Small-time Local Controllability Research Articles

Small-Time Local Controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term

The goal of this article is to contribute to a better understanding of the relations between the exact controllability of nonlinear PDEs and the control theory for ODEs based on Lie brackets, through a study of the Schrödinger PDE with bilinear control. We focus on the small-time local controllability (STLC) around an equilibrium, when the linearized system is not controllable. We study the second-order term in the Taylor expansion of the state, with respect to the control. For scalar-input ODEs, quadratic terms never recover controllability: they induce signed drifts in the dynamics. Thus proving STLC requires to go at least to the third order. Similar results were proved for the bilinear Schrödinger PDE with scalar-input controls. In this article, we study the case of multi-input systems. We clarify among the quadratic Lie brackets, those that allow to recover STLC. For ODEs, our result is a consequence of Sussmann’s sufficient condition S(θ), but we propose a new proof, designed to prepare an easier transfer to PDEs. This proof relies on a representation formula of the state inspired by the Magnus formula. By adapting it, we prove a new STLC result for the bilinear Schrödinger PDE.

Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition

We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known [K. Beauchard and M. Morancey, Math. Control Relat. Fields 4 (2014) 125-160, M. Bournissou, J. Diff. Equ. 351 (2023) 324−360] that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's S (θ) sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.

Small time local controllability of driftless nonholonomic systems in a task-space

Small time local controllability of driftless nonholonomic systems in a task-space

On the small-time local controllability

A class of polynomial control systems is considered. The local properties of the corresponding reachable sets are studied by using a general differential–geometrical approach based on the Campbell–Baker–Hausdorff (C–B–H) formula. The main result is a new sufficient condition of small-time local controllability. Two four-dimensional examples illustrate this result.

Quadratic behaviors of the 1D linear Schrödinger equation with bilinear control

We consider a 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear control. We study its controllability around the ground state when the linearized system is not controllable and wonder whether the quadratic term can help to recover the directions lost at the first order. More precisely, in this paper, we formulate assumptions under which the quadratic term induces a drift which prevents the small-time local controllability (STLC) of the system in appropriate spaces.For finite-dimensional systems, quadratic terms induce coercive drifts in the dynamic, quantified by integer negative Sobolev norms, along explicit Lie brackets which prevent STLC.In the context of the bilinear Schrödinger equation, the first drift, quantified by the H−1-norm of the control, was already observed in [8] and used to deny STLC with controls small in L∞. In this article, we improve this result by denying STLC with controls small in W−1,∞.Furthermore, for any positive integer n, we formulate assumptions under which one may observe a quadratic drift quantified by the H−n-norm of the control and we use it to deny STLC in suitable spaces.

Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates

<p style='text-indent:20px;'>We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [<xref ref-type="bibr" rid="b5">5</xref>], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.</p>

On the small-time local controllability of a KdV system for critical lengths

This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all noncritical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Crépeau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1; later Cerpa, and then Cerpa and Crépeau, established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is not small-time locally controllable.

On the Small-Time Local Controllability

On the Small-Time Local Controllability

Controllability and accessibility results for N-link horizontal planar manipulators with one unactuated joint

Previous work on controllability of underactuated serial horizontal planar manipulators mostly focused only on a specific number of links, whereas this paper considers a general N-link case. The main contribution of this paper is accessibility and small-time local controllability (STLC) results for N-link serial, horizontal, planar manipulators with one unactuated joint. An N-link (N≥3) manipulator with the first joint actuated is STLC for a subset of equilibrium points based on Sussmann’s general theorem for STLC. In contrast, if the first joint is not actuated, it is not accessible or STLC.

