In this note, an economically optimal and stabilizing feedback control paradigm for nonlinear systems is introduced. Specifically, an Economic Model Predictive Control (EMPC) that utilizes an equivalent bilinear prediction model derived from Koopman operator theory is developed. Such control strategy is very useful for real-time implementation of the EMPC due to the potential computation time reduction caused by replacing the underlying nonlinear prediction model with a bilinear model. To attain this control objective, we utilize the Lyapunov theory as well as the Koopman eignefunctions to design the proposed predictive controller. Unlike the Koopman-based tracking MPC, we generalize the steady-state operation of nonlinear systems using the Koopman operator to a time-varying operation regime where the closed-loop state can perpetually transition from any point within a predefined stability region to another. In addition, we utilize the Lyapunov theory to construct a stability region that is fully structured from the Koopman-based bilinear model. Finally, we demonstrate the efficacy of the proposed Koopman-based Economic MPC through a chemical process example.

First-order convex optimization algorithms are popular due to their computational attractiveness and applicability to a wide range of domains such as machine learning and control. Despite the substantial progress being made over the last few decades, some open questions related to their convergence remain unaddressed. In addition, majority of the first-order methods assume strong convexity to analyze both the stability of the method and derive an explicit convergence rate. In this manuscript, we relax the strong convexity condition, and then, lay out two main contributions. First, we provide a methodology where one can analyze the speed of convergence of the algorithm using the contractive theory and linear algebra. Second, we find explicit values of the tuning parameters of the Double Momentum Algorithm (which unifies many of the popular algorithms), ensuring stability for gradient L-Lipchitz functions. In this work, while an explicit convergence rate is not provided, the foundational results serve as a stepping stone in that direction, as we provide an explicit non-asymptotic rate. Furthermore, our numerical experiments demonstrate superior performance of the proposed method. Beyond optimization, we also apply our method to two-sided markets in non-cooperative game theory.

Epilepsy is a neurological disorder ranked as the second most serious neurological disease known to humanity, after stroke. Inter-ictal spiking is an abnormal neuronal discharge after an epileptic seizure. This abnormal activity can originate from one or more cranial lobes, often travels from one lobe to another, and interferes with normal activity from the affected lobe. The common practice for Inter-ictal spike detection of brain signals is via visual scanning of the recordings, which is a subjective and a very time-consuming task. Motivated by that, this article focuses on using machine learning for epileptic spikes classification in magnetoencephalography (MEG) signals. First, we used the Position Weight Matrix (PWM) method combined with a uniform quantizer to generate useful features from time domain and frequency domain through a Fast Fourier Transform (FFT) of the framed raw MEG signals. Second, the extracted features are fed to standard classifiers for inter-ictel spikes classification. The proposed technique shows great potential in spike classification and reducing the feature vector size. Specifically, the proposed technique achieved average sensitivity up to 87% and specificity up to 97% using 5-folds cross-validation applied to a balanced dataset. These samples are extracted from nine epileptic subjects using a sliding frame of size 95 samples-points with a step-size of 8 sample-points.