Controllability Lie Algebra Research Articles

Continuous Stabilizing Control for a Class Of Non-holonomic systems: Brockett System Example

A continuous adaptive controller is designed for a famous example of Brockett system which is an example of a class of non-holonomic systems. The controllability Lie Algebra of the Brockett’s system contains Lie brackets of depth one. It is shown that the closed loop system is globally uniformly asymptotically stable. The advantage of this method is that it does not require the conversion of the system model into a “chained form,” and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations.

Steering Control of an Underwater Vehicle using Adaptive Back Stepping Approach

This paper presents a simple and systematic approach to steer an underwater vehicle model by considering two different cases: (i) when all actuators are functional, and (ii) when one actuator is not working. In first case, the model of an underwater vehicle is steered by using adaptive Backstepping technique. The first actuator is necessary for the operation of the system so any of the other three actuators can be non-operational. So, the second case itself contains three different cases. Adaptive Backstepping is then used to steer the system with one non-working actuator. The synthesis method is general, in that it applies to a large class of drift free, completely controllable systems, for which the associated controllability Lie algebra is locally nilpotent.

Steering Control for a Rigid Body with two Torque Actuators using Adaptive Back Stepping

This paper presents a simple steering control algorithm for a rigid body model, which is a famous example of non-holonomic control systems with drift. The controllability Lie Algebra of a rigid body model contains Lie brackets of depth two. We propose a back-stepping-based adaptive controller design under the strict-feedback form. We analyze two cases for continuous steering. In the first case, the parameters of the model are assumed to be known while in the second case these are estimated by considering them unknown. This approach does not necessitate the conversion of the system model into a “chained form”, and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations and graceful stabilization.

Feedback Stabilization of Nonholonomic Drift-Free Systems Using Adaptive Integral Sliding Mode Control

This article presents a novel stabilizing control algorithm for nonholonomic drift-free systems. The control algorithm is based on adaptive integral sliding mode control technique. In order to utilize the benefit of integral sliding mode control, extended Lie bracket system is used as a nominal system which can easily be asymptotically stabilized. Firstly the original nonholonomic drift-free system is augmented by adding its missing Lie brackets and some unknown adaptive parameters. Secondly the controller and the adaptive laws are designed in such a way that the behaviour of the augmented system is similar to that of the nominal system on the sliding surface and the addition of missing Lie brackets in the original system can be compensated adaptively. The proposed method is applied on two nonholonomic drift-free systems including the Brockett’s system and the hopping robot in flight phase. The controllability Lie Algebra of the Brockett’s system has Lie brackets of depth one, whereas the hopping robot model in flight phase contains Lie brackets of depth one and two. The effectiveness of the proposed technique is verified through simulation studies.

Adaptive Integral Sliding Mode Stabilization of Nonholonomic Drift-Free Systems

This article presents adaptive integral sliding mode control algorithm for the stabilization of nonholonomic drift-free systems. First the system is transformed, by using input transform, into a special structure containing a nominal part and some unknown terms which are computed adaptively. The transformed system is then stabilized using adaptive integral sliding mode control. The stabilizing controller for the transformed system is constructed that consists of the nominal control plus a compensator control. The compensator control and the adaptive laws are derived on the basis of Lyapunov stability theory. The proposed control algorithm is applied to three different nonholonomic drift-free systems: the unicycle model, the front wheel car model, and the mobile robot with trailer model. The controllability Lie algebra of the unicycle model contains Lie brackets of depth one, the model of a front wheel car contains Lie brackets of depths one and two, and the model of a mobile robot with trailer contains Lie brackets of depths one, two, and three. The effectiveness of the proposed control algorithm is verified through numerical simulations.

