Keywords: Control and analysis of Nonlinear Systems

Keywords: Control and analysis of Nonlinear Systems, Machine Learning, Stability and Complex Dynamics
Abstract: This paper is concerned with the stability/instability analysis problem of feedback systems with rectified linear unit (ReLU) nonlinearities. Such feedback systems arise when we model dynamical (recurrent) neural networks (NNs) and NN-driven control systems where all the activation functions of NNs are ReLUs. In this study, we focus on the semialgebraic set representation characterizing the input-output properties of ReLUs. This allows us to employ a novel copositive multiplier in the framework of the integral quadratic constraint and, thus, to derive a new linear matrix inequality (LMI) condition for the stability analysis of the feedback systems. However, the infeasibility of this LMI does not allow us to obtain any conclusion on the system's stability due to its conservativeness. This motivates us to consider its dual LMI. By investigating the structure of the dual solution, we derive a rank condition on the dual variable certificating that the system at hand is unstable. In addition, we construct a hierarchy of dual LMIs allowing for improved instability detection. We illustrate the effectiveness of the proposed approach by several numerical examples.

Keywords: Stability and Complex Dynamics, Control of Nonlinear Systems, Control of Bifurcations, Oscillations and Chaos
Abstract: The definition of k-contraction promises a useful generalization of the classical notion of contraction for dynamical systems. However, most of the k-contraction literature focuses on continuous-time systems. In this work, we derive conditions for k-contractivity of discrete- time dynamics. Our first result follows traditional lines for k-contraction analysis, and provides Lyapunov-like sufficient conditions based on matrix compounds and state-dependent metrics. However, our subsequent results avoid the complexities related to matrix compounds. Inspired by recent findings in the context of k-contraction for continuous-time systems, we provide conditions on the system’s dynamics that rely on generalized Lyapunov inequalities and quadratic cone fields. The proposed conditions are also shown to be necessary for linear time-invariant systems.

Keywords: Stability and Complex Dynamics, Control and analysis of Nonlinear Systems, Process Control
Abstract: In this paper, we propose a change of coordinates that brings the state matrix of an autonomous linear system into a modified Jordan Block form. Such a change of coordinates allows us to obtain exact values of the scaling factor and the convergence rate of the exponential stability bound for linear systems. The analysis is then applied to a control design with requirements on the evolution of the state norm. Numerical examples are also provided to illustrate the effectiveness of the proposed approach.

Keywords: Modeling and Identification of Nonlinear Systems, Networked Nonlinear Systems, Control of Networks
Abstract: This paper suggests a model for evaluating the operational capability of a system to accomplish a mission. This model involves a dynamic equation defined on the space of multi-graphs, associated with a weak discrete metric. It could be analysed further using the framework of control theory. This modeling is based on a definition of the operational capability that includes three keywords: system, mission and ability. These keywords are defined and represented by appropriate mathematical objects. The paper focuses on justifying the use of the proposed mathematical objects to represent these keywords.

Keywords: Modeling and Identification of Nonlinear Systems
Abstract: Modeling of metal hydride storage tanks remains a challenging issue. In most of the literature, the charge and discharge of metal hydride storage tanks are controlled by the hydrogen flow rate. However, the case of controlling the charge and discharge of metal hydride tanks by pipe pressure has rarely been reported in the literature. Aiming at this problem, this work presents a two-dimensional state-space model that takes the pipe pressure as the system input and assumes that the tank temperature can be measured. Then, the First-order Trajectory Sensitivity Analysis method is used to analyze the parameter sensitivity of several selected unknown parameters. The results of the sensitivity analysis indicate that six parameters: entropy change for desorption Delta S_d, enthalpy change for desorption Delta H_d, activation energy for absorption E_a, activation energy for desorption E_d, plateau flatness coefficient varphi and plateau hysteresis coefficient beta have higher sensitivity to the model state.

Keywords: Control and analysis of Nonlinear Systems, Process Control
Abstract: Cold gas spray processing has gained increasing attention in the additive manufacturing community. It allows for coating and layer-by-layer printing through acceleration of the particles to the supersonic velocities against the substrate with the Laval nozzle. However, achieving efficient bonding requires careful control of particles impact velocity. In this paper, we developed a nonlinear control strategy for particle velocity based on direct Lyapunov method. We investigated a simplified model of the cold gas spray process for controller design and validated it using the full dynamical model based on Navier-Stokes equations. Our proposed controller achieves exponential convergence of particle velocity to the desired value while improving system performance.

