C. I. Byrnes, A. Isidori
In this paper we show how nonlinear internal models can be effectively used in the design of output regulators for nonlinear systems. This result provides a significant enhancement of the non-equilibrium theory for output regulation, which we have presented in the recent paper entitled "Limit Sets, Zero Dynamics, and Internal Models in the Problem of Nonlinear Output Regulation".
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Data-Driven Output Regulation via Internal Model Principle
针对未知线性系统的输出调节问题，本文基于内模原理将现有数据驱动算法扩展至MIMO系统，并通过分离学习与控制阶段的动力学，提出计算成本更低、可解性条件更弱的改进算法。
The internal model principle is a fundamental result stating a necessary and sufficient condition for a stabilizing controller to be robustly regulating. Its classical formulation is given in terms of coprime factorizations and the largest invariant factor of the signal generator which sets unnecessary restrictions for the theory and its applicability. In this article, the internal model principle is formulated using a general factorization approach and the generators of the fractional ideals generated by the elements of the signal generator. The proposed results are related to the classical o…
In this paper, we study the cooperative robust output regulation problem for linear uncertain multi-agent systems with both communication delay and input delay by the distributed internal model approach. The problem includes the leader-following consensus problem of linear multi-agent systems with time-delay as a special case. We first generalize the internal model design method to systems with both communication delay and input delay. Then, under a set of standard assumptions, we have obtained the solution of the problem via both the state feedback control and the output feedback control. In …
Data-Driven Output-Based Approach to the Output Regulation Problem of Unknown Linear Systems via Value Iteration
本文提出了一种基于数据的输出驱动方法，通过价值迭代解决未知线性系统的输出调节问题，并建立了状态反馈与输出反馈控制律在LQR意义上的关系。
In this paper, we study the cooperative robust output regulation problem for discrete-time linear multi-agent systems with both communication and input delays by distributed internal model approach. We first introduce the distributed internal model for discrete-time multi-agent systems with both communication and input delays. Then, we define so-called auxiliary system and auxiliary augmented system. Finally, we solve our problem by showing, under some standard assumptions, that if a distributed state feedback control or a distributed output feedback control solves the robust output regulation…
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…
We introduce general results on well-posedness and output regulation of regular linear systems with nonautonomous controllers. We present a generalization of the internal model principle for time-dependent controllers with asymptotically converging parameters. This general result is utilised in controller design for output tracking and disturbance rejection of harmonic signals with unknown frequencies. Our controller can be flexibly combined with different frequency estimation methods. The results are illustrated in rejection of unknown harmonic disturbances for a one-dimensional boundary cont…
This document contains the mathematical introduction to RORPack - a Python software library for robust output tracking and disturbance rejection for linear PDE systems. The RORPack library is open-source and freely available at https://github.com/lassipau/rorpack/ The package contains functionality for automated construction of robust internal model based controllers, simulation of the controlled systems, visualisation of the results, as well as a collection of example cases on robust output regulation of controlled heat and wave equations.
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…
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…
On forwarding techniques for stabilization and set-point output regulation of semilinear infinite-dimensional systems
本文针对级联半线性无穷维系统提出基于前馈技术的镇定器，推导了闭环稳定性充分条件与局部设定点输出调节结果，并验证了其在旋转关节柔性梁系统中的应用。
In this paper, we consider the event-triggered cooperative robust practical output regulation problem for a class of linear minimum-phase multi-agent systems. We first convert our problem into the cooperative robust practical stabilization problem of a well defined augmented system Based on the distributed internal model approach. Then, we design a distributed event-triggered output feedback control law together with a distributed output-based event-triggered mechanism to stabilize the augmented system, which leads to the solvability of the cooperative robust practical output regulation proble…
Output regulation of infinite-dimensional nonlinear systems: a forwarding approach for contraction semigroups
针对非线性压缩半群系统的鲁棒输出调节问题，论文采用加入积分作用的forwarding方法设计反馈律，建立局部指数稳定与全局渐近稳定条件，并通过半线性系统示例验证应用。
This paper considers the robust cooperative output regulation for a network of parabolic PDE systems. The solution of this problem is obtained by extending the cooperative internal model principle from finite to infinite dimensions. For a time-invariant digraph describing the communication topology, a two-step backstepping approach is presented to systematically design cooperative state feedback regulators. They allow to solve both the leader-follower and the leaderless output synchronization problem in the presence of disturbances and model uncertainty for a finite-dimensional leader. Solvabi…
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…
Output regulation for an unstable wave equation with output delay and one measurement only
本文针对具有输出延迟和未知扰动的不稳定波动方程，通过反步法设计前馈调节器，并利用单一测量构建误差反馈控制器，实现跟踪误差指数收敛，为分布式参数系统输出调节提供开创性方案。
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 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 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.
