Large-scale Nonlinear Systems Research Articles

Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom.

Predefined-time fuzzy adaptive decentralised control for fractional-order nonlinear large-scale systems by a cyclic-small-gain-based approach

The predefined-time fuzzy adaptive output feedback decentralised control problem is considered for fractional-order nonlinear large-scale (FONLS) systems. Firstly, by using fuzzy logic systems (FLSs) to model unknown nonlinear dynamics, a fuzzy decentralised state observer is designed to solve the immeasurable state problem. Secondly, the predefined-time stability criterion of fractional-order nonlinear systems is proposed, and then, based on the established predefined-time stability criterion of the fractional-order nonlinear systems, a predefined-time fuzzy adaptive output feedback decentralised control method is developed by combining the backstepping design technique with the cyclic-small-gain approach, and the semi-global practically predefined-time stable (SGPPTS) of the control system is proved. Finally, the numerical simulation is given and shows the effectiveness of the presented predefined-time control method.

Observer-Based Decentralized Control for Non-Strict-Feedback Fractional-Order Nonlinear Large-Scale Systems With Unknown Dead Zones.

This article addresses the output-feedback decentralized control issue for the fractional-order nonlinear large-scale nonstrict-feedback systems with states immeasurable and unknown dead zones. The unknown nonlinear functions are identified by neural networks (NNs), and immeasurable states are estimated by establishing an NNs' decentralized state observer. The algebraic loop issue is solved by using the property of NN basis functions and designing the fractional-order adaptation laws. In addition, the fractional-order dynamic surface control (FODSC) design technique is introduced in the adaptive backstepping control algorithm to avoid the issue of ``explosion of complexity.'' Then, by treating the nonsymmetric dead zones as the time-varying uncertain systems, an adaptive NNs' output-feedback decentralized control scheme is developed via the fractional-order Lyapunov stability criterion. It is proven that the controlled fractional-order systems are stable, and the tracking and observer errors can converge to a small neighborhood of zero. Two simulation examples are given to confirm the validity of the put forward control scheme.

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

Adaptive Decentralized Finite-Time Fuzzy Secure Control for Uncertain Nonlinear CPSs Under Deception Attacks

This article addresses the adaptive secure control problem for a class of uncertain nonlinear large-scale cyber-physical systems (CPSs) subjected to deception attacks. Specifically, a novel adaptive fuzzy control scheme for each subsystem of CPSs is designed to mitigate the effects of cyber-attacks that intentionally tamper with control signals from controllers to actuators and the state signals from sensors to controllers, and a Nussbaum function is proposed to tackle unknown time-varying control directions. Moreover, a finite-time converging secure control scheme is designed for compromised CPSs to guarantee all the states converge to a predetermined small set in finite time. It is shown that with the proposed control scheme, all the signals of the closed-loop system are proved to be semi-globally bounded. Two simulation examples are provided to illustrate the effectiveness of the proposed secure control method.

Prescribed Performance Bumpless Transfer Control for Switched Large-Scale Nonlinear Systems

This article presents a novel control protocol for a class of switched large-scale nonlinear systems subject to prescribed performance constraints and bumpless transfer constraints. The prescribed performance constraints enforce the output tracking error to converge to a predefined residual set, with a predefined maximum overshoot and minimum convergence rate. Meanwhile, the bumpless transfer constraints restrict the bump of the control input when switching between two adjacent modes. By incorporating the modification signals into the prescribed performance functions (PPFs), control-bump PPFs (CPPFs) are provided to proactively reduce the performance bounds to a reasonable level. The modification signals are associated with control bumps and generated by an auxiliary switched positive system. Furthermore, a model of control bump is developed to enhance the dynamic response, and switched disturbance observers are employed to estimate external disturbances. In the framework of dynamic surface control and multiple Lyapunov stability theory, a prescribed performance bumpless transfer control (PPBTC) approach is proposed to guarantee the prescribed performance and bumpless transfer performance. Finally, a practical example of the proposed control scheme is presented to demonstrate its validity and applicability.

Exponential Stabilization for Time-Delay Nonlinear Interconnected Systems With Unknown Control Directions and Unmodeled Dynamics

An exponential stable control scheme based on state transformation and Lyapunov functions is presented for a class of large-scale nonlinear time-delay interconnected systems subjected to unknown control directions and unmodeled dynamics. By regulating the transformed state, the original system state is regulated to the origin at an exponential rate. By introducing a class of type-B Nussbaum functions, the obstacle caused by multiple unknown control coefficients is successfully overcome. Dynamic gains technique and changing supply function idea are established to tackle time-delay terms and unmodeled dynamics in the system. It turns out that the system state converges to the origin at a prescribed speed, and all the closed-loop signals remain bounded. Finally, simulation results show the effectiveness of the proposed methodology.

