Publications about 'nonlinear control' |
Articles in journal or book chapters |
Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle. These approaches behave better when searching for saturated (bang-bang) solutions. In previous work by one of the authors, singular solutions were theoretically shown to exist for the time-optimal problem for two-link manipulators under hard torque constraints. The theoretical results gave explicit formulas, based on Lie theory, for singular segments of trajectories, but the global structure of solutions remains unknown. In this work, we show how to effectively combine these theoretically found formulas with the use of general-purpose optimal control softwares. By using the explicit formula given by theory in the intervals where the numerical solution enters a singular arcs, we not only obtain an algebraic expression for the control in that interval, but we are also able to remove artifacts present in the numerical solution. In this way, the best features of numerical algorithms and theory complement each other and provide a better picture of the global optimal structure. We showcase the technique on a 2 degrees of freedom robotic arm example, and also propose a way of extending the analyzed method to robotic arms with higher degrees of freedom through partial feedback linearization, assuming the desired task can be mostly performed by a few of the degrees of freedom of the robot and imposing some prespecified trajectory on the remaining joints. |
We consider a nonlinear SISO system that is a cascade of a scalar "bottleneck entrance" with a stable positive linear system. In response to any periodic inflow, all solutions converge to a unique periodic solution with the same period. We study the problem of maximizing the averaged throughput via controlled switching. We compare two strategies: 1) switching between a high and low value, and 2 ~using a constant inflow equal to the prescribed mean value. We show that no possible switching policy can outperform a constant inflow rate, though it can approach it asymptotically. We describe several potential applications of this problem in traffic systems, ribosome flow models, and scheduling at security checks. |
The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of ``error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject. |
The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of ``error'' signal). A very special case, when there are no inputs, is that of stability. This short and informal article provides an introduction to the subject. |
This paper shows that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since discontinuous feedbacks are allowed, solutions are understood in the ``sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). This work generalizes the CLSS Theorem, which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise. |
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. |
This paper introduces and studies a new definition of the minimum-phase property for general smooth nonlinear control systems. The definition does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the ``input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of minimum-phase systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control. |
We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed. |
(This is an expository paper prepared for a plenary talk given at the Second Nonlinear Control Network Workshop, Paris, June 9, 2000.) The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work. |
This note provides explicit algebraic stabilizing formulas for clf's when controls are restricted to certain Minkowski balls in Euclidean space. Feedbacks of this kind are known to exist by a theorem of Artstein, but the proof of Artstein's theorem is nonconstructive. The formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems. |
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. |
One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances. |
This paper provides a precise result which shows that insensitivity to small measurement errors in closed-loop stabilization can be attained provided that the feedback controller ignores observations during small time intervals. |
This paper proposes a generally applicable technique for the control of analytic systems with no drift. The method is based on the generation of "nonsingular loops" that allow linearized controllability. One can then implement Newton and/or gradient searches in the search for a control. A general convergence theorem is proved. |
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. |
This paper compares the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the known approximation theorems. The differences are not based on numerical accuracy or number of units needed, nor on capabilities for feature extraction, but rather on a much more basic classification into "direct" and "inverse" problems. The former correspond to the approximation of continuous functions, while the latter are concerned with approximating one-sided inverses of continuous functions - and are often encountered in the context of inverse kinematics determination or in control questions. A general result is given showing that nonlinear control systems can be stabilized using two hidden layers, but not in general using just one. |
This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations. |
This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result a Lyapunov-theoretic definition is proposed for bounded input bounded output stability. The main technical fact proved relates the notion of stabilizability studied in the state space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization. (Note: This is the original ISS paper, but the ISS results have been much improved in later papers. The material on coprime factorizations is still of interest, but the 89 CDC paper has some improvements and should be read too.) |
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. |
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. |
Development of an approach to nonlinear control based on mixtures of linear systems and finite automata. File obtained by scanning. |
Conference articles |
We consider a general class of translation-invariant systems with a specific category of output nonlinearities motivated by biological sensing. We show that no dynamic output feedback can stabilize this class of systems to an isolated equilibrium point. To overcome this fundamental limitation, we propose a simple control scheme that includes a low-amplitude periodic forcing function akin to so-called "active sensing" in biology, together with nonlinear output feedback. Our analysis shows that this approach leads to the emergence of an exponentially stable limit cycle. These findings offer a provably stable active sensing strategy and may thus help to rationalize the active sensing movements made by animals as they perform certain motor behaviors. |
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. |
Invited talk at the 1994 ICM. Paper deals with the notion of observables for nonlinear systems, and their role in realization theory, minimality, and several control and path planning questions. |
This paper proposes a technique for the control of analytic systems with no drift. It is based on the generation of "nonsingular loops" which allow linearized controllability. Once such loops are available, it is possible to employ standard Newton or steepest descent methods. The theoretical justification of the approach relies on results on genericity of nonsingular controls as well as a simple convergence lemma. |
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