In this paper, we consider the output synchronization problem for a network of heterogeneous diffusively-coupled nonlinear agents. Specifically, we show how the (non-identical) agents can be controlled in such a way that their outputs asymptotically track the output of a prescribed nonlinear exosystem. The problem is solved in two steps. In the first step, the problem of achieving consensus among (identical) nonlinear reference generators is addressed. In this respect, it is shown how the techniques recently developed to solve the consensus problem among linear agents can be extended to agents…
We study temperature and velocity output tracking problem for a two-dimensional room model with the fluid dynamics governed by the linearized translated Boussinesq equations. Additionally, the room model includes finite-dimensional models for actuation and sensing dynamics, thus the complete model dynamics are governed by an ODE-PDE-ODE system. As the main result, we design a low-dimensional internal model based controller for robust output racking of the room model. Efficiency of the controller is demonstrated through a numerical example of velocity and temperature tracking.
Output regulation for a reaction-diffusion system with input delay and unknown frequency
针对同时存在输入时滞和完全未知扰动的反应-扩散系统，提出结合模态分解与双观测器的自适应控制架构，实现状态与扰动实时估计，通过跟踪误差控制器达到输出指数收敛，并经数值仿真验证。
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…
Data-Driven Cooperative Output Regulation via Distributed Internal Model
本文研究未知线性多智能体系统在一般静态连通有向图和多输入多输出跟随者下的协同输出调节问题，利用简化线性系统与积分强化学习，并通过方程解耦降低计算成本、弱化满列秩可解性条件。
A Refined Algorithm for the Adaptive Optimal Output Regulation Problem
针对线性未知系统的自适应最优输出调节问题，本文将原有高维线性方程解耦为两个低维方程，降低了未知变量规模并显著弱化可解性条件。
Approximate robust output regulation of boundary control systems
针对边界控制与观测系统因内模原理需无穷维控制器的问题，提出近似鲁棒输出调节问题与有限维控制器结构，以环形域波动方程为例验证了方法有效性。
Output Regulation for Linear Parabolic Systems Using Finite-Dimensional Tracking-Error-Based Control
研究一维扩散反应系统在干扰和参考信号由不稳定外系统生成时的输出调节问题，提出基于有限维跟踪误差的调节器设计方法，通过推导组合装置转化问题，利用模态分解法刻画可观测性，证明观测器增益性质并给出LMI条件，分析了含未知时变测量延迟的情况。
This paper studies the event-triggered cooperative global robust output regulation problem for a class of nonlinear multi-agent systems via a distributed internal model design. We show that our problem can be solved practically in the sense that the ultimate bound of the tracking error can be made arbitrarily small by adjusting a design parameter in the proposed event-triggered mechanism. Our result offers a few new features. First, our control law is robust against both external disturbances and parameter uncertainties, which are allowed to belong to some arbitrarily large prescribed compact …
The paper deals with the problem of output regulation in a "non-equilibrium" context for a special class of multivariable nonlinear systems stabilizable by high-gain feedback. A post-processing internal model design suitable for the multivariable nature of the system, which might have more inputs than regulation errors, is proposed. Uncertainties in the system and exosystem are dealt with by assuming that the ideal steady state input belongs to a certain "class of signals" by which an appropriate model set for the internal model can be derived. The adaptation mechanism for the internal model i…
This paper considers the synchronized output regulation (SOR) problem of nonlinear multi-agent systems with switching graph. The SOR means that all agents regulate their outputs to synchronize on the output of a predefined common exosystem. Each agent constructs its local exosystem with the same dynamics as that of the common exosystem and exchanges the state information of the local exosystem. It is shown that the SOR is solvable under the assumptions same as that for nonlinear output regulation of a single agent, if the switching graph satisfies the bounded interconnectivity times condition.…
An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages
本文研究了在未知且可能随时间变化的负载变化下，如何通过最小化发电成本实现频率调节，采用增量被动性和分布式内部模型控制器解决输出一致性问题。
We consider the problem of designing a state feedback control law to achieve nonovershooting tracking for feedback linearisable multiple-input multiple-output nonlinear systems. The reference signal is assumed to be obtained from a linear exosystem. The design method adopts known methods for the nonovershooting tracking of linear systems and extends them to the output regulation of feedback linearisable nonlinear systems. The method accommodates arbitrary initial conditions and offers design choice of the tracking convergence speed.
