Distributed Parameter Systems Research Articles

Static boundary feedback stabilization of an anti-stable wave equation with both collocated and non-collocated measurements

In this paper, we consider the stabilization of a wave equation with an unknown anti-stable injection on the left boundary and the control input on the right boundary, where there are both collocated and non-collocated measurements. A static output feedback control law is designed to stabilize the wave equation. The value ranges of the feedback gains are given, such that all eigenvalues of the closed-loop system are shown to be inside the left-half complex plane by applying the Nyquist criterion for distributed parameter systems. Then the exponential stability of the closed-loop system is established. Numerical simulations are presented to verify the effectiveness of the proposed feedback control law.

Stabilization control of a flexible marine riser with failed and bounded actuator and time‐varying boundary constraints

AbstractIn this article, we investigate the stabilization control problem of a flexible marine riser system with a failed and bounded control action and time‐varying boundary constraints. The considered system is modeled as a distributed parameter system dominated by a partial differential equation. The involved problems are extended to some more general conditions. The saturation and failures of the actuator make the input and output of the actuator a nonlinear relationship, which leads to the normally designed control input invalid. By applying the Nussbaum gain technique, we report a synthetical control method to tackle the input nonlinearity. A piecewise barrier Lyapunov function (BLF) is introduced to reduce the boundary displacement in an asymmetric open set. Compared with the traditional BLFs, the initial boundary displacement is not forced to retain within the given region by integrating a shifting function into the BLF design. In addition, the adaptive method is combined with the hyperbolic tangent function to provide a smooth control solution to the unknown parameters from nonlinear input and external perturbation. Relying on a Lyapunov‐based analysis and a numerical simulation, it is proven that the actuated riser system is uniformly bounded and the boundary displacement always keeps in the limited set even subject to actuator failures and limitations.

Lyapunov based on-line model reduction and control of semilinear dissipative distributed parameter systems with minimum feedback information

We focus on the Lyapunov-based output feedback control problem for a class of distributed parameter systems with spatiotemporal dynamics described by input-affine linear and semilinear dissipative partial differential equations (DPDEs). The control problem is addressed via model order reduction. Galerkin projection is applied to discretize the DPDE and derive low-dimensional reduced order models (ROMs). The empirical basis functions needed for this discretization are recursively computed using adaptive proper orthogonal decomposition (APOD). To update the basis functions during process operation, APOD needs measurements of the system state’s complete profile (called snapshots) at revision times. This paper’s main objective is to minimize the demand for snapshots from the spatially distributed sensors by the control structure while maintaining closed-loop stability and performance. A control Lyapunov function is defined, and its value is monitored as the system evolves. Only when the value violates a closed-loop stability threshold, snapshots are requested for a brief period by APOD after which the ROM is updated, and the controller is reconfigured. The proposed approach is applied to stabilize the Kuramoto–Sivashinsky equation.

Extended Kalman Filter Design for Semilinear Distributed Parameter Systems with Application to Anaerobic Digestion

This paper is concerned with the online state estimation of a class of one-dimensional semilinear partial differential equation (PDE) systems considering piecewise measurements over the spatial domain. In the context of infinite-dimensional linear systems, it is well-known that the Kalman filter minimizes the mean square estimation error. For semilinear infinite-dimensional systems, the extended Kalman filter (EKF) is a widely used extension relying on successive linearizations of the estimation error dynamics. In this paper, we propose a computationally tractable implementation of the EKF using a sample-and-hold approach for which the optimal output injection operator associated to the proposed estimator is computed at each sampling time via the approximate solution of the infinite-dimensional Riccati equation. The performance of the observer is exemplified through numerical experiments which demonstrate the efficiency of the proposed approach.

Modeling and iterative learning control of spatially distributed parameter systems with sensing and actuation over a selected area of the domain

This paper gives new contributions to the development of iterative learning control for distributed parameter systems, based on using finite difference schemes to construct a finite-dimensional approximate model of the dynamics for control law design. To form a basis for the new results, systems whose dynamics are described by a fourth-order partial differential equation are considered together with the associated accuracy and numerical stability checks. Some previous control law designs use only a spatial variable as the control input, which can be a serious obstacle to practical implementation since many actuators and sensors must be deployed. This paper’s new design is based on spatially homogeneous sensing and excitation over a selected sub-area of the domain considered. Supporting numerical case studies are given to support the analysis.

