We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the L^2 norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to L^2 exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property.

This paper studies the decay of an objective functional using a new control technique within Pontryagin's framework. Convergence analysis is carried out on the infinite-dimensional space of Tokamak plasma dynamical state as described by weakly decoupled nonlinear partial differential equations. An adjoint-based optimal control is derived to minimize the deviation from a predefined dynamical trajectory leading to the desired target state at stationary regime, by turning Pontryagin's transversality conditions into a continuum of horizons. A feedback controller is proposed to steer the system efficiently in real time, as opposed to an open-loop controller resulting from the classical Pontryagin's setting. An algorithm synthesizing the constraint-free optimal controller is used for profile tracking based on experimental data.

Understanding the processes during gas trapping in ice is essential to accurately interpret the gas records in ice cores. As a consequence, it is very desirable to have firn core and firn air sampling campaigns associated with deep ice coring. We know that elemental fractionation occurs during bubble close-off, hence largely affecting the δO2/N2 measurements further used to date the ice cores on orbital timescales. A recent study also suggested that this elemental fractionation can be linked to surface characteristics (i.e. temperature and / or accumulation rate). The aim of this study is to investigate the elemental and isotopic fractionation of N2 and O2 during bubbles close-off at two sites of very different characteristics (D47 located at the edge of the East Antarctic plateau with high temperature and accumulation rate and Little Dome C at the center of the East Antarctic plateau with low accumulation and accumulation rate). For this study, we did measurements both in the open and closed porosity of the firn in the lock-in zone. The D47 lock-in zone extends over nearly 20 m and, over these 20 m, strong signals of increasing δO2/N2 (+ 7 permil) and decreasing δ15N (-0.05 permil) are observed with increasing depths. At Little Dome C, the site of the Beyond EPICA deep ice core, the lock-in depth is much thinner (a few meters thick only) and fractionation much smaller. We discuss how these signals relate to the signals measured in the closed porosity in both sites and present some perspectives for the interpretation of the gas records in the deep ice cores.

In this paper, we propose a feedback control strategy to protect vulnerable areas from wildfires. We consider a system of coupled partial differential equations (PDEs) that models heat propagation and fuel depletion in wildfires and study two cases. First, when the wind velocity is known, we design a Neumann-type boundary controller guaranteeing that the temperature of some protected region converges exponentially, in the L^2 norm, to the ambient temperature. Second, when the wind velocity is unknown, we design an adaptive Neumann-type boundary controller guaranteeing the asymptotic convergence, in the L^2 norm, of the temperature of the protected region to the ambient temperature. In both cases, the controller acts along the boundary of the protected region and relies solely on temperature measurements along that boundary. Our results are supported by numerical simulations.

We study in this paper boundary stabilization, in the L2 sense, of the perturbed Kuramoto–Sivashinsky (KS) equation subject to intermittent sensing. We assume that we measure the state on a given spatial subdomain during certain time intervals, while we measure the state on the remaining spatial subdomain during the remaining time intervals. We assign a feedback law at the boundary of the spatial domain and force to zero the value of the state at the junction of the two subdomains. Throughout the study, the equation’s destabilizing coefficient is assumed to be unknown and possibly space dependent but bounded. As a result, adaptive boundary controllers are designed under different assumptions on the perturbation. In particular, we guarantee input-to-state stability (ISS) when an upperbound on the perturbation’s size is known. Otherwise, only global uniform ultimate boundedness (GUUB) is guaranteed. In contrast, when the state is measured at every spatial point all the time (full state measurement), convergence to an arbitrarily-small neighborhood of the origin is guaranteed, even if the perturbation’s maximal size is unknown. Numerical simulations are performed to illustrate our results.

