Differentially Flat Systems

Hebertt Sira-Ramírez, Sunil K. Agrawal

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9780824754709
• eBook ISBN: 9781351830546
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

Illustrating the power, simplicity, and generality of the concept of flatness, this reference explains how to identify, utilize, and apply flatness in system planning and design. The book includes a large assortment of exercises and models that range from elementary to complex classes of systems. Leading students and professionals through a vast array of designs, simulations, and analytical studies on the traditional uses of flatness, Differentially Flat Systems contains an extensive amount of examples that showcase the value of flatness in system design, demonstrate how flatness can be assessed in the context of perturbed systems and apply static and dynamic feedback controller design techniques.

Chapter 1

Introduction

Differential flatness is a natural idea associated with under-determined systems of differential equations. It represents the possibility of completely parameterizing every system variable in terms of a privileged finite set of “free” variables in the system. In order to understand the basic features of flatness, consider a system of n linear algebraic equations in n + m unknowns ξ = (x₁, …, x_n, x_n+1, …, x_n+m) = (x, x_n+1, …, x_n+m) = (x, f), written as

$Ax+Bf=0,B\ne 0,\text{r}\text{a}\text{n}\text{k}[A,B]=n$

Assume that A is invertible and that B is full rank m. Clearly, all the solutions for x may be written in terms of the unknown vector f as

$x=-A^{-1}Bf$

i.e., all solutions are parameterizable in terms of f. On the other hand, since the n × m matrix B is full rank, the variable f is expressible in terms of the components of x, as in the formula

$f=-{(B^{T}B)}^{-1}{B}^{T}Ax$

The linear system [A, B]ξ = 0 exhibits then a privileged set of variables, namely f = (x_n+1, …, x_n+m), which is a set of internal variables to the system of equations (we will later prefer to use the adjective endogenous) capable of making x take on any arbitrary desired (finite) value. Suppose now that the matrix A is not invertible, say it is rank n – m. Then, in general, f is no longer free to take any value as it will become zero for any desired x which lies in the null space of A. As a consequence, it is no longer true that the components of x can be completely parameterized in terms of f. The property of being able to make x take on any desired value, by proper choice of f, is lost. Note that, in the above discussion, the rôle of the vector f may have been played, in principle, by another subset of m variables in ξ for which the corresponding A matrix is invertible and the corresponding B matrix is full rank.

Control systems are frequently represented in terms of linear, or nonlinear, sets of differential state equations with a certain number, m, of control inputs. Typically, the number of states n is larger than the number m of controls and the number of differential equations is equal to the number of state variables. Thus, the control inputs, which are unknown functions that usually require to be determined, constitute a set of additional variables that render the system of equations under-determined.

Flat systems, or differentially flat systems, exhibit a property which is highly reminiscent of the property exhibited by the previous elementary under-determined algebraic example. It is not surprising that such a property may exist in controlled systems as they constitute, generally speaking, under-determined systems of equations themselves. The number of control inputs being responsible for the under-determination. It is clear that the invertibility of A and the full rank of the matrix B in the linear algebraic example will find much more stringent, but still natural, conditions in the case of controlled differential equations. In essence, these conditions will be summarized by the controllability property of the given system.

Due to the fact that controllability is a fundamental desirable property of controlled dynamic systems (whether, continuous, discrete, linear, nonlinear, finite dimensional or not) then, the possibilities of finding the flatness property in a given dynamic system, will be tantamount of having the controlled system satisfy some form of the ubiquitous controllability property. Since, on the other hand, controllability is strongly related to being able to have the system state trajectory reasonably do whatever we want, within a finite interval of time, then, flatness will be strongly related to being able to off-line plan feasible state trajectories and to devising corresponding feedback controllers that make the system state precisely follows those desired trajectories. A striking advantage in the recognition of flatness is that, both, the trajectory planning and the controller specification tasks becomes surprisingly simple. Flatness is then related to the fact that the entire set of trajectories (solutions) of the system are in a smooth, one-to-one, correspondence with free trajectories lying in an m dimensional space, the space of the flat outputs. As a consequence, the specified desired trajectories for the flat outputs uniquely determine the state trajectories and the nominal control inputs behavior. Clearly, the solving of differential equations is sidestepped, at the planning stages, since differential expressions, involving the flat outputs, need to be evaluated for obtaining the state trajectories and the input variables behavior. These facts would have limited relevance if it were not true that, in the context of realistic physical examples, the flat outputs invariably have a specific and concrete meaning. Generally speaking, this set of fictitious outputs represents a key set of physical variables whose measurability is either granted or desirable.

