Analysis of exclusively kinetic two-link underactuated mechanical systems

Analysis of exclusively kinetic two-link underactuated mechanical systems is undertaken in this paper. It is first shown that such systems are not full-state feedback linearizable around any equilibrium point. Also, the equilibrium points for which the system is small-time locally controllable (STLC) is at most a one-dimensional submanifold. A concept less restrictive than STLC, termed the small-time local output controllability (STLOC) is introduced, the satisfaction of which guarantees that a chosen configuration output can be controlled at its desired value. It is shown that the class of systems considered is STLOC, if the inertial coupling between the input and output is nonzero. Also, in such a case, the system is nonminimum phase. An example section illustrates all the results presented.     

Local and global controllability for nonlinear systems

A technique to study local and global controllability properties for a wide class of nonlinear systems is presented. In addition to an extensive study of systems on ² we propose — as an application of our method — a new criterion for global controllability of systems with polynomial drift term, defined on ⁿ. In the case of linear systems, the criterion reduces to the classical Kalman condition.A study of local controllability at the equilibrium point of the drift term of systems defined on ² enables us not only to rederive a local controllability result derived by Hermes [4,5], but also to extend this result when no assumptions are made on the boundedness of the controls.     

Controllability of a planar body with unilateral thrusters

Investigates the minimal number of unilateral thrusters required for different versions of nonlinear controllability of a planar rigid body. For one to three unilateral thrusters, one gets a new property with each additional thruster: one thruster suffices for small-time accessibility on the body's state space TSE(2); two thrusters suffice for global controllability on TSE(2); and three thrusters suffice for small-time local controllability at zero velocity states     

Local controllability of nonlinear systems

The local controllability of control systems with an arbitrary number of controls is considered, first on an open set and then at a given point; necessary conditions are derived concerning the Lie algebra T′ generated by the input vector fields, and applied to gradient systems.The main results are a geometric sufficient condition for local controllability at a point, and an equivalent condition based on the computation of Lie brackets, assuming T′ to be (n − 1)-dimensional.     

Local controllability of scalar input systems on 3-manifolds

Local controllability of generic C^k pairs of vector fields in a three dimensional connected manifold M is studied, showing that in general a necessary condition is also sufficient on an open dense subset of the set of points where it is verified. The results obtained are used to determine local controllability for scalar input affine control systems.     

Null controllability of the Burgers system with distributed controls

This paper is devoted to present some positive and negative controllability results for the viscous Burgers equation. More precisely, in the context of null controllability with distributed controls, we present sharp estimates of the minimal time of controllability T(r) of initial data of L²-norm less or equal to r. In particular, we see that (global) null controllability does not hold in general (unless the control is exerted everywhere). The same results apply to similar boundary controllability systems with one boundary control.     

Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation

The concepts of local controllability and observability of nonlinear discrete-time systems with the Caputo-, Riemann–Liouville- and Grünwald–Letnikov-type h-difference fractional order operators are studied. The Implicit Function Theorem is used in order to show that the nonlinear systems are locally observable or controllable in a finite number of steps if their linear approximations are observable or controllable, respectively, in the same number of steps.     

On the controllability of linear time-delay systems with piecewise constant inputs †

An important problem in system theory is whether a dynamic system is controllable with inputs from the class of admissible controls. This paper treats the case whore the dynamic system is described by differential-difference equations and the class of admissible control inputs consists of piecewise constant functions. This situation arises in practice in the on-line control of some industrial processes. Euclidean space and function space controllability are studied, and controllability conditions which are both necessary and sufficient are derived for R ⁿ and R ⁿ null controllability. Simple algebraic conditions are also given for two special cases. In the study of function space null controllability the concept of j-controllability is introduced and a sufficiency theorem is stated in terms of this new concept.     

Controllability of multi-agent systems with periodically switching topologies and switching leaders

This paper considers controllability of multi-agent systems with periodically switching topologies and switching leaders. The concept of m-periodic controllability is proposed, and a criterion for m-periodic controllability is established. The effect of the duration of subsystems on controllability is analysed by utilising a property of analytic functions. In addition, the influence of switching periods on controllability is investigated, and an algorithm is proposed to search for the fewest periods to ensure controllability. A necessary condition for m-periodic controllability is obtained from the perspective of eigenvectors of the subsystems’ Laplacian matrices. For a system with switching leaders, it is proved that switching-leader controllability is equivalent to multiple-leader controllability. Furthermore, both the switching order and the tenure of agents being leaders have no effect on the controllability. Some examples are provided to illustrate the theoretical results.     

