Lie algebra is an area of mathematics that is largely used by electrical engineer students, mainly at post-graduation level in the control area. The purpose of this paper is to illustrate the use of Lie algebra to control nonlinear systems, essentially in the framework of mobile robot control. The study of path following control of a mobile robot using an input-output feedback linearization controller is performed. The effectiveness of the nonlinear controller is illustrated with simulation examples.

Several industrial systems are characterized by high nonlinearities with wide operating ranges and large set point changes. Identification and representation of these systems represent a challenge, especially for control engineers. Multimodel technique is one effective approach that can be used to describe nonlinear systems through the combination of several submodels, where each is contributing to the output with a certain degree of validity. One major concern in this technique, especially for systems with unknown operating conditions, is the partitioning of the system's operating space and thus the identification of different submodels. This paper proposes a three-stage approach to obtain a multimodel representation of a nonlinear system. A reinforced combinatorial particle swarm optimization and hybrid K-means are used to determine the number of submodels and their respective parameters. The proposed method automatically optimizes the number of submodels with respect to the submodel complexity. This allows operating space partition and generation of a parsimonious number of submodels without prior knowledge. The application of this approach on several examples, including a continuous stirred tank reactor, demonstrates its effectiveness.

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. Illustrative examples show that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the stability analysis and the solution of robust feedback stabilization problems.

We consider the problem of state and parameter estimation for a class of nonlinearoscillators defined as a system of coupled nonlinear ordinary differential equations.Observable variables are limited to a few components of state vector and an input signal.This class of systems describes a set of canonic models governing the dynamics of evokedpotential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,and Morris-Lecar models. We consider the problem of state and parameter reconstruction forthese models within the classical framework of observer design. This framework offerscomputationally-efficient solutions to the problem of state and parameter reconstructionof a system of nonlinear differential equations, provided that these equations are in theso-called adaptive observer canonic form. We show that despite typical neural oscillatorsbeing locally observable they are not in the adaptive canonic observer form. Furthermore,we show that no parameter-independent diffeomorphism exists such that the originalequations of these models can be transformed into the adaptive canonic observer form. Wedemonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,parameter-dependent coordinate transformations can be used to render these systems intothe adaptive observer canonical form. This allows reconstruction, at least partially andup to a (bi)linear transformation, of unknown state and parameter values with exponentialrate of convergence. In order to avoid the problem of only partial reconstruction and atthe same time to be able to deal with more general nonlinear models in which the unknownparameters enter the system nonlinearly, we present a new method for state and parameterreconstruction for these systems. The method combines advantages of standardLyapunov-based design with more flexible design and analysis techniques based on thenotions of positive invariance and small-gain theorems. We show that this flexibilityallows to overcome ill-conditioning and non-uniqueness issues arising in this problem.Effectiveness of our method is illustrated with simple numerical examples.

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploitsthe action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, withStokes equations governing the surrounding fluid.The action of thecilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a systemis generically controllable when the space of controlled velocity fields is at least three-dimensional. We also provide a complete characterization of controllable systems in the case in whichthe organism has a spherical shape. Finally, we offer a complete picture of controllable and non-controllable systems under the additional hypothesis thatthe organism and the fluid have densities of the same order of magnitude.

This paper presents a new approach for the design of variable structure control (VSC) of nonlinear systems. The approach is based on estimation of joint acceleration signals with introduction of load estimation with the asymptotic observer. The control system is insensitive to parameter variations for a chosen switching hypersurface in conditions when it is reached by the dynamic motion with the required dynamics. The parameter insensitive response provided by this control method is demonstrated on the model of the SCARA robot. Simulation results confirm the validity of accurate tracking capability and the robust performance.

This work presents, first, a complete dynamic model of a unicycle-like mobile robot that takes part in a multi-robot formation. A linear parameterization of this model is performed in order to identify the model parameters. Then, the robot model is input-output feedback linearized. On a second stage, for the multi-robot system, a model is obtained by arranging into a single equation all the feedback linearized robot models. This multi-robot model is expressed in terms of formation states by applying a coordinate transformation. The inverse dynamics technique is then applied to design a formation control. The controller can be applied both to positioning and to tracking desired robot formations. The formation control can be centralized or decentralized and scalable to any number of robots. A strategy for rigid formation obstacle avoidance is also proposed. Experimental results validate the control system design.

This paper examines three methods of adaptive output feedback control for robotic manipulators. Implementing output feedback control allows use of only the position information, which can be measured quite accurately. Velocity and acceleration measurements can get corrupted by noise. A method proposed by K. W. Lee and H. K. Khalil [Adaptive output feedback control of robot manipulators using high-gain observer, Int. J. Control, 6, 869–886 (1997)] using a high-gain observer, one proposed by J. J. Craig, P. Hsu and S. S. Sastry [Adaptive control of mechanical manipulators, Int. J. Robot. Res., 6(2), 16–27 (1987)] with the addition of a linear observer that we propose, and a method proposed by R. Gourdeau and H. M. Schwartz [Adaptive control of robotic manipulators: Experimental results, Proceedings of the 1991 IEEE International Conference on Robotics and Automation (Apr. 1991) pp. 8–15] using an Extended Kalman Filter are examined. The methods are implemented in simulation and experimentally on a direct-drive robot. The performance of each of the algorithms is compared.

In this paper, we prove the genericity of the observability for discrete-time systems with more outputs than inputs.

In this paper, we consider general nonlinear systems with observations, containing a (single) unknown function φ. We study the possibility to learn about this unknown function via the observations: if it is possible to determine the [values of the] unknown function from any experiment [on the set of states visited during the experiment], and for any arbitrary input function, on any time interval, we say that the system is “identifiable”. For systems without controls, we give a more or less complete picture of what happens for this identifiability property. This picture is very similar to the picture of the “observation theory” in [7]: Contrarily to the case of the observability property, in order to identify in practice, there is in general no hope to do something better than using “approximate differentiators”, as show very elementary examples. However, a practical methodology is proposed in some cases. It shows very reasonable performances.
As an illustration of what may happen in controlled cases, we consider the equations of a biological reactor, [2,4], in which a population is fed by some substrate. The model heavily depends on a “growth function”, expressing the way the population grows in presence of the substrate. The problem is to identify this “growth function”. We give several identifiability results, and identification methods, adapted to this problem.

We construct explicitly an homogeneous feedback for a class of single input, two dimensional and homogeneous systems.

This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known [4-6] that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems. Corresponding canonical forms are presented and sufficient conditions which permit the design of constant and high gain observers for these systems are given.

On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).

Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains – as a prototype subclass – all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary $\mathbb{R}^M$-valued reference signal r of class W1,∞ (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ${\varphi}(t)\| e(t)\| < 1$ for all t ≥ 0, where φ a prescribed real-valued function of class W1,∞ with the property that φ(s) > 0 for all s > 0 and $\liminf_{s\rightarrow\infty}{\varphi}(s)>0$. A simple (neither adaptive nor dynamic) error feedback control of the form $u(t)=- \alpha ({\varphi}(t)\|e(t)\|)e(t)$ is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain $\alpha ({\varphi}(\cdot )\|e(\cdot )\|)$.