Inexact free derivative quasi-Newton method for large-scale nonlinear system of equations

In this work, we propose a free derivative quasi-Newton method for solving large-scale nonlinear systems of equation. We introduce a two-stage linear search direction and develop its global convergence theory. Besides, we prove that the method enjoys superlinear convergence rate. Finally, numerical experiments illustrate that the proposed method is competitive with respect to Newton-Krylov methods and other well-known methods for solving large-scale nonlinear systems of equations.

Adaptive backstepping control for nonlinear interconnected systems with prespecified-performance-driven output triggering

This paper considers a class of nonlinear large-scale interconnected systems and proposes an event-triggered decentralized controller by using adaptive output feedback backstepping. To save the communication resource, the output measurements are transmitted to the controller based on a novel double-events-triggered (DET) mechanism driven by transient performance request. Due to the use of triggered mechanism, only discontinuous output signals can be utilized such that the derived virtual controls are discontinuous, of which the derivatives cannot be computed as often done in standard backstepping design procedure. To overcome this difficulty, a new error signal decomposition and compensation mechanism is proposed to quantify and compensate for the difference between the virtual controls derived by using the event-triggered measurement signals and the real-time system outputs. Through a nonsmooth stability analysis where the closed-loop stability is considered under different conditions of output data transmissions, it is shown that the tracking error remains within a prespecified performance bound all the time. In contrast to the existing backstepping results addressing event-triggered outputs, the designed DET mechanism does not require the virtual control signals to be reversely transmitted to the sensor side from the controller side and guarantees the communication burden can be reduced under tracking performance guarantees.

Observer-based Decentralized Fuzzy Control for Discrete-time Nonlinear Large-scale Systems

Observer-based Decentralized Fuzzy Control for Discrete-time Nonlinear Large-scale Systems

Feasibility Conditions-Free Prescribed Performance Decentralized Fault-Tolerant Neural Control of Constrained Large-Scale Systems

This article investigates the command filter-based decentralized prescribed performance adaptive fault-tolerant compensation control strategy for uncertain nonlinear large-scale systems subject to asymmetric time-varying full-state constraints. Via integrating the performance function with command filter-based backstepping technique, the prescribed performance control problem is addressed, under which the complexity of controller design is reduced. Under the prescribed performance control framework, the nonlinear transformed function is constructed so as to ensure that the asymmetric time-varying full-state constraints free from feasibility conditions imposed on virtual control signals are not violated. Besides, the effect of infinite number of time-varying actuator faults is compensated with the aid of projection adaption design. Furthermore, based on the piecewise Lyapunov function analysis, it is rigorously testified that entire involved signals are bounded, desired constraints are not breached and tracking errors within the predefined domains. Finally, the effective performances of the developed control algorithm are confirmed by some simulation results.

Finite-time adaptive neural resilient DSC for fractional-order nonlinear large-scale systems against sensor-actuator faults

Finite-time adaptive neural resilient DSC for fractional-order nonlinear large-scale systems against sensor-actuator faults

Event-triggered impulsive control of lower-triangular large-scale nonlinear systems based on gain scaling technique

This paper investigates the event-triggered impulsive control problem for a class of large-scale nonlinear systems in lower-triangular form. Based on gain scaling technique and impulsive control theory, a novel decentralized event-triggered impulsive control strategy is first put forward by introducing a static scaling gain, where no control input exists between two consecutive triggering points. Moreover, when the large uncertainties exist in system nonlinearities, we further develop a new control strategy by introducing a time-varying scaling gain. It is proved that the proposed closed-loop control strategies exclude the Zeno behavior without sacrificing the global convergence of system states. Compared with the existing results, it is the first time to apply impulsive control to lower-triangular large-scale nonlinear systems, and the advantages of event-triggered impulsive control and gain scaling technique are subtly combined in the proposed control strategies. Finally, two simulation examples are used to demonstrate the effectiveness of the proposed schemes.

Plug-and-Play Distributed Control of Large-Scale Nonlinear Systems.

