We address the problem of dynamic output feedback stabilization at an unobservable target point. The challenge lies in according the antagonistic nature of the objective and the properties of the system: the system tends to be less observable as it approaches the target. We illustrate two main ideas: well chosen perturbations of a state feedback law can yield new observability properties of the closed-loop system, and embedding systems into bilinear systems admitting observers with dissipative error systems allows to mitigate the observability issues. We apply them on a case of systems with linear dynamics and nonlinear observation map and make use of an ad hoc finite-dimensional embedding. More generally, we introduce a new strategy based on infinite-dimensional unitary embeddings. To do so, we extend the usual definition of dynamic output feedback stabilization in order to allow infinite-dimensional observers fed by the output. We show how this technique, based on representation theory, may be applied to achieve output feedback stabilization at an unobservable target.
For nonlinear affine in the control systems, a Lie algebraic sufficient condition for sampled-data feedback semi-global stabilization is established. We use this result, in order to derive sufficient conditions for sampled-data feedback stabilization for a couple of three-dimensional systems.
In the present work, sufficient conditions for global stabilization of nonlinear uncertain systems by means of discrete-delay static output feedback are presented. Illustrating examples show the efficiency of the proposed control strategy.
Sufficient conditions are established for sampled-data feedback global asymptotic stabilization for nonlinear autonomous systems. One of our main results is an extension of the well known Artstein-Sontag theorem on feedback stabilization concerning affine in the control systems. A second aim of the present work is to provide sufficient conditions for sampled-data feedback asymptotic stabilization for two interconnected nonlinear systems. Lie algebraic sufficient conditions are derived for the case of affine in the control interconnected systems without drift terms.
Output feedback stabilization of control systems is a crucial issue in engineering. Most of these systems are not uniformly observable, which proves to be a difficulty to move from state feedback stabilization to dynamic output feedback stabilization. In this paper, we present a methodology to overcome this challenge in the case of dissipative systems by requiring only target detectability. These systems appear in many physical systems and we provide various examples and applications of the result.
This paper is concerned with the stabilization of parallel-flow heat exchangers equation with boundary control and arbitrarily delayed observation. The observer and predictor-based method, developed by [B.Z. Guo, K.Y. Yang, Automatica, 2009, 45(6): 1468-1475], is used to stabilize equation. The exponentially stabilization of such system by estimated state feedback control law is proved.
For a general time-varying system, we prove that existence of an "Output Robust Control Lyapunov Function" implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well-known result towards feedback stabilization due to J. M. Coron and L. Rosier concerning stabilization of autonomous systems by means of time-varying periodic feedback.
The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in AB to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances.
It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variat…
We construct a patchy feedback for a general control system on $\R^n$ which realizes practical stabilization to a target set $Σ$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $Σ$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in "patchy" form, a particular class of piecewise smooth controls which ensure the existence of local Carathéodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with r…
We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete $L^2-$stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by …
We are interested in the control of forming processes for nonlinear material models. To develop an online control we derive a novel feedback law and prove a stabilization result. The derivation of the feedback control law is based on a Laypunov analysis of the time-dependent viscoplastic material models. The derivation uses the structure of the underlying partial differential equation for the design of the feedback control. Analytically, exponential decay of the time evolution of perturbations to desired stress--strain states is shown. We test the new control law numerically by coupling it to …
It is shown that an oblique projection based feedback control is able to stabilize the state of the Kuramoto-Sivashinsky equation, evolving in rectangular domains, to a given time-dependent trajectory. The number of actuators is finite and consists of a finite number of indicator functions supported in small subdomains. Simulations are presented, in the one-dimensional case, showing the stabilizing performance of the feedback control.
An oblique projections based feedback stabilizability result in the literature is extended to a larger class of reaction-convection terms. A discussion is presented including a comparison between explicit oblique projections base feedback controls and Riccati based feedback controls. Advantages and limitations of each type of feedback are addressed as well as their finite-elements implementation. Results of numerical simulations are presented comparing their stabilizing performances for the case of time-periodic dynamics. It is shown that the solution of the periodic Riccati based feedback can…
A learning approach for optimal feedback gains for nonlinear continuous time control systems is proposed and analysed. The goal is to establish a rigorous framework for computing approximating optimal feedback gains using neural networks. The approach rests on two main ingredients. First, an optimal control formulation involving an ensemble of trajectories with 'control' variables given by the feedback gain functions. Second, an approximation to the feedback functions via realizations of neural networks. Based on universal approximation properties we prove the existence and convergence of opti…
We use the backstepping method to study the stabilization of a 1-D linear transport equation on the interval (0, L), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized exponentially in that space and, for any given decay rate, we give an explicit feedback law that achieves that decay rate.
The aim of this work is to design an explicit finite dimensional boundary feedback controller of sampled-data form for locally exponentially stabilizing the equilibrium solutions to semilinear parabolic equations. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any sampling rate, not necessary to be small enough, and it tends to the continuous-times version when the sampling period tends to zero.
