Linear 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.

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

Connected systems with switching between three linear discrete-time subsystems are considered, and a new frequency-domain criterion for the existence of a quadratic Lyapunov function ensuring the stability of such systems under arbitrary switching is proposed. The application of this criterion is demonstrated on an example of a third-order system.

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

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.

Tomographic entanglement indicators in a coupled oscillator model

We study entanglement in a simple model comprising two coupled linear harmonic oscillators of the same natural frequency. The system is separable in the center of mass (COM) and relative coordinates into two oscillators of frequency ω c and ω r. We compute standard entanglement measures (subsystem linear entropy and subsystem von Neumann entropy) as well as several tomographic entanglement indicators (Bhattacharyya distance, Kullback–Leibler divergence, and inverse participation ratio) as functions of the frequency ratio η = ω c/ω r, keeping the COM oscillator in its ground state. We demonstrate that, overall, the entanglement indicators reflect quite faithfully the variations in the standard measures. The entanglement is shown to be minimum at η = 1 and maximum at η → 0 or ∞.

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.

Topological equivalence of nonhyperbolic nonautonomous systems

Backes, Dragičević and Palmer proved in [1, J. Differential Equations 2021] a C0 linearization result for nonautonomous coupled systems, whose linear part is nonhyperbolic but the second subsystem in their equations remains unchanged. In this paper, we establish a topological equivalence sending the solutions of the semilinear evolution equationx′=A(t)x+f(t,x,y),y′=r(t,x,y), onto those of its nonhyperbolic linearized partx′=A(t)x,y′=0. We require that one of the linear subsystem x′=A(t)x admits a nonuniform strong exponential dichotomy, but the other is a critical case, i.e., its linear part is a null operator. Then the difficulty arises: how to implement the linearization of the second subsystem? We will overcome the difficulty by introducing an implicit function, which enable us to construct equivalent functions to achieve the topological conjugacy between nonhyperbolic systems.

Research on the coordinated control of oxygen excess ratio and air pressure for PEMFC's air supply system

The proton exchange membrane fuel cell (PEMFC) air supply system is a complex nonlinear system with strong coupling of air flow and pressure. To avoid oxygen starvation and unstable output performance during rapid load changes, the oxygen excess ratio (OER) and air pressure need to be controlled effectively. To cope with the nonlinearity, coupling and uncertainty of the PEMFC air system, this paper puts forward a coordinated controller regulating OER and air pressure based on the input-output linearization and sliding mode methods. Firstly, a control-oriented third-order model is proposed to describe the dynamic behavior of the air supply system. Secondly, this paper decouples the flow and pressure of the PEMFC air system by using an input-output linearization method to obtain two independent linear subsystems. Moreover, a sliding mode method, effectively coping with the uncertainty of the system model, is designed to regulate the OER and air pressure of the PEMFC, which enhances the robustness of the control system. The overall simulation outcomes show that the proposed controller regulates OER and cathode pressure with smaller root mean square error (RMSE) and mean absolute percentage error (MAPE) compared with the combined proportional integral (PI) and feedforward controller.

Experimental nonlinear model of a set of connecting elements in view of nonlinear modal coupling

The development process of mechanical systems involves the evaluation of its modes of vibrations in the frequency range of interest. In general, a linear modal analysis is sufficient to determine whether the system can operate in dynamic conditions. However, in some cases the assembly is composed of many subsystems connected through nonlinear connections which make the response depend on the amplitude and frequency of the excitation. In those cases, Linear Normal Modes (LNMs) are not sufficient to fully describe the dynamics of the system and Nonlinear Normal Modes (NNMs) must be used. Using substructuring techniques it is possible to treat nonlinear joints as independent subsystems. However, a reliable nonlinear model is needed to use this approach. The experimental characterization is the only way to correctly estimate the nonlinear behavior of the connection. Thus, the aim of this work is to develop and validate the experimental characterization to build an experimental nonlinear modal model of a strongly nonlinear element that can be used to connect different linear subsystems and can be regarded as a localized source of nonlinearity. The NNMs identification is performed using the single-point single-harmonic phase resonance method, and the experimental nonlinear modal model of the NLCE is obtained by retaining only the first harmonic term of each NNM in order to get nearly uncoupled modal equations.

