Admissible Initial Conditions Research Articles

Event-triggered synchronization of discrete-time neural networks: A switching approach

This paper investigates the event-triggered synchronization control of discrete-time neural networks. The main highlights are threefold: (1) a new event-triggered mechanism (ETM) is presented, which can be regarded as a switching between the discrete-time periodic sampled-data control and a continuous ETM; (2) a saturating controller which is equipped with two switching gains is designed to match the switching property of the proposed ETM; (3) a dedicated switching Lyapunov–Krasovskii functional is constructed, which takes the sawtooth constraints of control input into account. Based on these ingredients, the synchronization criteria are derived such that the considered error systems are locally stable. Whereafter, two co-design problems are discussed to maximize the set of admissible initial conditions and the triggering threshold, respectively. Finally, the effectiveness and advantages of the proposed method are validated by two numerical examples.

A Generalized Minimum Phase Property for Finite-Dimensional Continuous-Time MIMO LTI Systems with Additive Disturbances

In this paper, we further generalize the definition of the extended zero dynamics to finite-dimensional continuous-time MIMO LTI systems with additive disturbances; and then introduce the concept of minimum phase for this class of systems. We show that the extended zero dynamics is invariant under the application of dynamic extension to its input, and therefore, a minimum phase system remains minimum phase after a finite number of steps of dynamic extension. We further introduce the extended zero dynamics canonical form for square MIMO LTI systems with uniform vector relative degree. We prove that a system is minimum phase according to our definition if its zero dynamics is asymptotically stable. The converse of the statement holds if the system is stabilizable from the control input.The objective of this research is to solve the model reference robust adaptive control problem for finite-dimensional continuous-time square MIMO LTI systems that is minimum phase according to the generalized definition using an appropriately vectorized version of Pan and Başar (2000), and results here constitute important building blocks. In a subsequent paper, Başar and Pan (2019), we further connect the dots: starting with a square MIMO LTI system that is minimum phase with respect to admissible initial conditions and admissible disturbance waveforms, we must obtain a true system representation that admits both the extended zero dynamics canonical form representation and the strict observer canonical form representation. Toward this end, we need to be able to extend the given system to one with uniform vector relative degree and with uniform observability indices without changing its minimum phase property, and this has been done in Başar and Pan (2019), fully resolving this issue.

Event-triggered control co-design for rational systems

This work deals with the problem of designing stabilizing event-triggered state-feedback controllers for rational systems. Using differential algebraic representations and Lyapunov theory techniques, LMI-based conditions are derived to ensure regional asymptotic stability of the origin. These conditions are then cast into a convex optimization problem to the co-design of the event generator parameters and the state-feedback gain in order to reduce the controller updates while ensuring the asymptotic stability of the origin with respect to a given set of admissible initial conditions. The proposed methodology is illustrated by means of a numerical example.

On the optimization of actuator saturation limits for LTI systems: an LMI-based invariant ellipsoid approach

This paper considers the problem of optimal actuator dimensioning for LTI systems, in the sense of choosing appropriate saturation limits for a given set of admissible initial conditions and for a predefined integral state-feedback control law. By using an invariant ellipsoid argument, it is shown that this problem can be described as a linear matrix inequality (LMI)-based optimization that can be solved efficiently. Moreover, the paper shows that the optimal actuator dimensioning is connected to the choice of the initial conditions of the integral states of the controller, which can be included in the overall optimization to improve further the results. Two different methods are described and analyzed by means of numerical simulation.

New Observer-Based Controller Design for Delayed Systems Subject to Input Saturation and Disturbances

A new methodology to design an $${H}_{\infty }$$ observer-based controller for discrete-time delayed systems subject to input saturation and $${\mathcal {L}}_{2}$$-norm disturbances is proposed, motivated by a benchmark problem. The stabilizing observer-based controller gains are obtained from an LMI formulation using a Lyapunov–Krasovskii functional, free weighting matrices, and an effective approach to overcome the obstacle of the bilinearity. To select the most adequate controller, an iterative optimization algorithm is developed to maximize the set of admissible initial conditions. The proposed results are illustrated by a practical application using the quadruple tank process to visualize different control theory notions and phenomena in laboratory conditions.

A Gaussian moment method for polydisperse multiphase flow modelling

The accurate prediction of multiphase flow when particles are differentiated by a set of “internal” variables, such as size or temperature, can pose modelling and numerical challenges. Although Lagrangian particle methods can provide predictive simulations of a wide spectrum of complex multiphase problems, they can become prohibitively expensive as the number of particles becomes large. Alternatively, Eulerian approaches have the potential to improve the computational efficiency of multiphase flows, but classical methods produce modelling artifacts or do not properly treat the local statistical dependence between the particle velocities and internal variables, or between the internal variables themselves. In this paper an extension is proposed of the classical Gaussian ten-moment model from gaskinetic theory to a model for the treatment of a dilute particle phase with an arbitrary number of internal variables based on an entropy-maximization argument. Unlike previous formulations, this new model provides a set of first-order robustly-hyperbolic balance laws that include a direct treatment for the local statistical variance of each variable, as well as the covariance between the internal variables or the internal variables and particle velocity. A study of the wave speeds of the general hyperbolic system is presented. To demonstrate an example application, the model is then specialized for polydisperse flows that are subject to viscous fluid drag as well as gravitational and buoyancy forces. The complete eigenstructure of this fifteen-moment polydisperse Gaussian model (PGM) is presented and the PGM is shown to maintain a physically realizable distribution function for all admissible initial conditions. Finally, several illustrative low-dimensional problems are studied to demonstrate the predictive capabilities of the new model.