We propose to explore switching methods in order to recover some properties of Kalmanlike observers for output feedback stabilization of state-affine systems that may present observability singularities. The self-tuning gain matrix in Kalman-like observers tend to be singular in the case of non-uniformly observable systems. We show in the case of state-affine systems with observable target that it can be prevented by dynamically monitoring observability of the system, and switching the control when it becomes critical.
The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and…
Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in \cite{PBT}. In this paper we have addressed stabilization problems related to sabra shell model of turbulence. We have studied internal stabilization via finite dimensional controller. Moreover we have also studied optimal robust control problem by solving…
This paper studies the periodic feedback stabilization for a class of linear $T$-periodic evolution equations.Several equivalent conditions on the linear periodic feedback stabilization are obtained. These conditions are related with the following subjects: the attainable subspace of the controlled evolution equation under consideration; the unstable subspace (of the evolution equation with the null control) provided by the Kato projection; the Poincar$\acute{e}$ map associated with the evolution equation with the null control; and two unique continuation properties for the dual equations on d…
The aim of this work is to design an explicit finite dimensional boundary feedback controller for locally exponentially stabilizing the equilibrium solutions to Fisher's equation in both $L^2(0,1)$ and $H^1(0,1)$. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any level of instability, which extends the result of \cite{02} for nonlinear parabolic equations. The effectiveness of the approach is illustrated by a numerical simulation.
For a general class of dynamical systems (of which the canonical continuous and uniform discrete versions are but special cases), we prove that there is a state feedback gain such that the resulting closed-loop system is uniformly exponentially stable with a prescribed rate. The methods here generalize and extend Gramian-based linear state feedback control to much more general time domains, e.g. nonuniform discrete or a combination of continuous and discrete time. In conclusion, we discuss an experimental implementation of this theory.
In this work feedback control laws are designed for achieving three-axis attitude stabilization of inertial pointing spacecraft using only magnetic torquers. The designs are based on an almost periodic model of geomagnetic field along the spacecraft's orbit. Both attitude plus attitude rate feedback, and attitude only feedback are proposed. Both feedback laws achieve local exponential stability robustly with respect to large uncertainties in the spacecraft's inertia matrix. The latter properties are proved using general averaging and Lyapunov stability. Simulations are included to validate the…
The value function associated with an optimal control problem subject to the Navier-Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated.
The objective of this document is to design continuous feedback controls for global asymptotic stabilization (GAS) of affine systems, with control restricted to a compact and convex set (CVS). This stabilization problem is solved based on a design of a feedback function restricted to the hyperbox and obtained by means of the CLF theory. By "normalizing" this feedback, the continuous stabilizer restricted to CVS is obtained.
Motivated by recent applications in control theory, we study the feedback stabilizability of switched systems, where one is allowed to chose the switching signal as a function of $x(t)$ in order to stabilize the system. We propose new algorithms and analyze several mathematical features of the problem which were unnoticed up to now, to our knowledge. We prove complexity results, (in-)equivalence between various notions of stabilizability, existence of Lyapunov functions, and provide a case study for a paradigmatic example introduced by Stanford and Urbano.
This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in $L^2(0,\infty)$ but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feed…
In this paper, we investigate the estimator-based output feedback control problem of multi-delay systems. This work is an extension of recently developed operator-value LMI framework for infinite-dimensional time-delay systems. Based on the optimal convex state feedback controller and generalized Luenberger observer synthesis conditions we already have, the estimator-based output feedback controller is designed to contain the estimates of both the present state and history of the state. An output feedback controller synthesis condition is proposed using SOS method, which is expressed in a set …
We consider the problem of boundary feedback control of single-input-single-output (SISO) one-dimensional linear hyperbolic systems when sensing and actuation are anti-located. The main issue of the output feedback stabilization is that it requires dynamic control laws that include delayed values of the output (directly or through state observers) which may not be robust to infinitesimal uncertainties on the characteristic velocities. The purpose of this paper is to highlight some features of this problem by addressing the feedback stabilization of an unstable open-loop system which is made up…
In this paper we study the state-feedback stabilization of a discrete-time Markov jump linear system when the observation of the Markov chain of the system, called the Markov state, is time-randomized by another Markov chain. Embedding the Markov state into an extended Markov chain, we transform the given system with time-randomized observations to another one having the enlarged Markov-state space but with so-called cluster observations of Markov states. Based on this transformation we propose linear matrix inequalities for designing stabilizing state-feedback gains for the original Markov ju…
The paper addresses the stabilization of nonlinear systems with semi-quadratic cost: quadratic with respect to controls and nonlinear for state variables. Paper presents the effective new feedback synthesis procedure. The novel feedback design procedure is based on the ideas borrowed from nonlinear optics and the theory of semi-classical asymptotics.