Diffusivity Control of Heat Transfer Process Using Optimality Conditions

In this paper, a distributed parameter system expressed as a parabolic partial differential equation governed by a diffusivity control is considered. A modal space expansion approach is used to convert the distributed parameter system into a lumped parameter system. Thereafter, Pontryagin’s maximum principle is used to compute the optimal control function that leads to a nonlinear two-point boundary value problem (TPBVP). An iterative numerical technique, variation of extremals is used to solve the nonlinear TPBVP. The feasibility and applicability of the proposed solution is demonstrated by numerical simulations generated in MATLAB.

Modified High-Order SVD for Spatiotemporal Modeling of Distributed Parameter Systems

Modeling high-spatial dimensional (high-D) distributed parameter systems (DPSs) is very difficult because of the spatially distributed characteristic and complex spatiotemporal coupling. In this article, a new framework based on high-order singular vector decomposition (HOSVD) is proposed to model a high-D DPS. A modified HOSVD is designed for separation of time and multiple spatial variables. It can well preserve the spatially distributed nature of the high-D DPS. It also considers the interaction across different spatial modes, which is an important part of spatiotemporal coupling in the high-D DPS. Experiments of a two-spatial dimensional thermal curing process are used to verify the effectiveness of the proposed method. An average prediction error near 1% of the curing temperature can be achieved.

Anti-collision and Obstacle Avoidance of Mobile Sensor-plus-actuator Networks over Distributed Parameter Systems with Time-varying Delay

This paper addresses the anti-collision problem among mobile sensors-actuators, and settles the obstacle avoidance trouble between dynamic sensors-actuators and erratic obstacles in mobile sensor-actuator networks, which are based on a class of distributed parameter systems with time-varying delay. Initially, the radar obstacle avoidance technology is evolved into an obstacle avoidance function, combined with the anti-collision function. Subsequently, the static output feedback controller of distributed parameter systems is established. Then, by using the abstract development equation theory, operator semigroup approach and Lyapunov stability arguments, the stability analysis of the distributed parameter systems with time-varying delay is carried out. Moreover, an iterative and continuous control force on account of Newton’s second law is constructed, which makes anti-collision and obstacle avoidance control of mobile sensors-actuators be realized, and accelerates the state of this delayed system to be stable. Finally, numerical simulation results indicate that the proposed control strategy is effective.

Dissimilarity Analysis-Based Multimode Modeling for Complex Distributed Parameter Systems

For complex distributed parameter systems (DPSs) with strong nonlinearities and time-varying dynamics, the conventional spatiotemporal modeling methods become ill-suited since the elementary assumption that the process data follow a unimodal Gaussian distribution usually becomes invalid. In this paper, a multimode method is proposed for modeling of such systems. First, the original operating space is partitioned along the time dimension into several subspaces via modified dissimilarity analysis. Each subspace represents the local spatiotemporal characteristics of the original system. Second, the Karhunen–Loeve decomposition (KLD)-based spatiotemporal modeling approach is applied to approximate the local dynamics of each subspace. Finally, an ensemble model is obtained using the soft weighting sum of the local ones, where the corresponding weights are calculated by principal component regression. By properly decomposing the original space into several local parts, the ensemble model is capable of handling the strong nonlinearities and time-varying dynamics of the system. The validity and efficiency of the proposed method are verified on two representative applications: 1) a one-dimensional parabolic catalytic rod and 2) a two-dimensional curing thermal process. The experimental results show that the proposed method provides a superior performance regarding modeling accuracy compared to several baselines.

Modelling and boundary optimal control design of hybrid column flotation

AbstractA three‐phase continuous hybrid flotation column that seeks to obtain the benefits of both mechanical cells and flotation columns is modelled as the interconnection of a CSTR representing the well‐mixed zone and two plug‐flow reactors (PFR) representing pulp and froth zones. The plant model accounts for the micro‐scale processes such as bubble‐particle collision and attachment and the appearance and breakage of bubbles. This complex distributed parameter system (DPS) is described by sets of nonlinear coupled conservation counter‐current hyperbolic partial differential equations (PDEs) and one set of ordinary differential equations (ODEs). The dynamic conservation law‐based model for the continuous hybrid flotation column including well‐stirred, pulp (bubbly), and froth zones, is utilized in an optimal model‐based controller design. This linear quadratic regulator (LQR)‐based controller accounts for optimality, stability, and performance. The controller design utilizes a linear model obtained by linearization at operating steady states of interest. A full‐state optimal feedback control law is designed and controller performance has been demonstrated through a numerical simulation of physically meaningful and relevant plant operating conditions. The LQR‐based optimal controller outperforms proportional‐integral (PI)‐based control by more than an order of magnitude in terms of a return to steady state after a perturbation in the initial condition.

