Nonlinear Subsystems Research Articles

A reduced-order two-grid method based on POD technique for the semilinear parabolic equation

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis.

Bidirectional pendulum-type tuned mass damper inerter for synchronously controlling the alongwind and crosswind responses of super-tall buildings

In the context of slender super-tall buildings, which have a high height-to-width ratio and low damping, significant bidirectional vibrations can occur under strong winds. Traditional TMDI device is mainly focused on controlling unidirectional wind-induced vibration responses. To address this issue, the present study introduces a bidirectional pendulum-type tuned mass damper inerter (BPTMDI) designed to synchronously mitigate alongwind and crosswind responses. The study derives nonlinear equations of motion for a multiple degrees of freedom (MDOF) structure integrated with the BPTMDI, using the Euler-Lagrange method. These equations capture the three-dimensional motion characteristics of the BPTMDI. Additionally, linear equations of motion are derived for a simplified planar model. The time-domain responses of the MDOF structure coupled with the BPTMDI are calculated by dividing the system into linear and nonlinear subsystems and applying the state space method and Runge-Kutta method, respectively. The effectiveness of the BPTMDI is evaluated through time history response analysis of a 400 m tall case-study building subjected to bidirectional wind loadings. Results show that the BPTMDI can significantly reduce both alongwind and crosswind responses, with reductions of at least 29.75 % in displacement and 48.64 % in acceleration, regardless of wind direction. Furthermore, the BPTMDI demonstrates improved control efficacy compared to BPTMD systems. Lastly, the simplified planar model of the BPTMDI shows basic similarity to the nonlinear planar-spherical model.

Parameter learning of multi‐input multi‐output Hammerstein system with measurement noises utilizing combined signals

SummaryIn this article, the parameter learning scheme for the multi‐input multi‐output (MIMO) Hammerstein nonlinear systems under measurement noises is studied, which is derived by exploiting the correlation analysis and data filtering technique. The coupled MIMO Hammerstein system presented involves a static nonlinear subsystem modeled by neural fuzzy model (NFM), and a dynamic linear subsystem established by autoregressive moving average with extra input (ARMAX) model. To learn the unknown parameter of the MIMO Hammerstein system, the combined signals are designed to realize that identification of the nonlinear subsystem is separated from that of linear subsystem. First, the correlation properties of separable signals in a nonlinear system are analyzed, then the parameters of the linear subsystem are estimated utilizing correlation analysis, which can deal with the issue of unmeasured intermediate variable in the Hammerstein system. Second, the data filtering technique is introduced to derive the data filtering‐based recursive least squares technique for learning the nonlinear subsystem parameter, which can reduce the impact of the moving average noise and improve the precision of parameter estimation. Finally, the effectiveness and feasibility of the proposed identification scheme is proved by numerical simulation and nonlinear pH process.

Analysis of bifurcation and chaotic behavior of the micro piezoelectric pipe-line robot drive system with stick - slip mechanism

Pipeline robots using the conventional driving mode have encountered a bottleneck in miniaturization. To address this problem, a micro piezoelectric pipeline robot based on the inertia stick-slip driving principle is proposed in this paper. The robot is well suited to the inspection needs of micro pipes. However, undesirable design parameters found during the structural optimization phase can lead to unstable operation and reduced load capacity of the robot, and can even cause the drive system to stop with chaotic behavior. Therefore, the nonlinear dynamics model of the four degree of freedom discrete system of the pipeline robot is established. The Runge-Kutta method is used to solve the dynamic response of each nonlinear subsystem. The nonlinear dynamical behavior of the system is analyzed through the bifurcation diagram, time domain diagram, phase diagram, Poincaré map and power spectrum diagram of each subsystem response. The stable operation intervals of the machine and electrical parameters in the drive system are given, and the conditions for the occurrence of chaotic behavior are determined. This research can be applied to most of the drive systems consisting of piezoelectric stacks and flexure hinges. It helps to determine and optimize the structural parameters of the piezoelectric actuator, thus avoiding chaotic behavior and ensuring that the actuator maintains good operational performance.