Keywords: Stability and Complex Dynamics, Control and analysis of Nonlinear Systems, Modeling and Identification of Nonlinear Systems
Abstract: This paper aims to formulate a new approach to dissipative modeling of nonlinear positive systems with interior equilibria. Processing positivity separately from stability becomes unnecessary. For asymmetric state spaces on which systems evolve, this paper proposes a supply rate of non-gain-type asymmetry and develops formulas of their aggregation, which is not as trivial as aggregating L 2 gain and passivity supply rates. The novelty of their proofs is that the involved treatment of asymmetric functions is reduced to the evaluation of directed graphs carrying balance information between variables. The dissipativity on asymmetric spaces and the aggregation formulas qualify a single logarithmic storage function to establish stability and positivity simultaneously for an interconnected system. The usefulness is illustrated by examples.

Keywords: Nonlinear Systems with Time Delay, Control of Nonlinear Systems
Abstract: We announce a local version of an approach to proving asymptotic stability based on Halanay's inequality. Our result is amenable to nonlinear systems that contain input and state delays. It provides robustness estimates for systems containing actuator uncertainty, in the sense of input-to-state stability. Our numerical example illustrates how our method leads to useful bounds on the allowable uncertainties and on the basin of attraction.

Keywords: Control and analysis of Nonlinear Systems, Control of Bifurcations, Oscillations and Chaos, Mechatronics and Robotics
Abstract: This paper focuses on the analysis of the nonlinear automatic gain control (NLAGC) in order to make a mass oscillate at its natural frequency with a specified amplitude. Despite their widespread use in Micro Electro Mechanical Sensor gyroscopes, ensuring the stability of this oscillation remains challenging. In this paper, we investigate the stability of two versions of the NLAGC: an ideal and an implementable one, using appropriate change of variables and transverse contraction criterion from orbital stability theory. The results are validated through a numerical example obtained from a real gyroscope system.

Keywords: Control and analysis of Distributed Parameter Systems

Keywords: Control and analysis of Distributed Parameter Systems
Abstract: In this paper, the stability analysis of a class of hyperbolic systems with non-diagonalizable principal part and nonuniform coefficients is addressed. We give a stability condition and express the convergence rate that can be obtained. Two cases will be treated independently: with and without a source term. A numerical scheme is designed to illustrate the performance of the Lyapunov stability analysis.

Keywords: Control of Nonlinear Systems
Abstract: This paper addresses the stabilization of a wave equation coupled with non-expansive systems with input constraints. The open-loop dynamic of the wave equation is conservative, while the non-expansive system is marginally stable. The proposed feedback control law with integral action is derived through the forwarding-Lyapunov method, utilizing a weak Lyapunov functional and a Sylvester equation. This paper establishes the well-posedness and global asymptotic stability of the closed-loop system. Notably, the theoretical analysis includes an approximate observability condition and employs LaSalle's invariance principle. Numerical simulation results are presented to illustrate the effectiveness of the proposed methodology.

Keywords: Control and analysis of Distributed Systems, Control of Networks
Abstract: In this paper, we design an output-feedback controller to stabilize n + m hetero-directional transport partial differential equations (PDEs) coupled on both domain boundaries to ordinary differential equations (ODEs). This class of systems can represent, for instance, actuator and load dynamics at the boundaries of a hyperbolic system. The actuator is located at the connection point between the PDE and one of the ODEs, and we consider anti-collocated PDE measurements. We first design a state-observer by combining the backstepping methodology with time-delay system approaches. We then introduce a state feedback controller using analogous techniques before designing the wanted output-feedback control law.

Keywords: Modeling and Identification of Nonlinear Systems, Control and analysis of Nonlinear Systems
Abstract: This paper presents an identification approach for feedforward controller design dedicated to nonlinear systems when their structure is known. We propose a model-based approach, leveraging Block-Oriented models, which allows a fast identification of the system when its structure is known. The controller is then designed based on the identified model using a criterion close to traditional identification criteria. Additionally, a costless iterative model identification and controller design approach is also proposed to improve the performance of the controller if necessary. The effectiveness of the approach is demonstrated through the design of a feedforward controller for a Wiener system.

Keywords: Observers Design, Biomedical Systems, Control and analysis of Nonlinear Systems
Abstract: This article analyzes the possibility of using observer-based approaches for the detection of cardiovascular anomalies through a representation of the model written in a quadratic observability normal form. This particular representation allows to design an observer that is sensitive to changes in the cardiovascular mitral and aortic valves. The idea of the present result is based on the design of a residual generator that allows analyzing when a failure occurs in the heart valves. In addition, numerical simulations are performed in order to show the performance and capacity of the proposed strategy.

Keywords: Control of Networks