This paper investigates the output feedback setpoint regulation control of a reaction-diffusion equation by means of boundary control. The considered reaction-diffusion plant may be open-loop unstable. The proposed control strategy consists of the coupling of a finite-dimensional observer and a PI controller in order to achieve the boundary setpoint regulation control of various system outputs such as the Dirichlet and Neumann traces. In this context, it is shown that the order of the finite-dimensional observer can always be selected large enough, with explicit criterion, to achieve both the stabilization of the plant and the setpoint regulation of the system output.
The cooperative output regulation problem of linear multi-agent systems was formulated and studied by the distributed observer approach in [20, 21]. Since then, several variants and extensions have been proposed, and the technique of the distributed observer has also been applied to such problems as formation, rendezvous, flocking, etc. In this chapter, we will first present a more general formulation of the cooperative output regulation problem for linear multi-agent systems that includes some existing versions of the cooperative output regulation problem as special cases. Then, we will descr…
This paper is concerned with the regulation control of a one-dimensional reaction-diffusion equation in the presence of a state-delay in the reaction term. The objective is to achieve the PI regulation of the right Dirichlet trace with a command selected as the left Dirichlet trace. The control design strategy consists of the design of a PI controller on a finite dimensional truncated model obtained by spectral reduction. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the proposed control design procedure achieves both the exponential …
In this paper, we first present an adaptive distributed observer for a discrete-time leader system. This adaptive distributed observer will provide, to each follower, not only the estimation of the leader's signal, but also the estimation of the leader's system matrix. Then, based on the estimation of the matrix S, we devise a discrete adaptive algorithm to calculate the solution to the regulator equations associated with each follower, and obtain an estimated feedforward control gain. Finally, we solve the cooperative output regulation problem for discrete-time linear multi-agent systems by b…
This paper investigates the output feedback boundary control of reaction-diffusion equations with either distributed or boundary measurement by means of a finite-dimensional observer. A constructive method dealing with the design of finite-dimensional observers for the feedback stabilization of reaction-diffusion equations was reported in a recent paper in the case where either the control or the observation operator is bounded and also satisfies certain regularity assumptions. In this paper, we go beyond by demonstrating that a finite-dimensional state-feedback combined with a finite-dimensio…
In this paper, we study the cooperative output regulation problem for the discrete linear time-delay multi-agent systems by distributed observer approach. In contrast with the same problem for continuous-time linear time-delay multi-agent systems, the problem has two new features. First, in the presence of time-delay, the regulator equations for discrete-time linear systems are different from those for continuous-time linear systems. Second, under the standard assumption on the connectivity of the communication graph, a distributed observer for any continuous-time linear leader system always e…
This paper shows how the theory of nonlinear adaptive observers can be effectively used in the design of internal models for nonlinear output regulation. The theory substantially enhances the existing results in the context of {\em adaptive} output regulation, by allowing for not necessarily stable zero dynamics of the controlled plant and by weakening the standard assumption of having the steady state control input generated by a linear system.
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…
The general context of this work is the feedback control of an infinite-dimensional system so that the closed-loop system satisfies a fading-memory property and achieves the setpoint tracking of a given reference signal. More specifically, this paper is concerned with the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. In this setting, the studied reaction-diffusion equation might be either open-loop stable or unstable. The proposed control strategy goes as follows. First, …
This paper considers the cooperative output regulation problem for linear multi-agent systems with a directed communication graph, heterogeneous linear subsystems, and an exosystem whose output is available to only a subset of subsystems. Both the cases with nominal and uncertain linear subsystems are studied. For the case with nominal linear subsystems, a distributed adaptive observer-based controller is designed, where the distributed adaptive observer is implemented for the subsystems to estimate the exogenous signal. For the case with uncertain linear subsystems, the proposed distributed o…
This paper revisits a well studied leader-following consensus problem of linear multi-agent systems, while aiming at follower nodes' transient performance. Conventionally, when not all follower nodes have access to the leader's state information, distributed observers are designed to estimate the leader's state, and the observers are coupled via communication network. Then each follower node only needs to track its observer's state independently, without interacting with its neighbors. This paper deliberately introduces certain coupling effect among follower nodes, such that the follower nodes…
This paper shows how the theory of adaptive observers can be effectively used in the design internal models for nonlinear output regulation. The main result obtained in this way is a new method for the synthesis of adaptive internal models which substantially enhances the existing theory of adaptive output regulation, by allowing nonlinear internal models and more general classes of controlled plants.