Feedback control of a redundant wheeled snake mechanism using transverse functions SO(4)

The Transverse Function (TF) approach is applied to the control of a nonholonomic three-segments/snake-like wheeled mechanism, similar to the planar low-dimensional version of Hirose's Active Cord Mechanism (ACM) previously studied by the authors with the same control approach, but with two additional internal degrees of freedom (d.o.f.) whose actuation yields more flexible and efficient control solutions. From a theoretical point of view, these complementary d.o.f. modify the Control Lie Algebra of the system so that only first-order Lie brackets of the control vector fields are needed to satisfy the Lie Algebra Rank Condition (LARC). The fact that four independent (angular velocity) control inputs are used also implies for this system the existence of Transverse Functions (TF) defined on the six-dimensional special orthogonal group(4). Several examples of mechanisms whose control involve TF defined on(3) have been pointed out in the past. Beyond the specific control problem addressed here, a motivation for the present study is to illustrate for the first time how functions defined on the larger set(4) can be determined and used for the control of a physical system. This study is complemented with recalls concerning the parametrization of(4) by pairs of isoclinic quaternions and with the derivation of complementary differential calculus relations associated with this parametrization.

Generic nonlinear stabilization of systems with matching algebraic structure

The paper demonstrates that using algebraic methods for the construction of time varying stabilizing controls for general controllable systems which are affine in the control is not only computationally feasible, but delivers generic feedback laws. A single feedback control law can be stabilizing for all systems which have the same algebraic structure and also for systems that can be adequately approximated by this structure. The systems considered are not limited to those whose controllability Lie algebra is nilpotent or even finite dimensional. The stabilizing controls are constructed by the help of an open-loop control problem on an associated Lie group which is posed as a trajectory interception problem in the logarithmic coordinates of flows.

Constructive Decomposition of the Controllability Lie Algebra for Quantum Systems

In this technical note, we show how to use the analysis of the controllability Lie algebra of a quantum mechanical system to study its dynamics and facilitate the design of controls. We give algorithms to decompose the dynamics and describe an example of application to two coupled spin 1/2's.

Switched Feedback Control for a class of First-order Nonholonomic Driftless Systems

This paper is concerned with stabilizing feedback control for a class of nonholonomic driftless systems, whose controllability Lie algebra rank condition is satisfied by up to first-order Lie brackets. We propose a switched feedback law which drives all the initial states to the origin with bounded control inputs (as opposed unbounded, division-by-zero-type discontinuous control). The discontinuity of the feedback law takes place on a subspace defined by the ‘parallelism’ condition for the base and the fiber vectors in ℝ 3 (or simply q × ϕ = 0). We also show that the complement of this discontinuity region is homotopic to SO (3) which is also isomorphic to S 2 × S . The proposed control law is examined by numerical simulations.

Homogeneous flow field effect on the control of Maxwell materials

The controllability of viscoelastic fields is a fundamental concept that defines some essential capabilities and limitations of the resulting materials. In this paper, we study the controllability of different homogeneous flow fields of viscoelastic fluids governed by the upper convected Maxwell model. The approach is largely based on the nonlinear geometric control theory. Through the analysis of the control Lie algebra, we find the submanifolds in the state space on which the homogeneous flow fields are weakly controllable. Our approach can be generalized to more complicated systems.

Controllability and Posture Control of a Wheeled Pendulum Moving on an Inclined Plane

The paper considers a specific class of wheeled mobile robots, namely, mobile wheeled pendulums (MWPs). Robots pertaining to this class are composed of two wheels rotating about a central body. The main feature of MWP pertains to the central body, which can rotate about the wheel axis. As such motion is undesirable, the problem of the stabilization of the central body in an MWP is crucial. The novelty of the work reported here resides in the construction of: 1) the system controllability Lie algebra for the purpose of a rigorous controllability analysis and the computation of the largest feedback-linearizable subsystem; 2) a controller by input-output linearization of the system for achieving the desired steering rate of the robot while stabilizing the central body; 3) a controller based on the internal properties of the system to achieve the desired heading velocity of the robot; and 4) a controller based on the sliding-mode approach for controlling both the position and the orientation of the robot. The entire control structure that permits full control of the robot posture comprises three imbricated loops. Simulations showing good performance of the controlled system are provided. Preliminary tests performed on an experimental platform confirm the validity of the controller.