On controllability of a class of discrete‐time homogeneous bilinear systems with solvable controls

This paper studies the controllability of a class of discrete‐time homogeneous bilinear systems. Necessary and sufficient conditions for the controllability are obtained. In particular, an algorithm for computing the controls, which achieve given states transition, for the controllable systems is given with examples. Furthermore, the necessary and sufficient conditions are applied to present four cases of the change of the controllability for Euler discretization, which show that Euler discretization may and may not change the controllability of nonlinear systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Driving a linear constant system by a piecewise constant control

Recent studies have shown that complete controllability by means of all admissible controls is equivalent to complete controllability by means of piecewise constant controls with n preassigned switching times. This paper addresses the problem of determining the switching times and provides a simple algorithm for the derivation of a piecewise constant control function which accomplishes the transfer of the system from its current position to a desired reachable target. A discussion on the applications of the results obtained in this study to design problems, especially to computer-controlled systems, is given.     

On the relation between local controllability and stabilizabilityfor a class of nonlinear systems

The problem of local stabilizability of locally controllable nonlinear systems is considered. It is well known that, contrary to the linear case, local controllability does not necessarily imply stabilizability. A class of nonlinear systems for which local controllability implies local asymptotic stabilizability using continuous static-state feedback is described, as for this class of systems the well-known Hermes controllability condition is necessary and sufficient for local controllability     

On the constrained small-time controllability of linear systems

This note presents a necessary and sufficient condition for small-time controllability of a linear system with respect to a cone. This result extends the controllability conditions for linear systems to the case of control and phase constraints.     

Constrained controllability of perturbed discrete-time systems

In this paper we study the global null controllability with bounded controls of perturbed linear discrete-time systems in Rⁿ . More explicitly, a globally null-controllable autonomous discrete-time system is perturbed by ‘suitably small’ terms V(k) and B(k) to obtain a system of the form x(k + 1) = [A + V(k)]x(k) + [B + B(k)]u(k) and sufficient conditions are given to ensure the global null controllability of the resulting perturbed system.     

Stabilizability of Discrete Systems over Rings

Linear nonautonomous discrete single-input systems in K ⁿ , where K is a ring with unit, are studied. Total controllability is defined and every totally controllable system is shown to be representable in canonical form (i.e., as an nth-order scalar equation with coefficients in K). Therefore, every totally controllable system is stabilizable in the sense that all solutions vanish for some linear feedback, beginning from a finite instant.     

Approximative piecewise constant bang-bang principle for differential repetitive processes

In the article, we consider a point controllability over all (infinite many) passes for linear differential repetitive processes. We prove the following approximative piecewise constant bang-bang principle: the set of points that can be reached with the aid of measurable controls on passes, taking the values in a convex compact set M ? ? ^r coincides with the closure of the set of points that can be reached with the aid of piecewise constant controls on passes, taking the values in the set of extreme points of M.     

Robust Controllability of T–S Fuzzy-Model-Based Control Systems With Parametric Uncertainties

The robust controllability problem for the Takagi–Sugeno (T–S) fuzzy-model-based control systems is studied in this paper. Under the assumption that the nominal T–S fuzzy-model-based control systems are locally controllable (i.e., each fuzzy rule of the nominal T–S fuzzy-model-based control systems has a full row rank for its controllability matrix), a sufficient condition is proposed to preserve the assumed property when the parameter uncertainties are added into the nominal T–S fuzzy-model-based control systems. The proposed sufficient condition can provide the explicit relationship of the bounds on parameter uncertainties to preserve the assumed property. Besides, a robustly global controllability condition and the related robustly global stabilizability condition of the uncertain T–S fuzzy-model-based control systems are also presented in this paper. A nonlinear mass–spring–damper mechanical system with parameter uncertainties is given as an example to illustrate the application of the proposed sufficient conditions.      

Necessary and sufficient conditions for quadratic controllability of a class of uncertain systems

We introduce a controllability notion for uncertain systems, namely, quadratic controllability. For a specific class of norm bounded uncertain systems, we show that this notion is equivalent to controllability of a nominal system and the satisfaction of a certain structural condition by the uncertainty. For systems in which the uncertain parameters affect all control inputs, this structural condition is simply the familiar matching condition commonly encountered in the literature on robust and adaptive control of uncertain systems.     

Necessary conditions for exact controllability with a finite-dimensional input space

We derive necessary conditions for exact controllability of infinite-dimensional systems described by , where the generator A has a Riesz basis of eigenvectors and the input space is finite-dimensional. These conditions are in terms of the eigenvalues of A and the degree of unboundedness of B, but no further information about B is needed. Our results easily imply lack of exact controllability for a wide range of distributed parameter systems. It is known from partial differential equation (PDE) examples that the addition of damping into an exactly controllable second order system can destroy exact controllability. We discuss the effect of damping on such systems using our general conditions. We also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension.     

Controllability of right invariant systems on real simple Lie groups of typeF 4,G 2,C n,andB n *

We deal with controllability of right-invariant systems for some real simple Lie groups ofF ₄,G ₂,C _n , andB _n types. We prove that the so-calledcontrollability rank condition is a necessary and sufficient condition for controllability for an open class of systems. In other papers, analogous results were obtained for Lie groups of the remaining types (i.e.,E ₆,E ₇,E ₈,A _n , andD _n ) using a special property of the root systems of their Lie algebras.