Nonlinear Distributed Parameter Systems Research Articles

Adaptive resilient control for a class of nonlinear distributed parameter systems with actuator faults

This paper presents a new model-based fault resilient control scheme for a class of nonlinear distributed parameter systems (DPS) represented by parabolic partial differential equations (PDE) in the presence of actuator faults. A Luenberger-like observer on the basis of nonlinear PDE representation of DPS is developed with boundary measurements. A detection residual is generated by taking the difference between the measured output of the DPS and the estimated one given by the observer. Once a fault is detected, an unknown actuator fault parameter vector together with a known basis function is utilized to adaptively estimate the fault dynamics. A novel tuning algorithm is derived to estimate the unknown actuator fault parameter vector. Next, in order to achieve resilience, the controller from the healthy scenario is adjusted to mitigate the faults by using both estimated fault dynamics and a secondary measurement. Subsequently, an explicit formula is developed to estimate the time-to-resilience (TTR). Finally, a nonlinear example is utilized to illustrate the effectiveness of the proposed scheme.

Computation-Efficient Fault Detection Framework for Partially Known Nonlinear Distributed Parameter Systems.

Fault detection for distributed parameter systems (DPSs) generally requires the complete model information to be known so far. However, for numerous industrial applications, it is common that accurate first-principles physical models are extremely difficult to obtain. Hence, the applicability of traditional model-based methods is being restricted. To pave the way, an adaptive neural network (AdNN) is constructed to simultaneously estimate the state variable and the unknown nonlinearity for a class of partially known nonlinear DPSs. Moreover, considering that full-state measurement is unrealistic in applications, the proposed adaptive neural observer is based on a reduced-order model, which also increases the computation efficiency. Then, the residual generation and evaluation are conducted using the output estimation error of the proposed adaptive neural observer. Bearing the effects of the neglected fast dynamics in mind, a data-driven threshold generation scheme is proposed. Extensive experimental results are presented and analyzed to validate the effectiveness of the proposed method.

Fuzzy Boundary Sampled-Data Control for Nonlinear Parabolic DPSs.

For a nonlinear parabolic distributed parameter system (DPS), a fuzzy boundary sampled-data (SD) control method is introduced in this article, where distributed SD measurement and boundary SD measurement are respected. Initially, this nonlinear parabolic DPS is represented precisely by a Takagi-Sugeno (T-S) fuzzy parabolic partial differential equation (PDE) model. Subsequently, under distributed SD measurement and boundary SD measurement, a fuzzy boundary SD control design is obtained via linear matrix inequalities (LMIs) on the basis of the T-S fuzzy parabolic PDE model to guarantee exponential stability for closed-loop parabolic DPS by using inequality techniques and a LF. Furthermore, respecting the property of membership functions, we present some LMI-based fuzzy boundary SD control design conditions. Finally, the effectiveness of the designed fuzzy boundary SD controller is demonstrated via two simulation examples.

A multi-kernel-based spatiotemporal modeling approach for energy transfer of complex thermal processes and its applications

Most of actual thermal processes work in a large-scale operation region. In each region, the processes are along with complex energy transfer and featured by nonlinear distributed parameter system (DPS). Also, the dynamics at different regions is different, and the energy transfer between neighboring regions is complex. All these factors increase the modeling difficulty of thermal processes. In this paper, a multi-kernel spatiotemporal modeling approach is proposed to reconstruct dynamics of this kind of processes. First, a multi-kernel global spatial function construction method is developed to extract the global spatial features of thermal dynamics. For each local region, it constructs the corresponding spatial kernel function to represent the energy exchange on space. In order to reflect the global spatial feature, these spatial kernel functions are integrated by using Gaussian mixture strategy to form a global spatial function. Then, the temporal dynamics can be obtained and modeled by projecting the spatiotemporal data on this global spatial function. Integrating the global spatial function and temporal model, the spatiotemporal model is constructed for this process with large-scale operation region. The performance of this method is verified by theoretical analysis. Using experiments, its modeling ability for thermal processes are demonstrated in comparison with two commonly used modeling methods.

