Institute for Mathematics and its Applications (University of Minnesota-Twin Cities)
NSF-funded mathematicians have developed a breakthrough in control theory that has the potential to radically improve auto safety.
Imagine you are driving at night on a twisty mountain road. A deer jumps out of nowhere. You swerve, slam on the brakes, and all four wheels hit an icy patch. Without a computer-controlled safety system, a serious accident is likely. Instead, the safety system reads your direction and speed and measures all four tire speeds. It applies braking and distributes torque differentially to each wheel, takes over the steering, and brings the car safely to rest. Such safety technology requires sophisticated algorithms to apply controlling mechanisms based on sensor readings.
Dan Bates of the Institute of Mathematics and its Applications, together with visiting scientists Ioannis Fotiou and Philipp Rostalski from the Swiss Federal Institute of Technology (ETH Zurich), came up with a breakthrough algorithm. Instead of devising a control law--a rule by which a system responds to sensor readings--they devised a computer algorithm that takes into account a succession of sensor readings and activates controlling mechanisms optimally. Such an algorithm can bring a system under control in real time quickly and efficiently.
Their key tool comes, surprisingly, from the often abstract mathematical field of algebraic geometry--a discipline dating to the 17th century and related to the problem of finding roots of a polynomial. They implemented a mathematical technique called 'continuation' so that the optimal control can be found rapidly.
Progress on the application of algebraic geometric methods to optimal control and algorithmic development for continuation is moving at a rapid pace. It is a matter of time before these advanced methods make their way into the automotive arena, leading to smarter and safer cars.
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