Understanding the processes during gas trapping in ice is essential to accurately interpret the gas records in ice cores. As a consequence, it is very desirable to have firn core and firn air sampling campaigns associated with deep ice coring. We know that elemental fractionation occurs during bubble close-off, hence largely affecting the δO2/N2 measurements further used to date the ice cores on orbital timescales. A recent study also suggested that this elemental fractionation can be linked to surface characteristics (i.e. temperature and / or accumulation rate).  The aim of this study is to investigate the elemental and isotopic fractionation of N2 and O2 during bubbles close-off at two sites of very different characteristics (D47 located at the edge of the East Antarctic plateau with high temperature and accumulation rate and Little Dome C at the center of the East Antarctic plateau with low accumulation and accumulation rate). For this study, we did measurements both in the open and closed porosity of the firn in the lock-in zone. The D47 lock-in zone extends over nearly 20 m and, over these 20 m,  strong signals of increasing δO2/N2 (+ 7 permil) and decreasing δ15N (-0.05 permil) are observed with increasing depths. At Little Dome C, the site of the Beyond EPICA deep ice core, the lock-in depth is much thinner (a few meters thick only) and fractionation much smaller. We discuss how these signals relate to the signals measured in the closed porosity in both sites and present some perspectives for the interpretation of the gas records in the deep ice cores.

Nonlinear convection, the source of turbulence in fluid flows, may hold the key to stabilizing turbulence by solving a specific cubic polynomial equation. We consider the incompressible Navier-Stokes equations in a two-dimensional channel. The tangential and normal velocities are assumed to be periodic in the streamwise direction. The pressure difference between the left and right ends of the channel is constant. Moreover, we consider no-slip boundary conditions, that is, zero tangential velocity, at the top and bottom walls of the channel, and normal velocity actuation at the top and bottom walls. We design the boundary control inputs to achieve global exponential stabilization, in the L2 sense, of a chosen Poiseuille equilibrium profile for an arbitrarily large Reynolds number. The key idea behind our approach is to select the boundary controllers such that they have zero spatial mean (to guarantee mass conservation) but non-zero spatial cubic mean. We reveal that, because of convection, the time derivative of the L2 energy of the regulation error is a cubic polynomial in the cubic mean of the boundary inputs. Regulation is then achieved by solving a specific cubic equation, using the Cardano root formula. The results are illustrated via a numerical example.

We propose the first generalization of Sontag s universal controller to systems not affine in the control, particularly, to PDEs with boundary actuation. We assume that the system admits a control Lyapunov function (CLF) whose derivative, rather than being affine in the control, has either a depressed cubic, quadratic, or depressed quartic dependence on the control. For each case, a continuous universal controller that vanishes at the origin and achieves global exponential stability is derived. We prove our result in the context of convectionreaction-diffusion PDEs with Dirichlet actuation. We show that if the convection has a certain structure, then the L2 norm of the state is a CLF. In addition to generalizing Sontag s formula to some non-affine systems, we present the first general Lyapunov approach for boundary control of nonlinear PDEs. We illustrate our results via a numerical example.