Multiple models for decentralised adaptive control of discrete‐time systems

AbstractThis paper investigates decentralised adaptive control of discrete‐time systems consisting of linear time‐invariant systems and a class of nonlinear systems. The parameters of the subsystems and the interconnection strengths between the subsystems are assumed unknown. Multiple adaptive prediction models with switching are used in this paper to address the relatively poor transient performance that typically results from a single prediction model. However, the application of this methodology requires indirect model reference discrete‐time adaptive control, and there appears to be very little focus in the literature on this. The principal contribution of this paper is to fill this void by arriving at proof of global stability of such decentralised adaptive systems using the theory of Lyapunov and properties of square‐summable sequences. The chosen representation of the system enables the same results to be directly applicable to both linear time‐invariant and linear‐in‐the‐parameters nonlinear subsystems. We consider two parametric update algorithms, one of which has a normalisation factor. Simulation studies included in this paper demonstrate the improvement in transient performance.

Robust H2 nonlinear fuzzy decentralized control design for a VSC‐HVDC link

SummaryA Robust Nonlinear Fuzzy Decentralized Controller ( RNFDC) is proposed to improve the overall robustness and tracking ability of a VSC‐HVDC link. The studied system is considered as composed by two overlapping nonlinear subsystems. The nonlinear interaction between the two subsystems is treated as a disturbance rejection problem by fuzzy control techniques. First, the Takagi Sugeno (TS) fuzzy model is adopted for fuzzy modeling of the uncertain nonlinear system. Next, new stability conditions for a generalized class of uncertain HVDC systems are derived from robust control techniques based on Linear Matrix Inequalities (LMIs). The design method employs the so‐called Parallel Distributed Compensation (PDC) to obtain RNFDC gains based on LMIs.The efficiency and robustness of the proposed controllers are analytically proven and tested through validation simulations. The main contributions of this paper are: (i) resilience: in case of failure of one converter or loss of measures or controls, the control of the other converter is not affected; (ii) robustness is improved in order to provide good responses in case of network variations and HVDC line parameters changes; (iii) fuzzy techniques are adopted to handle nonlinearities and changes of operating conditions. The proposed RNFDC is compared with Fuzzy H Decentralized State Feedback Control ( NFDC) and Linear Decentralized Control ( LDC) in simulation to illustrate the control synthesis and its effectiveness.

Adaptive decentralized fuzzy compensation control for large optical mirror processing systems

PurposeLarge optical mirror processing systems (LOMPSs) consist of multiple subrobots, and correlated disturbance terms between these robots often lead to reduced processing accuracy. This abstract introduces a novel approach, the nonlinear subsystem adaptive dispersed fuzzy compensation control (ADFCC) method, aimed at enhancing the precision of LOMPSs.Design/methodology/approachThe ADFCC model for LOMPS is developed through a nonlinear fuzzy adaptive algorithm. This model incorporates control parameters and disturbance terms (such as those arising from the external environment, friction and correlation) between subsystems to facilitate ADFCC. Error analysis is performed using the subsystem output parameters, and the resulting errors are used as feedback for compensation control.FindingsExperimental analysis is conducted, specifically under the commonly used concentric circle processing trajectory in LOMPS. This analysis validates the effectiveness of the control model in enhancing processing accuracy.Originality/valueThe ADFCC strategy is demonstrated to significantly improve the accuracy of LOMPS output, offering a promising solution to the problem of correlated disturbances. This work holds the potential to benefit a wide range of practical applications.