We address the output regulation problem for a general class of linear stochastic systems. Specifically, we formulate and solve the ideal full-information and output-feedback problems, obtaining perfect, but non-causal, asymptotic regulation. A characterisation of the problem solvability is deduced. We point out that the ideal problems cannot be solved in practice because they unrealistically require that the Brownian motion affecting the system is available for feedback. Drawing from the ideal solution, we formulate and solve approximate versions of the full-information and output-feedback pr…
This paper is concerned with the output feedback stabilization of a reaction-diffusion equation by means of bounded control inputs in the presence of saturations. Using a finite-dimensional controller composed of an observer coupled with a finite-dimensional state-feedback, we derive a set of conditions ensuring the stability of the closed-loop plant while estimating the associated domain of attraction in the presence of saturations. This set of conditions is shown to be always feasible for an order of the observer selected large enough. The stability analysis relies on Lyapunov functionals al…
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…
Optimal Output Feedback Learning Control for Discrete-Time Linear Quadratic Regulation
School of Automation, Beijing Institute of Technology, Beijing 100081, China（自动化学院，北京理工大学，北京）；Department of Chemical Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada（化学工程系，皇后大学，加拿大）
One of the most interesting questions in control theory is that of constructing observers. Observers compute estimates of the internal states of a dynamical system, using data provided by measurement probes or partial state information. For linear systems, Luenberger observers (also known as "deterministic Kalman filters" since they amount to Kalman filters designed without regard to the statistics of measurement noise) provide a general solution, but, for nonlinear systems, establishing generally applicable conditions for existence and convergence of observers is an open and active area of re…
This paper studies the design of a finite-dimensional output feedback controller for the stabilization of a reaction-diffusion equation in the presence of a sector nonlinearity in the boundary input. Due to the input nonlinearity, classical approaches relying on the transfer of the control from the boundary into the domain with explicit occurrence of the time-derivative of the control cannot be applied. In this context, we first demonstrate using Lyapunov direct method how a finite-dimensional observer-based controller can be designed, without using the time derivative of the boundary input as…
We present three dynamic error feedback controllers for robust output regulation of regular linear systems. These controllers are (i) a minimal order robust controller for exponentially stable systems (ii) an observer-based robust controller and (iii) a new internal model based robust controller structure. In addition, we present two controllers that are by construction robust with respect to predefined classes of perturbations. The results are illustrated with an example where we study robust output tracking of a sinusoidal reference signal for a two-dimensional heat equation with boundary co…
This paper provides observer-based sampled-data and event-triggered boundary control strategies for a class of reaction-diffusion PDEs with collocated sensing and Robin actuation. Infinite-dimensional backstepping design is used as the underlying control approach. It is shown that the continuous-time output feedback boundary control applied in a sample-and-hold fashion ensures global closed-loop exponential stability, provided that the sampling period is sufficiently small. Further, robustness to perturbations of the sampling schedule is guaranteed. For the event-triggered implementation of th…
In this paper, we study the event-triggered global robust practical output regulation problem for a class of nonlinear systems in output feedback form with any relative degree. Our approach consists of the following three steps. First, we design an internal model and an observer to form the so-called extended augmented system. Second, we convert the original problem into the event-triggered global robust practical stabilization problem of the extended augmented system. Third, we design an output-based event-triggered control law and a Zeno-free output-based event-triggered mechanism to solve t…
This paper is concerned with the cooperative output regulation problem for a network of agents with different dynamics described by parabolic PDEs subject to spatially- and time-varying parameters. Firstly, a networked controller is designed achieving output synchronization for identical finite-dimensional reference models, which deliver the state of the global reference model required for the synchronization to the parabolic agents. The latter can be subject to local disturbances acting in-domain, on all boundaries and on the anti-collocated output to be controlled. The cooperative output reg…