A method to design plug-and-play (PnP) distributed controllers for large-scale nonlinear systems represented by interconnected Takagi-Sugeno fuzzy models with nonlinear consequent is presented in this article. From the combination of techniques to use multiple fuzzy summations and to explore the chordal decomposition of the interconnection graph associated with the large-scale nonlinear system, sufficient conditions for distributed stabilization are derived in terms of linear matrix inequalities (LMIs). Conditions specially designed to allow seamless subsystems plugging-in and unplugging operations from the large-scale system, without requiring the redesign of all previously tuned distributed controllers, are provided. The approach can be used together with fault detection and isolation (FDI) systems, and also in the context of mixed distributed and decentralized controllers operating in a network of interconnected systems. To illustrate the effectiveness of the proposed PnP approach, a network of nonlinearly coupled and heterogeneous Van der Pol oscillators is used in the numerical experiments.

Decentralized Adaptive Control of Large-Scale Nonlinear Systems With Time-Delay Interconnections and Asymmetric Dead-Zone Input

In this article, a decentralized adaptive control strategy is proposed for a class of large-scale nonlinear systems with interconnections. Each subsystem contains multiple state delays, asymmetric dead-zone inputs, and bounded time-varying disturbances. Every error subsystem is first decomposed according to input matrix assumption. Then, the decentralized adaptive controller is developed to handle uncertain functions in the subsystem. The nonlinear control gain function is constructed and used in adaptive control design. Two robust adaptive control schemes are, respectively, addressed for the cases of matched and mismatched time-delay nonlinearities. It is proved that all closed-loop signals are bounded, and the tracking error converges to an adjustable region. A simulation example is presented to show the effectiveness of the proposed method.

On sampling Kaczmarz–Motzkin methods for solving large-scale nonlinear systems

On sampling Kaczmarz–Motzkin methods for solving large-scale nonlinear systems

A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: Application to shape optimization

We present a numerical method to efficiently solve optimization problems governed by large-scale nonlinear systems of equations, including discretized partial differential equations, using projection-based reduced-order models accelerated with hyperreduction (empirical quadrature) and embedded in a trust-region framework that guarantees global convergence. The proposed framework constructs a hyperreduced model on-the-fly during the solution of the optimization problem, which completely avoids an offline training phase. This ensures all snapshot information is collected along the optimization trajectory, which avoids wasting samples in remote regions of the parameters space that are never visited, and inherently avoids the curse of dimensionality of sampling in a high-dimensional parameter space. At each iteration of the proposed algorithm, a reduced basis and empirical quadrature weights are constructed precisely to ensure the global convergence criteria of the trust-region method are satisfied, ensuring global convergence to a local minimum of the original (unreduced) problem. Numerical experiments are performed on two fluid shape optimization problems to verify the global convergence of the method and demonstrate its computational efficiency; speedups over 18× (accounting for all computational cost, even cost that is traditionally considered “offline” such as snapshot collection and data compression) relative to standard optimization approaches that do not leverage model reduction are shown.

Data‐based learning control for optimization of nonlinear systems

Data‐based learning control for optimization of nonlinear systems

Economic model predictive control based on lattice trajectory piecewise linear model for wastewater treatment plants

Economic model predictive control (EMPC) is an effective control strategy for wastewater treatment plants (WWTPs), which are crucial for preventing water pollution and improving the water quality. However, industrial processes typically involve large-scale nonlinear and strongly coupled systems, and it is computationally expensive to solve nonlinear optimization problems based on nonlinear prediction models such as EMPC. In this study, to facilitate the application of EMPC to large-scale nonlinear systems such as WWTPs, the lattice trajectory piecewise linear (PWL) model is used to approximate the nonlinear system with a predefined error bound. The resulting optimization problem can be expressed as a continuous PWL programming problem if the cost function of the EMPC is linear. Therefore, an iterative descent algorithm is proposed to transform the PWL programming problem into a series of linear programming problems. The stability of the nonlinear system operating under EMPC is analyzed. The EMPC scheme based on the lattice trajectory PWL model is applied to a WWTP benchmark problem, in which the state is 78-dimensional and the EMPC cost is linear. The proposed strategy outperforms the EMPC schemes based on the nonlinear prediction model and trajectory PWL prediction model. Overall, the EMPC scheme based on the lattice trajectory PWL prediction model can improve the computational efficiency of optimization problems while ensuring control performance.

Online neural network-based model reduction and switching fuzzy control of a nonlinear large-scale fractional-order system

Online neural network-based model reduction and switching fuzzy control of a nonlinear large-scale fractional-order system