Identification of continuous-time Hammerstein model using improved Archimedes optimization algorithm

Although various optimization algorithms have been widely employed in multiple applications, the traditional Archimedes optimization algorithm (AOA) has presented imbalanced exploration with exploitation phases and the propensity for local optima entrapment. Therefore, this article identified various continuous-time Hammerstein models based on an improved Archimedes optimization algorithm (IAOA) to address these concerns. The proposed algorithm employed two principal modifications to mitigate these issues and enhance identification accuracy: (i) exploration and exploitation phase recalibrations using a revised density decreasing factor and (ii) local optima entrapment alleviation utilizing safe experimentation dynamics. Various advantages were observed with this proposed algorithm, including a lower number of coefficient criteria, improved accuracy in Hammerstein model identification, and diminished processing demands by reducing gain redundancy between nonlinear and linear subsystems. This proposed algorithm also discerned linear and nonlinear subsystem variables within a continuous-time Hammerstein model utilizing input and output data. The process was evaluated using a numerical example and two practical experiments [twin-rotor system (TRS) and electro-mechanical positioning system (EMPS)]. Several parameters were then analyzed, such as the convergence curve of the fitness function, frequency and time domain-related responses, variable deviation index, and Wilcoxon's rank-sum test. Consequently, the proposed algorithm reliably determined the most optimal design variables during numerical trials, demonstrating 54.74 % mean fitness function and 75.34 % variable deviation indices enchantments compared to the traditional AOA. Improved mean fitness function values were also revealed in the TRS (11.63 %) and EMPS (69.63 %) assessments, surpassing the conventional algorithm. This proposed algorithm produced solutions with superior accuracy and consistency compared to various established metaheuristic strategies, including particle swarm optimizer, grey wolf optimizer, multi-verse optimizer, AOA, and a hybrid optimizer (average multi-verse optimizer-sine-cosine algorithm).

Distributed Lebesgue Approximation Model for Distributed Continuous-Time Nonlinear Systems.

Approximation models play a crucial role in model-based methods, as they enhance both accuracy and computational efficiency. This article studies distributed and asynchronous discretized models to approach continuous-time nonlinear systems. The considered continuous-time system consists of some distributed but physically coupled nonlinear subsystems that exchange information. We propose two Lebesgue approximation models (LAMs): 1) the unconditionally triggered LAM (CT-LAM) and 2) the CT-LAM. In both approaches, a specific LAM approximates an individual subsystem. The iteration of each LAM is triggered by either itself or its neighbors. The collection of different LAMs executing asynchronously together form the approximation of the overall distributed continuous-time system. The aperiodic nature of LAMs allows for a reduction in the number of iterations in the approximation process, particularly when the system has slow dynamics. The difference between the unconditionally and CT-LAMs is that the latter checks an "importance" condition, further reducing the computational effort in individual LAMs. Furthermore, the proposed LAMs are analyzed by constructing a distributed event-triggered system which is proved to have the same state trajectories as the LAMs with linear interpolation. Through this specific event-triggered system, we derive conditions on the quantization sizes in LAMs to ensure asymptotic stability of the LAMs, boundedness of the state errors, and prevention of Zeno behavior. Finally, simulations are carried out on a quarter-car suspension system to show the advantage and efficiency of the proposed approaches.

Identification of switched dynamic system for electric multiple unit train modeling

A new iterative algorithm is proposed to identify the high-speed train model which is described as a switched dynamic system including a fast switched nonlinear static subsystem followed by a non-switched linear dynamic subsystem. The non-switched property of the linear part leads to a more complex dynamics of the whole system, compared to the switched Hammerstein system which linear part switches synchronously with its nonlinear part. The proposed iterative algorithm alternates between identifying the switched nonlinear static subsystem and the linear dynamic subsystem. The switched nonlinear subsystem identification incorporates a quadratic error bound constraint with respect to nominal values of model parameters, and is formulated as a second-order cone program. Properties of convergence and asymptotic unbiasedness of the obtained estimates are proved under some necessary assumptions. With real data and simulation examples, the proposed iterative algorithm produces convergent and asymptotically unbiased estimates. In the simulation results, incorporating the quadratic error bounds results in a fast convergence rate in the presence of a small noise variance, and also produces a smaller identification error in the presence of colored noises with a non-zero mean and a large variance.

