An algorithm for orbital feedback linearization of single-input control affine systems

Abstract

Feedback linearization is an effective design and analysis tool used in the study of nonlinear control systems. However, cases arise where linearizability conditions cannot be met. For systems that are not linearizable by classical techniques, orbital feedback linearization (or feedback linearization by state-dependent time scaling) has been proposed to relax these conditions. Unfortunately, approaches proposed to date have led to conditions that tend to be more difficult to check then conditions for state-feedback linearization. In this paper, necessary and sufficient conditions for orbital feedback linearizability is presented for a class of single-input nonlinear systems. The conditions are simple and can be checked directly from the data of the problem. Using an exterior calculus approach, a simple algorithm is developed to compute state-dependent time scaling that yield state-feedback linearizable systems. It is shown that orbital feedback linearizability generalizes the concept state-feedback linearizability to deal with locally weakly accessible control systems.

Introduction

Feedback linearization by state-feedback has been a standard technique in nonlinear control system design. Starting with the work of Brockett [2], Jakubckyk and Respondek [10] and Hunt et al. [9], the problem of feedback linearization has been of great concern for both theoreticians and practitioners. For example, the chemical engineering literature contains a number of applications of this technique for the design of nonlinear controllers for bioreactors [7], [8], polymerization reactors [15], [17] and chemical reactors [16]. Although the technique can be useful in some situations, the conditions for linearizability of nonlinear systems as given by Jakubckyk and Respondek [10] impose serious restrictions on the structure of the nonlinear control system. Such conditions are seldom met in practice. To overcome this problem, alternative approaches to the problem have been presented in the control literature to generalize the standard feedback linearization techniques. A number of alternative mathematical frameworks have emerged such as exterior calculus approaches [1], [6] and differential algebraic techniques [5] in an attempt to refine the conditions derived from the Lie-algebraic approach to feedback linearization and to facilitate its generalization to larger classes of systems. For nonlinearizable systems, approximate feedback linearization [12] has been considered to derive state-space and feedback transformations that can approximately linearize, up to some pre-specified order, a nonlinear control system. The stabilization of such systems can be achieved by robust techniques that can account for the evolution of approximation errors. Another approach used to handle nonlinearizable systems is partial feedback linearization (see [13] and the references therein) and, in particular, input–output linearization. Such techniques are useful and can be very successful for the control and trajectory generation of minimum phase systems (see [11] for a chemical process example).

One possibility that has been studied by a number of researchers is the use of state-dependent time scaling transformations. This problem was first considered and solved by Sampei and Furuta [18]. Their development introduced the concept of linearizing the nonlinear systems with respect to a new state-dependent time scale. They developed a set of necessary and sufficient conditions that can be used to obtain linearizing transformations. Unfortunately, conditions involve the solution of nonlinear PDEs to obtain the time scaling transformations which cannot be readily checked from the data of the problem. For the treatment of so-called nonflat systems, Fliess and co-workers [3] introduced the concept of orbital flatness. They considered a more general framework for studying linearizability based on the application of Lie–Backlünd transformations in which time scaling transformations can be included naturally. Recently, Respondek [14] developed necessary and sufficient conditions for the orbital feedback linearization of a single-input systems. These conditions can be used to verify conditions for equivalence of single-input systems to linear normal forms. In this paper, we consider the problem of orbital feedback linearization using an exterior calculus approach. A necessary and sufficient condition is obtained for control-affine nonlinear systems. The conditions yield an algorithm that can be used to compute families of orbital feedback transformations through algebraic manipulations including the computation of the derived flag of the associated Pfaffian system and integration of Frobenius system of dimension of, at most, two. Similar to the approach of Respondek [14], the conditions are easily checked from the data of the problem. Applications to two physical examples demonstrate the use of the method and its generality.

The paper is as follows: in Section 2, conditions for feedback linearizability are introduced. The main result is presented in Section 3. Two examples are presented in Section 4, followed by brief conclusions in Section 5.