Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.Comment: Submitted to IEEE Transactions on Automatic Contro

Robust feedback control of Rayleigh-Bénard convection

We investigate the application of linear-quadratic-Gaussian (LQG) feedback control, or, in modern terms, H2 control, to the stabilization of the no-motion state against the onset of Rayleigh-Bénard convection in an infinite layer of Boussinesq fluid. We use two sensing and actuating methods: The planar sensor model (Tang & Bau 1993, 1994), and the shadowgraph model (Howle 1997a). By extending the planar sensor model to the multi-sensor case, it is shown that a LQG controller is capable of stabilizing the no-motion state up to 14.5 times the critical Rayleigh number. We characterize the robustness of the controller with respect to parameter uncertainties, unmodelled dynamics. Results indicate that the LQG controller provides robust performances even at high Rayleigh numbers

Optimization based control design techniques for distributed parameter systems

The study presents optimization based control design techniques for the systems that are governed by partial differential equations. A control technique is developed for systems that are actuated at the boundary. The principles of dynamic inversion and constrained optimization theory are used to formulate a feedback controller. This control technique is demonstrated for heat equations and thermal convection loops. This technique is extended to address a practical issue of parameter uncertainty in a class of systems. An estimator is defined for unknown parameters in the system. The Lyapunov stability theory is used to derive an update law of these parameters. The estimator is used to design an adaptive controller for the system. A second control technique is presented for a class of second order systems that are actuated in-domain. The technique of proper orthogonal decomposition is used first to develop an approximate model. This model is then used to design optimal feedback controller. Approximate dynamic programming based neural network architecture is used to synthesize a sub-optimal controller. This control technique is demonstrated to stabilize the heave dynamics of a flexible aircraft wings. The third technique is focused on the optimal control of stationary thermally convected fluid flows from the numerical point of view. To overcome the computational requirement, optimization is carried out using reduced order model. The technique of proper orthogonal decomposition is used to develop reduced order model. An example of chemical vapor deposition reactor is considered to examine this control technique --Abstract, page iii

Anticollocated backstepping observer design for a class of coupled reaction-diffusion PDEs

The state observation problem is tackled for a system ofncoupled reaction-diffusion PDEs, possessing the same diffusivity parameter and equipped with boundary sensing devices. Particularly, a backstepping-based observer is designed and the exponential stability of the error system is proven with an arbitrarily fast convergence rate. The transformation kernel matrix is derived in the explicit form by using the method of successive approximations, thereby yielding the observer gains in the explicit form, too. Simulation results support the effectiveness of the suggested design

Machine Learning Accelerated PDE Backstepping Observers

State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.Comment: Accepted to the 61st IEEE Conference on Decision and Control (CDC), 202

Interpretable PID Parameter Tuning for Control Engineering using General Dynamic Neural Networks: An Extensive Comparison

Modern automation systems rely on closed loop control, wherein a controller interacts with a controlled process, based on observations. These systems are increasingly complex, yet most controllers are linear Proportional-Integral-Derivative (PID) controllers. PID controllers perform well on linear and near-linear systems but their simplicity is at odds with the robustness required to reliably control complex processes. Modern machine learning offers a way to extend PID controllers beyond their linear capabilities by using neural networks. However, such an extension comes at the cost of losing stability guarantees and controller interpretability. In this paper, we examine the utility of extending PID controllers with recurrent neural networks-namely, General Dynamic Neural Networks (GDNN); we show that GDNN (neural) PID controllers perform well on a range of control systems and highlight how they can be a scalable and interpretable option for control systems. To do so, we provide an extensive study using four benchmark systems that represent the most common control engineering benchmarks. All control benchmarks are evaluated with and without noise as well as with and without disturbances. The neural PID controller performs better than standard PID control in 15 of 16 tasks and better than model-based control in 13 of 16 tasks. As a second contribution, we address the lack of interpretability that prevents neural networks from being used in real-world control processes. We use bounded-input bounded-output stability analysis to evaluate the parameters suggested by the neural network, thus making them understandable. This combination of rigorous evaluation paired with better interpretability is an important step towards the acceptance of neural-network-based control approaches. It is furthermore an important step towards interpretable and safely applied artificial intelligence

Mini-Workshop: Recent Developments on Approximation Methods for Controlled Evolution Equations

This mini-workshop brought together mathematicians engaged in partial differential equations, functional analysis, numerical analysis and systems theory in order to address a number of current problems in the approximation of controlled evolution equations