Sensitivity of Piezoelectric Stack Actuators

This paper investigates the properties of a mass−attached piezoelectric stack actuator and analyzes its sensitivity, which is defined as the spectrum of the driving force (the output) caused by a single−frequency voltage (the input). The force spectrum is utilized because of the nonlinear hysteresis effect of the piezoelectric stack. The sensitivity analysis shows that the nonlinear dynamics of the actuator can be interpreted as a cascade of two subsystems: a nonlinear hysteresis subsystem and a linear mechanical subsystem. Analytical solutions of the nonlinear differential equations are proposed, which show that the nonlinear transformation can be described by a steady−state mapping of a single−frequency voltage input to a multiple−frequency driving force at the driving frequency and its odd harmonics. The steady−state sensitivity is then determined by the response of the mechanical subsystem to the line spectrum of the driving force. The maximum sensitivity can be achieved by setting the frequency of the input voltage close to the natural frequency of the mechanical subsystem. The analytical model is also validated by a numerical model and experimental results and it may be used for the analysis and design of piezoelectric actuators with different structural configurations.

High-Bandwidth Repetitive Trajectory Tracking Control of Piezoelectric Actuators via Phase-Hysteresis Hybrid Compensation and Feedforward-Feedback Combined Control.

Piezoelectric actuators (PEAs) are widely used in many nano-resolution manipulations. A PEA's hysteresis becomes the main factor limiting its motion accuracy. The distinctive feature of a PEA's hysteresis is the interdependence between the width of the hysteresis loop and the frequency or rate of the control voltage. Generally, the control voltage is first amplified using a voltage amplifier (VA) and then exerted on the PEA. In this VA-PEA module, the linear dynamics of the VA and the nonlinearities of the PEA are coupled. In this paper, it is found that the phase lag of the VA also contributes to the rate dependence of the VA-PEA module. If only the PEA's hysteresis is considered, it will be difficult to achieve high-frequency modeling and control. Consequently, great difficulties arise in high-frequency hysteresis compensation and trajectory tracking, e.g., in the fast scanning of atomic force microscopes. In this paper, the VA-PEA module is modeled to be the series connection of a linear subsystem and a nonlinear subsystem. Subsequently, a feedforward phase-dynamics compensator is proposed to compensate for both the PEA's hysteresis and the phase lag of the VA. Further, an unscented Kalman-filter-based proportional-integral-derivative controller is adopted as the feedback controller. Under this feedforward-feedback combined control scheme, high-bandwidth hysteresis compensation and trajectory tracking are achieved. The trajectory tracking results show that the closed-loop trajectory tracking bandwidth has been increased to the range of 0-1500 Hz, exhibiting excellent performance for fast scanning applications.

Machine learning interfaces for modular modelling and operation-based design optimization of solar thermal systems in process industry

The trend of introducing solar thermal systems (STSs) in process industries has resulted in a new energy paradigm– an interactive platform where there are economic benefits and motivations to address sustainable development. On the other hand, this paradigm has also introduced fluctuations and uncertainties not previously seen on the energy system, and is challenging the industry. Industrial STSs are composed of large number of interacting, nonlinear and uncertain subsystems and, therefore are complex systems. This complexity of heterogenous, stochastic, and dynamic behavior make designing of industrial STSs as well as evaluation of their management strategies challenging tasks for the existing tools. In an attempt to support integration decisions, always inefficient and insufficient, machine learning (ML) is leveraged in this research. Following the ML approach, some important findings were made. Firstly, it permits for different releases of the same/different module, which can be easily embedded, coupled exogenously or run in parallel. Thus, the operation-based design approach, both for process and supply level integration of STSs, were scalable and tractable. Distinguished from past work, the approach also allowed testing of an improved STS sizing scheme, while at the same time, enabling optimization of its control concept for adaptive and flexible operation. In the considered case study, inclusion of time-varying heat demand, process temperature and solar irradiation in STS design reduces over-dimensioning by 10–30%. Furthermore, optimizing the control concept of the overall system led to improved energy efficiency of solar field and storage respectively by 23.7% and 8.3% over the state-of-the-art methods.

SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations

Complex nonlinear partial differential equations (PDEs) can be decomposed into subsystems comprising certain linear and nonlinear sub-operators. We usually have fundamental solutions to the linear subsystems and have a good knowledge of their uniqueness and regularity. The nonlinear subsystems have more straightforward formulas than the original complex nonlinear PDEs, even with reduced dimensions. Solutions of the complex nonlinear PDEs based on simpler subsystems are investigated in this study. We derive a proper subsystem decomposition of the nonlinear PDEs into linear and nonlinear subsystems to get unique solutions. We propose the sub-operator learning enhanced neural networks (SONets) based on this proper subsystem decomposition to solve nonlinear PDEs. The solutions of nonlinear subsystems with generated force terms are modeled via operator learning. In contrast, only the cumbersome integration and series parts of fundamental solutions of linear subsystems are modeled. The complex nonlinear PDEs are then solved by finding the force terms to minimize the physics-informed loss functions. It is also convenient to generalize SONets for a new PDE parameter, e.g., the diffusion coefficient. Next, we extend the SONets to a meta version (meta-SONets) for parametric PDEs by transfer learning. In meta-SONets, the initial conditions are explicitly encoded via operator learning neural networks (ENNs) to output force terms, followed by the whole SONets to solve PDEs. In the pre-training stage of meta-SONets, parametric PDEs given randomly generated initial conditions are solved by training ENNs to minimize the physics-informed loss functions. At the fine-tuning stage, the trainable parameters of ENNs are fine-tuned to solve the PDE given a new initial condition. Numerical experiments are conducted to solve the Burgers equation, the nonlinear diffusion-reaction system, and the Allen-Cahn equation. The numerical results indicate that SONets with neural network-parameterized force terms obtain solutions with good accuracy. With transfer learning, the computational cost of meta-SONets for solving the nonlinear PDEs with a new initial condition is dramatically reduced.

Parameter estimation of Wiener-Hammerstein system based on multi-population self-adaptive differential evolution algorithm

PurposeThe Wiener-Hammerstein nonlinear system is made up of two dynamic linear subsystems in series with a static nonlinear subsystem, and it is widely used in electrical, mechanical, aerospace and other fields. This paper considers the parameter estimation of the Wiener-Hammerstein output error moving average (OEMA) system.Design/methodology/approachThe idea of multi-population and parameter self-adaptive identification is introduced, and a multi-population self-adaptive differential evolution (MPSADE) algorithm is proposed. In order to confirm the feasibility of the above method, the differential evolution (DE), the self-adaptive differential evolution (SADE), the MPSADE and the gradient iterative (GI) algorithms are derived to identify the Wiener-Hammerstein OEMA system, respectively.FindingsFrom the simulation results, the authors find that the estimation errors under the four algorithms stabilize after 120, 30, 20 and 300 iterations, respectively, and the estimation errors of the four algorithms converge to 5.0%, 3.6%, 2.7% and 7.3%, which show that all four algorithms can identify the Wiener-Hammerstein OEMA system.Originality/valueCompared with DE, SADE and GI algorithm, the MPSADE algorithm not only has higher parameter estimation accuracy but also has a faster convergence speed. Finally, the input–output relationship of laser welding system is described and identified by the MPSADE algorithm. The simulation results show that the MPSADE algorithm can effectively identify parameters of the laser welding system.

Partition‐based distributed extended Kalman filter for large‐scale nonlinear processes with application to chemical and wastewater treatment processes

AbstractIn this article, we address a partition‐based distributed state estimation problem for large‐scale general nonlinear processes by proposing a Kalman‐based approach. First, we formulate a linear full‐information estimation design within a distributed framework as the basis for developing our approach. Second, the analytical solution to the local optimization problems associated with the formulated distributed full‐information design is established, in the form of a recursive distributed Kalman filter algorithm. Then, the linear distributed Kalman filter is extended to the nonlinear context by incorporating successive linearization of nonlinear subsystem models, and the proposed distributed extended Kalman filter approach is formulated. We conduct rigorous analysis and prove the stability of the estimation error dynamics provided by the proposed method for general nonlinear processes consisting of interconnected subsystems. A chemical process example is used to illustrate the effectiveness of the proposed method and to justify the validity of the theoretical findings. In addition, the proposed method is applied to a wastewater treatment process for estimating the full‐state of the process with 145 state variables.