Quantized-Output-Based Exponential Stabilization of Fuzzy DPSs With Time-Varying Delay and Nonlinearities

The exponential stabilization of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this paper. With the employment of the Takagi-Sugeno (T-S) fuzzy model, the nonlinear DPSs are described as a fuzzy parabolic partial differential equation (PDE). Then, with the consideration of the measurement output and control input information quantized by a kind of dynamic quantizers, fuzzy static and dynamic output feedback controllers are proposed. The time-delayed feedback controllers are also provided in the forms of static and dynamic. On the basis of the derived T-S fuzzy PDE model, Lyapunov-based control design strategies are proposed with respect to the effect of quantization. Moreover, by constructing appropriate Lyapunov functionals, sufficient conditions for designing the output feedback control gains and the quantizers' adjusting parameters which guarantee the exponential stability of the fuzzy closed-loop system are provided in terms of standard linear matrix inequalities (LMIs). Finally, we explore the output feedback controllers into the FitzHugh-Nagumo (FHN) equation and Fisher equation as applications. The efficiency of the design strategies is indicated through the simulation results.

A spatiotemporal fuzzy control approach for complex distributed parameter system with application to thermal processes

Many industrial processes belong to distributed parameter systems (DPSs) with unknown spatiotemporal dynamics and complex boundary conditions. However, attentions to both temporal and spatial dynamics in the control law are less paid in the existing control methods, which often makes them challenging to achieve accurate control. Here, a spatiotemporal fuzzy control approach with consideration of both temporal and spatial dynamics is proposed for unknown and complex DPSs. First, a spatiotemporal fuzzy model is developed to reconstruct dynamics of nonlinear DPS. Since it constructs a two-layer fuzzy rules (i.e., temporal fuzzy rule and spatial fuzzy rule) to consider both the nonlinear dynamics on time and the nonlinear energy transfer on space, it can effectively achieve the model of DPSs. On this basis, a spatiotemporal fuzzy controller is developed according to the developed two-layer fuzzy rules, which means that this derived control law considers both temporal and spatial dynamics. Then, a two-step solution method is designed for the spatiotemporal control law. Stability conditions of the control system are rigorously deduced by transforming the solving process to a series of linear matrix inequalities (LMIs). The stability and effect of the proposed spatiotemporal control strategy are demonstrated by mathematical proof and case studies. Comparison results demonstrate that the proposed method has better control performance (relative errors and mean absolute percentage errors are bounded in 4%).

A temporal–spatiotemporal domain transformation-based modeling method for nonlinear distributed parameter systems

Abstract Complex nonlinear distributed parameter systems (DPSs) exist widely in advanced industrial thermal processes. The modeling of such highly nonlinear systems is a challenge for traditional time/space-separation-based methods since they employ linear methods for the model reduction and spatiotemporal reconstruction, which may lead to an inefficient application of the nonlinear spatial structure features represented by the spatial basis functions. To overcome this problem, a novel spatiotemporal modeling framework composed of nonlinear temporal domain transformation and nonlinear spatiotemporal domain reconstruction is proposed in this paper. Firstly, local nonlinear dimension reduction based on the locally linear embedding technique is utilized to perform nonlinear temporal domain transformation of the spatiotemporal output of nonlinear DPSs. In this step, the original spatiotemporal data can be directly transformed into low-order time coefficients. Then, the extreme learning machine (ELM) method is utilized to establish a temporal model. Finally, through the spatiotemporal domain reconstruction based on the kernel-based ELM method, the prediction of the temporal dynamics obtained from the temporal model can be reconstructed back to the spatiotemporal output. The effectiveness and performance of the proposed method are demonstrated in experiments on the thermal processes of a snap curing oven and a lithium-ion battery.

