Sufficient conditions for global stabilization of nonlinear systems with delayed input by means of approximate predictors are presented. An approximate predictor is a mapping which approximates the exact values of the stabilizing input for the corresponding system with no delay. A systematic procedure for the construction of approximate predictors is provided for globally Lipschitz systems. The resulting stabilizing feedback can be implemented by means of a dynamic distributed delay feedback law. Illustrating examples show the efficiency of the proposed control strategy.
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.
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…
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 …
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…
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…
Sampling arises simultaneously with input and output delays in networked control systems. When the delay is left uncompensated, the sampling period is generally required to be sufficiently small, the delay sufficiently short, and, for nonlinear systems, only semiglobal practical stability is generally achieved. For example, global stabilization of strict-feedforward systems under sampled measurements, sampled-data stabilization of the nonholonomic unicycle with arbitrarily sparse sampling, and sampled-data stabilization of LTI systems over networks with long delays, are open problems. In this …
We study a feedback scheme to stabilize an arbitrary photon number state in a microwave cavity. The quantum non-demolition measurement of the cavity state allows a non-deterministic preparation of Fock states. Here, by the mean of a controlled field injection, we desire to make this preparation process deterministic. The system evolves through a discrete-time Markov process and we design the feedback law applying Lyapunov techniques. Also, in our feedback design we take into account an unavoidable pure delay and we compensate it by a stochastic version of a Smith predictor. After illustrating …
This paper studies the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition while the system output is selected as a either Dirichlet or Neumann boundary trace. The control strategy takes the form of a finite-dimensional observer-based controller with feedback and observer gains that are computed in order to dominate the state delayed term. For any arbitrarily given value of the state delay, we show the exponential stability of the resulting closed-loop s…
This paper addresses the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs with delayed boundary measurement. The output takes the form of a either Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component which is employed to obtain the control input to be applied at current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided th…
Recently, [2] proved that the closed-loop system resulting from the output proportional feedback stabilization of a class of delayed neural fields is input-to-state stable (ISS) for sufficiently high gain, subject to the existence of an equilibrium point for the closed-loop system. In the present paper, we show that a sufficient condition for such an equilibrium to exist is that the activation functions are bounded.
The problem of vibration suppression in helicopter rotors is the main topic of this paper. We revisit a strategy for the stabilization of such systems based on delayed feedback. By means of an improved analysis, made possible by recently developed mathematical tools, we present insights on how the delay can be used as stabilization tool for this problem. The results are supported with numerical simulations. For simplicity, only the case of hovering flight is considered.
This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of a either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input del…
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…
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）
Recently a constructive method was introduced for finite-dimensional observer-based control of 1D parabolic PDEs. In this paper we present an improved method in terms of the reduced-order LMIs (that significantly shorten the computation time) and introduce predictors to manage with larger delays. We treat the case of a 1D heat equation under Neumann actuation and non-local measurement, that has not been studied yet. We apply modal decomposition and prove $L^2$ exponential stability by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension $N$ and r…
This paper deals with the problem of predicting the future state of discrete-time input-delayed systems in the presence of unknown disturbances that can affect both the state and the output equations of the plant. Since the disturbance is unknown, computing an exact prediction of the future plant states is not possible. To circumvent this problem, we propose using a high-order extended Luenberger-type observer for the plant states, disturbances, and their finite difference variables, combined with a new equation for computing the prediction based on Newton's series from the calculus of finite …
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…
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 note, we present an effective solution to the stabilization of linear input delay systems subject to dissipative constraints while all the effect of input delay is compensated by a controller with novel structure. The method is inspired by the recent development in the mathematical treatment of distributed delays and predictor controllers, which are critical for the derivation of the solution. An important conceptual innovation is the use of a parameterized dynamical state feedback controller (DSFC), where the dimension of the controller equals the dimension of the control input. A suf…
Much recent progress has been achieved for stabilization of linear and nonlinear systems with input delays that are long and dependent on either time or the plant state---provided the dependence is known. In this paper we consider the delay variations as unknown and study robustness of nominal constant-delay predictor feedbacks under delay variations that depend on time and the state. We show that when the delay perturbation and its rate have sufficiently small magnitude, the local asymptotic stability of the closed-loop system, under the nominal predictor-based design, is preserved. For the s…
