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.
This paper deals with the problem of boundary stabilization of first-order n\times n inhomogeneous quasilinear hyperbolic systems. A backstepping method is developed. The main result supplements the previous works on how to design multi-boundary feedback controllers to realize exponential stability of the original nonlinear system in the spatial H^2 sense.
We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.
Hyperbolic systems in one dimensional space are frequently used in modeling of many physical systems. In our recent works, we introduced time independent feedbacks leading to the finite stabilization for the optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work, we present Lyapunov's functions for these feedbacks and use estimates for Lyapunov's functions to rediscover the finite stabilization results.
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.
In this paper, we present Lyapunov-based robust and adaptive controllers for the finite time stabilization of a perturbed chain of integrators with bounded uncertainties. The proposed controllers can be designed for integrator chains of any arbitrary length. The uncertainty bounds are known in the robust control problem whereas they are unknown in the adaptive control problem. Both controllers are developed from a class of finite time stabilization controllers for pure integrator chains. Lyapunov-based design permits to calculate upper bound on convergence time.
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 address the question of robustness of critical bit rates for the stabilization of networked control systems over digital communication channels. For a deterministic nonlinear system, the smallest bit rate above which practical stabilization (in the sense of set-invariance) can be achieved is measured by the invariance entropy of the system. Under the assumptions of chain controllability and a uniformly hyperbolic structure on the set of interest, we prove that the invariance entropy varies continuously with respect to system parameters. Hence, in this case the critical bit rat…
The notion of the relaxed Robust Control Lyapunov Function (relaxed RCLF) is introduced and is exploited for the design of robust feedback stabilizers for nonlinear systems. Particularly, it is shown for systems with input constraints that relaxed RCLFs can be easily obtained, while RCLFs are not available. Moreover, it is shown that the use of relaxed RCLFs usually results to different feedback designs from the ones obtained by the use of the standard RCLF methodology. Using the relaxed RCLFs feedback design methodology, a simple controller that guarantees robust global stabilization of a per…
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…
This paper studies the cooperative global robust stabilization problem for a class of nonlinear multi-agent systems. The problem is motivated from the study of the cooperative global robust output regulation problem for the class of nonlinear multi-agent systems in normal form with unity relative degree which was studied recently under the conditions that the switching network is undirected and some nonlinear functions satisfy certain growth condition. We first solve the stabilization problem by using the multiple Lyapunov functions approach and the average dwell time method. Then, we apply th…
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…
This paper considers the problem of robust stability and stabilization for linear fractional-order system with nonlinear uncertain parameters, with fractional order 0<a<2. A dynamic output feedback controller, with predetermined order, for asymptotically stabilizing such uncertain fractional-order systems is designed. The derived stabilization conditions are in LMI form. Simulation results of two numerical examples illustrate that the proposed sufficient theoretical results are applicable and effective for tackling robust stabilization problems. Keywords: Fractional-order system, nonlinear unc…
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…
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…
We aim at providing a characterization of the ability to maintain a stochastic coupled system with porous media components in a prescribed set of constraints by using internal controls. This property is proven via a quasi-tangency local-in-time condition in the spirit of Euler approximation schemes. In particular, by employing one of the components of the system as asymptotic supervisor, we give conditions guaranteeing the exponential asymptotic stabilizability of controlled porous media equations.
Results on the problem of stabilizing a nonlinear continuous-time system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some significant control problems.
In this paper we present a general tool to handle the presence of zero dynamics which are asymptotically but not locally exponentially stable in problems of robust nonlinear stabilization by output feedback. We show how it is possible to design locally Lipschitz stabilizers under conditions which only rely upon a partial detectability assumption on the controlled plant, by obtaining a robust stabilizing paradigm which is not based on design of observers and separation principles. The main design idea comes from recent achievements in the field of output regulation and specifically in the desig…
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.
This paper investigates the robust stability and stabilization analysis of interval fractional-order systems with time-varying delay. The stability problem of such systems is solved first, and then using the proposed results a stabilization theorem is also included, where sufficient conditions are obtained for designing a stabilizing controller with a predetermined order, which can be chosen to be as low as possible. Utilizing efficient lemmas, the stability and stabilization theorems are proposed in the form of LMIs, which is more suitable to check due to various existing efficient convex opt…
In this paper, we study the stabilization problem for a hyperbolic type Stokes system posed on a bounded domain. We show that when the damping effects are restricted to a subdomain satisfying the geometrical control condition the system decays exponentially. The result is a consequence of a new quasi-mode estimate for the Stokes system.
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…
Output-based controllers are known to be fragile with respect to model uncertainties. The standard $\mathcal{H}_{\infty}$-control theory provides a general approach to robust controller design based on the solution of the $\mathcal{H}_{\infty}$-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with…
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…
Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linear…
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…
We study the problem of target stabilization with robust obstacle avoidance in robots and vehicles that have access only to vision-based sensors for the purpose of realtime localization. This problem is particularly challenging due to the topological obstructions induced by the obstacle, which preclude the existence of smooth feedback controllers able to achieve simultaneous stabilization and robust obstacle avoidance. To overcome this issue, we develop a vision-based hybrid controller that switches between two different feedback laws depending on the current position of the vehicle using a hy…
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 …
This paper proposes a procedure to control an uncertain discrete-time networked control system through a limited stabilizing input information. The system is primarily affected by the time-varying, norm bounded, mismatched parametric uncertainty. The input information is limited due to unreliability of communicating networks. An event-triggered based robust control strategy is adopted to capture the network unreliability. In event-triggered control the control input is computed and updated at the system end only when a pre-specified event condition is violated. The robust control input is deri…
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…
We consider control strategies for large-scale interacting agent systems under uncertainty. The particular focus is on the design of robust controls that allow to bound the variance of the controlled system over time. To this end we consider $\mathcal{H}_\infty$ control strategies on the agent and mean field description of the system. We show a bound on the $\mathcal{H}_\infty$ norm for a stabilizing controller independent on the number of agents. Furthermore, we compare the new control with existing approaches to treat uncertainty by generalized polynomial chaos expansion. Numerical results a…
In this paper, we consider N-level quantum angular momentum systems interacting with electromagnetic fields undergoing continuous-time measurements. We suppose unawareness of the initial state and physical parameters, entailing the introduction of an additional state representing the estimated quantum state. The evolution of the quantum state and its estimation is described by a coupled stochastic master equation. Here, we study the asymptotic behavior of such a system in presence of a feedback controller. We provide sufficient conditions on the feedback controller and on the estimated paramet…
We analyze robust stability, in an input-output sense, of switched stable systems. The primary goal (and contribution) of this paper is to design switching strategies to guarantee that input-output stable systems remain so under switching. We propose two types of {\em supervisors}: dwell-time and hysteresis based. While our results are stated as tools of analysis they serve a clear purpose in design: to improve performance. In that respect, we illustrate the utility of our findings by concisely addressing a problem of observer design for Lur'e-type systems; in particular, we design a hybrid ob…
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）
This paper presents new results that allow one to address the discrete-time general nonlinear robust control problem. The uncertain system is described by a general nonlinear function set characterized by the nominal model and the corresponding modeling error bound. Traditional synthesis methods design parameters of a structured robust controller. The key aim of this paper is to find an unstructured robust controller set in the state-control space, which enlarges the estimate of the closed-loop robust domain of attraction (RDOA). Based on the interval analysis arithmetic, a numerical method to…
This paper discusses the in-domain feedback stabilization of reaction-diffusion PDEs with Robin boundary conditions in the presence of an uncertain time- and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable…