Nonlinear Controllability Assessment of Aerial Manipulator Systems using Lagrangian Reduction

This paper analyzes the nonlinear Small-Time Local Controllability (STLC) of a class of underatuated aerial manipulator robots. We apply methods of Lagrangian reduction to obtain their lowest dimensional equations of motion (EOM). The symmetry-breaking potential energy terms are resolved using advected parameters, allowing full $SE(3)$ reduction at the cost of additional advection equations. The reduced EOM highlights the shifting center of gravity due to manipulation and is readily in control-affine form, simplifying the nonlinear controllability analysis. Using Sussmann's sufficient condition, we conclude that the aerial manipulator robots are STLC near equilibrium condition, requiring Lie bracket motions up to degree three.

Small-Time Local Controllability of Switched Nonlinear Systems

This article considers the small-time local controllability (STLC) of switched nonlinear systems (SNSs). First, the original SNS is associated with a nilpotent Lie algebra, on which the solutions of the SNS are approximated by the hybrid Lie power series. Second, the concepts of high-order control variations and bad Lie brackets are defined for SNSs from a geometrical point of view. This helps to establish a new high-order sufficient STLC condition that is closely related to “neutralized” bad Lie brackets. Finally, it is shown that the obtained general condition is compatible with the existing controllability conditions of switched linear systems and nonswitched linear and nonlinear systems.

A necessary condition for small-time local controllability

A nonlinear control system whose right-hand side is Lipschitz continuous with respect to the state–space variables is considered. A new necessary small-time local controllability condition is proved under natural assumptions. A nonlinear three-dimensional example illustrates the applicability of the obtained result.

Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations

We consider scalar-input control systems in the vicinity of an equilibrium, at which the linearized systems are not controllable. For finite dimensional control systems, the authors recently classified the possible quadratic behaviors. Quadratic terms introduce coercive drifts in the dynamics, quantified by integer negative Sobolev norms, which are linked to Lie brackets and which prevent smooth small-time local controllability for the full nonlinear system.In the context of nonlinear parabolic equations, we prove that the same obstructions persist. More importantly, we prove that two new behaviors occur, which are impossible in finite dimension. First, there exists a continuous family of quadratic obstructions quantified by fractional negative Sobolev norms or by weighted variations of them. Second, and more strikingly, small-time local null controllability can sometimes be recovered from the quadratic expansion. We also construct a system for which an infinite number of directions are recovered using a quadratic expansion.As in the finite dimensional case, the relation between the regularity of the controls and the strength of the possible quadratic obstructions plays a key role in our analysis.

On Small-Time Local Controllability

In this paper, we study small-time local controllability of real analytic control-affine systems under small perturbations of their vector fields. Consider a real analytic control system X which is small-time locally controllable and whose reachable sets shrink with the polynomial rate of order N with respect to time. We will prove a general theorem which states that any real analytic control-affine system whose vector fields are perturbations of the vector fields of X with polynomials of order higher than N is again small-time locally controllable. In particular, we show that this result connects two long-standing open conjectures about small-time local controllability of systems.

Two-Stage Switching Hybrid Control Method Based on Improved PSO for Planar Three-Link Under-Actuated Manipulator

The planar three-link passive-active-active (PAA) under-actuated manipulator does not satisfy the small-time local controllability (STLC), which makes it more difficult to control than other planar under-actuated manipulators. This paper presents a two-stage switching hybrid control method based on improved particle swarm optimization (PSO) for the PAA under-actuated manipulator. According to the model reduction strategy, the control process is divided into two stages. The workspace of the manipulator is analyzed to guarantee the target position within the reachable region. In order to reduce the probability of falling into local optimum, we introduce the Metropolis criterion of simulated annealing algorithm and crowding factor of artificial fish swarm algorithm into PSO algorithm to calculate the target angles of two active joints. At each control stage, we design the Lyapunov function to maintain the angle of an active joint unchanged while we build the sliding surface and select the power function as reaching law to make the other active joint converge to the target angle. The control law switches to next stage when the rotated joint converges to the target angle. The proposed method not only retains the rapidity of sliding mode control, but also reduces the total chattering of the system by the Lyapunov function method. Simulation results demonstrate that the proposed method has shorter steady-state adjustment time than that of existing typical control methods, and its overshoot is almost zero.

