Differential flatness and sliding mode control

Abstract

A system is said to be differentially flat if there exists a set of independent differential functions of the state (i.e., they do not satisfy any differential equations and, additionally, they are functions of the state and of a finite number of their time derivatives), called the flat outputs. The set of flat outputs exhibits the same number of elements as that found on the input set. The nature of the flat outputs is such that all variables in the system: i.e., states, outputs, and inputs, are, in turn, expressible as differential functions of the flat output. Flatness was introduced, by M. Fliess and his colleagues in a series of remarkable articles ([5–7]) where the reader is referred for theoretical issues and many illustrative examples. Contrary to unwarranted belief, flatness is not just another way to do feedback linearization. For SISO systems, indeed, flatness and feedback linearizability are equivalent but flatness goes beyond feedback linearization, specially in the MIMO case. Generally speaking, flatness is, in fact, a structural property of the system that allows one to establish all the salient features which are needed for the application of a particular feedback controller design technique (like back-stepping, passivity, sliding, and, of course, feedback linearization). Thus flatness is also an analysis tool naturally related to equilibria, control limitations, state restrictions, and singularity avoidance. Flatness, in its more popular conception, is a property that readily trivializes the exact linearization problem in a nonlinear system, whether or not the system is mono-variable. Moreover, flatness may be present on any type of nonlinear controlled system, regardless of the nonlinear, or affine, nature of the control inputs in the system equations. Flatness, thanks to its relations with invertibility, immediately yields the required open loop (nominal) behavior of the system for a particular desired trajectory tracking task. It is, therefore, most suitable for trajectory planning, controller saturation avoidance, the handling of state restrictions and predictive control, specially for those cases involving non-minimum phase outputs (see [27] and [9]). One of the distinctive features of flatness lies in the possibilities of differentially parameterizing all system variables. States, inputs, actual system’s (non-flat) outputs are all expressible as functions of the flat outputs and a finite number of their time derivatives. In mono-variable cases, this allows for a natural specification of the sliding surface coordinate function in terms of a stable linear differential polynomial acting on the flat output. In multi-variable systems, flatness naturally leads to inputs-to-flat outputs decoupling via static or dynamic feedback (see Charlet et al. [2], Rouchon [9]). Flat outputs are, generally speaking, physically meaningful variables in the system. Thus, their control to specific values or reference time functions is immediately related to a control objective whose feasibility may be readily assessed. We shall assume that the flat output variables are all measurable for feedback purposes.