Temperature field prediction of lithium-ion batteries using improved local tangent space alignment

The temperature field prediction of lithium-ion batteries (LIBs) plays a crucial role in the safety of electric vehicles and their lifetime. However, it is essentially a nonlinear distributed parameter system (DPS), and suitable partial differential equations are difficult to obtain. Further, the existing methods are weak in handling nonlinear sparse data. To address this problem, a nonlinear DPS modeling approach based on kernel and weighted local tangent space alignment (KWLTSA) is proposed in this paper. Firstly, the nonlinear dimensionality reduction method of KWLTSA is used to construct basic functions (BFs) by using spatiotemporal data. Secondly, the low-order temporal coefficients are obtained according to the spatial BFs. Subsequently, the coefficients are fed into a backpropagation neural network (BP-NN) with the input signals for training. Finally, the temperature distribution of LIBs is obtained by spatiotemporal reconstruction. Comparative experiments are designed to estimate the model performance using existing evaluation metrics. The root mean square error (RMSE) is 0.1000 and the R-square (R2) is 0.9972 of the temperature prediction by the proposed scheme. The experimental results indicate that the proposed method is more robust than the traditional dimensionality reduction method.

Fuzzy Output Regulation of Nonlinear Time-Delay Distributed Parameter Systems With Quantization and Actuator Failures: Application to Hot Strip Mill Cooling Process

The output regulation of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this article. By employing the Takagi–Sugeno (T–S) fuzzy model, the nonlinear time-delay DPSs are described as a fuzzy partial differential equation. To deal with the limited capability of communication channels, the state information and output regulation error are quantized by a kind of dynamic quantizer. The actuator failures which occur frequently in practice are considered in two cases: to be unknown and to follow the Bernoulli stochastic distribution. Then, the boundary-control-based output regulation controllers are designed based on the derived T–S fuzzy model. And fuzzy controller design strategies are proposed to ensure the system state and the output regulation error are finite-time bounded. By constructing appropriate Lyapunov functionals, sufficient conditions for the control gains and the quantizer’s adjusting parameters are provided in forms of linear matrix inequalities. Finally, an application of the output regulation strategies into the cooling process in a hot strip mill is explored. The feasibility and effectiveness of the design strategies are indicated through the simulation results.

Latent-Variable-Modeling-Based Fault Diagnosis for Unknown Nonlinear Distributed Parameter Systems with Application to Snap Curing Oven

A novel fault detection method based on time/space separation and latent variable model is proposed for unknown nonlinear distributed parameter systems in sensor-constrained environments. By performing time/space separation, the augmented matrix formed by the spatio-temporal distribution data of the DPSs can be split into a spatial basis function (SBFs) and a time series model, and the dimensionality reduction capability of the SBF is further utilized to obtain a low-order temporal model. Then, the temporal model is further extracted by using a dynamic latent variable modeling method to obtain the dominant time components and establish the corresponding monitoring statistics. Utilizing the appropriate kernel density function, confidence bounds are selected for the monitoring statistics when the system is normal. As a data-based fault diagnosis method, it requires only the past data records of the system and no reliance on complex mathematical models. Two sets of experiments performed on a snap curing oven verified the effectiveness of the proposed method.

Nonparametric identification based on Gaussian process regression for distributed parameter systems

This paper proposes a nonparametric identification method based on Gaussian process regression (GPR) for completely unknown nonlinear distributed parameter systems (DPSs). Inspired by linear parameter-varying (LPV) modelling approach, an interpolated spatio-temporal Volterra model is developed to represent the DPSs in nonparametric form, in which local Volterra models are interpreted as Gaussian processes. According to the empirical Bayesian approach, we design the third-order stable kernel structure used for embedding prior knowledge and derive the estimation of hyperparameters. The hyperparameters included in local weighting functions and kernel functions are determined by the maximum likelihood method. By utilising the nonparametric identification approach to avoid model structure selection, the proposed method can improve identification result for completely unknown distributed parameter systems. Finally, two case studies validate the effectiveness of the proposed identification method.

