Efficient approximation of the Koopman operator for large-scale nonlinear systems

A successful implementation of Model Predictive Control (MPC) requires appropriately tuned parameters. This paper presents a novel tuning approach based on Artificial Neural Network (ANN). To build the data learning base of the ANN, we adopted the Particle Swarm Optimisation (PSO) method, and we used the reliable algorithm, Online Sequential Extreme-Learning-Machine (OS-ELM) to learn the ANN. The objective of this work is to show that good tuning of MPC parameters makes it possible to reach closed-loop stability and ensure robustness against disturbances and sensor noises, without using robustification approaches in addition to MPC. The effectiveness of our approach is brought to light by comparing the obtained performances to other MPC tuning approaches without disturbances, and also to a robustified Generalized Predictive Control (GPC) using Youla parametrisation in the presence of disturbances.

International audience

This paper presents a real-time implementation of model predictive control (MPC) for HVAC systems in an ice-cream factory building. The target building consists of two large open spaces served by two HVAC systems. We developed four artificial neural network (ANN) models that predict the thermal states of the supply air and indoor air of the two thermal zones and prove to be accurate enough (MBE = 2.65, CVRMSE = 9.43). The control variables employed in this study are the number of operating chillers, frequency of supply-air fan inverter and outdoor-air intake ratio. The objective function minimizes total energy use, and a constraint was set to maintain average indoor air temperatures close to set points. Real-time MPC was implemented at a sampling time of 20 min from 3 August to 30 August 2021 and could save approximately 31.7% of electricity when compared to the existing simple rule-based control.

Model predictive control (MPC) as a nonlinear control method, is being studied for a wide variety of power electronics applications targeting various power ranges. The fast dynamic response, robustness under different load conditions, multi-objective control, and simplicity of implementation are the main features of MPC. However, the implementation of finite control set MPC (FCS-MPC) or direct MPC in its basic form for multilevel converters (MLCs) demonstrates the inability to generate the staircase multilevel output voltage waveform in a voltage sourced inverter (VSI) application. Adding a reference evaluation block to the basic FCS-MPC method can significantly improve its multilevel voltage generation capability. In this paper, an enhanced model predictive nearest level control (EMP-NLC) is proposed and implemented on a 5-Level flying capacitor multilevel converter (FCMC). The proposed method is simulated under different load conditions, while a comparison against existing MPC techniques is also presented. Experimental results verified the efficacy of the proposed controller.

In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, Proper Orthogonal Decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model, is obtained via linear interpolation between two of these autonomous systems. Using a recent convergence result for Extended Dynamic Mode Decomposition (EDMD), convergence to the true optimum can be proved. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1D Burgers Equation and the 2D Navier-Stokes Equations as examples. Finally, an extension for online adaptivity is presented.

Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

A key issue in distributed MPC control of Large Scale Systems (LSS) is how shared variables among the different MPC controller in charge of controlling each system partition (subsystems) are handled. When these connections represent control variables, the distributed control has to be
\nconsistent for both subsystems and the optimal value of these variables will have to accomplish a common goal.
\nIn order to achieve this, the present work combines ideas from Distributed Artificial Intelligence (DAI), Reinforcement Learning (RL) and Model Predictive Control (MPC) in order to provide an approach based on negotiation, cooperation and learning techniques.
\nResults of the application of this approach to a small drinking water network show that the resulting trajectories of the levels in tanks (control variables) can be acceptable compared to the centralized solution. The application to a real network (the Barcelona case) is currently under development.

The model predictive control (MPC) of voltage source pulse width modulation (PWM) rectififier is extremely dependent on system model in control process. System model mismatch will make the prediction results deviate from the actual value, seriously affect the system control results, even caused the system to be unstable. In addition, MPC is a control method based on state space model and predictive state of controlled system. In each control cycle, it performs an ergodic optimization to solve the online optimization, and thus determines optimal driving signal of the switching device. However, to obtain the optimal control variables online usually leads to a high amount of computation, which is one of the difficulties in the implementation of MPC on PWM rectifier. To solve these problems, neural network is used to approximate the unknown functions in system model. A finite set model predictive path integral control (FCS-MPPIC) based on neural network is proposed to effectively decrease computational complexity of control algorithm under premise of ensuring the distortion rate of the grid side current. At the same time, dynamic response speed of the system is improved. Finally, the availability of this method is verified through the simulation results.

