Flat Systems

Control systems are ubiquitous in modern technology. The use of feedback control can be found in systems ranging from simple thermostats that regulate the temperature of a room, to digital engine controllers that govern the operation of engines in cars, ships, and planes, to ight control systems for high performance aircraft. The rapid advances in sensing, computation, and actuation technologies is continuing to drive this trend and the role of control theory in advanced (and even not so advanced) systems is increasing. A typical use of control theory in many modern systems is to invert the system dynamics to compute the inputs required to perform a speci c task. This inversion may involve nding appropriate inputs to steer a control system from one state to another or may involve nding inputs to follow a desired trajectory for some or all of the state variables of the system. In general, the solution to a given control problem will not be unique, if it exists at all, and so one must trade o the performance of the system for the stability and actuation e ort. Often this tradeo is described as a cost function balancing the desired performance objectives with stability and e ort, resulting in an optimal control problem. This inverse dynamics problem assumes that the dynamics for the system are known and xed. In practice, uncertainty and noise are always present in systems and must be accounted for in order to achieve acceptable performance of this system. Feedback control formulations allow the system to respond to errors and changing operating conditions in real-time and can substantially