On the Simulation of Lower Order Control Strategies for Higher Order Systems

In model-based control of dynamical systems the order of the necessary controller is that of the time-derivative of the generalized coordinate of the system that immediately can be set by the controller. It is determined by the “physics” of the system to be controlled. In finite time-resolution approach the digital controllers model the derivatives by finite differences, and the simplest numerical integration via Euler’s method also uses such differences. Whenever some fractional order derivative of the tracking error is fed back, this long memory term allows the realization of such control strategies for various integer order systems by appropriately masking” and weighting certain elements of longer sequences. On this mathematical basis the idea of controlling a higher order system by the application of a lower order strategy can be formulated, too. The main benefit may be achieving monotonic decrease in the components of the tracking error. In this paper the first order control of a popular second order system often used for benchmark applications, the van der Pol oscillator is considered. The realization of a simple proportional feedback first order controller in two variants, and a PID-type Computed Torque Control are considered in the fixed point iteration-based adaptive framework to tackle the problems caused by modeling imprecisions. The applicability and the limits of the investigated approaches are concluded.