Koopman operator-based model reduction for switched-system control of PDEs

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier--Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641

Identification and Control of Nonlinear Singularly Perturbed Systems Using Multi-time-scale Neural Networks

Many industrial systems are nonlinear with "slow" and "fast" dynamics because of the presence of some ``parasitic" parameters such as small time constants, resistances, inductances, capacitances, masses and moments of inertia. These systems are usually labeled as "singularly perturbed" or ``multi-time-scale" systems. Singular perturbation theory has been proved to be a useful tool to control and analyze singularly perturbed systems if the full knowledge of the system model parameters is available. However, the accurate and faithful mathematical models of those systems are usually difficult to obtain due to the uncertainties and nonlinearities. To obtain the accurate system models, in this research, a new identification scheme for the discrete time nonlinear singularly perturbed systems using multi-time-scale neural network and optimal bounded ellipsoid method is proposed firstly. Compared with other gradient descent based identification schemes, the new identification method proposed in this research can achieve faster convergence and higher accuracy due to the adaptively adjusted learning gain. Later, the optimal bounded ellipsoid based identification method for discrete time systems is extended to the identification of continuous singularly perturbed systems. Subsequently, by adding two additional terms in the weight's updating laws, a modified identification scheme is proposed to guarantee the effectiveness of the identification algorithm during the whole identification process. Lastly, through introducing some filtered variables, a robust neural network training algorithm is proposed for the system identification problem subjected to measurement noises. Based on the identification results, the singular perturbation theory is introduced to decompose a high order multi-time-scale system into two low order subsystems -- the reduced slow subsystem and the reduced fast subsystem. Then, two controllers are designed for the two subsystems separately. By using the singular perturbation theory, an adaptive controller for a regulation problem is designed in this research firstly. Because the system order is reduced, the adaptive controller proposed in this research has a simpler structure and requires much less computational resources, compared with other conventional controllers. Afterward, an indirect adaptive controller is proposed for solving the trajectory tracking problem. The stability of both identification and control schemes are analyzed through the Lyapunov approach, and the effectiveness of the identification and control algorithms are demonstrated using simulations and experiments

Control of non-minimum-phase nonlinear systems through constrained input-output linearization

Proceedings of the 2006 American Control Conference, pp. 4522-4527.This paper presents a novel control method that provides optimal output-regulation with guaranteed closedloop asymptotic stability within an assessable domain of attraction. The closed-loop stability is ensured by requiring state variables to satisfy a hard, second-order Lyapunov constraint. Whenever input-output linearization alone cannot ensure asymptotic closed-loop stability, the closed-loop system evolves while being at the hard constraint. Once the closed-loop system enters a state-space region in which input-output linearization can ensure asymptotic stability, the hard constraint becomes inactive. Consequently, the nonlinear control method is applicable to stable and unstable processes, whether non-minimum- or minimum-phase. The control method is implemented on a chemical reactor with multiple steady states, to show its application and performance

Digital Control and Monitoring Methods for Nonlinear Processes

The chemical engineering literature is dominated by physical and (bio)-chemical processes that exhibit complex nonlinear behavior, and as a consequence, the associated requirements of their analysis, optimization, control and monitoring pose considerable challenges in the face of emerging competitive pressures on the chemical, petrochemical and pharmaceutical industries. The above operational requirements are now increasingly imposed on processes that exhibit inherently nonlinear behavior over a wide range of operating conditions, rendering the employment of linear process control and monitoring methods rather inadequate. At the same time, increased research efforts are now concentrated on the development of new process control and supervisory systems that could be digitally implemented with the aid of powerful computer software codes. In particular, it is widely recognized that the important objective of process performance reliability can be met through a comprehensive framework for process control and monitoring. From: (i) a process safety point of view, the more reliable the process control and monitoring scheme employed and the earlier the detection of an operationally hazardous problem, the greater the intervening power of the process engineering team to correct it and restore operational order (ii) a product quality point of view, the earlier detection of an operational problem might prevent the unnecessary production of o-spec products, and subsequently minimize cost. The present work proposes a new methodological perspective and a novel set of systematic analytical tools aiming at the synthesis and tuning of well-performing digital controllers and the development of monitoring algorithms for nonlinear processes. In particular, the main thematic and research axis traced are: (i) The systematic integrated synthesis and tuning of advanced model-based digital controllers using techniques conceptually inspired by Zubov’s advanced stability theory. (ii) The rigorous quantitative characterization and monitoring of the asymptotic behavior of complex nonlinear processes using the notion of invariant manifolds and functional equations theory. (iii) The systematic design of nonlinear state observer-based process monitoring systems to accurately reconstruct unmeasurable process variables in the presence of time-scale multiplicity. (iv) The design of robust nonlinear digital observers for chemical reaction systems in the presence of model uncertainty

Advances in PID Control

Since the foundation and up to the current state-of-the-art in control engineering, the problems of PID control steadily attract great attention of numerous researchers and remain inexhaustible source of new ideas for process of control system design and industrial applications. PID control effectiveness is usually caused by the nature of dynamical processes, conditioned that the majority of the industrial dynamical processes are well described by simple dynamic model of the first or second order. The efficacy of PID controllers vastly falls in case of complicated dynamics, nonlinearities, and varying parameters of the plant. This gives a pulse to further researches in the field of PID control. Consequently, the problems of advanced PID control system design methodologies, rules of adaptive PID control, self-tuning procedures, and particularly robustness and transient performance for nonlinear systems, still remain as the areas of the lively interests for many scientists and researchers at the present time. The recent research results presented in this book provide new ideas for improved performance of PID control applications