Imagine you are driving at night on a twisty mountain road. A deer jumps out of nowhere ahead of you. You turn your car to avoid it, and all four wheels of your car hit an icy patch. You instinctively slam on the brakes. Without a computer-controlled safety system this would most likely lead to a serious accident. Instead, the safety system kicks in. It reads the direction and speed of your car, measures the tire speeds on the four wheels, and takes over control of the car from you. It applies braking and distributes torque differentially to each wheel, takes over the steering, and brings the car to rest safely. Such safety technology requires sophisticated algorithms to apply controlling mechanisms based on sensor readings.

IMA postdoctoral fellow Dan Bates and visiting scientists Ioannis Fotiou and Philipp Rostalski from ETH in Zurich came up with a breakthrough in control theory which has the potential to radically improve auto safety technology. Instead of devising a control law, a rule by which a system responds to sensor readings to achieve control, they came up with a computer algorithm that takes into account a succession of sensor readings and activates controlling mechanisms in an optimal way. Such an algorithm can have a great advantage since it can bring a system under control faster and more efficiently. Indeed their method is guaranteed to find the fastest way to control a system whereas other methods can only find a way to control it.

The breakthrough comes from the fact that their algorithm can solve the optimal control problem to within acceptable accuracy in real time, thus moving this holy grail technology closer to reality. Their key tool comes, surprisingly, from the often abstract mathematical field of algebraic geometry. This is a discipline that dates back to the 17th century and is 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 are 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.