E. Cartan and D. Hilbert are the forefathers of flatness from the context of under-determined sets of differential equations. In their work, they either searched for nonlinear space, and time, coordinate transformations which rendered the studied system easily integrable, or for a particular set of variables which completely parameterized the system solutions without solving differential equations ([1], [2] [19]).

The precise formulation of differential flatness in the control systems context is due to the work of Professor Michel Fliess and his colleagues: Jean Levine, Philippe Martin and Pierre Rouchon. The first fundamental articles, and developments, appeared a decade ago in 1993. The first journal article [8] written by the team, in French, is devoted to flatness of nonlinear systems and the associated idea of defect (i.e., the lack of flatness). The setting of the contribution is that of differential algebra, a topic not easily found in engineering curricula throughout the world. In this article the idea of flatness appears as a natural outcome of the equivalence problem formulated in a differential algebraic context. The article clearly shows, through a non-trivial multivariable non linear system example describing an overhead crane, that flatness is suitable for dealing with physical systems even if they are not described in traditional differential equations form but as a set of differential equations subject to a set of algebraic restrictions. The notion of defect, or the lack of flatness, is explored in [9] along with a set of interesting physical examples such as the Kapitsa pendulum, the ball and beam and many others. A high frequency control approach is proposed to uncover the flatness of the average system in a variety of these examples. A complete exposition of all the developments concerning flatness, defect and a collection of some of the many challenging physical examples, that the new theory was capable of handling, appeared in an article by Fliess, Levine, Martin and Rouchon (FLMR) in 1995 [10]. It soon became clear that the differential algebraic setting could be recast in a purely differential geometric setting involving infinite jet spaces, diffieties (an abbreviation for “differential varieties”) and Cartan fields. This generalization naturally englobed the idea of space and time coordinate transformations and brought to the attention the relevance of Lie-Bäcklund transformations in dynamic systems equivalence problems and in feedback linearization. The complete recasting of flatness in this new context appeared in an article by FLMR in [11]. An earlier version of this approach had already appeared in 1993 (See Fliess et al. [12]). Several conference, or workshop, papers also appeared around that time exploiting different aspects of the theory with numerous examples. An independent contribution, in a similar line of thought, can be found in Pomet [25]. A closely related approach is represented by the work of Rathinam [28] which studies flatness from an absolute equivalence setup due to E. Cartan. In this contribution by Rathinam the flatness associated to Lagranian systems with one control input less than the degrees of freedom of the system is characterized via the use of configuration variables. The author addresses such flatness as configuration flatness (See also Rathinam and Murray [29]). Contributions to feedback linearization, from the viewpoint of absolute equivalence, were also furnished by Sluis [32]. The ideas of absolute equivalence are also used, in a Pfaffian setup, for the characterization of flatness in van Nieuwstadt et al. in [35].

It is fair to say that the insight into the algebraic and geometric formulations of flatness was greatly motivated from the algebraic theory of linear systems based on modules. This trend was initiated by Fliess in [5]. The idea of relating controllability to the freeness of the system module is intimately tied to the notion of a module basis. The basis of a free module is central to the flatness property, and to the characterization of flat outputs, in any linear time invariant, or time varying, system. The relation is particularly clear from the developments found in the article by Fliess [6] where it is shown that Willems’ concept of controllability, which, roughly speaking, consists in being able to smoothly tie past state trajectories to future state trajectories ¹, is equivalent to the freeness of the corresponding system module. In turn, the module theoretic formulation of linear systems arose as a particular application area of the Differential Algebraic approach in the study of continuous and discrete (nonlinear) dynamic systems (See Fliess [3]). Complete accounts of this algebraic line of thought can be found in some subsequent works by Fliess of which we mention [4], and the tutorial effort represented by the work of Fliess and Glad [7].

A key contribution in the history of flatness is the establishment of a general necessary condition for the assessment of flatness. This characterization is due to Rouchon [31] and has become popularly known as the ruled manifold condition. This condition, of geometric flavor, and quite easy to check, says that at each point of the state space, the set of state velocity vectors obtained for all possible control inputs must be a ruled submanifo...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Series Introduction
• Preface
• Acknowledgments
• 1 Introduction
• 2 Linear Time-Invariant SISO Systems
• 3 Linear Time-Invariant MIMO Systems
• 4 Time-Varying Linear Systems
• 5 Discrete-Time Linear Systems
• 6 Infinite Dimensional Linear Systems
• 7 SISO Nonlinear Systems
• 8 Multivariable Nonlinear Systems
• 9 Mobile Robots
• 10 Flatness and Optimal Trajectories
• 11 Optimal Planning with Constraints
• 12 Non-Differentially Flat Systems
• Index