State and Feedback Linearizations of Single-Input Control Systems

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

Linearizable Feedforward Systems: A Special Class

We address the problem of linearizability of systems in feedforward form. In a recent paper [22] 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 in [9] and we show that this class is the unique linearizable among the class of quasi-linear strict feedforward systems (see Definition III.1). 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 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

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

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

Feedback and Partial Feedback Linearization of Nonlinear Systems: A Tribute to the Elders

Arthur Krener and Roger Brockett pioneered the feedback linearization problem for control systems, that is, the transforming of a nonlinear control system into linear dynamics via change of coordinates and feedback. While the former gave necessary and sufficient conditions to linearize a system under change of coordinates only, the latter introduced the concept of feedback and solved the problem for a particular case. Their work was soon extended in the earlier eighties by Jakubczyk and Responder, and Hunt and Su who gave the conditions for a control system to be linearizable by change of coordinates and feedback (full rank and involutivity of the associated distributions). It turned out that those conditions are very restrictive; however, it was showed later that systems that fail to be linearizable can still be transformed into two interconnected subsystems: one linear and the other nonlinear. This fact is known as partial feedback linearization. For input-output systems with well-defined relative degree, coordinates can be found by differentiating the outputs. For systems without outputs, necessary and sufficient geometric conditions for partial linearization have been obtained in terms of the Lie algebra of the system; however, both results of linearization and partial feedback linearization lack practicability. Until recently, none has provided a way to actually compute the linearizing coordinates and feedback. In this paper, we propose an algorithm allowing to find the linearizing coordinates and feedback if the system is linearizable, and in the contrary, to decompose a system (without outputs) while achieving the largest linear subsystem. Those algorithms are built upon successive applications of the Frobenius theorem. Examples are provided to illustrate

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

Time-Invariant Quadratic Hamiltonians via Generalized Transformations

In this paper we give necessary and sufficient conditions for achieving a quadratic positive definite time-invariant Hamiltonian for time-varying generalized Hamiltonian control systems using canonical transformations. Those necessary and sufficient conditions form a system of partial differential equations that reduces to the matching conditions obtained earlier in the literature for time-invariant systems. Their theoretical solvability is proved via the Cauchy-Kowalevskaya theorem and their practical solvability discussed in some particular cases. Systems with time-invariant positive definite Hamiltonian are known to yield a passive input-output map and can be stabilized by unity feedback, which underlines the importance of achieving the positive definiteness and time-invariancy for the Hamiltonian. We illustrate the results with few examples including the rolling coin

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 [6] and illustrate our algorithm by various examples mostly taken from [5], [6]