Nonovershooting Cooperative Output Regulation for Linear Multi-Agent Systems
本文提出分布式动态输出反馈方法，解决线性多智能体系统的协同输出调节问题，并确保所有智能体在暂态响应中无超调跟踪参考信号。
A Numerical Example about the Geometric Approach to the Output Regulation Problem with Stability for Linear Switching Systems
本文通过数值例子展示如何利用几何方法解决线性切换系统的输出调节问题，并保证二次稳定性。
We investigate the problem of practical output regulation, i.e., to design a controller that brings the system output in the vicinity of a desired target value while keeping the other variables bounded. We consider uncertain systems that are possibly nonlinear and the uncertainty of their linear parts is modeled element-wise through a parametric family of matrix boxes. An optimization-based design procedure is proposed that delivers a continuous-time control and estimates the maximal regulation error. We also analyze an event-triggered emulation of this controller, which can be implemented on …
Distributed Output Regulation for a Class of Nonlinear Multi-Agent Systems with Unknown-Input Leaders
本文研究了一类不确定非线性多智能体系统在局部扰动下的分布式输出调节问题，针对领导者未知输入和动态差异的情况，设计分布式反馈控制器使智能体全局或半全局跟踪不确定领导者。
Output regulation via input-output data
本文研究了如何利用输入输出数据设计反馈控制器，以消除系统输出对未知外信号的影响，核心方法是通过半正定规划设计控制器，并通过辅助系统与原系统的分析关系实现迁移。
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.
Cooperative Output Regulation of Linear Multi-agent Systems with Communication Constraints
本文研究了存在通信约束的异构线性多智能体系统的协同输出调节问题，提出了一种基于间歇性和异步离散时间信息交换的分布式控制算法，能够在未知时变延迟和信息丢失下实现协同输出调节。
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…
We consider controller design for robust output tracking and disturbance rejection for continuous-time periodic linear systems with periodic reference and disturbance signals. As our main results we present four different controllers: A feedforward control law and a discrete-time dynamic error feedback controller for output tracking and disturbance rejection, a robust discrete-time feedback controller, and finally a discrete-time feedback controller that achieves approximate robust output tracking and disturbance rejection. The presented constructions are also new for time-invariant finite and…
Nonparametric Steady-State Learning for Nonlinear Output Feedback Regulation
研究一般非线性输出反馈系统的非参数和鲁棒输出调节问题，利用非自适应框架转化问题，纳入非参数学习框架确保非线性映射可行性，通过三个实例验证该方法在复杂系统中保证误差收敛的有效性。
Optimal Output Regulation for Square, Over-Actuated and Under-Actuated Linear Systems
本文研究了线性系统轨迹跟踪控制中的两个问题：控制非唯一性以实现最节能的输入能量效率，以及在精确跟踪不可行时实现最准确的控制。本文将这两个目标作为最优输出调节问题进行建模，并贡献了两组新的调节方程，为输出调节理论提供了新的解决方案。
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 …
This paper is about the nonlinear local error feedback regulator problem. The plant is a nonlinear finite-dimensional system with a single control input and a single output and it is locally exponentially stable around the origin. The plant is driven, via a separate disturbance input, by a Lyapunov stable exosystem whose states are nonwandering. The reference signal that the plant output must track is a nonlinear function of the exosystem state. The local error feedback regulator problem is to design a dynamic feedback controller, with the tracking error as its input, such that (i) the closed-…
Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals
本文针对非周期非光滑外生信号的线性系统，提出基于状态反馈、误差反馈及两种鲁棒内部模型设计的输出调节方法，并给出非共振条件，通过电路实例验证有效性。
We study the robust output regulation of linear boundary control systems by constructing extended systems. The extended systems are established based on solving static differential equations under two new conditions. We first consider the abstract setting and present finite-dimensional reduced order controllers. The controller design is then used for particular PDE models: high-dimensional parabolic equations and beam equations with Kelvin-Voigt damping. Numerical examples will be presented using Finite Element Method.
