Parabolic Distributed Parameter Systems Research Articles

Event-triggered iterative learning control for discrete parabolic distributed parameter systems

Iterative learning control (ILC) is an intelligent control strategy which can achieve the desired trajectory tracking by continuously renewing the control signal based on the previous experience. In this article, an event-triggered condition that can save system resources by reducing the number of iterative updates is proposed. Moreover, an event-triggered ILC algorithm is designed for linear discrete parabolic distributed parameter systems, the system input is updated with the learning law only when the derived triggered condition is met. Through rigorous mathematical analysis, the tracking error can converge to zero when the iterative batch approaches infinity under the given sufficient condition. Then, we also extended the above ideas to nonlinear case. By simulation, the effectiveness of the designed algorithm are illustrated.

Boundary Optimal Control for Parabolic Distributed Parameter Systems With Value Iteration.

A reinforcement learning-based boundary optimal control algorithm for parabolic distributed parameter systems is developed in this article. First, a spatial Riccati-like equation and an integral optimal controller are derived in infinite-time horizon based on the principle of the variational method, which avoids the complex semigroups and operator theories. Using state data along the system trajectory, a value iteration algorithm via the Bellman optimality principle is proposed to obtain the solution of the spatial Riccati-like equation and the optimal control law. The convergence of the value iteration algorithm is proved. Subsequently, an approximation scheme based on weighted residuals is developed to implement the value iteration algorithm, where radial basis functions are chosen as the basic functions to approximate the solution of the spatial Riccati-like equation. Simulations on the diffusion-reaction process demonstrate the effectiveness of the developed 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.

Self-Correcting Iterative Learning-Based Fault Estimation for Parabolic Distributed Parameter Systems

Self-Correcting Iterative Learning-Based Fault Estimation for Parabolic Distributed Parameter Systems

Robust Control for Semilinear Parabolic Distributed Parameter Systems With External Disturbances via Mobile Actuators and Sensors.

This article presents a robust feedback compensator design approach for semilinear parabolic distributed parameter systems (DPSs) with external disturbances via mobile actuators and sensors. An performance constraint is introduced to deal with the external disturbances from the model and measurement noise. Two types of feedback compensators are designed in terms of the collocated and noncollocated mobile actuators and sensors. By the Lyapunov direct technique, some sufficient conditions based on LMI constraints are proposed for the exponential stability under performance constraints in the L²-norm. Moreover, the open-loop and closed-loop well-posedness of the semilinear DPSs with external disturbances are analyzed via the C0 -semigroup theory approach. Finally, extensive numerical simulation results for semilinear DPSs with external disturbances via collocated and noncollocated mobile actuators and sensors are shown to verify the effectiveness of the proposed method.

Formula omitted] fuzzy intermittent boundary control for nonlinear parabolic distributed parameter systems

For nonlinear space-varying parabolic distributed parameter systems, this paper introduces an H∞ fuzzy intermittent boundary control, where the output measurements are only available at some specified boundary position (i.e., boundary measurements). Initially, a Takagi–Sugeno fuzzy parabolic partial differential equation is used to precisely describe the nonlinear space-varying parabolic distributed parameter system. Then, under boundary measurements, an H∞ fuzzy intermittent boundary control design based on the Takagi–Sugeno fuzzy parabolic partial differential equation model ensuring the exponential stability with an H∞ performance for closed-loop space-varying distributed parameter system is subsequently obtained via spatial linear matrix inequalities by employing inequality techniques and piecewise switching-time-dependent Lyapunov function. Furthermore, in order to solve the H∞ fuzzy intermittent boundary controller design of nonlinear space-varying parabolic distributed parameter systems under boundary measurements, we express spatial linear matrix inequalities as linear matrix inequalities and further present some linear matrix inequality based fuzzy intermittent boundary control design conditions respecting the property of membership functions. Finally, two simulation examples are offered to demonstrate the effectiveness of the proposed H∞ fuzzy intermittent boundary control method.

Data‐based learning control for optimization of nonlinear systems

Data‐based learning control for optimization of nonlinear systems

Global fuzzy adaptive asymptotic tracking control for unknown semi-linear parabolic distributed parameter systems

This paper is mainly devoted to investigating the globally asymptotic tracking problem of unknown semi-linear parabolic distributed parameter systems (SLPDPSs) with uncertain external disturbance. Firstly, an analysis of the well-posedness of the system solution is given based on the operator semigroup theory. Then, from the Lyapunov direct method, a distributed adaptive fuzzy controller (AFC) is designed so that the unknown SLPDPSs can asymptotically track target systems. More importantly, the globally asymptotic tracking condition can be obtained, which overcomes the semi-global stability results in the existing literatures. Finally, several simulation cases are introduced to verify the availability and practicability of the proposed method.