In this work, model-predictive control (MPC) was combined for the first time with singular perturbation theory, and an original plasma kinetic control method based on extremely simple data-driven models and a two-time-scale MPC algorithm has been developed. A comprehensive review is presented in this paper. Slow and fast semi-empirical models are identified from data, by considering the fast kinetic plasma dynamics as a singular perturbation of a quasi-static equilibrium, which itself is governed, on the slow time scale, by the flux diffusion equation. This control technique takes advantage of the large ratio between the time scales involved in magnetic and kinetic plasma transport. It is applied here to the simultaneous control of the safety factor profile, q(x), and of several kinetic variables, such as the poloidal beta parameter, beta(p), and the internal inductance parameter, l(i), on the EAST tokamak. In the experiments, the available control actuators were lower hybrid current drive (LHCD) and co-current neutral beam injection (NBI) from different sources. Ion cyclotron resonant heating (ICRH) and electron cyclotron resonant heating (ECRH) are used as additional actuators in control simulations. In the controller design, an observer provides, in real time, an estimate of the system states and of the mismatch between measured and predicted outputs, which ensures robustness to model errors and offset-free control. Based on the observer information, the controller predicts the behavior of the system over a given time horizon and computes the optimal actuation by solving a quadratic programming optimization problem that takes the actuator constraints into account. A number of control applications are described in the paper, either in nonlinear simulations with EAST-like parameters or in real experiments on EAST. The simulations were performed with a fast plasma simulator (METIS) using either two control actuators (LHCD and ICRH) in a low density scenario, or up to four actuators at higher density: LHCD, ECRH, and two NBI systems driven in a on/off pulse-width-modulation (PWM) mode, with different injection angles. The control models are identified with the prediction-error method, using datasets obtained from open loop simulations in which the actuators are modulated with pseudo-random binary sequences. The simulations with two actuators show that various q(x) profiles and beta(p) waveforms can be tracked without offset, within times that are consistent with the resistive and thermal diffusion time scales, respectively. In simulations with four actuators, simultaneous tracking of time-dependent targets is shown for q(x) at two normalized radii, x = 0 and x = 0.4, and for beta(p). Due to the inherent mismatch between the optimal NBI power request and the delivered PWM power, the kinetic controller performs with reduced accuracy compared with simulations that do not use the NBI/PWM actuators. The first experimental tests using this new control algorithm were performed on EAST when the only available actuator was the LHCD system at 4.6 GHz. The algorithm was thus used in its simplest single-input-single-output version to track time-dependent targets for the central safety factor, q(0), or for beta(p). In the closed loop control experiments, the q(0) targets were tracked in about one second, consistently with the plasma resistive time constant. Excellent tracking of a piecewise linear beta(p) target waveform was also achieved. When the NBI system became controllable in real time by the EAST plasma control system, new experiments were dedicated to multiple-input-multiple-output MPC control with three actuators: LHCD and two NBI actuators using the PWM algorithm. Given that the minimum time allowed between NBI on/off switching was 0.1 s, i.e. larger than the characteristic time of the fast plasma dynamics, a reduced version of the MPC controller based only on the slow model was used. Various controller configurations were tested during a single experimental session, with up to three controlled variables chosen among q(0) = q(x= 0), q(1) = q(x= 0.5), beta(p) and l(i). The main difficulty encountered during this session was the unavailability of the full baseline ICRH and ECRH powers that were used in the reference scenario, and from which the plasma model was identified. This often led to the saturation of one or several actuators, which prevented some targets selected in advance from being accessible. Nevertheless, in cases that were free from actuator saturation, q(0) and q(1) targets were successfully reached, in a time that is consistent with the resistive diffusion time of the model and with small oscillations that are characteristic of the PWM operation of the neutral beams. During the simultaneous control of q(0) and beta(p), the ICRH power was too low and, in addition, the plasma density was much larger than the reference one. The q(0) targets were not accessible in this high-density/low-power case, but beta(p) control was successful. Finally, the simultaneous control of q(0) and l(i) was satisfactory and, during the simultaneous control of, q(0), beta(p) and l(i), the tracking of beta(p) and l(i) was satisfactory but q(0) was too large due to the lack of ICRH power and to NBI saturation. In conclusion, the extensive nonlinear simulations described in this paper have demonstrated the relevance of combining MPC, data-driven models and singular perturbation methods for plasma kinetic control. This technique was also assessed experimentally on EAST, although some tests were perturbed by undesired parameter changes with respect to the reference scenario.

We consider the problem of inverse optimal control design for systems that are not affine in the control. In particular, we consider some classes of partial differential equations (PDEs) with quadratic convection and counter-convection, for which the L2 norm is a control Lyapunov function (CLF) whose derivative has either a depressed cubic or a quadratic dependence in the boundary control input. We also consider diffusive PDEs with or without linear convection, for which a weighted L2 norm is a CLF whose derivative has a quadratic dependence in the control input. For each structure on the derivative of the CLF, we achieve inverse optimality with respect to a meaningful cost functional. For the case where the derivative of the CLF has a depressed cubic dependence in the control, we construct a cost functional for which the unique minimizer is the unique real root of a cubic polynomial: the Cardano-Lyapunov controller. When the derivative of the CLF is quadratic in the control, we construct a cost functional that is minimized by two distinct feedback laws, that correspond to the two distinct real roots of a quadratic equation. We show how to switch from one root to the other to reduce the control effort.

