Controllability and stabilization of programmed motions of an automobile-type transport robot

A non-linear model of the motion of an automobile-type transport robot (TR) with absolutely rigid wheels, a steering device and actuators based on DC motors, is considered. Such a model for TR motion is a non-holonomic electromechanical system and, if the dynamics of the actuators and the steering device (forces of elasticity and attenuation in its elements) is ignored, corresponds to the model of automobile motion devised by Lineikin [1]. Non-linear canonical transformations of the state and control space coordinates are constructed which reduce the initial equations of motion of the TR to a simpler canonical form, convenient for the analysis and synthesis of control systems for the TR. These transformations are used to find the conditions for the controllability of the TR as a controlled object. Algorithms are given for constructing programmed controls and programmed motions of the TR. Stabilizing control laws are synthesized that make the programmed motions of the TR asymptotically stable and guarantee that the transients will have preassigned properties     

Controllability and stabilization of programmed motions of reversible mechanical and electromechanical systems

Problems of controllability and methods of stabilizing programmed motions of a large class of mechanical and electromechanical systems which are reversible with respect to the control are considered. Criteria of the controllability and stabilizability of reversible systems are obtained. Programmed motions and algorithms of programmed control are designed in analytical form and algorithms of programmed motions for non-linear reversible systems are synthesized.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Controllability and asymptotic stabilization of the Camassa-Holm equation

We investigate the problems of exact controllability and asymptotic stabilization of the Camassa-Holm equation on the circle, by means of a distributed control. The results are global, and in particular the control prevents the solution from blowing up.     

Controllability and stabilization for linear systems of algebraic and differential equations

In this paper, we study several reachability and controllability definitions for linear, autonomous systems of degenerate ordinary differential equations. We try to identify the most interesting among them; we prove the connections with stabilizability and full stabilizability properties of the systems, which are also studied. In particular, we show the connection of the stabilizability property with the existence of special solutions of a suitable version of the Riccati equation.This paper was written under the auspices of GNAFA-CNR (Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche), Rome Italy.     

Controllability and quadratic stability quadratic stabilization of discrete-time interval systems--an LMI approach

^** Email: qlzhang{at}mail.neu.edu.cn This paper presents necessary and sufficient conditions fordiscrete-time interval systems to be quadratically stable orquadratically stabilizable, and sufficient conditions for discrete-timeinterval systems to be controllable. All the results are obtainedin terms of linear matrix inequality (LMI). With the LMI toolbox,the problems can be conveniently solved. Illustrative numericalexamples show that the results are effective and less conservativeto check the robust controllability and stability and to designthe stabilizing controller for discrete-time interval systems.     

Classification of three-parametric spatial motions with a transitive group of automorphisms and three-parametric robot manipulators

The paper deals with the differential geometry of submanifolds of the kinematical space of Euclidean space kinematics, which is a six-dimensional pseudo-Riemannian symmetric space of signature (3, 3). The main result is in the proof of the classification theorem for three-dimensional Euclidean space motions with a transitive group of automorphisms. All of them are products (in the group multiplication) of homogeneous spaces and their list is provided. All three-parametric robot manipulators with constant invariants are found as an application of the classification theorem.     

Stability of steady motions of a mobile robot with roller-carrying wheels and a displaced centre of mass

The motion of a mobile three-wheel robotic vehicle on a horizontal surface is investigated. Passive rollers are fastened along the rim of each wheel, enabling each wheel not only to roll in the usual manner, but also to move perpendicular to its plane. Each of these wheels, as well as the ordinary wheels, is equipped with one drive, which rotates the wheel about its axis. The vehicle equipped with roller-carrying wheels can move in any direction with any orientation. The motion of the robot on a horizontal surface is studied in the case where the centre of mass of the robot deviates from the geometric centre of the triangular platform, and there is no slip at the points of contact of the rollers with the supporting surface. In the case of free motion of the robot, an additional first integral is pointed out and the exact solution found is analysed. An equation for specifying steady motions, under which a constant voltage is supplied to the DC motors that drive the wheels, is constructed. The stability of the rectilinear motion of the robot is investigated.     

Controllability and observability in the problem of stabilizing steady motions of non-holonomic mechanical systems with cyclic coordinatest

The approach to the solution of stabilization problems for steady motions of holonomic mechanical systems [1, 2] based on linear control theory, combined with the theory of critical cases of stability theory, is used to solve the analogous problems for non-holonomic systems. It is assumed that the control forces may affect both cyclic and positional coordinates, where the number r of independent control inputs may be considerably less than the number n of degrees of freedom of the system, unlike in many other studies (see, e.g., [3–5]), in which as a rule r = n. Several effective new criteria of controllability and observability are formulated, based on reducing the problem to a problem of less dimension. Stability analysis is carried out for the trivial solution of the complete non-linear system, closed by a selected control. This analysis is a necessary step in solving the stabilization problem for steady motion of a non-holonomic system (unlike holonomic systems), since in most cases such a system is not completely controllable.     

The stabilization of program motions of controlled nonlinear mechanical systems

We consider a controlled nonlinear mechanical system described by the Lagrange equations. We determine the control forcesQ ₁ and the restrictions for the perturbationsQ ₂ acting on the mechanical system which allow to guarantee the asymptotic stability of the program motion of the system. We solve the problem of stabilization by the direct Lyapunov's method and the method of limiting functions and systems. In this case we can use the Lyapunov's functions having nonpositive derivatives. The following examples are considered: stabilization of program motions of mathematical pendulum with moving point of suspension and stabilization of program motions of rigid body with fixed point.     

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.     

Justification of the stabilization method for a mathematical model of charge transport in semiconductors

An initial boundary value problem for a quasilinear system of equations is studied and effectively applied to numerically determine, by the stabilization method, stationary solutions of a hydrodynamic model describing the motion of electrons in the silicon transistor MESFET (metal semiconductor field effect transistor).     

Tabu search algorithms for job-shop problems with a single transport robot

We consider a generalized job-shop problem where the jobs additionally have to be transported between the machines by a single transport robot. Besides transportation times for the jobs, empty moving times for the robot are taken into account. The objective is to determine a schedule with minimal makespan.We present local search algorithms for this problem where appropriate neighborhood structures are defined using problem-specific properties. An one-stage procedure is compared with a two-stage approach and a combination of both. Computational results are presented for test data arising from job-shop benchmark instances enlarged by transportation and empty moving times.     

Controllability of a multichannel system

We consider the system consisting of K coupled acoustic channels with the different sound velocities $c_j$. Channels are interacting at any point via the pressure and its time derivatives. Using the moment approach and the theory of exponential families with vector coefficients we establish two controllability results: the system is exactly controllable if(i) the control $u_j$ in the jth channel acts longer than the double travel time of a wave from the start to the end of the j-th channel;(ii) all controls $u_j$ act more than or equal to the maximal double travel time.     

Subgridscale stabilization of time-dependent convection dominated diffusive transport

In [W.J. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002) 147-157], a variationally consistent eddy viscosity discretization is given for the stationary convection diffusion equation. We further develop this discretization to include the time-dependent problem. We give comprehensive stability and error analysis of the semi-discrete case. We also state the stability and error results for the fully discrete algorithm with a Crank-Nicholson time discretization. The error bound is near optimal and independent of the diffusion coefficient, ?. Finally, we give guidance on optimal parameter selection for some common finite element spaces.