power series solutions in a neighborhood of an operating point and linear algorithms can compute the lowest terms of these series. This yields a solution on some patch of the state space containing the operating point. We have extended the solution on other patches encircling the original patch. These patchy methods have proven to be fast and accurate. We have tested them on a nonlinear optimal control problem where the exact solution the Hamilton Jacobi Bellman partial differential equation is known and they are over 50 times more accurate than standard engineering practice.
One of the difficulties of our patchy approach in higher dimensions is to define the patch structure. We propose to ameliorate this using a partition of unity in place of the patches. We shall also extend the patchy technique to compute the families of optimal trajectories in collaboration with colleagues at the University of Rome. We shall also extend the patchy method to optimal control problems with state and control constraints.
We believe that our patchy techniques will be particularly useful in speeding up model predictive control techniques. Model predictive control has proven extremely useful in chemical processing applications where the time constants are long enough so that the necessary calculations can be done in real time. In order to use model predictive control for faster processes such as modem fighter aircraft the calculations must be speeded up. We believe that patchy techniques can do this.
|Publications||Publications, theses (not shown) and data repositories will be added to the portal record when information is available in FAIRS and brought back to the portal|
|Data||Publications, theses (not shown) and data repositories will be added to the portal record when information is available in FAIRS and brought back to the portal|