Filter-Based Fault Detection and Isolation in Distributed Parameter Systems Modeled by Parabolic Partial Differential Equations

This paper covers model-based fault detection and isolation for linear and nonlinear distributed parameter systems (DPS). The first part mainly deals with actuator, sensor and state fault detection and isolation for a class of DPS represented by a set of coupled linear partial differential equations (PDE). A filter based observer is designed based on the linear PDE representation using which a detection residual is generated. A fault is detected when the magnitude of the detection residual exceeds a detection threshold. Upon detection, several isolation estimators are designed using filters whose output residuals are compared with predefined isolation thresholds. A fault on a linear DPS is declared to be of certain type if the corresponding isolation estimator output residual is below its isolation threshold while the other fault isolation estimator output residual is above its threshold. Next, the fault location is determined when a state fault is identified. The second part of this paper focuses on fault detection and isolation of nonlinear DPS by using a Luenberger type observer. Here fault isolation framework is introduced to isolate actuator, sensor and state faults with isolability condition by using additional boundary measurements and without filters. Finally, the effectiveness of the proposed fault detection and isolation schemes for both linear and nonlinear DPS are demonstrated through simulation.

Robustly Linearized Model Predictive Control for Nonlinear Infinite-Dimensional Systems

This work presents a computationally efficient approach to robustly linearized model predictive control for nonlinear affine-in-control evolution equations on infinite-dimensional system state. In this setting, robust linearization refers to a need to account for the approximation errors in linearization and discretization in the model predictive control law, such that the original output constraints are not violated on the true system, a feature that present model predictive control frameworks lack. The main purpose of this work is to enable tractable model predictive control for nonlinear distributed parameter systems while accounting for these approximation errors by means of output constraints. These output constraints are derived using tight integral inequalities that rest on mild assumptions on the nonlinear system dynamics, and are easy to evaluate in real-time. Using our method, linearization and discretization errors are explicitly accounted for, producing for the first time a model predictive control law that is robust to approximation errors. This approach hence enables a trade-off between computational efficiency and strictness of output constraints, much akin to robust control methods. We demonstrate our method on a nonlinear distributed parameter system, namely a one-dimensional heat equation with a velocity-controlled moveable heat source, motivated by autonomous energy-based surgery.

A Dual Adaptation-Based Spatial Model Predictive Control for Nonlinear Distributed Parameter Systems

A Dual Adaptation-Based Spatial Model Predictive Control for Nonlinear Distributed Parameter Systems

Output regulation for general heterodirectional linear hyperbolic PDEs coupled with nonlinear ODEs

This paper considers general heterodirectional linear hyperbolic PDEs with boundary actuation and collocated measurement, that are bidirectionally coupled with nonlinear ODEs at the unactuated boundary. An output feedback regulator is designed to have the control output track a reference in the presence of disturbances, with a nonlinear signal model generating the reference and the disturbances. This leads to new challenges for both the observer and controller design. The regulator design makes use of results from output regulation theory for nonlinear lumped-parameter and distributed-parameter systems. The derivation of the state feedback regulator requires solving a new type of regulator equations, which consist of a Cauchy problem for linear and semilinear hyperbolic PDEs. A key component of the novel observer design is a nonlinear retarded observer for a finite-dimensional subsystem. Both the controller and the observer design are systematic in nature and derived in several successive steps, that include backstepping transformations and, most notably, predictions of PDE and ODE states. The latter are shown to be the unique solution of general nonlinear Volterra integro-differential equations. Combining the observer with the state feedback regulator is proven to achieve output regulation for the closed-loop system. A numerical example illustrates and confirms the theoretical results.

Iterative learning based consensus control for distributed parameter type multi-agent differential inclusion systems with time-delay

This paper investigates consensus control problem for a class of distributed parameter type multi-agent differential inclusion systems with state time-delay by utilizing iterative learning control (ILC). Unlike most ILC literature of nonlinear distributed parameter systems that require an identical virtual leader, the virtual leader is iteratively varying in this work and can be described by any follower agent. Both P -type closed-loop and D -type open-loop ILC are proposed for the distributed parameter type multi-agent differential inclusion systems, which enable the output trajectories of each agent to achieve consensus in the sense of W 1 , 2 norm as the iteration number increases. Meanwhile, consensus control theorems for these protocols are established by using Steiner-type selector, Green's formulas and some technical inequalities. Finally, numerical simulations testify the validity of the proposed methods.