Boundary control and observation of coupled parabolic PDEs

Reaction-diffusion equations are parabolic Partial Differential Equations (PDEs) which often occur in practice, e.g., to model the concentration of one or more substances, distributed in space, under the in uence of different phenomena such as local chemical reactions, in which the substances are transformed into each other, and diffusion, which causes the substances to spread out over a surface in space. Certainly, reaction-diffusion PDEs are not confined to chemical applications but they also describe dynamical processes of non-chemical nature, with examples being found in thermodynamics, biology, geology, physics, ecology, etc. Problems such as parabolic Partial Differential Equations (PDEs) and many others require the user to have a considerable background in PDEs and functional analysis before one can study the control design methods for these systems, particularly boundary control design. Control and observation of coupled parabolic PDEs comes in roughly two settingsdepending on where the actuators and sensors are located \in domain" control, where the actuation penetrates inside the domain of the PDE system or is evenly distributed everywhere in the domain and \boundary" control, where the actuation and sensing are applied only through the boundary conditions. Boundary control is generally considered to be physically more realistic because actuation and sensing are nonintrusive but is also generally considered to be the harder problem, because the \input operator" and the "output operator" are unbounded operators. The method that this thesis develops for control of PDEs is the so-called backstepping control method. Backstepping is a particular approach to stabilization of dynamic systems and is particularly successful in the area of nonlinear control. The backstepping method achieves Lyapunov stabilization, which is often achieved by collectively shifting all the eigenvalues in a favorable direction in the complex plane, rather than by assigning individual eigenvalues. As the reader will soon learn, this task can be achieved in a rather elegant way, where the control gains are easy to compute symbolically, numerically, and in some cases even explicitly. In addition to presenting the methods for boundary control design, we present the dual methods for observer design using boundary sensing. Virtually every one of our control designs for full state stabilization has an observer counterpart. The observer gains are easy to compute symbolically or even explicitly in some cases. They are designed in such a way that the observer error system is exponentially stabilized. As in the case of finite-dimensional observer-based control, a separation principle holds in the sense that a closed-loop system remains stable after a full state stabilizing feedback is replaced by a feedback that employs the observer state instead of the plant state

Feedback stabilisation of pool-boiling systems : for application in thermal management schemes

The research scope of this thesis is the stabilisation of unstable states in a pool-boiling system. Thereto, a compact mathematical model is employed. Pool-boiling systems serve as physical model for practical applications of boiling heat transfer in industry. Boiling has advantages over conventional heat-transfer methods based on air- or single-phase liquids by enabling extremely high heat-transfer rates at isothermal conditions. This o¿ers solutions to thermal issues emerging in cutting-edge technologies as semi-conductor manufacturing and electric vehicles (EVs). Continuous miniaturisation in micro-electronics is pushing heat-¿ux densities beyond the limits of standard cooling schemes and growing architecture complexity makes thermal uniformity during chip manufacturing increasingly critical. Further development of EVs may bene¿t equally from boiling heat transfer by its utilisation for actuator cooling and thermal conditioning of battery packs. A pool-boiling system consists of a heater that is submerged in a pool of boiling liquid. The theater is the to-be-cooled device (or a thermally conducting element between the device and the boiling liquid) and is heated at its bottom. On top of the heater, heat is extracted by the boiling liquid. In order to exploit boiling to its fullest e¿ciency, unstable modes need to be stabilised to avoid the formation of a thermally-insulating vapour ¿lm on the heater that causes collapse of the cooling capacity and that heralds a dangerous and ine¿cient mode of boiling. The pool-boiling model comprises a partial di¿erential equation (PDE), i.e. the well- known heat equation, and corresponding boundary conditions that represent adiabatic sidewalls, a uniform heat supply at the bottom, and a nonuniform and nonlinear heat extraction at the heater top. This nonlinear boundary condition renders the entire model nonlinear, resulting in multiple equilibria and complex and exciting dynamics. Restriction to uniform temperature distributions within the heater admits description by a model of one spatial dimension (1D). The 1D model is investigated mathematically and the results are compared with those found by the analyses of spatial-discretisations of the model. Two spatial-discretisation schemes, based on a ¿nite-di¿erence method and a spectral method, are investigated. The latter shows far better convergence properties than the former. Moreover, application of full state feedback of the spectral modes (modal control) results in signi¿cantly better properties than by regulation via standard P-control. In practical applications, the heater temperature can only be measured at the heater top. Consequently, an observer is implemented that estimates the spectral modes of the temperature within the heater, which are subsequently used in the feedback-law. The e¿ciency and performance of this controller-observer combination is examined by numerical simulations. A pool-boiling system with an electrically heated wire as heater can be represented by the model as described above, but now with two spatial dimensions (2D). The 2D model can be analysed mathematically only for uniform equilibria, i.e. the equilibria that exist also for the 1D system. For nonuniform equilibria, the mathematical analysis becomes too complex and a spatial discretisation is required to obtain results. A 1D characteristic equation on the ¿uid-heater interface can be obtained by analytical reduction of the 2D eigenvalue problem using the method of separation of variables. The system poles follow from spatially discretising this equation. Because of its outstanding performance for the 1D model, the 2D model is again stabilised by a modal controller (full state feedback) in combination with an observer. Simulations are again performed to determine the e¿ciency of the controller-observer combination. If a thermally conducting foil is considered as heater, the three-dimensional (3D) form of the model must be investigated. This involves essentially the same methodology as described above, resulting in a 2D characteristic equation on the ¿uid-heater interface. However, spatial discretisation of this equation yields large system matrices and requires excessive calculation times. Hence, the 3D system is analysed only at moderate discretisation orders. The above modal control strategy is, as before, applied in combination with an observer to stabilise unstable equilibria and the evolution of the nonlinear system is again investigated and demonstrated by way of simulations. Finally, a series of exploratory experiments, to investigate the application of pool-boiling to thermally condition battery cells in EVs, is considered. Experiments are performed to investigate the ability for thermal homogenisation of the boiling process and the ability to manipulate the boiling process via the pressure in the boiling chamber. Furthermore, the application of pool-boiling to overcome thermal issues in high-end technologies is investigated by numerical simulations