Qualitative Analysis of the Dynamic for the Nonlinear Korteweg–de Vries Equation with a Boundary Memory

This paper addresses the impact of the presence of a boundary memory term in the third-order Korteweg–de Vries equation in a bounded interval $$[0,\ell ]$$ . First, an overall literature review is provided. Indeed, a comprehensive discussion on the literature constitutes a survey part of the current paper. Thereafter, it is shown that the system under consideration possesses a unique solution under a smallness assumption on the initial data and an appropriate condition on the parameters and the kernel involved in the memory term. Last but not least, we demonstrate that the zero solution is exponentially stable as long as the length $$\ell $$ is small enough by means of Lyapunov method, which permits to provide an estimate of the exponential decay rate. These findings improve and complement those of Zhang (in: Desch W, Kappel F, Kunisch K (eds) Proceedings of international conference on control and estimation of distributed parameter systems: nonlinear phenomena, International Series of Numerical Mathematics, vol 118, Birkhauser, Basel, pp 371–389, 1994) (resp. Baudouin et al. in IEEE Trans Autom Control 64:1403–1414, 2019), where no memory term is present (resp. a delay occurs instead of memory).

Robust H ∞ Feedback Compensator Design for Linear Parabolic DPSs with Pointwise/Piecewise Control and Pointwise/Piecewise Measurement

In this paper, a robust H ∞ control problem of a class of linear parabolic distributed parameter systems (DPSs) with pointwise/piecewise control and pointwise/piecewise measurement has been investigated via the robust H ∞ feedback compensator design approach. A unified Lyapunov direct approach is proposed in consideration of the pointwise/piecewise control and point/piecewise measurement based on the distributions of the actuators and sensors. A new type of Luenberger observer is developed on the continuous interval of space domain to track the state of the system, and an H ∞ performance constraint with prescribed H ∞ attenuation levels is proposed in this paper. By utilizing Lyapunov technique, mathematical inequalities, and integration theory, a sufficient condition based on LMI for the exponential stability of the corresponding closed-loop coupled system under an H ∞ performance constraint is presented. Finally, the effectiveness of the proposed design method is verified by numerical simulation results.

A distributed-parameter control system using electromagnetic neural stimulation for human–machine perception interface

ABSTRACT This article presents a new design of a distributed-parameter control system for human–machine perception interface, which is capable of precisely stimulating the neural systems with electromagnetic fields. By discretising the neural systems, a state-space representation of the electromagnetic stimulation is developed to facilitate the following design and analysis. A forward controller with multiple inputs and multiple outputs is consequently designed to estimate the excitation current. This novel approach enables the applications of the well-established control theory to analyse the system. The feasibility and accuracy of the control system are numerically illustrated and validated with the applications of Transcranial Magnetic Stimulation (TMS) and retinal stimulation. The results indicate that the newly designed control system can not only generate electromagnetic stimulation with better attenuation and focality than the most widely used Figure-8 coil, but also transmit the patterns extracted from images with electromagnetic stimulations to human retinas.

Event-generator-based H∞ control of fuzzy distributed parameter systems

This paper studies the adaptive event-based H∞ control problem for a class of T-S fuzzy distributed parameter system with parameter uncertain and actuator faults. To overcome the drawback of the period control scheme, an event-triggered control scheme with adaptive threshold is proposed to minimize the number of unnecessary sampled data transmission and to reduce the update frequency of the controller. Furthermore, a T-S fuzzy controller based on the adaptive event-triggered sampled data is designed to ensure the stochastic exponential stability of the distributed parameter system with H∞ disturbance attenuation performance. Based on a new Lyapunov functional and inequality technique, the stochastic exponential stability criterion of the closed-loop system is obtained, and the controller parameters are designed. The new developed inequality can reduce the conservatism of the stability criterion. Finally, the effectiveness of the theoretical calculation results is verified by numerical simulation, and the results are compared with the relevant literature in the simulation, showing that the methods and results in this paper are less conservative.

Reconstructing a gradient state by constructing an asymptotical gradient observer on a boundary region

The objective of this paper is devoted to development an approach to analyze the asymptotical regional gradient boundary observer connected with the selection of strategic gradient sensors on a boundary area for infinite distributed parameter system in a Sobolev space. More accurately, the main target behind introducing this notion is that the possibility to design a dynamic system (may be called regional gradient boundary observer) which enable to estimate the unknown system state gradient.

