Publications about 'controllability' |
Articles in journal or book chapters |
We discuss several issues related to the stabilizability of nonlinear systems. First, for continuously stabilizable systems, we review constructions of feedbacks that render the system input-to-state stable with respect to actuator errors. Then, we discuss a recent paper which provides a new feedback design that makes globally asymptotically controllable systems input-to-state stable to actuator errors and small observation noise. We illustrate our constructions using the nonholonomic integrator, and discuss a related feedback design for systems with disturbances. |
The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well. |
In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controllability, The Problem of Stabilization, Obstructions to Continuous Stabilization, Control-Lyapunov Functions and Artstein's Theorem, Discontinuous Feedback, Nonsmooth CLF's, Insensitivity to Small Measurement and Actuator Errors, Effect of Large Disturbances: Input-to-State Stability, Comments on Notions Related to ISS. |
This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set. |
This paper studies controllability properties of recurrent neural networks. The new contributions are: (1) an extension of the result in "Complete controllability of continuous-time recurrent neural networks" to a slightly different model, where inputs appear in an affine form, (2) a formulation and proof of a necessary and sufficient condition, in terms of local-local controllability, and (3) a complete analysis of the 2-dimensional case for which the hypotheses made in previous work do not apply. |
For analytic discrete-time systems, it is shown that uniform forward accessibility implies the generic existence of universal nonsingular control sequences. A particular application is given by considering forward accessible systems on compact manifolds. For general systems, it is proved that the complement of the set of universal sequences of infinite length is of the first category. For classes of systems satisfying a descending chain condition, and in particular for systems defined by polynomial dynamics, forward accessibility implies uniform forward accessibility. |
This paper provides an exposition of some recent results regarding system-theoretic aspects of continuous-time recurrent (dynamic) neural networks with sigmoidal activation functions. The class of systems is introduced and discussed, and a result is cited regarding their universal approximation properties. Known characterizations of controllability, observability, and parameter identifiability are reviewed, as well as a result on minimality. Facts regarding the computational power of recurrent nets are also mentioned. |
It is shown that every asymptotically controllable system can be stabilized by means of some (discontinuous) feedback law. One of the contributions of the paper is in defining precisely the meaning of stabilization when the feedback rule is not continuous. The main ingredients in our construction are: (a) the notion of control-Lyapunov function, (b) methods of nonsmooth analysis, and (c) techniques from positional differential games. |
This paper presents a characterization of controllability for the class of control systems commonly called (continuous-time) recurrent neural networks. The characterization involves a simple condition on the input matrix, and is proved when the activation function is the hyperbolic tangent. |
This paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability along suitable trajetories. Once such loops are available, it is possible to employ standard Newton or steepest descent methods, as classically done in numerical control. The theoretical justification of the approach relies on recent results establishing the genericity of nonsingular controls, as well as a simple convergence lemma. |
Controllability questions for discrete-time nonlinear systems are addressed in this paper. In particular, we continue the search for conditions under which the group-like notion of transitivity implies the stronger and semigroup-like property of forward accessibility. We show that this implication holds, pointwise, for states which have a weak Poisson stability property, and globally, if there exists a global "attractor" for the system. |
For analytic systems satisfying the strong accessibility rank condition, generic inputs produce trajectories along which the linearized system is controllable. Applications to the steering of systems without drift are briefly mentioned. |
The notion of controllability was identified by Kalman as one of the central properties determining system behavior. His simple rank condition is ubiquitous in linear systems analysis. This article presents an elementary and expository overview of the generalizations of this test to a condition for testing accessibility of discrete and continuous time nonlinear systems. |
Results are given on the integrability of certain distributions which arise from smoothly parametrized families of diffeomorphisms acting on manifolds. Applications to control problems and in particular to the problem of sampling are discussed. Pages 42-50 apply the results to the control of continuous time systems; this is an exposition of some of the basic results of the Lie algebraic accessibility theory. |
This paper presents a geometric study of controllability for discrete-time nonlinear systems. Various accessibility properties are characterized in terms of Lie algebras of vector fields. Some of the results obtained are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear. |
This note establishes a result linking algebraically coprime factorizations of transfer matrices of delay systems to approximately coprime factorizations in the sense of distributions. The latter have been employed by the second author in the study of function-space controllability for such systems. |
This paper studies accessibility (weak controllability) of bilinear systems under constant sampling rates. It is shown that the property is preserved provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case, and which, for observability, results in the classical Nyquist theorem. |
The present article compares the difficulties of deciding controllability and accessibility. These are standard properties of control systems, but complete algebraic characterizations of controllability have proved elusive. We show in particular that for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard. |
This paper concerns itself with the existence of open-loop control generators for nonlinear (continuous-time) systems. The main result is that, under relatively mild assumptions on the original system, and for each fixed compact subset of the state space, there always exists one such generator. This is a new system with the property that the controls it produces are sufficiently rich to preserve complete controllability along nonsingular trajectories. General results are also given on the continuity and differentiability of the input to state mapping for various p-norms on controls, as well as a comparison of various nonlinear controllability notions. |
A nonlinear controllable plant, under mild technical conditions, admits a precompensator with the following property: along control trajectories joining pairs of states, the composite system (precompensator plus plant) is, up to first order, isomorphic to a parallel connection of integrators. |
Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case. |
We continue here our investigation into the preservation of structural properties under the sampling of nonlinear systems. The main new result is that, under minimal hypothesis, a controllable system always satisfies a strong type of approximate sampled controllability. |
In this note we present an algebraic approach to the proof that a linear system with matrices (A,B) is null-controllable using bounded inputs iff it is null-controllable (with unbounded inputs) and all eigenvalues of A have nonpositive real parts (continuous time) or magnitude not greater than one (discrete time). We also give the analogous results for the asymptotic case. Finally, we give an interpretation of these results in the context of local nonlinear controllability. |
This note studies the preservation of controllability (and other properties) under sampling of a nonlinear system. More detailed results are obtained in the cases of analytic systems and of systems with finite dimensional Lie algebras. |
It is shown that a control system in Rn is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls. |
A paper that introduces a separation principle for general finite dimensional analytic continuous-time systems, proving the equivalence between existence of an output regulator (which is an abstract dynamical system) and certain "0-detectability" and asymptotic controllability assumptions. |
Conference articles |
This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type) is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context. |
It is shown that the existence of a continuous control-Lyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finite-dimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative (upper contingent derivative). This result generalizes to the non-smooth case the theorem of Artstein relating closed-loop feedback stabilization to smooth CLF's. It relies on viability theory as well as optimal control techniques. A "non-strict" version of the results, analogous to the LaSalle Invariance Principle, is also provided. |
This paper shows how to extend recent results of Colonius and Kliemann, regarding connections between chaos and controllability, from continuous to discrete time. The extension is nontrivial because the results all rely on basic properties of the accessibility Lie algebra which fail to hold in discrete time. Thus, this paper first develops further results in nonlinear accessibility, and then shows how a theorem can be proved, which while analogous to the one given in the work by Colonius and Klieman, also exhibits some important differences. A counterexample is used to show that the theorem given in continuous time cannot be generalized in a straightforward manner. |
This paper shows the existence of (nonlinear) smooth dynamic feedback stabilizers for linear time invariant systems under input constraints, assuming only that open-loop asymptotic controllability and detectability hold. |
It has been known for a long time that certain controllability properties are more difficult to verify than others. This article makes this fact precise, comparing controllability with accessibility, for a wide class of nonlinear continuous time systems. The original contribution is in formalizing this comparison in the context of computational complexity. (This paper placed here by special request.) |
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