Harnessing fluctuations to discover dissipative evolution equations

Continuum modeling of dissipative processes in materials often relies on strong phenomenological assumptions, as their derivation from underlying atomistic/particle models remains a major long-standing challenge. Here we show that the continuum evolution equations of a wide class of dissipative phenomena can be numerically obtained (in a discretized form) from fluctuations via an infinite-dimensional fluctuation-dissipation relation. A salient feature of the method is that these continuum equations can be fully pre-computed, enabling macroscopic simulations of arbitrary admissible initial conditions, without the need of any further microscopic simulations. We test this coarse-graining procedure on a one-dimensional non-linear diffusive process with known analytical solution, and obtain an excellent agreement for the density evolution. This illustrative example serves as a blueprint for a new multiscale paradigm, where full dissipative evolution equations — and not only parameters — can be numerically computed from lower scale data.

Improved Synchronization Criteria for Chaotic Neural Networks with Sampled-data Control Subject to Actuator Saturation

In this paper, the synchronization problem for chaotic neural networks with sampled-data control and actuator saturation is investigated. By constructing a suitable time-dependent function and utilizing a modified free-matrix-based integral inequality, a sampled-data synchronization criterion for chaotic neural networks is derived as the framework of linear matrix inequalities. Based on the first result, a design method of sampled-data controller subject to actuator saturation for chaotic neural networks is introduced through an optimization method which enlarges the set of admissible initial conditions. The superiority and validity of the proposed results will be verified through comparing with the existing results in a numerical example.

Further results on sampled-data synchronization control for chaotic neural networks with actuator saturation

The problem of local synchronization control for chaotic neural networks with sampled-data and saturating actuators is investigated in this paper. The intervals from the sampling instant tk to its next instant tk+1 are assumed to be within a sampling interval. By taking advantage of characteristic information on the whole sampling interval, a new two-sided sampling-interval-dependent discontinuous Lyapunov functional is first constructed, which depends on the available information of both the intervals from tk to t and from t to tk+1. Then, by utilizing the Lyapunov functional, a novel sampling-interval-dependent stability condition is derived, rendering the synchronization error systems stable. Moreover, an optimization approach is provided to design desired sampled-data controllers such that the set of admissible initial conditions can be maximized. Finally, the effectiveness and benefits of the presented method is verified by numerical simulation.

Multiclass AQM on a TCP/IP router: A control theory approach

SummaryActive queue management (AQM) is a well‐known technique to improve routing performance under congested traffic conditions. It is often deployed to regulate queue sizes, thus aiming for constant transmission delay. This work addresses AQM using an approach based on control theory ideas. Compared with previous results in the literature, the novelty is the consideration of heterogeneous traffic, ie, multiclass traffic. Thus, each traffic class may have different discarding policies, queue sizes, and bandwidth share. This feature brings the proposal nearer to real network management demands than previous approaches in the literature. The proposed technique assumes that each class already has a simple controller, designed a priori, and focuses on designing a static state‐feedback controller for the multiclass system, where the design is based on using LMIs for the calculations. For this, optimization problems with LMI constraints are proposed to compute the state‐feedback gains that ensure stability for a large set of admissible initial conditions. These conditions ensure not only closed‐loop stability but also some level of performance. As far as we know, this is the first control theory based approach for the AQM problem on TCP/IP routers that allows a multiclass AQM while also considering time‐varying delays and input saturation. This is an important step to frame AQM in a more formal, yet realistic context, enabling it to address important service level agreement (SLA) directives. The proposal is tested on a simulated system at the end of this paper, showing the feasibility and performance of the approach in the presence of multiclass traffic.

On the collapse of trial solutions for a damped-driven nonlinear Schrödinger equation

We consider the focusing 2D nonlinear Schrödinger equation, perturbed by a damping term, and driven by multiplicative noise. We show that a physically motivated trial solution does not collapse for any admissible initial condition although the exponent of the nonlinearity is critical. Our method is based on the construction of a global solution to a singular stochastic Hamiltonian system used to connect trial solution and Schrödinger equation.

Quantized feedback stabilization of continuous time-delay systems subject to actuator saturation

In this paper, the problem of quantized feedback stabilization is investigated for continuous time-delay systems subject to actuator saturation. By utilizing two different methods, delay-independent conditions are obtained to guarantee the existence of a region of admissible initial conditions from which all trajectories of the resulting closed-loop system converge to a neighborhood of the equilibrium. Furthermore, delay-dependent conditions are developed based on the delay partitioning idea and the Lyapunov–Razumikhin functional approach, respectively. Finally, several examples are provided to demonstrate the effectiveness of the proposed approaches.