State Estimation for a Class of Distributed Parameter Systems with Time-Varying Delay over Mobile Sensor–Actuator Networks with Missing Measurements

This work proposes a state estimation strategy over mobile sensor–actuator networks with missing measurements for a class of distributed parameter systems (DPSs) with time-varying delay. Initially, taking advantage of the abstract development equation theory and operator semigroup method, this kind of delayed DPSs described by partial differential equations (PDEs) is derived for evolution equations. Subsequently, the distributed state estimators including consistency component and gain component are designed; the purpose is to estimate the original state distribution of the delayed DPSs with missing measurements. Then, a delay-dependent guidance approach is presented in the form of mobile control forces by constructing an appropriate Lyapunov function candidate. Furthermore, by applying Lyapunov stability theorem, operator semigroup theory, and a stochastic analysis approach, the estimation error systems have been proved asymptotically stable in the mean square sense, which indicates the estimators can approximate the original system states effectively when this kind of DPS has time-delay and the mobile sensors occur missing measurements. Finally, the correctness of control strategy is illustrated by numerical simulation results.

Locally Weighted Principal Component Analysis-Based Multimode Modeling for Complex Distributed Parameter Systems.

Global principal component analysis (PCA) has been successfully introduced for modeling distributed parameter systems (DPSs). In spite of the merits, this method is not feasible due to parameter variations and multiple operating domains. A novel multimode spatiotemporal modeling method based on the locally weighted PCA (LW-PCA) method is developed for large-scale highly nonlinear DPSs with parameter variations, by separating the original dataset into tractable subsets. This method implements the decomposition by making full use of the dependence among subset densities. First, the spatiotemporal snapshots are divided into multiple different Gaussian components by using a finite Gaussian mixture model (FGMM). Once the components are derived, a Bayesian inference strategy is then applied to calculate the posterior probabilities of each spatiotemporal snapshot belonging to each component, which will be regarded as the local weights of the LW-PCA method. Second, LW-PCA is adopted to calculate each locally weighted snapshot matrix, and the corresponding local spatial basis functions (SBFs) can be generated by the PCA method. Third, all the local temporal models are estimated using the extreme learning machine (ELM). Thus, the local spatiotemporal models can be produced with local SBFs and corresponding temporal model. Finally, the original system can be approximated using the sum form of each local spatiotemporal model. Unlike global PCA, which uses global SBFs to construct a global spatiotemporal model, LW-PCA approximates the original system by multiple local reduced SBFs. Numerical simulations verify the effectiveness of the developed multimode spatiotemporal model.

Nonlinear model predictive control for distributed parameter systems by time–space‐coupled model reduction

AbstractNonlinear high‐dimensional distributed parameter systems (DPSs) described by sets of parabolic partial different equations (PDEs) exhibit a dominant, low‐dimensional slow behavior that can be captured using model reduction. A time–space‐coupled model reduction architecture combining encoder–decoder networks with recurrent neural networks (RNNs) was presented in our previous work, for modeling the spatiotemporal dynamics of DPSs without recourse to the governing equations. In this work, we further understand the stability of the training dynamics of the deep architecture by using the Lyapunov exponents (LEs). Subsequently, we construct nonlinear model predictive control (MPC) formulations for the DPS based on the learned, dimensional‐reduced model. We use a path‐integral optimal control algorithm for MPC implementation to avoid any analytic derivatives of the dynamics. The effectiveness of integration of a deep neural network‐based model with MPC is demonstrated in a tubular reactor with recycle cases. The results of the simulation also show that the LE can serve as a readout of training stability for the learned dynamical model.

Iteration-based parameter identification and its applications about distributed parameter systems

This paper addresses a general problem to parameter identification in distributed parameter systems (DPSs), as well as handles a secure communication issue by the extended identification method. First, an iterative approach is put forward based on the approximator and historical information. Meanwhile, an execution scheme for the iterative approach is provided by an optimization policy. Then, combined with the reference model and measurement data, an extended identification method is applied in secure communication. Subsequently, a quantum-leading-following-based optimization with mutation strategy (MQLFBO) algorithm is proposed as the optimization policy. Next, the global convergence and computational complexity are respectively discussed for MQLFBO algorithm in theory. Finally, simulation experiments are performed on parameter identification for DPSs and its application in secure communication, which verify the fast convergence and high precision of the developed method.

Iterative learning control with high-order internal model for first-order hyperbolic systems

This paper studies the iterative learning control (ILC) algorithm for first-order hyperbolic systems. Unlike most of the ILC literature of distributed parameter systems, in the iteration domain, that require identical desired trajectories. Here the desired trajectories are iteratively varying and described by a high-order internal model (HOIM). The HOIM-based P-type ILC design is firstly introduced in this paper to the first-order hyperbolic systems, which enable the systems to achieve the perfect tracking for the iteration-varying desired trajectories on L2 space. Meanwhile, the convergence theorem of the proposed algorithm is established for first-order time-delay hyperbolic systems. Finally, simulation results testify the validity of the algorithm.