A new Takagi–Sugeno–Kang model for time series forecasting

A fuzzy inference system consists of a machine learning concept that combines accuracy and interpretability. They are divided into two main groups: Mamdani and Takagi–Sugeno-Kang. While Mamdani models favor interpretability, Takagi–Sugeno-Kang models provide more accurate results because of their ability to approximate a nonlinear system through a collection of linear subsystems. The evolving Takagi Sugeno model inspired a new class of Takagi–Sugeno-Kang models classified as an evolving fuzzy system, which can update their structure and functionality to adapt themselves to changes in the data. However, these models may provide several rules that reduce the interpretability. Furthermore, as they usually present many hyper-parameters, it can be challenging to obtain results that satisfy specific requirements in terms of the accuracy-interpretability trade-off. To overcome such shortcomings, this paper introduces a new model in which the only hyper-parameter is the maximum number of rules. Consequently, the user can define the number of rules considering the accuracy-interpretability trade-off. The introduced models are evaluated using benchmark time series, three well-known financial series, S&P 500, NASDAQ, and TAIEX, and renewable energy datasets. The results are compared with other state-of-the-art machine learning models, such as classical models and some rule-based evolving Fuzzy Systems. The results are evaluated regarding error metrics and the number of final rules. The proposed model obtained similar or equal performance in the simulations to the compared models with increased interpretability. The code of the proposed model is available at https://github.com/kaikerochaalves/NTSK.git.

Extended State Observer Based Robust Feedback Linearization Control Applied to an Industrial CSTR

In the chemical and petrochemical industries, the Continuous Stirred Tank Reactor (CSTR) is, without doubt, one of the most popular processes. From a control point of view, the mathematical model describing the temporal evolution of the CSTR has a strongly nonlinear cross-coupled character. Moreover, modeling errors such as external disturbances, neglected dynamics, and parameter variations or uncertainties make its control task a very difficult challenge. This problem has been the subject of a wide number of control strategies. This article attempts to propose a viable, robust nonlinear decoupling control scheme. The idea behind the proposed approach lies in the design of two nested control loops. The inner loop is responsible for the compensation of the nominal model's nonlinear cross-coupled terms via a static nonlinear feedback; while the outer loop, designed around an Extended State Observer (ESO), which the additional state gathers the global effect of modeling errors, is charged with instantaneously estimating and then compensating the ESO extended state. This way, the CSTR complex dynamics are reduced to a series of decoupled linear subsystems easily controllable using a simple Proportional-Integral (PI) linear control to ensure the robust pursuit of reference signals respecting the desired performance. The presented control validation was performed numerically by an objective comparison to a classical PID controller.

Decentralised controller design for robust stabilisation and L 2 gain of nonlinear interconnected systems

In this paper, a decentralised controller is extended for robust stabilisation and L 2 gain analysis of a class of nonlinear interconnected systems via dynamic output feedback control within the LMI framework. This large-scale system is composed of linear subsystems coupled by nonlinear time-varying interconnections satisfying quadratic constraints. An LMI approach is sufficiently proposed as a feasibility problem of a BMI with respect to controller gains via dynamic output feedback structure. An example of inverted pendulums is used to illustrate the applicability of extended approach.

Online Guaranteed Reachable Set Approximation for Systems With Changed Dynamics and Control Authority

This work presents a method of efficiently computing inner and outer approximations of forward reachable sets for nonlinear control systems with changed dynamics and control authority, given an a priori computed reachable set for the nominal system. The method functions by shrinking or inflating a precomputed reachable set based on prior knowledge of the system's trajectory deviation growth dynamics, depending on whether an inner approximation or outer approximation is desired. These dynamics determine an upper bound on the minimal deviation between two trajectories emanating from the same point that are generated by control inputs from the nominal and diminished set of control inputs, respectively. The dynamics depend on the given Hausdorff distance bound between the nominal set of admissible controls and the possibly unknown impaired space of admissible controls. Because of its computational efficiency compared to direct computation of the off-nominal reachable set, this procedure can be applied to on-board fault-tolerant path planning and failure recovery. In addition, the proposed algorithm does not require convexity of the reachable sets unlike our previous work, thereby making it suitable for general use. We raise a number of implementational considerations for our algorithm, and we present three illustrative examples, namely an application to the heading dynamics of a ship, a lower triangular dynamical system, and a system of coupled linear subsystems.

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).

Disturbance Rejection Event-Triggered Robust Model Predictive Control for Tracking of Constrained Uncertain Robotic Manipulators.

A novel hierarchical control framework combining computed-torque-like control (CTLC) with disturbance-observer-based event-triggered robust model predictive control (DO-ET-RMPC) is proposed for the trajectory tracking control of robotic manipulators with bounded disturbances and state and control input constraints. The CTLC approach is first used to cancel the exact nonlinear dynamics of the original tracking error system to obtain a set of decoupling linear tracking error subsystems, thus reducing the optimization complexity of model predictive control (MPC). The composite DO-ET-RMPC scheme is then developed based on the so-called dual-mode MPC approach to robustly stabilize the tracking error subsystems, which could improve the robustness of MPC and save its computational resources simultaneously. The continuous-time theoretical properties of the DO-ET-RMPC scheme, considering disturbances and state and control input constraints simultaneously, are provided for the first time, including the avoidance of Zeno behavior, robust constraint satisfaction, recursive feasibility, and stability. In the end, the superiorities of the proposed control scheme are verified by the comparative simulations.

