ABSTRACT
In this chapter, we explore the concept of flatness in the context of linear time invariant dynamic systems provided with a single input and a single output. These systems are commonly addressed as SISO systems (from Single Input, Single Output). Usually, it is desired to stabilize the output of the system or to have it track a desired reference trajectory. This is greatly facilitated if the system is flat, regardless of the nature of the internal dynamics associated with the output variable (zero dynamics, residual dynamics). In the very particular context of linear systems, the connection between flatness and the concept of controllability is, perhaps, the clearest one: A linear time invariant system is flat if and only if the system is controllable. Linear SISO systems may be represented in terms of rational proper transfer functions or in matrix state space form. We show that for either type of representation, the concept of flatness is equivalent to that of controllability. The identification of the flat output in a linear system becomes of particular importance since the corresponding differential parametrization associated with the flat output allows one to reduce any stabilization or tracking problem to a corresponding problem defined on the flat output. A challenging problem is that of having a non-minimum phase output track a desired reference. For rest-to-rest maneuvers flatness is particularly helpful and provides an elegant indirect solution to the problem. For other more complicated problems, such as tracking arbitrary output trajectories in a non-minimum phase flat system, the problem may prove to be difficult. An interesting feature of flatness is that all system properties, pertaining input to state, state to output and input-output relations, may be inferred from the differential parametrization of the system variables. In some instances this may be easier than handling the original system equations. Systems are always affected by external (exogenous) perturbations. The flatness property may still 12be suitably exploited in the regulation of such prevailing class of systems. For this, one relies on the fact that for sufficiently benign perturbations, the flatness property always results in the possibilities of having a “matched” system.