Trident snake robot: Locomotion analysis and control

In this paper, we propose a novel kind of wheeled mobile robot, called “trident snake robot”. From a viewpoint of nonlinear control theory, this robot is classified as a nonholonomic system with two generators and higher order Lie brackets, whose controllability structure is quite complicated (in comparison to chained systems, for example). Based on analysis of the controllability Lie algebra and corresponding holonomy, we clarify the locomotion principle of the robot and realize rotation and translation control using periodic inputs. The proposed method is illustrated by numerical simulations.

Quantum control in infinite dimensions

Accurate control of quantum evolution is an essential requirement for quantum state engineering, laser chemistry, quantum information and quantum computing. Conditions of controllability for systems with a finite number of energy levels have been extensively studied. By contrast, results for controllability in infinite dimensions have been mostly negative, stating that full control cannot be achieved with a finite-dimensional control Lie algebra. Here we show that by adding a discrete operation to a Lie algebra it is possible to obtain full control in infinite dimensions with a small number of control operators.

SWITCHING FEEDBACK CONTROL OF NON-HOLONOMIC SYSTEMS WITH MULTI-GENERATORS

In this paper, we propose a switching state feedback control algorithm for a class of non-holonomic symmetric affine systems with multi-generators. The controllability Lie algebra of a multi-generator system is structurally different from that of single-generator systems, such as conventional chained form systems. A multi-generator dynamics is partially considered a single-generator system and each subsystem can be stabilized by any existing controller proposed for chained systems. We propose a switching control algorithm, in that each generator is chosen in sequence and corresponding sub-controllers are applied, where each sub-controller is designed by existing methods for chained systems. The efficiency of the proposed strategy is evaluated via numerical simulations.

Symmetric Affine Systems with Two-Generators: Topology and Discontinuous Feedback Control

In this paper, we investigate a discontinuous state feedback control law for a symmetric affine system with two-generators, whose controllability Lie algebra is essentially different from conventional chained(single-generator) structures. At first, we provide a topological principle to design a discontinuous surface to get rid of the nonexistence of continuous stabilizing feedback, with a pair of illustrative examples of single-generator case. Based on this concept and nonlinear controllability analysis, we propose a discontinuous control strategy for a 2-input 5-state two-generator system.

Applying optimal control methods in the optimization of smooth functions

We study the optimization problems of smooth functions on the reachable set of a nonlinear control system. By means of the optimal control theory and Lie algebra, we not only derive the basic formulation solving the problem, but also approximate the solution by Lie series and give a computational way to deal with the optimization.

Exponential stabilization of driftless nonlinear control systems using homogeneous feedback

This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers.

Nilpotent Infinitesimal Approximations to a Control LIE Algebra

We show that a general control system, that is, a connected manifold M with vector fields X 1 ,…, X k , may be approximated in some precise sense, in the neighbourhood of each point x , by a nilpotent control system. This one consists of T x M and vector fields X ^ 1 , … , X ^ n wich generate a nilpotent Lie algebra g x . If x is a regular point, g x has the same dimension as M , and T x M may be identified with a nilpotent Lie group G x having g x as its Lie algebra. At singular points, T x M may only be identified to a coset space G x / H x . The construction of the Lie algebra Q X is based on a notion of vanishing order for functions at some point x , relative to the vector fields X 1 ,…, X k . In suitable adapted coordinates u 1 ,…, u n , this notion turns out to coincide with the notion of quasi-homogeneous degree obtained by attributing some weights w 1 , ..., w n to the variables u 1 , ..., u n . These notions originate from the work of Rothschild and Stein about hypoelliptic differential equations [4], but here they are given a new, more intrinsic, presentation. As an example of the use of g x as an infinitesimal, or tangent, model, and of the notion of quasi-homogeneity, which plays an essential role in its definition, we compute a bound for the degree of nonholonomy for a controllable polynomial system control on R n .

A remark on nonlinear observability

The observability of a nonlinear system is related to a Lie subalgebra of the control Lie algebra.