Subpredictor-Based Fuzzy Control Design for a Reaction–Diffusion Equation

This article addresses the stabilization of the nonlinear reaction–diffusion equation with an unknown but bounded delay. Based on Takagi–Sugeno (T–S) fuzzy rules, a T–S fuzzy partial differential equation (PDE) model is constructed to describe the nonlinear distributed parameter system. More importantly, we contribute to the large delay compensation and the future state prediction by introducing a chain of subpredictors for the T–S fuzzy PDE model, and the convergence of observation errors is guaranteed as well. Different from the conventional approaches, the proposed observer can compensate a larger delay. To stabilize the system, for the first time, a new subpredictor-based T–S fuzzy control law is suggested by using the proposed observer, which is composed of elementary observers connected in series. The implementation of the subpredictor feedback is simpler than the classical methods that only one of the elementary observers is employed. Via Lyapunov–Krasovskii approach, sufficient exponential stability conditions are established in the framework of linear matrix inequalities. A numerical example is provided to illustrate the effectiveness of theoretical results.

State prediction of distributed parameter systems based on multi-source spatiotemporal information

To predict the process states of nonlinear distributed parameter systems (DPS) using various process variables, a novel multi-source spatiotemporal modeling method is proposed in this paper to improve conventional temporal modeling methods. To tackle the challenge of modeling industrial process data of DPS with multi-source spatiotemporal characteristics, a multi-source spatiotemporal network (MS-STN) model, which is the integration of a long short-term memory (LSTM) network to extract information containing temporal characteristics and a convolutional long short-term memory (ConvLSTM) network to extract information containing spatiotemporal characteristics, is constructed. The comprehensive use of various process information is beneficial to improving the fitting accuracy of the model, and the generalization ability of the model can be enhanced at the same time. To prevent the gradient vanishing or gradient explosion in presenting complex spatiotemporal data due to the increase of network layers, a model structure of the residual network based on ConvLSTM is proposed in performing model training. Finally, the industrial ethylene oxychlorination reaction process is taken as an example. The experimental results of predicting the temperature of the reaction tube well demonstrate the effectiveness of the proposed method.

Mobile Sensor Networks for Finite-Time Distributed H∞ Consensus Filtering of 3D Nonlinear Distributed Parameter Systems with Randomly Occurring Sensor Saturation

This paper is concerned with designing a distributed bounded H∞ consensus filter to estimate an array of three-dimensional (3D) nonlinear distributed parameter systems subject to bounded perturbation. An optimization framework based on mobile sensing is proposed to improve the performance of distributed filters. The measurement output is obtained from a mobile sensor network, where a phenomenon of randomly occurring sensor saturation is taken into account to reflect the reality in a mobile networked environment. A sufficient condition is established by utilizing operator-dependent Lyapunov functional for the filtering error system to be finite-time bounded. Note that the velocity law of each mobile sensor is included in this condition. The effect from the exogenous perturbation to the estimation accuracy is guaranteed at a given level by means of H∞ consensus performance constraint. Finally, simulation examples are presented to demonstrate the applicability of the theoretical results.

Low-Dimensional-Approximate Model Based Improved Fuzzy Non-Singular Terminal Sliding Mode Control for Rigid-Flexible Manipulators

The dynamic characteristics of rigid-flexible manipulators involve complex rigid-flexible coupling phenomena, which essentially comprise a nonlinear distributed parameter system with infinite degrees of freedom. Consequently, it results in challenges to the manipulator’s precise positioning. This study combined a state feedback module and a fuzzy non-singular terminal sliding mode to suppress vibration and deformation. Using the improved fuzzy strategy and non-singular terminal sliding mode control, an adaptive dynamic supplementary control law is proposed. The results based on MATLAB simulation and a built hardware experiment show that this method is effective and superior. While realizing the accurate positioning of the end of the manipulator, the vibration of the end of the flexible arm is significantly suppressed. This method has a high tracking performance, which enables accurate positioning of the manipulator terminal and provides strong robustness under the action of bounded external interference.