This paper presents a learning-based control strategy for non-linear throttle valves with an asymmetric hysteresis, leading to a near-optimal controller without requiring any prior knowledge about the environment. We start with a carefully tuned Proportional Integrator (PI) controller and exploit the recent advances in Reinforcement Learning (RL) with Guides to improve the closed-loop behavior by learning from the additional interactions with the valve. We test the proposed control method in various scenarios on three different valves, all highlighting the benefits of combining both PI and RL frameworks to improve control performance in non-linear stochastic systems. In all the experimental test cases, the resulting agent has a better sample efficiency than traditional RL agents and outperforms the PI controller.

Electric throttle valves represent a challenge for control design, as their dynamics involve strong nonlinearities, characterized by an asymmetric hysteresis. Carrying experiments on multiple valves, a large variability in the characteristics of each valve and erratic steady-state behaviors can also be noticed, impairing classical model-based control strategies. Nevertheless, local data-driven linear models can be obtained and simple proportional-integral (PI) controllers, tuned individually for each valve with the appropriate data set, provide good tracking performance. As these controllers cannot be transposed from one valve to another, a robust strategy and an adaptive controller (using identification in closed-loop and controller re-design) may be necessary to propose a general method. This work aims at promoting control education on a simple yet challenging process, going from frequency analysis and linear design to an adaptive control method implemented with an online recursive algorithm.

Electric throttle valves represent a challenge for control design, as their dynamics involves strong nonlinearities, characterized by an asymmetric hysteresis. Large variability in the characteristics of each valve and erratic steady-state behaviors can also be noticed by carrying out experiments on multiple valves, impairing classical model-based control strategies. Nevertheless, local data-driven linear models can be obtained by system identification, and simple proportional-integral (PI) digital controllers can be tuned individually for each valve, providing good tracking performance. As these controllers cannot be transposed from one valve to another, a robust control design is considered. Taking into account the variability of electric throttle valves, a real-time data-driven strategy is then proposed, using identification in closed-loop and controller re-design. This methodology is necessary if control performance is a key issue, and can be embedded on a low-cost controller board (Arduino & REG; Mega 2560). Experimental results going from frequency analysis and linear design to real-time data-driven control illustrate the methodology presented in the paper.

Recent advances in the use of Artificial Intelligence to control complex systems make it suitable for profile plasma control. In this work, we propose an algorithm based on Deep Reinforcement Learning to control the safety factor profile with a feedback design. For this purpose, we first derive a device-specific control-oriented model with fast simulation time. Then, in order to enhance robustness with respect to external disturbances and model errors, we include an error time integrator into the controller. A cascade of the kinetic and magnetic models with the error time integrator is used in the learning procedure of the feedback controller. Finally, to illustrate the efficiency of the proposed design procedure, the obtained controller is tested in a reference plasma simulator, the Raptor simulator.

We study in this paper the one-dimensional Kuramoto-Sivashinsky equation (KS), subject to intermittent sensing. Namely, we measure the state on a sub-interval of the spatial domain during certain intervals of time, and we measure the state on the remaining sub-interval of space during the remaining intervals of time. As a result, we assign an active control at the boundaries of the spatial domain, and we set a zero boundary condition at the junction of the two spatial subintervals. Under the assumption that the destabilizing coefficient is unknown, we design adaptive boundary controllers that guarantee global exponential stability (GES) of the trivial solution in the L 2 norm. Numerical simulations are performed to illustrate our results.