In this paper, stability analysis of time delay systems is considered based on decomposition of the systems to subsystems. The decomposition process needs matrices of these systems to be simultaneously block triangularize. We show that a finite set of matrices are simultaneously block triangularize if and only if they have a common invariant subspace. The importance of the decomposition is highlighted by some systems that their characteristic equation has repeated roots on the imaginary axis. For such systems, the cluster treatment method and the direct method cannot be employed to analyze the…
Anantharam showed in 1985 the existence of a model in which some stabilizable plants do not have its right-/left-coprime factorizations. In this paper, we give a condition of the nonexistence of the right-/left-coprime factorizations of stabilizable plants as a generalization of Anantharam's result. As examples of the models which satisfy the condition, we present two models. We illustrate the construction of a stabilizing controller of stabilizable single-input single-output plants of such models.
In this paper, we provide a systematic approach for the design of stabilizing feedback controllers for nonlinear control systems using the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. …
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…
We describe a general multiplier method to obtain boundary stabilization of the wave equation by means of a (linear or quasi-linear) Neumann feedback. This also enables us to get Dirichlet boundary control of the wave equation. This method leads to new geometrical cases concerning the "active" part of the boundary where the feedback (or control) is applied. Due to mixed boundary conditions, the Neumann feedback case generate singularities. Under a simple geometrical condition concerning the orientation of the boundary, we obtain a stabilization result in linear or quasi-linear cases.
On classical solutions in the stabilization problem for nonholonomic control systems with time-varying feedback laws
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine（乌克兰国家科学院应用数学与力学研究所）；Department of Mathematics, University of Klagenfurt（克拉根福大学数学系）
We consider a one-dimensional controlled reaction-diffusion equation, where the control acts on the boundary and is subject to a constant delay. Such a model is a paradigm for more general parabolic systems coupled with a transport equation. We prove that this is possible to stabilize (in H 1 norm) this process by means of an explicit predictor-based feedback control that is designed from a finite-dimensional subsystem. The implementation is very simple and efficient and is based on standard tools of pole-shifting. Our feedback acts on the system as a finite-dimensional predictor. We compare o…
We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in $L^2[0,\infty)…
Dynamic output-feedback stabilization of uncertain linear dynamics via digital twins
Johann Radon Institute for Computational and Applied Mathematics, ÖAW（奥地利科学院计算与应用数学约翰·拉德诺研究所）；Institute of Mathematics and Scientific Computing, Karl-Franzens University of Graz（格拉茨大学数学与科学计算研究所）；Departament of Mathematics, FCT, NOVA University of Lisbon（里斯本NOVA大学FCT数学系）
In this paper, we study the boundary feedback stabilization of a quasilinear hyperbolic system with partially dissipative structure. Thanks to this structure, we construct a suitable Lyapunov function which leads to the exponential stability to the equilibrium of the $H^2$ solution. As an application, we also obtain the feedback stabilization for the Saint-Venant-Exner model under physical boundary conditions.
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…
This paper provides an alternative approach referred to as pseudo-predictor feedback (PPF) for stabilization of linear systems with multiple input delays. Differently from the traditional predictor feedback which is from the model reduction appoint of view, the proposed PPF utilizes the idea of prediction by generalizing the corresponding results for linear systems with a single input delay to the case of multiple input delays. Since the PPF will generally lead to distributed controllers, a truncated pseudopredictor feedback (TPPF) approach is established instead which gives finite dimensional…
We develop a predictor-feedback control design for multi-input nonlinear systems with distinct input delays, of arbitrary length, in each individual input channel. Due to the fact that different input signals reach the plant at different time instants, the key design challenge, which we resolve, is the construction of the predictors of the plant's state over distinct prediction horizons such that the corresponding input delays are compensated. Global asymptotic stability of the closed-loop system is established by utilizing arguments based on Lyapunov functionals or estimates on solutions. We …
A switched equilibrium of a switched system of two subsystems is a such a point where the vector fields of the two subsystems point strictly towards one another. Using the concept of stable convex combination that was developed by Wicks-Peleties-DeCarlo (1998) for linear systems, Bolzern-Spinelli (2004) offered a design of a state feedback switching rule that is capable to stabilize an affine switched system to any switched equilibrium. The state feedback switching rule of Bolzern-Spinelli gives a nonlinear (quadratic) switching threshold passing through the switched equilibrium. In this paper…
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 presents a first mathematical convergence analysis of a Fock states feedback stabilization scheme via single-photon corrections. This measurement-based feedback has been developed and experimentally tested in 2012 by the cavity quantum electrodynamics group of Serge Haroche and Jean-Michel Raimond. Here, we consider the infinite-dimensional Markov model corresponding to the ideal set-up where detection errors and feedback delays have been disregarded. In this ideal context, we show that any goal Fock state can be stabilized by a Lyapunov-based feedback for any initial quantum state …
In this paper, we discuss the feedback stabilization of bilinear systems under weak observation properties. In this case, the uniform stability is not guaranteed. Thus we provide an explicit weak decay rate for all regular initial data. Applications to Schrödinger and wave equations are provided.
For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be …