On Domination and 2-packing Numbers in Intersecting Linear Systems

A linear system is a pair \((P,\mathcal{L})\) where \(\mathcal{L}\) is a finite family of subsets on a finite ground set \(P\) such that any two subsets of \(\mathcal{L}\) share at most one element. Furthermore, if for every two subsets of \(\mathcal{L}\) share exactly one element, the linear system is called intersecting. A linear system \((P,\mathcal{L})\) has rank \(r\) if the maximum size of any element of \(\mathcal{L}\) is \(r\). By \(\gamma(P,\mathcal{L})\) and \(\nu_2(P,\mathcal{L})\) we denote the size of the minimum dominating set and the maximum 2-packing of a linear system \((P,\mathcal{L})\), respectively. It is known that any intersecting linear system \((P,\mathcal{L})\) of rank \(r\) is such that \(\gamma(P,\mathcal{L})\leq r-1\). Li et al. in [S. Li, L. Kang, E. Shan and Y. Dong, The finite projective plane and the 5-Uniform linear intersecting hypergraphs with domination number four, Graphs and 34 Combinatorics (2018) , no.~5, 931–945.] proved that every intersecting linear system of rank 5 satisfying \(\gamma(P,\mathcal{L})=4\) can be constructed from a 4-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order 3 satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=4\), where \(\tau(P^\prime,\mathcal{L}^\prime)\) is the transversal number of \((P^\prime,\mathcal{L}^\prime)\). In this paper, we give an alternative proof of this result given by Li et al., giving a complete characterization of these 4-uniform intersecting linear subsystems. Moreover, we prove a general case, that is, we prove if $q$ is an odd prime power and \((P,\mathcal{L})\) is an intersecting linear system of rank \((q+2)\) satisfying \(\gamma(P,\mathcal{L})=q+1\), then this linear system can be constructed from a spanning \((q+1)\)-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order \(q\) satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=q+1\).

Data‐driven parallel Koopman subsystem modeling and distributed moving horizon state estimation for large‐scale nonlinear processes

AbstractIn this article, we consider a state estimation problem for large‐scale nonlinear processes in the absence of first‐principles process models. By exploiting process operation data, both process modeling and state estimation design are addressed within a distributed framework. By leveraging the Koopman operator concept, a parallel subsystem modeling approach is proposed to establish interactive linear subsystem process models in higher‐dimensional subspaces, each of which correlates with the original nonlinear subspace of the corresponding process subsystem via a nonlinear mapping. The data‐driven linear subsystem models can be used to collaboratively characterize and predict the dynamical behaviors of the entire nonlinear process. Based on the established subsystem models, local state estimators that can explicitly handle process operation constraints are designed using moving horizon estimation. The local estimators are integrated via information exchange to form a distributed estimation scheme, which provides estimates of the unmeasured/unmeasurable state variables of the original nonlinear process in a linear manner. The proposed framework is applied to a chemical process and an agro‐hydrological process to illustrate its effectiveness and applicability. Good open‐loop predictability of the linear subsystem models is confirmed, and accurate estimates of the process states are obtained without requiring a first‐principles process model.

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.

NONLINEAR STABILITY CONTROL OF INVERTED PENDULUM ON A CART USING LQR-BASED T-S FUZZY CONTROL

The inverted pendulum on a cart is a challenging and widely studied control problem in the fields of control systems and robotics. To address the need for more effective control strategies, this paper presents a novel approach that combines the strengths of two powerful control techniques: Linear-Quadratic Regulator (LQR) and Takagi-Sugeno (T-S) fuzzy control. LQR control is well-established for stabilizing linear systems, but it faces limitations with nonlinear systems like the inverted pendulum. On the other hand, T-S fuzzy control excels in handling nonlinearities by approximating the system's behavior with local linear models. Our proposed approach leverages T-S fuzzy systems to approximate complex nonlinearities, while using LQR control for each local linear subsystem. The combined method's efficacy is substantiated by simulation results, specifically considering criteria such as stability under disturbance conditions, variations in the initial angle of the pendulum, and a comparative analysis between stability-only scenarios and those involving both swing-up and stability control.

Finite-time stabilization of discrete-time conewise linear systems

The finite-time stabilizing control design problem for discrete-time conewise linear systems is tackled in this paper. Such a class of systems consists of the union of ordinary linear time-invariant subsystems, whose dynamics are defined in prescribed conical regions, constituting a conical partition of the state space. By imposing some cone-copositivity properties to a suitable piecewise quadratic function, two sufficient conditions are preliminarily derived concerning the system’s finite-time stability. By building on them, novel results are then presented for the system’s finite-time stabilization through a piecewise linear output feedback controller. Such results are based on the solution of feasibility problems involving sets of Linear Matrix Inequalities (LMIs). A numerical example illustrates the effectiveness of the proposed approach.