This paper presents a new Python software for the parametric design of stabilizing feedback laws with time delays, called Partial Pole Placement via Delay Action (P3$δ$). After an introduction recalling recent theoretical results on the multiplicity-induced-dominancy (MID) and coexisting real roots-induced-dominancy (CRRID) properties and their use for the feedback stabilization of control systems operating under time delays, the paper presents the current version of P3$δ$, which relies on the MID property to compute delayed stabilizing feedback laws for scalar differential equations with a si…
We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a const…
New trajectory-based small-gain results are obtained for nonlinear feedback systems under relaxed assumptions. Specifically, during a transient period, the solutions of the feedback system may not satisfy some key inequalities that previous small-gain results usually utilize to prove stability properties. The results allow the application of the small-gain perspective to various systems which satisfy less demanding stability notions than the Input-to-Output Stability property. The robust global feedback stabilization problem of an uncertain time-delayed chemostat model is solved by means of th…
We study a sampled-data implementation of linear controllers that depend on the output and its derivatives. First, we consider an LTI system of relative degree $r\ge 2$ that can be stabilized using $r-1$ output derivatives. Then, we consider PID control of a second order system. In both cases, the Euler approximation is used for the derivatives giving rise to a delayed sampled-data controller. Given a derivative-dependent controller that stabilizes the system, we show how to choose the parameters of the delayed sampled-data controller that preserves the stability under fast enough sampling. Th…
We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas the structure displacement satisfies the damped plate equation. We consider here the Navier slip boundary conditions. The main result of this work is the feedback stabilization of the strong solutions of the corresponding system around a stationary state for any exponential decay rate by means of a time delayed control localized on the fixed fluid bound…
In this paper, we consider boundary stabilization for a cascade of ODE-heat system with state delay under actuator saturation. To stabilize the system, we design a state feedback controller via the backstepping method and find a bound on the domain of attraction. The latter bound is based on Lyapunov method, whereas the exponential stability of the delayed cascaded systems is proved by using Halanay's inequality. Numerical examples illustrate the efficiency of the method.
This paper investigates state feedback control laws for Markov jump linear systems with state and mode-observation delays. An assumption in this study is that the delay of mode observation obeys an exponential distribution. Also, we raise an unknown time-varying state delay applied in the composition of the state feedback controller. A method of remodeling the closed-loop system as a standard Markov jump linear system with state delay is shown. Furthermore, on the basis of this remodeling, several Linear Matrix Inequalities (LMI) for designing feedback gains for stabilization and mixed $H_2/H_…
The problem of stabilization of unstable periodic orbits of discrete nonlinear systems is considered in the article. A new generalization of the delayed feedback, which solves the stabilization problem, is proposed. The feedback is represented as a convex combination of nonlinear feedback and semilinear feedback introduced by O. Morgul. In this article, the O. Morgul method was transferred from the scalar case to the vector one. It is shown that the additional introduction of the semilinear feedback into the equation makes it possible to significantly reduce the length of the prehistory used i…
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 …
We present a methodology for stabilization of general nonlinear systems with actuator dynamics governed by general, quasilinear, first-order hyperbolic PDEs. Since for such PDE-ODE cascades the speed of propagation depends on the PDE state itself (which implies that the prediction horizon cannot be a priori known analytically), the key design challenge is the determination of the predictor state. We resolve this challenge and introduce a PDE predictor-feedback control law that compensates the transport actuator dynamics. Due to the potential formation of shock waves in the solutions of quasili…
Multi-objective LQR with Optimum Weight Selection to Design FOPID Controllers for Delayed Fractional Order Processes
School of Electronics and Computer Science, University of Southampton（南安普顿大学电子与计算机科学学院）；Department of Power Engineering, Jadavpur University（贾达普大学动力工程系）；Department of Earth Science and Engineering, Imperial College London（帝国理工学院地球科学与工程系）；Reactor Control Division, Bhabha Atomic Research Centre（巴哈原子研究中心反应堆控制部）
This paper presents the software Partial Pole Placement via Delay Action, or P3$δ$ for short. P3$δ$ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays, thanks to two properties of the distribution of quasipolynomials' zeros, called multiplicity-induced-dominancy and coexisting real roots-induced-dominancy. After recalling recent theoretical results on these properties and their use for the feedback stabilization of control systems operating under time delays, the paper presents the main features of the current version of …
This paper studies the problem of designing sampled-data observers and observer-based, sampled-data, output feedback stabilizers for systems with both discrete and distributed, state and output time-delays. The obtained results can be applied to time delay systems of strict-feedback structure, transport Partial Differential Equations (PDEs) with nonlocal terms, and feedback interconnections of Ordinary Differential Equations with a transport PDE. The proposed design approach consists in exploiting an existing observer, which features robust exponential convergence of the error when continuous-…