Model-based network-wide optimizations in real-time emergency control of power systems are largely bottlenecked by the fast computational requirements. Furthermore, the operational aspects of modern power and energy systems are going through transformative changes to tackle global renewable energy expansion-related issues. Decentralized operations in power systems are becoming essential with the influx of distributed energy resources (DERs), dynamic loads, and electric vehicles (EVs). This research considers a well-known model-based strategy, Model Predictive Control (MPC), for real-time operation and controls of power systems and proposes novel methods utilizing machine learning, data-driven methods, and operator theory to solve computational challenges by making those real-time for the first time. Additionally, we develop a distributed MPC framework to address the recent needs of power systems' operation and control. In Part I, we first present a machine learning (ML)-accelerated MPC implementation in emergency voltage control problems. This method computes an optimal control strategy for the nominal system model offline and performs successive online control corrections at each control instant to adapt the offline computed controls for real-time scenarios. In the online phase, the required voltage trajectory prediction and its sensitivity computation to control inputs are achieved by machine learning-based approaches, thereby accelerating the overall control computation multiple times. Next, we present an alternative data-learned linear Koopman embedding approach for model predictive emergency voltage control in a power grid. This approach involves a "basis-dictionary free" method to lift the nonlinear power system dynamics to a higher dimensional linear space where MPC is exercised in a real-time fashion. We propose a novel data-driven encoder-decoder architecture, Koopman-inspired deep neural network (KDNN), which supports end-to-end learning from trajectory data. The MPC optimization in linear space makes the problem scalable and rapid for real-time implementations. Part II of this research focuses on distributed MPC design for quasi-static power flow-based voltage control problems under short-term (15-minute) variations of load and generation profiles. To realize the proposed distributed control, first, a centralized voltage control problem assuming grid observability is formulated and next transferred to the distributed versions based on both bus-wise and area-wise decomposition of the network. Distributed versions are solved via the alternating direction method of multipliers (ADMM), utilizing only neighbor-to-neighbor communications to attain optimal solutions. The implementation, efficacy, performance, and robustness of proposed research approaches are validated with IEEE test system benchmarks.

The optimal control of nonlinear time-varying systems, particularly when the mathematical model of the system is unavailable or inexact, is an interesting and difficult control problem. This paper outlines a methodology for the design of an intelligent controller to perform optimal control of a nonlinear system adaptively, using emerging technologies of fuzzy logic (FL) and artificial neural networks (ANN). FL is utilized to incorporate the available knowledge into the control system, and ANN technology is applied to adaptively provide an optimal control strategy based on some performance criteria. The technique is tested on a system that consists of a DC motor (a linear time-invariant (LTI) system) and a thermal system (a time-varying nonlinear system). Performance criteria such as tracking accuracy, cost, robustness, are considered, and the results are presented in this paper. >

This paper deals with a novel Plug-and-Play (PnP) architecture for the control and monitoring of Large-Scale Systems (LSSs). The proposed approach integrates a distributed Model Predictive Control (MPC) strategy with a distributed Fault Detection (FD) architecture and methodology in a PnP framework. The basic concept is to use the FD scheme as an autonomous decision support system: once a fault is detected, the faulty subsystem can be unplugged to avoid the propagation of the fault in the interconnected LSS. Analogously, once the issue has been solved, the disconnected subsystem can be re-plugged-in. PnP design of local controllers and detectors allow these operations to be performed safely, i.e., without spoiling stability and constraint satisfaction for the whole LSS. The PnP distributed MPC is derived for a class of nonlinear LSSs and an integrated PnP distributed FD architecture is proposed. Simulation results in two paradigmatic examples show the effectiveness and the potential of the general methodology.

Model predictive control (MPC) has emerged as a promising approach to control a modular multilevel converter (MMC). With the help of a cost function, the control objectives of an MMC are achieved easily by using an MPC approach. However, the MPC has several technical challenges and issues including the need of accurate system models, computational complexity, and variable switching frequency operation and weighting factor selection, when it comes to the control of an MMC. In the past few years, several research studies are conducted to address some of the challenges and issues in an MPC and developed several model predictive algorithms for an MMC. In this paper, the importance of each challenge and its impact on the system performance is discussed. Also, the MMC mathematical models used in the implementation of MPC are presented. Furthermore, some of the popular MPC algorithms are discussed briefly, and their features and performance are highlighted through case studies. Finally, summary and future trends of MPC for an MMC are presented.

In this paper, we consider learning-based modeling and predictive control of nonlinear systems subject to multimodal uncertainties. A Gaussian mixture Koopman operator, which learns the evolution of the observable distribution in a higher-dimensional space, is proposed to characterize the dynamics of the underlying nonlinear systems. Based on the learned probabilistic Koopman model, a stochastic model predictive control method with chance constraints is introduced for the nonlinear systems with state constraints. Finally, a simulated chemical process is used to illustrate the effectiveness of the proposed method.

Model Predictive Control (MPC) is a proven control concept with many applications in the process industry. Popularity of the framework is mainly due to its ability to optimize behavior of the process while respecting physical and economical constraints. The major challenge of implementing MPC in real time on low-cost hardware is the inherent computational complexity. To address this goal, it is proposed to solve a given MPC problem using parametric programming, which encodes the optimal control moves as a lookup table. A great advantage being that such tables can then be processed even with low computational resources and therefore allow MPC to be deployed to low cost control devices. In the paper we present a unique software tool which allows MPC problems to be designed with low human effort, and is capable to automatically generate real-time executable code for various target platforms.

The applicability of linear model predictive control to the 2-butene metathesis process is studied. Similarly to industrial practice, the model predictive controller is configured on a supervisory level, providing set points to basic process controllers. The development of the process model is based on open-loop identification from input–output data extracted from dynamic simulation performed in Aspen Plus Dynamics. The model predictive controller, designed using MATLAB tools, supervises a system consisting of two inputs (feed rate and reaction temperature) and two outputs (ethylene and propylene production rates). The performance of the model-based control strategy is assessed by Aspen Plus Dynamics-Simulink co-simulation and compared to regulatory control through several indexes (mean square error, integral square error, peak error, and integral absolute error). The model predictive controller outperforms the feedback controller. Considerations regarding the workflow for the implementation of model predictive control in an industrial environment are provided.