Issa Amadou Tall

State and feedback linearizations of single-input control systems

Abstract In this paper we address the problem of state (resp. feedback) linearization of nonlinear single-input control systems using state (resp. feedback) coordinate transformations. Although necessary and sufficient geometric conditions have been provided in the early eighties, the problems of finding the state (resp. feedback) linearizing coordinates are subject to solving systems of partial differential equations. We will provide here a solution to the two problems by defining algorithms allowing to compute explicitly the linearizing state (resp. feedback) coordinates for any nonlinear control system that is indeed linearizable (resp. feedback linearizable). Each algorithm is performed using a maximum of n − 1 steps ( n being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. We illustrate with several examples borrowed from the literature.

Strict feedforward form and symmetries of nonlinear control systems

We establish a relation between strict feedforward form and symmetries of nonlinear control systems. We prove that a system is feedback equivalent to the strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry and, secondly, it is the factor system of the preceding one, i.e., is reduced from the preceding one by its symmetry. We also propose a strict feedforward normal form and prove that a smooth strict feedforward system can be smoothly brought to that form.

On linearizability of strict feedforward systems

In this paper we address the problem of linearizability of systems in strict feedforward form. We provide an algorithm, along with explicit transformations, that linearizes a system by change of coordinates when some easily checkable conditions are met. Those conditions turn out to be necessary and sufficient, that is, if one fails the system is not linearizable. We revisit type I and type II classes of linearizable strict feedforward systems provided by Krstic in (M. Krstic, 2004) and illustrate our algorithm by various examples mostly taken from (M. Krstic, 2004).

Feedback Linearizable Feedforward Systems: A Special Class

The problem of feedback linearizability of systems in feedforward form is addressed and an algorithm providing explicit coordinates change and feedback given. At each step, the algorithm replaces the involutive conditions of feedback linearization by some, easily checkable. We also reconsider type II subclass of linearizable strict feedforward systems introduced by Krstic and we show that it constitutes the only linearizable among the class of quasilinear strict feedforward systems. Our results allow an easy computation of the linearizing coordinates and thus provide a stabilizing feedback controller for the original system among others. We illustrate by few examples including the VTOL.

Smooth and Analytic Normal and Canonical Forms for Strict Feedforward Systems

Recently we proved that any smooth (resp.analytic) strict feedforward system can be brought into its normal form via a smooth (resp. analytic) feedback transformation. This will allow us to identify a subclass of strict feedforward systems, called systems in special strict feedforward form, shortly (SSFF), possessing a canonical form which is an analytic counterpart of the formal canonical form. For (SSFF)-systems, the step-by-step normalization procedure of Kang and Krener leads to smooth (resp. convergent analytic) normalizing feedback transformations. We illustrate the class of (SSFF)-systems by a model of an inverted pendulum on a cart.

State linearization of control systems: An explicit algorithm

In this paper we address the problem of linearization of nonlinear control systems using coordinate transformations. Although necessary and sufficient geometric conditions have been provided in the early eighties, the problem of finding the linearizing coordinates is subject to solving a system of partial differential equations and remained open 30 years later. We will provide here a complete solution to the problem by defining an algorithm allowing to compute explicitly the linearizing state coordinates for any nonlinear control system that is indeed linearizable. Each algorithm is performed using a maximum of n − 1 steps (n being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. The problem of feedback linearization is addressed in a companion paper. A possible implementation via software like mathematica/matlab/maple using simple integrations, derivations of functions might be considered.

Linearizable feedforward systems: A special class

We address the problem of linearizability of systems in feedforward form. In a recent paper, we completely solved the linearizability for strict feedforward systems. We extend here those results to a special class of feedforward systems. We provide an algorithm, along with explicit transformations, that linearizes the system by change of coordinates when some easily checkable conditions are met. We also re-analyze type II class of linearizable strict feedforward systems provided by Krstic and we show that this class is the unique linearizable among the class of quasi-linear strict feedforward systems. Our results allow an easy computation of the linearizing coordinates and thus provide a stabilizing feedback controller for the original system. They can also be implemented via software like Mathematica/Matlab/Maple using simple integrations, derivations of functions.

Feedback Classification of Multi-Input Nonlinear Control Systems

We study the feedback group action on multi-input nonlinear control systems with uncontrollable mode. We follow slightly an approach proposed in Kang and Krener [W. Kang and A. J. Krener, SIAM J. Control. Optim., 30 (1992), pp. 1319--1337] which consists of analyzing the system and the feedback group step by step. We construct a normal form which generalizes, on one hand, the results obtained in the single-input case and, on the other hand, those recently obtained by the same author in the controllable case. We illustrate our results by studying the Caltech Multi-Vehicle Wireless Testbed (MVWT) and the prototype of Planar Vertical TakeOff and Landing aircraft {(PVTOL)}. We also study the notion of bifurcation of controllability for systems with one nonzero uncontrollable mode. We first show that the equilibria for those systems is a p-dimensional submanifold (p equals number of inputs). Provided that one term in their normal form is nonzero, we show that these systems are linearly controllable, hence stabilizable, at any nearby equilibrium point of the origin.

Explicit feedback linearization of control systems

This paper addresses the problem of feedback linearization of nonlinear control systems via state and feedback transformations. Necessary and sufficient geometric conditions were provided in the early eighties but finding the feedback linearizing coordinates is subject to solving a system of partial differential equations and had remained open since then. We will provide in this paper a complete solution to the problem (see the companion paper where the state linearization has been addressed) by defining an algorithm that allows to compute explicitly the linearizing state coordinates and feedback for any nonlinear control system that is truly feedback linearizable. Each algorithm is performed using a maximum of n − 1 steps (n being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. A possible implementation via software like mathematica/matlab/maple using simple integrations, derivations of functions might be considered.

Strict feedforward control systems, linearisability, and convergent normal forms

This article discusses the feedback equivalence of multi-inputs feedforward control systems via smooth (resp. analytic) feedback transformations. We first address the state (resp. feedback) linearisation problem, and provide easily computable algorithms that yield explicit state (resp. feedback) linearising coordinates for systems in strict feedforward form. The application of the algorithms does not require checking the commutativity (resp. involutivity) of the distributions associated with the system, and the algorithms fail after few steps if the system is not linearisable. In the latter case, the algorithms are extended to provide coordinate systems bringing the system into a normal form which is a smooth (resp. analytic) counterpart of Kangs formal normal form. Illustrative examples for both the linearisation and convergent normal form include the vertical take off and landing aircraft, the multi-vehicle wireless testbed among others.