Robust stabilization using a sampled-data strategy of uncertain neutral state-delayed systems subject to input limitations

Abstract Stabilization of neutral systems with state delay is considered in the presence of uncertainty and input limitations in magnitude. The proposed solution is based on simultaneously characterizing a set of stabilizing controllers and the associated admissible initial conditions through the use of a free weighting matrix approach. From this mathematical characterization, state feedback gains that ensure a large set of admissible initial conditions are calculated by solving an optimization problem with LMI constraints. Some examples are presented to compare the results with previous approaches in the literature.

Event-Triggered Stabilization of Neural Networks With Time-Varying Switching Gains and Input Saturation.

This paper investigates the event-triggered stabilization of neural networks (NNs) subject to input saturation. The main core lies in the design of a novel controller with time-varying switching gains and the associated switching event-triggered condition (ETC). The ETC is essentially a switching between the aperiodic sampling and continuous event trigger. The control gains of the designed controller are composed of an exponentially decaying term and two gain matrices. The two gain matrices are required to be switched when the switching between the aperiodic sampling and continuous event trigger is met. By employing the generalized sector condition and switching Lyapunov function, several sufficient conditions that ensure the local exponential stability of the NNs are formulated in terms of linear matrix inequalities (LMIs). Both the exponentially decaying term and switching gains improve the feasible region of LMIs, and then they are helpful to enlarge the set of admissible initial conditions, the threshold in ETC, and the average waiting time. Together with several optimization problems, two numerical examples are employed to validate the effectiveness of our results.

Control of Nonlinear Systems Subject to Amplitude Bounded Disturbances Using a N-fuzzy Strategy

Abstract The dynamic output feedback stabilization problem of nonlinear discrete-time systems subject to amplitude bounded disturbances by means of Takagi-Sugeno (T-S) N-fuzzy models is considered in this paper. The linear matrix inequality (LMI) approach is applied to design a dynamic output feedback fuzzy controller such that the system trajectories are ultimately bounded, that is, the state trajectory starting in a set of admissible initial conditions is guaranteed to converge in finite time to a nearby region of the state space origin. Furthermore, the proposed approach enables to handle with nonlinearities that are not function of the measured outputs but are sector bounded. A numerical example is considered to illustrate the effectiveness of the developed technique.

Design of robust H∞ controllers for congestion control in data networks

A novel approach for internet congestion control is presented here, using a static feedback law. A congestion controller for Network TCP/IP Routers is derived, to ensure stability for a large set of admissible initial conditions. From a formal point of view, we propose a static state-feedback for discrete-time delayed systems, with both saturating inputs and link capacity disturbances, that ensures simultaneously closed-loop stability and a prescribed level of performance. The effectiveness of the proposed approach is verified via simulation, comparing the results to PI and RED controllers.

Control Based on Saturated Time-Delay Systems Theory of Mach Number in Wind Tunnels

A proposal for the regulation of the Mach number in wind tunnels using static state feedback for saturated systems with delays is presented here. As these systems can be precisely represented by a time-delay model with saturating inputs, a general solution for discrete delayed systems with saturating input is first derived. This general solution is based on modeling the saturation using a Lyapunov functional, using free weighting matrices and maximizing the set of admissible initial conditions. The application of this solution to the control of the Mach number in a wind tunnel is then presented, illustrating the design procedures.

Local Stabilization of Nonlinear Discrete-Time Systems Subject to Amplitude Bounded Disturbances

The stabilization problem of nonlinear discrete-time systems subject to amplitude bounded disturbances by means of Takagi-Sugeno (T-S) fuzzy models is considered in this paper. The linear matrix inequality (LMI) approach is applied to design a state feedback fuzzy controller such that the system trajectories are ultimately bounded, that is, the state trajectory starting in a set of admissible initial conditions is guaranteed to converge in finite time to a nearby region of the state space origin. Numerical examples are considered to illustrate the approach demonstrating the effectiveness of the proposed technique for the control synthesis of nonlinear discrete-time systems.

Sampled-data synchronization control for chaotic neural networks subject to actuator saturation

In this paper, the sampled-data control is applied to synchronize chaotic neural networks subject to actuator saturation. By employing a time-dependent Lyapunov functional that captures the characteristic information of actual sampling pattern, we derive a local stability condition for the synchronization error systems. By this condition, we design a sampled-data controller to regionally synchronize the drive neural networks and response neural networks subject to actuator saturation. Moreover, an optimization method is given to design the desired sampled-data controller such that the set of admissible initial conditions is maximized. A numerical example is given to demonstrate the effectiveness and merits of the proposed design technique.

Robust stabilization using LMI techniques of neutral time-delay systems subject to input saturation

The robust stabilization of uncertain saturated neutral systems with state delay is solved in this paper: based on a free weighting matrix approach, sufficient conditions are obtained via an LMI formulation. From these conditions, state feedback gains that ensure stability for the largest set of admissible initial conditions can be calculated solving optimization problems with LMI constraints. Some applications of this methodology to feedback control are then presented and compared with previous results in the literature.