Output Regulation for Impedance Passive Systems with Input Saturation
研究具有输入饱和的抽象无限维系统的输出跟踪和干扰抑制，采用含稳定误差反馈与前馈项的控制律解决问题，并据此为二维热方程和一维波动方程设计控制律。
Sampled-data Output Regulation of Unstable Well-posed Infinite-dimensional Systems with Constant Reference and Disturbance Signals
针对具有恒定参考与扰动信号的不稳定适定无限维系统，研究了输出跟踪与扰动抑制的采样数据控制问题，通过Nevanlinna-Pick插值方法提出了有限维控制器设计并给出了存在充分条件。
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 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…
Coordinated Output Regulation of Heterogeneous Linear Systems under Switching Topologies
本文提出了一种框架，用于研究多个异构线性系统的协调输出调节问题，通过设计基于观测器的反馈控制器，在有限切换拓扑下实现系统协调输出调节。
On the Solvability of Quasi-Regulator Equations in Non-smooth Output Regulation
针对非光滑非周期外生信号下的线性系统输出调节问题，研究准调节方程的可解性，通过将其重述为微分代数方程并引入非光滑非共振条件，给出了可解性的充要刻画。
Output Regulation of Stochastic Sampled-Data Systems with Post-processing Internal Model
本文研究了随机采样测量下LTI系统的输出调节问题，提出了一种含混合观测器和后处理内模型的调节器，利用类Lyapunov定理给出均方指数稳定的充分条件，并通过LMI实现数值可解设计。
Multi-Partite Output Regulation of Multi-Agent Systems
本文提出了一种简单且不依赖图的视角，用于划分图的节点集，并为多智能体系统提供超越合作与二分划分的目标。通过引入k-划分变换，解决了异构线性MASs的多部输出调节问题，展示了其在合作输出调节和二分输出调节中的应用，并提出两种设计策略以提高可扩展性。
We present a model predictive control (MPC) framework to solve the constrained nonlinear output regulation problem. The main feature of the proposed framework is that the application does not require the solution to classical regulator (Francis-Byrnes-Isidori) equations or any other offline design procedure. In particular, the proposed formulation simply minimizes the predicted output error, possibly with some input regularization. Instead of using terminal cost/sets or a positive definite stage cost as is standard in MPC theory, we build on the theoretical results by Grimm et al. 2005 using a…
Nonlinear Bipartite Output Regulation with Application to Turing Pattern
本文针对符号网络下的非线性多智能体系统，提出非线性分布式观测器解决二分输出调节问题，并应用于领导者跟随二分一致性与Turing图案生成。
Nonadaptive Output Regulation of Second-Order Nonlinear Uncertain Systems
针对带未知 exosystem 的二阶非线性不确定系统，本文提出基于 generic internal models 和坐标变换的非自适应鲁棒控制方法，避免 bursting phenomenon，实现输出调节并应用于 Duffing 系统。
Nonadaptive Learning in Robust Nonlinear Output Regulation
针对相对阶任意高的输出反馈非线性系统，提出结合输入驱动滤波器、通用内模与递归反步律的非自适应调节方法，通过杜芬系统验证其可实现全局渐近调节及误差收敛。
A nonparametric learning framework for nonlinear robust output regulation
针对非线性鲁棒输出调节问题，提出非参数学习框架，放宽稳态生成器假设、简化实现降复杂度，结合动态增益实现无先验知识镇定，还可用于未知频率正弦信号估计与线性输出调节求解，经仿真验证有效。
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