Stability and nonlinear controllability analysis of a quadrotor-like autonomous underwater vehicle considering variety of cases

A quadrotor-like autonomous underwater vehicle that is similar to, yet different from quadrotor unmanned aerial vehicles, has been reported recently. This article investigates the stability and nonlinear controllability properties of the vehicle. First, the 12-degree-of-freedom model of the vehicle deploying an X shape actuation system is developed. Then, a stability property is investigated showing that the vehicle cannot be stabilized by a time invariant smooth state feedback law. After that, by adopting a nonlinear controllability analysis tool in geometric control theory, the small-time local controllability of the vehicle is analyzed for a variety of cases, including the vertical plane motion, the horizontal plane motion, and the three-dimensional space motion. Finally, different small-time local controllability conditions for different cases are developed. The result shows that the small-time local controllability holds for vertical plane motion and horizontal plane motion. However, the full degree of freedom kinodynamics model (i.e. 12 states) of the vehicle does not satisfy the small-time local controllability from zero-velocity states.

Design for three-dimensional stabilization control of underactuated autonomous underwater vehicles

In order to realize three-dimensional stable control of underactuated autonomous underwater vehicles (AUVs), this study analyses nonlinear hydrodynamic characteristics of AUVs based on the mathematical Taylor series. The nonholonomic control system properties of AUVs are investigated using nonholonomic system theory under three-dimensional control input. The constraint of underactuated AUVs is proved not to be integrated by the local integrability theorem, and the controllability of underactuated AUVs is verified through small-time local controllability (STLC). In order to simplify the trigonometric terms in underactuated AUV motion functions, quaternion theory is applied to transform the function to develop a continuous time-varying controller. Results of a simulation experiment show that this control law is effective in achieving three-dimensional stabilization from arbitrary initial positions.

Quadratic obstructions to small-time local controllability for scalar-input systems

We consider nonlinear finite-dimensional scalar-input control systems in the vicinity of an equilibrium.When the linearized system is controllable, the nonlinear system is smoothly small-time locally controllable: whatever m>0 and T>0, the state can reach a whole neighborhood of the equilibrium at time T with controls arbitrary small in Cm-norm.When the linearized system is not controllable, we prove that: either the state is constrained to live within a smooth strict manifold, up to a cubic residual, or the quadratic order adds a signed drift with respect to it. This drift holds along a Lie bracket of length (2k+1), is quantified in terms of an H−k-norm of the control, holds for controls small in W2k,∞-norm and these spaces are optimal. Our proof requires only C3 regularity of the vector field.This work underlines the importance of the norm used in the smallness assumption on the control, even in finite dimension.

Addendum to “Local Controllability of the Two-Link Magneto-Elastic Microswimmer”

In the above mentioned note (<hal-01145537>, <arXiv:1506.05918>, published in IEEE Trans. Autom. Cont., 2017), the first and fourth authors proved a local controllability result around the straight configuration for a class of magneto-elastic micro-swimmers.That result is weaker than the usual small-time local controllability (STLC), and the authors left the STLC question open. The present addendum closes it by showing that these systems cannot be STLC.

Local controllability and attitude stabilization of multirotor UAVs: Validation on a coaxial octorotor

This paper addresses the attitude controllability problem for a multirotor unmanned aerial vehicle (UAV) in case of one or several actuators failures. The small time local controllability (STLC) of the system attitude dynamics is analysed using the nonlinear controllability theory with unilateral control inputs. This analysis considers different actuators configurations and compares their fault tolerance capabilities regarding actuators failures. Analytical results are then validated experimentally on a coaxial octorotor. A stabilization control law is applied on the coaxial configuration under one, two, three and four motors failures, when the system is controllable. Real-time experimental results demonstrate the effectiveness of the applied strategy.