In this paper, the problem of partial stabilization of nonlinear systems along a given trajectory is considered. This problem is treated within the framework of stability of a family of sets. Sufficient conditions for the asymptotic stability of a one-parameter family of sets using time-dependent control in the form of trigonometric polynomials are derived. The obtained results are applied to a model mechanical system.
Global uniform asymptotic stabilization and k-exponential trajectory tracking of underactuated surface ships with non-diagonal inertia/damping matrices
In this paper we consider the problem of stabilization and tracking of desired state trajectory for a wide range of nonlinear control problems with disturbances. We present the sufficient conditions for the existence of $C^k$ state feedback controllers and the process of their mathematical designing is described.
Given a nonstationary trajectory of the Navier-Stokes system, a finite-dimensional feedback boundary controller stabilizing locally the system to the given trajectory is derived. Moreover the controller is supported in a given open subset of the boundary of the domain containing the fluid. In a first step a controller is derived that stabilizes the linear Oseen-Stokes system "around the given trajectory" to zero; for that a corollary of a suitable truncated boundary observability inequality, the regularizing property for the system, and some standard techniques of the optimal control theory ar…
We present a framework of sequential action control (SAC) for stabilization of systems of partial differential equations which can be posed as abstract semilinear control problems in Hilbert spaces. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyze the closed-loop system obtained from the SAC feedback for the linear problem with quadratic stage costs. We apply this theory to a prototypical example of an unstable heat equation and provide numerical results as the verification …
This article presents a state feedback control design strategy for the stabilization of a vehicle along a reference collision avoidance maneuver. The stabilization of the vehicle is achieved through a combination of steering, acceleration and braking. A Linear Parameter-Varying (LPV) model is obtained from the linearization of a non-linear model along the reference trajectory. A robust state feedback control law is computed for the LPV model. Finally, simulation results illustrate the stabilization of the vehicle along the reference trajectory.
This paper focuses on the problem of constructing time-varying feedback laws that asymptotically stabilize a given part of the state variables for nonlinear control-affine systems. It is assumed that the class of systems under consideration satisfies nonlinear controllability conditions with respect to the stabilizable variables. Under these assumptions, a time-periodic feedback control is constructed explicitly by using the inversion of the matrix composed of the control vector fields and their Lie brackets. The proposed control design scheme is applied to solving the leader-following problem…
We consider a possible application of the Ważewski topological method to feedback control systems and to more general dynamical systems. We show how this method can be used to prove the impossibility of global stabilization in such problems. Moreover, we give sufficient conditions for the existence of a solution such that its trajectory never leaves a subset of the extended phase space of the system and does not tend asymptotically to a given equilibrium. We illustrate our result with various real-life systems including the Furuta pendulum and the wheeled inverted pendulum.
This paper addresses the problem of stabilizing a part of variables for control systems described by stochastic differential equations of the Ito type. The considered problem is related to the asymptotic stability property of invariant sets and has important applications in mechanics and engineering. Sufficient conditions for the asymptotic stability of an invariant set are proposed by using a stochastic version of LaSalle's invariance principle. These conditions are applied for constructing the state feedback controllers in the problem of single-axis stabilization of a rigid body. The cases o…
The problem of partial stabilization for nonlinear control systems described by the Ito stochastic differential equations is considered. For these systems, we propose a constructive control design method which leads to establishing the asymptotic stability in probability of the trivial solution of the closed-loop system with respect to a part of state variables. Mechanical examples are presented to illustrate the efficiency of the obtained controllers.
The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schrödinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=λ|u|^2u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exa…
Feedback asymptotic stabilization of control systems is an important topic of control theory and applications. Broadly speaking, if the system $\dot{x} = f(x,u)$ is locally asymptotically stabilizable, then there exists a feedback control $u(x)$ ensuring the convergence to an equilibrium for any trajectory starting from a point sufficiently close to the equilibrium state. In this paper, we develop a reasonably natural and general composition operator approach to stabilizability. To begin with, we provide an extension of the classical Hautus lemma to the generalized context of composition opera…
We consider the problem of stabilizing the bilayer \textit{Saint-Venant} model, which is a coupled system of two rightward and two leftward convecting transport partial differential equations (PDEs). In the stability proofs, we employ a Lyapunov function in which the parameters need to be successively determined. To the best of the authors' knowledge, this is the first time this kind of Lyapunov function is employed, and this result is the first one on the stabilization of the linearized bilayer \textit{Saint-Venant} model. Numerical simulations of the bilayer \textit{Saint-Venant} problem are…
Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetri…
In this paper, we study the stabilization problem for a food extrusion process in the isothermal case. The model expresses the mass conservation in the extruder chamber and consists of a hyperbolic Partial Differential Equation (PDE) and a nonlinear Ordinary Differential Equation (ODE) whose dynamics describes the evolution of a moving interface. By using a Lyapunov approach, we obtain the exponential stabilization for the closed-loop system under natural feedback controls through indirect measurements.
The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. An finite-dimensional delay controller is computed for the unstable part, and it is shown th…
A system of partial differential equations describing the spatial oscillations of an Euler-Bernoulli beam with a tip mass is considered. The linear system considered is actuated by two independent controls and separated into a pair of differential equations in a Hilbert space. A feedback control ensuring strong stability of the equilibrium in the sense of Lyapunov is proposed. The proof of the main result is based on the theory of strongly continuous semigroups.
This paper proposes a line integral Lyapunov function approach to stability analysis and stabilization for Itô stochastic T-S models. Unlike the deterministic case, stability analysis of this model needs the information of Hessian matrix of the line integral Lyapunov function which is related to partial derivatives of the basis functions. By introducing a new method to handle these partial derivatives and using the property of state-dependent matrix with rank one, the stability conditions of the underlying system can be established via a line integral Lyapunov function. These conditions obtain…
Stabilizing an unknown dynamical system is one of the central problems in control theory. In this paper, we study the sample complexity of the learn-to-stabilize problem in Linear Time-Invariant (LTI) systems on a single trajectory. Current state-of-the-art approaches require a sample complexity linear in $n$, the state dimension, which incurs a state norm that blows up exponentially in $n$. We propose a novel algorithm based on spectral decomposition that only needs to learn "a small part" of the dynamical matrix acting on its unstable subspace. We show that, under proper assumptions, our alg…