Symmetric actor–critic deep reinforcement learning for cascade quadrotor flight control

Even though deep reinforcement learning (DRL) has been extensively applied to quadrotor flight control to simplify parameter adjustment, it has some drawbacks in terms of control performance, such as instability and asymmetry. To address these problems, we propose an odd symmetric actor to achieve stable and symmetric control performance, and an even critic to stabilize the training process. Concretely, the bias of neural networks is eliminated, and the absolute value operation is adopted to construct the activation function. Furthermore, we devise a cascade architecture, where each module trained with DRL controls a symmetric subsystem of the quadrotor. Comparative simulations have verified the effectiveness of the proposed control scheme, which shows superiority in dealing with high-dimensional, nonlinear subsystems and disadvantage in dealing with low-dimensional, linear subsystems.

Positivity and decentralized [formula omitted] control of nonlinear interconnected switched positive systems under MDDT constraint

The article is focused on the problems of positivity and decentralized L1 control for nonlinear interconnected switched positive systems(NISPSs) under mode-dependent dwell time(MDDT) constraint, which includes mode-dependent constant dwell time(MDCDT), mode-dependent minimum dwell time(MDMDT) and mode-dependent ranged dwell time(MDRDT) constraints. Both nonlinear subsystems and nonlinear interconnections are included in the considered NISPSs. First, a sufficient and necessary condition of positivity is presented for NISPSs. Next, by applying a novel discretized linear copositive Lyapunov function method, a sufficient condition of exponential stability with L1-gain performance is provided for NISPSs under MDCDT, MDMDT and MDRDT constraints, respectively. Further, by applying the matrix decomposition technology to the decentralized controller gain matrix, an effective mode-dependent piece-wise design scheme of decentralized L1 controller in the solvable linear programming(LP) form is proposed for NISPSs under MDCDT, MDMDT and MDRDT constraints, respectively. The proposed decentralized L1 controller design scheme for NISPSs under MDDT constraint can be degenerated into the one in mode-independent constraint case accordingly. At last, four examples with comparisons are given to illustrate the importance and effectiveness of the obtained results.

Robust Approach for Global Stabilization of a Class of Underactuated Mechanical Systems in Presence of Uncertainties

Underactuated mechanical systems offer complex dynamic behavior and poses control challenges, especially in the presence of uncertainties in the system. To cope with such systems, control mechanisms are required, which needs to be robust. In this research, an algorithm based on sliding mode control (SMC) is presented. The algorithm design offer a methodical way to handle underactuation, while the robustness properties of SMC suppresses the effect of norm bounded uncertainties and external disturbances. To proceed with the design, an underactuated system is converted into cascaded subsystems, a linear one and reduced-order nonlinear subsystem. The proposed SMC design is backed by rigorous mathematical presentation and based on Lyapunov theory, so that the global stabilization of overall system is ensured. Numerical simulations are performed, on the laboratory test bench underactuated systems (Inertial Wheel, Furuta Pendulum, TORA, and an Overhead Crane), to validate the efficacy of the proposed design. In addition, a novel sliding surface is presented for Inertial Wheel and Furuta Pendulum to achieve swingup control and global stabilization subjected to uncertainties.

A Multiobjective Feedback Linearization Control of a Cuk Converter

Cuk converter is one of the most common power electronic converters, but its controller design has always been a challenging problem due to its fourth-order nonlinear and nonminimum phase characteristics. In order to solve this problem, this paper proposes a multi-objective feedback linearization control method, which selecting the variables of interest to rebuild the output function to linearize the nonlinear system into a combination of a linear subsystem and a nonlinear subsystem. According to the structural characteristics of these two subsystems, the linear optimal control theory is adopted for the control design of the linear subsystem to make the original nonlinear system have a good output performance, while for the nonlinear subsystem, the coefficient of the output function is adjusted to ensure the stability of the original nonlinear system. The results of simulation and experiment show the feasibility and superiority of using the multi- objective feedback linearization control method to design the nonlinear control law of the Cuk converter.