ProgramThe workshop will be held at: CentraleSupélec, 9 Rue Joliot Curie, 91190 GifsurYvette Bâtiment Bouygues, Amphithéatre Joël Rousseau (e.070) Thursday 16/119h3010h30: Pierre Riedinger, Harmonicbased modeling and control The presentation focuses on the application of harmonicbased modeling and control design for periodic systems. We first introduce the concept of sliding Fourier decomposition, which transforms periodic systems into infinite timeinvariant systems, offering a different perspective from the conventional timedomain approach. Indeed, recent advances in the harmonic framework have made it a viable and effective tool for designing controls for periodic systems. These advances include consistent methods for solving the harmonic Lyapunov, Sylvester and Riccati equations up to an arbitrarily small error, harmonic pole placement for Linear Time Periodic systems, and the resolution of harmonic LMIs paving the way for various control design syntheses. The ability of these harmonicbased methods to handle the infinite dimensionality inherent in harmonic equations is crucial and ensures that the originalperiodic system performs as desired. We provide an example of harmonic LQR synthesis to solve the periodic trajectory tracking problem for LTP systems and end with a case study of a threephase bridge rectifier in which harmonicbased control has been successfully applied to reject unwanted harmonic content.
11h12h: Jean Auriol, Output regulation for linear ODEhyperbolic PDEODE systems In this talk, we focus on the stabilization and output regulation of a class of interconnected systems. The class of systems under consideration consists of a linear hyperbolic Partial Differential Equations (PDE) system coupled at both ends with Ordinary Differential Equations (ODEs). The proximal ODE system represents actuator dynamics, while the distal ODE system represents load dynamics. The control objective is to ensure, in the presence of a disturbance signal that a virtual output exponentially converges to zero. By doing so, we can ensure that a state component of the distal ODE state robustly converges towards a known reference trajectory (output tracking problem) even in the presence of a disturbance with a known structure. The proposed approach combines the backstepping methodology and frequency analysis techniques. We first map the original system to a simpler target system using an invertible integral change of coordinates. From there, we design an adequate fullstate feedback controller in the frequency domain. Following a similar approach, we propose a state observer that estimates the state and reconstructs the disturbance from the available measurement. Combining the fullstate feedback controller with the state estimation results in a dynamic outputfeedback control law. Finally, existing filtering techniques guarantee the closedloop system robustness properties. 12h12h30: Pietro Lorenzetti, Solving the constrained output regulation problem using projected dynamical systems Projected dynamical systems are a mathematical tool used to describe the evolution of dynamical systems when solutions are constrained in a set. Originally, they have been developed for studying variational inequalities in the field of economics, optimization, and game theory. Motivated by power system applications, we borrow this theory to solve the constrained output regulation problem for constant references. For this, we propose a novel antiwindup proportionalintegral controller for stable multiinput multioutput nonlinear plants. We use tools from projected dynamical systems theory to force the integrator state to remain in a desired (compact and convex) region, such that the plant input steadystate values satisfy the operational constraints of the problem. Under suitable monotonicity assumptions on the plant steadystate inputoutput map, we use singular perturbation theory results to prove the existence of a sufficiently small controller gain ensuring closedloop (local) exponential stability and reference tracking for a feasible set of constant references. We suggest a particular controller design, which embeds (when possible) the right inverse of the plant steadystate inputoutput map. The relevance of the proposed controller scheme is validated through an application in the power systems domain, namely, the output (active and reactive) power regulation for a gridconnected synchronverter.
In control, common approaches to dealing with system nonlinearities are to cancel them, e.g. through feedback linearization, to suppress them through a highgain controller design, or do nothing about them if they do not affect stability. Although these approaches may lead to the stability of the closedloop system, they do not necessarily lead to good (in some sense) controller performance. The latter includes practical issues like the energy efficiency of the controller or its sensitivity to measurement noise. This is particularly relevant for nonstationary control problems like synchronization or output regulation. In this presentation, we will present a controller design method that allows one to improve controller performance using nonlinearities in the controller design. The approach is based on a novel notion of incremental feedback passivity with a nonlinear gain. We will illustrate the approach with examples from the controlled synchronization problem that demonstrate the methodology and the achieved performance improvements in energy efficiency and noise sensitivity of the controller.
