This paper is concerned with the output feedback exponential stabilization for a flexible beam with tip mass. When there is no disturbance, it is shown that only one non-collocated measurement is enough to exponentially stabilize the original system by constructing an infinite-dimensional Luenberger state observer to track the state and designing an estimated state based output feedback control law. This essentially improves the existence result in [F. Conrad and O. Mörgül, SIAM J. Control Optim., 36 (1998), 1962-1986] where two collocated measurements including high order feedback were adopted. In the case that boundary internal uncertainty and external disturbance are considered, an infinite-dimensional disturbance estimator is constructed to estimate the state and total disturbance in real time. By virtue of the estimated state and estimated total disturbance, an output feedback control law is designed to exponentially stabilize the original system while guaranteeing the boundedness of the closed-loop system. Some illustration simulations are presented.
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.
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)…
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 solving the problem of asymptotic tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected must be generated by an exosystem such as a finite dimensional exosystem with pure imaginary eigenvalues. The objective of this paper is to solve such a tracking problem for the unstable Burgers' equation without this assumption. Our treatment of this problem is straightforward. Using the method of variable transform, the tracking problem is split into two separate problems: a simple Neumann boundary stabilization problem a…
In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \mathbb{N}$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b$ of the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed ba…
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.
In this expository paper, which covers material presented at the NATO Advanced Study Institute "Nonlinear Analysis, Differential Equations, and Control" (Montreal, Jul/Aug 1998), 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 Controll…
We study the output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary. With only one measurement, we construct an infinite-dimensional state observer to trace the state and design an estimated state based controller to exponentially stabilize the original system. This is an essentially important improvement for the existence literature [Ö. Morgül, B.P. Rao and F. Conrad, IEEE Transactions on Automatic Control, 39(10) (1994), 2140-2145] where two measurements including the high order angular velocity feedback were adopted. When a control matched nonlinear internal unce…
Adaptive stabilization of a leaderless bearing-constrained formation with disturbances
East Asia University of Technology（东亚理工大学）；University of Southern California（南加州大学）；Ghost Robotics（幽灵机器人公司）；Hanoi University of Science and Technology（河内科技大学）
We consider nonlinear control systems of the so-called generalized triangular form (GTF) with time-varying and periodic dynamics which linearly depends on some external disturbances. Our purpose is to construct a feedback controller which provides the global input-to-state stability of the corresponding closed-loop w.r.t. the disturbances. To do this, we combine the method proposed in the earlier work \cite{pavlichkov_ge_2009} devoted the the global asymptotic stabilization of the GTF systems without disturbances with the ISS theory for time-varying systems proposed in \cite{wang}. Following t…
Yet Another Tutorial of Disturbance Observer: Robust Stabilization and Recovery of Nominal Performance
ASRI, Department of Electrical and Computer Engineering, Seoul National University（首尔国立大学电气与计算机工程系自动系统研究室）；Department of Electrical Engineering and Computer Science, University of Central Florida（中佛罗里达大学电气与计算机科学系）；School of Robotics, Kwangwoon University（光云大学机器人学院）；Department of Electrical Engineering, Soongsil University（崇实大学电气工程系）
In this paper, the control problem for underactuated systems in the presence of external disturbances and model uncertainties is considered. An adaptive fuzzy sliding mode controller (AFSMC) is proposed to solve the problem, satisfying the robustness properties against uncertainties and disturbances. The designed controller can be applied to a wide class of underactuated systems and have fewer restrictions, compared with many pervious works. To evaluate the performance of proposed algorithm, it is applied to an underactuated inverted pendulum and a Translational Oscillator/Rotational Actuator …
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.
In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.