The present paper addresses the problem of existence of an (output) feedback law to the purposes of asymptotically steering to zero a given controlled variable, while keeping all state variables bounded, for any initial conditions in a given compact set. The problem can be viewed as an extension of the classical problem of semi-globally stabilizing the trajectories of a controlled system to a compact set. The problem also encompasses a version of the classical problem of output regulation. Assuming only the existence of a feedback law that keeps the trajectories of the zero dynamics of the con…
This paper is concerned with the state estimation problem for genetic regulatory networks with time-varying delays and reaction-diffusion terms under Dirichlet boundary conditions. It is assumed that the nonlinear regulation function is of the Hill form. The purpose of this paper is to design a state observer to estimate the concentrations of mRNA and protein through available measurement outputs. By introducing new integral terms in a novel Lyapunov--Krasovskii functional and employing Wirtinger-based integral inequality, Wirtinger's inequality, Green's identity, convex combination approach, …
A small-gain approach is proposed to analyze closed-loop stability of linear diffusion-reaction systems under finite-dimensional observer-based state feedback control. For this, the decomposition of the infinite-dimensional system into a finite-dimensional slow subsystem used for design and an infinite-dimensional residual fast subsystem is considered. The effect of observer spillover in terms of a particular (dynamic) interconnection of the subsystems is thoroughly analyzed for in-domain and boundary control as well as sensing. This leads to the application of a small-gain theorem for interco…
We study robust output regulation for parabolic partial differential equations and other infinite-dimensional linear systems with analytic semigroups. As our main results we show that robust output tracking and disturbance rejection for our class of systems can be achieved using a finite-dimensional controller and present algorithms for construction of two different internal model based robust controllers. The controller parameters are chosen based on a Galerkin approximation of the original PDE system and employ balanced truncation to reduce the orders of the controllers. In the second part o…
We present a novel methodology for designing output-feedback backstepping boundary controllers for an unstable 1-D diffusion-reaction partial differential equation with spatially-varying reaction. Using "folding" transforms the parabolic PDE into a 2X2 coupled parabolic PDE system with coupling via folding boundary conditions. The folding approach is novel in the sense that the design of bilateral controllers can be generalized to centering around arbitrary points, which admit additional design parameters for both the state-feedback controller and the state observer. The design can be selectiv…
This paper is aimed at presenting a short survey of velocity-free control schemes for robot manipulators. A salient feature of this work is the implementation of this control schemes on a simulation model of the 3-DOF industrial PHANToM 1.5 A robot. For both the regulation and tracking problems, two classes of control algorithms without velocity measurements are considered, namely, those based on observer-controller combination and those based on approximate derivation through an n-order filter. The latter was shown to be more advantageous from different aspects. First the controllers based on…
An adaptive funnel control method is considered for the regulation of the output for a class of nonlinear infinite-dimensional systems on real Hilbert spaces. After a decomposition of the state space and some change of variables related to the Byrnes-Isidori form, it is shown that the funnel controller presented in (Berger et al., 2020) achieves the control objective under some assumptions on the nonlinear system dynamics, like well-posedness and Bounded-Input-State Bounded-Output (BISBO) stability. The theory is applied to the regulation of the temperature in a chemical plug-flow tubular reac…
We investigate the regulator problem (tracking and disturbance rejection) for a system (plant) described by a boundary controlled anti-stable linear one-dimensional Schrodinger equation, using the backstepping approach. The output to be controlled is not required to be measurable and its observation operator is assumed to be admissible for a certain operator semigroup that is related to the operator semigroup of the original plant. We consider both the state feedback and the output feedback regulator problem. In the latter case, the measurement from the Schrodinger equation is taken at the bou…