We analyze in this paper the effect of the well-known intelligent proportional controller on the stability of linear control systems. Inspired by the literature on neutral time-delay systems and advanced-type systems, we derive sufficient conditions on the order of the control system, under which, the used controller fails to achieve exponential stability. Furthermore, we obtain conditions, relating the system's and the control parameters, such that the closed-loop system is either unstable or not exponentially stable. After that, we provide cases where the used controller achieves exponential stability. The obtained results are illustrated on an experimental benchmark that consists of an electronic throttle valve.

Electric throttle valves represent a challenge for control design, as their dynamics involves strong nonlinearities, characterized by an asymmetric hysteresis. Large variability in the characteristics of each valve and erratic steady-state behaviors can also be noticed by carrying out experiments on multiple valves, impairing classical model-based control strategies. Nevertheless, local data-driven linear models can be obtained by system identification, and simple proportional–integral (PI) digital controllers can be tuned individually for each valve, providing good tracking performance. As these controllers cannot be transposed from one valve to another, a robust control design is considered. Taking into account the variability of electric throttle valves, a real-time data-driven strategy is then proposed, using identification in closed-loop and controller re-design. This methodology is necessary if control performance is a key issue, and can be embedded on a low-cost controller board (Arduino® Mega 2560). Experimental results going from frequency analysis and linear design to real-time data-driven control illustrate the methodology presented in the paper.

The problem of estimating the temperatures and the heat transfer coefficient of a concentric tube heat exchanger coupled with a heater is considered in this work. Measurements collected from the extremities of the exchanger tube are used to estimate the heat distribution over the length of the exchanger, which induces a boundary estimation problem. This system, which is part of any standard cooling plant, is particularly challenging due to the distributed nature of its variables. It is modeled by a system of (2 × 2) hyperbolic PDEs, coupled with an ODE at the boundary. To solve the estimation problem, we consider a general class of systems consisting of a (2 × 2) hyperbolic system coupled with a set of nX linear time-varying (LTV) ODEs at the boundary. Both the PDE and the ODEs have uncertain parameters to be estimated. The objective is to estimate the PDE states, the ODE states, and the parameters simultaneously with no assumption on the ODEs stability. We design a Luenberger state observer, and our method is mainly based on the decoupling of the PDE estimation error states from that of the ODEs via swapping design. We then derive the observer gains from the Lyapunov analysis of the decoupled system after proving the boundedness of the swapping filters. We give sufficient conditions of the exponential convergence of the adaptive observer through differential Lyapunov inequalities (DLIs). Finally, we apply the developed theory on the coupled heat exchanger–heater model to evaluate the performance of the observer in numerical simulations.

In this paper, two boundary controllers are proposed to stabilize the origin of the nonlinear Kuramoto-Sivashinsky equation under intermittent measurements. More precisely, the spatial domain is divided into two sub-domains. The state of the system on the first sub-domain is measured along a given interval of time, and the state on the remaining sub-domain is measured along another interval of time. Under the proposed sensing scenario, we control the considered equation by designing the value of the state at three isolated spatial points, the two extremities of the spatial domain plus one inside point. Furthermore, we impose a null value for the spatial gradient of the state at these three locations. Under such a control loop, we propose two types of controllers and we analyze the stability of the resulting closed-loop system in each case. The paper is concluded with some discussions and future works.

This work is devoted to the modeling and structural analysis of ventilation networks in small-scale mines using a physically oriented modeling method that ensures power conservation. Small-scale mines are common in the mineral extraction industry of underdeveloped countries and their physical characteristics are taken into account in the modeling process. The geometrical topology of the ventilation network in addition with the conservation laws of the fluid distribution along the network are considered in order to obtain a simple modeling methodology. Non-linear characteristics of the interconnected fluid dynamics represent a challenge to determine significant features of the system from a control point of view. Observability and controllability properties are analyzed by considering the structural systems approach. An structural analysis provides information based on the network topology independently of the mine parameters allowing the number of sensors and actuators to be reduced while also preserving the observability and controllability of the ventilation system. Experimental results are provided by building a small-scale ventilation network benchmark to evaluate the proposed model and its properties.