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 …
In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is …
This work addresses the problem of transient stabilization of a power grid, following a destabilizing disturbance. The model considered is the cascade interconnection of seven New England test models with the disturbance (e.g., a powerline failure) occurring in the first grid and propagating forward, emulating a wide-area blackout. We consider a data-driven control framework based on the Koopman operator theory, where a linear predictor, evolving on a higher dimensional (embedded) state-space, is built from observed data and subsequently used within a model predictive control (MPC) framework, …
One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying (LPV) plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation (LFT) models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified $L^{2}$-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to z…
This study focuses on the problem of optimal mismatched disturbance rejection control for uncontrollable linear discrete-time systems. In contrast to previous studies, by introducing a quadratic performance index such that the regulated state can track a reference trajectory and minimize the effects of disturbances, mismatched disturbance rejection control is transformed into a linear quadratic tracking problem. The necessary and sufficient conditions for the solvability of this problem over a finite horizon and a disturbance rejection controller are derived by solving a forward-backward diffe…
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…
In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on the Arstein's theory of control Liapunov functions and related Sontag's formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distrib…
Optimal Universal Controllers for Roll Stabilization
Oceaneering AGV Systems B.V.（Oceaneering AGV系统有限公司）；University of Groningen（格罗宁根大学）；Politecnico di Torino（都灵理工大学）；Institute for Problems of Mechanical Engineering（机械工程学院问题研究所）
This paper deals with the exponential input-to-state stabilization with respect to boundary disturbances of a class of diagonal infinite-dimensional systems via delay boundary control. The considered input delays are uncertain and time-varying. The proposed control strategy consists of a constant-delay predictor feedback controller designed on a truncated finite-dimensional model capturing the unstable modes of the original infinite-dimensional system. We show that the resulting closed-loop system is exponentially input-to-state stable with fading memory of both additive boundary input perturb…
In this work, we propose a methodology for the expression of necessary and sufficient Lyapunov-like conditions for the existence of stabilizing feedback laws. The methodology is an extension of the well-known Control Lyapunov Function (CLF) method and can be applied to very general nonlinear time-varying systems with disturbance and control inputs, including both finite- and infinite-dimensional systems. The generality of the proposed methodology is also reflected upon by the fact that partial stability with respect to output variables is addressed. In addition, it is shown that the generalize…
Robust global stabilization of nonlinear systems by observer-based feedback controllers is a challenging task. This article investigates the problem of designing observer-based stabilizing controllers for incrementally quadratic nonlinear systems with external disturbances. The nonlinearities considered in the system model satisfy the incremental quadratic constraints, which are characterized by incremental multiplier matrices and encompass many common nonlinearities. The simultaneous search for the observer and the controller gain matrices is formulated as a feasibility problem of linear matr…
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.
In this paper, we consider issues relative to prescribed time stabilisation of a chain of integrators of arbitrary length, either pure (i.e., where there is no disturbance) or perturbed. In a first part, we revisit the proportional navigation feedback (PNF) approach and we show that it can be appropriately recasted within the framework of time-varying homogeneity. As a first consequence, we first recover all previously obtained results on PNF with simpler arguments. We then apply sliding mode inspired feedbacks to achieve prescribed stabilisation with uniformly bounded gains. However, all thes…
Road traffic crashes have been the leading cause of death among young people. Most of these accidents occur when the driver becomes distracted and a loss-of-control situation occurs. Steer-by-Wire systems were recently proposed as an alternative to mitigate such accidents. This technology enables the decoupling of the front wheel steering angles from the driver hand wheel angle and, consequently, the measurement of road/tire friction limits and the development of novel control systems capable of ensuring vehicle stabilization and safety. However, vehicle safety boundaries are highly dependent …
This paper discusses the boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we sh…
In this paper, we are interested in the relation between the solutions of the control system $\dot x=f(x,u)$ and the solutions of its (potentially unknown) perturbation $\dot x=f(x,u)+w(x,t).$ Under the assumption that the linear part of the unperturbed system at the point $(0,0)$ is controllable and that disturbance $w(x,t)$ is asymptotically sufficiently small, there exists a state-feedback controller of the form $u=-Kx$ such that the perturbed system preserves the local asymptotic stability of the zero solution of unperturbed system. The main result of this paper gives the sufficient condit…
H$_1$-ISS Analysis and Boundary Control Design for Coupled Linear ODEs and Hyperbolic PDEs
Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学）；Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，格勒诺布尔国立理工学院，GIPSA实验室）
This paper is concerned with the point torque boundary feedback stabilization of a damped Euler-Bernoulli beam model in the presence of a time-varying state-delay. First, a finite-dimensional truncated model is derived by spectral reduction. Then, for a given stabilizing state-feedback of the delay-free truncated model, an LMI-based sufficient condition on the maximum amplitude of the state-delay is employed to guarantee the stability of the closed-loop state-delayed truncated model. Second, we assess the exponential stability of the resulting closed-loop infinite-dimensional system under the …
This work is concerned with practical stabilization of nonlinear systems by means of inf-convolution-based sample-and-hold control. It is a fairly general stabilization technique based on a generic non-smooth control Lyapunov function (CLF) and robust to actuator uncertainty, measurement noise, etc. The stabilization technique itself involves computation of descent directions of the CLF. It turns out that non-exact realization of this computation leads not just to a quantitative, but also qualitative obstruction in the sense that the result of the computation might fail to be a descent directi…
Global Stabilization of Nonlinear Delay Systems With a Compact Absorbing Set
National Technical University of Athens（雅典国立技术大学）；University of California, San Diego（加州大学圣地亚哥分校）；Laboratoire GREYC CNRS-ENSICAEN（GREYC国家科学研究中心-ENSICAEN实验室）；Centre National de la Recherche Scientifique, CNRS-EECI SUPELEC（法国国家科学研究中心，CNRS-EECI苏佩莱克）