In this talk the cooperative output regulation problem is considered for multiagent systems (MAS) with agents both described by coupled hyperbolic PDEODE systems and ODE systems. In the first part of the talk the leaderfollower synchronization problem for distributedparameter MAS with a finitedimensional leader in the presence of disturban ces and nondestabilizing model uncertainty is solved. For the networked controller design incorporating a cooperative internal model of the synchronization dynamics an extensi on of the backstepping method is presented. The usefulness of the new approach for the networked control of PDE agents is demonstrated for the cooperative load transport with heavy ropes. In the second part of the talk the containment control problem is solved for ODE agents using continuum models, which amounts to ensure the convergence of the agents into a prescribed timevarying containment area. The application of a continuum model allows to reformulate the cooperative output regulation problem as a classical out put regulation problem using feedforward control. Thereby, it is shown that containment can be guaranteed by making use of Bézier curves to describe the agent formation. A network controller is designed based on bilateral backstepping, which allows to distribute the control effort between the boundary agents. The theoretical results are validated for the formation control of ODE agents. Work supervised by Joachim Deutscher.
Since Francis and Wonham's seminal works introducing the internal model principle in control, regulation theory has always focused on autonomous systems. Motivated by a research question in neurophysiology, this talk explores the extension of the theory to a class of open systems where the exosystem and the controller depend on an external unmodeled and measured input. In particular, the regulation problem is formalized as an asymptotic behavior assignment problem, and some early results on the topic are overviewed. First, it is shown that an internal model principle can be proved also for open systems if "internal model" is interpreted as a (robust) factorization property of the controller. Next, a constructive design principle based on the addition of a "synchrony detector" on the regulated output is presented. Finally, the topic of modularity and the possible future directions and open problems are discussed.
Friday 17/119h3010h30: Lassi Paunonen, The abstract approach to internal model control of PDE systems [Presentation] In this presentation we study robust output regulation for systems described by linear partial differential equations, such as wave equations, convectiondiffusion equations, or hyperbolic systems. We focus on the "abstract approach", where the PDE model is first reformulated as an infinitedimensional linear system, and the controller design utilises the internal model based controllers that are available for the relevant class of abstract systems. This approach is powerful and it especially allows us to invoke suitable versions of the Internal Model Principle as part of our controller design. However, the controller design in the abstract setting can be tedious and technically demanding  especially in the case of PDE systems with boundary control and observation. In this talk we focus on illustrating and discussing the key steps in the internal model controller design for selected PDE models. In particular, we demonstrate how the questions regarding the tuning of the controller parameters can be converted from the operatortheoretic setting back into the PDE domain. We also discuss the various benefits and disadvantages of the abstract approach compared to other available options.
10h3011h: Coffee break
The necessity of the internal model principle for robust regulation suggests that regulation is not possible without calibration: a trajectory from the environment can only be regulated exactly with an exact internal model of that trajectory. This calibration principle is a key “analog” limitation of our current theory of regulation, synchronization, and online adaptation. It is however at odds with animal behavior, which provides ample evidence of the key role of sloppy internal models in uncertain and changing environments. In this talk, I will explore the idea that the eventbased nature of animal behaviors is at the core of what allows for exact regulation without exact calibration. Using the modelling framework of mixed feedback systems, it will be shown that many existing results from regulation and synchronisation theory can be made robust and calibrationfree provided we specialise them to systems equipped with neuromorphic excitability thresholds and synaptic couplings.
Even hundreds of years after its first appearance in engineering applications, the ProportionalIntegral (PI) controller, is still one of the most used control techniques. If the theory for linear systems is almost entirely welldeveloped, a lot has still to be said for nonlinear ones. In this presentation, we focus on a global output setpoint tracking and constant disturbance rejection problem for a class of continuoustime multiinput multioutput inputaffine nonlinear systems. With the word "global'', we mean that we allow the references and the disturbances to be arbitrarily large and the initial conditions of the system to range in the fullstate space. Following the linear paradigm, we first rely on the common approach of extending the system with an integral action processing the regulation error. Then, we look for a feedback control law for the (extended) system. To avoid the explicit use of normal forms, we cast the problem in the contraction framework and exploit incremental stability properties. We present a set of sufficient conditions for the design of a statefeedback control law able to make the resulting closedloop system incrementally inputtostate stable, uniformly with respect to the references and the disturbances. Such a property guarantees the existence of a unique attractive equilibrium on which setpoint tracking is achieved. For the feedback design, we take advantage of the system's structure and we develop an incremental version of forwardingbased control techniques. To conclude, we discuss possible extensions of the presented results.

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