Iterative learning feedback control for linear parabolic distributed parameter systems with multiple collocated piecewise observation

This paper presents a novel iterative learning feedback control method for linear parabolic distributed parameter systems with multiple collocated piecewise observation. Multiple actuators and sensors distributed at the same position of the spatial domain are utilized to perform collocated piecewise control and measurement operations. The advantage of the proposed method is that it combines the iterative learning algorithm and feedback technique. Not only can it use the iterative learning algorithm to track the desired output trajectory, but also the feedback control approach can be utilized to achieve real-time online update. By utilizing integration by parts, triangle inequality, mean value theorem for integrals and Gronwall lemma, two sufficient conditions based on the inequality constraints for the convergence analysis of the tracking error system are presented. Some simulation experiments are provided to prove the effectiveness of the proposed method.

Robust piecewise adaptive control for an uncertain semilinear parabolic distributed parameter systems

In this study, we focus on designing a robust piecewise adaptive controller to globally asymptotically stabilize a semilinear parabolic distributed parameter systems (DPSs) with external disturbance, whose nonlinearities are bounded by unknown functions. Firstly, a robust piecewise adaptive control is designed against the unknown nonlinearity and the external disturbance. Then, by constructing an appropriate Lyapunov–Krasovskii functional candidate (LKFC) and using the Wiritinger’s inequality and a variant of the Agmon’s inequality, it is shown that the proposed robust piecewise adaptive controller not only ensures the globally asymptotic stability of the closed-loop system, but also guarantees a given performance. Finally, two simulation examples are given to verify the validity of the design method.

Backstepping-Based Distributed Abnormality Detection for Nolinear Parabolic Distributed Prameter Systems

In this paper, we proposed a model-based abnormality detection scheme for a class of nonlinear parabolic distributed parameter systems (DPSs). The proposed methodology consists of the design of an observer and an abnormality detection filter (ADF) based on the backstepping technique and a limited number of in-domain measurements plus one boundary measurement. By taking the difference between the measured and estimated outputs from observer, a residual signal is generated for fault detection. For the detection purpose, the residual is evaluated in a lumped manner and we propose an explicit expression for the time-varying threshold. The convergence properties of the PDE observer and the residual are analyzed by Lyapunov stability theory. Eventually, the proposed abnormality detection scheme is demonstrated on a nonlinear DPS.

Fixed-time stabilization of parabolic distributed parameter systems with spatially and temporally varying reactivity

This paper concerns the problem of boundary time-varying feedback controller for fixed-time stabilization of a linear parabolic distributed parameter system with spatially and temporally varying reactivity. By utilizing the continuous backstepping approach, the invertible Volterra integral transformation with the time-dependent gain kernel is introduced to convert the closed-loop system into a target system with a time-dependent coefficient. Meanwhile, the convergence of the target system within the prescribed time is guaranteed via the Lyapunov method. The well-posedness of the resulting kernel partial differential equations is also proven by exploiting the method of successive approximation. In addition, the growth-in-time of the kernel functions is estimated by applying the generalized Laguerre polynomials and the modified Bessel functions. Subsequently, the fixed-time stability of the closed-loop system under state feedback control within the prescribed time is proven by using the fixed-time stability of the target system and the time-varying kernel functions. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control method.

Backstepping-based distributed abnormality localization for linear parabolic distributed parameter systems

Abnormality localization for industrial processes described by distributed parameter systems (DPSs) is vital but seldom studied to the best of our knowledge. In this paper, a systematic framework is proposed to address this problem for a class of linear parabolic DPSs using a limited number of in-domain measurements plus one boundary measurement rather than the full-state measurement. The proposed methodology consists of an abnormality detection filter (ADF) and an abnormality localization filter (ALF) design based on the backstepping techniques and eigenspectrum. For the detection purpose, the residual is evaluated in a lumped manner; For the localization purpose, a distributed residual is first constructed and then evaluated in a distributed manner. Numerical simulations on a heat transfer rod are conducted to demonstrate the effectiveness of the proposed method.

Closed loop variable gain ILC for a class of nonlinear parabolic distributed parameter systems with moving boundaries

AbstractThis article researched the output tracking problem of a class of nonlinear parabolic distributed parameter systems with moving boundaries. Considering the large deviation of the initial state reset of the system, a closed‐loop variable gain accelerated iterative learning control algorithm is designed. And sufficient condition for the convergence of the tracking error is given. Through rigorous theoretical analysis guaranteed that the tracking error of the system converges to zero in the positive direction of the iteration axis in the sense of the norm. Finally, the effectiveness of the designed algorithm is verified by a numerical simulation.

Iterative learning control for linear parabolic distributed parameter systems with multiple point sensors

AbstractThis article presents an iterative learning control (ILC) approach for linear parabolic distributed parameter systems with multiple actuators and multiple sensors. The distribution functions of actuators and sensors are chosen as delta function to produce pointwise control and pointwise measurement. A P‐type ILC law is proposed based on the iterative inputs and outputs to ensure the iterative process of the system is convergent under the ILC law. By utilizing integration by parts, triangle inequality, property of delta function, and Gronwall lemma, a sufficient condition based on an inequality constraint for the convergence analysis of the track error system is presented. Finally, the effectiveness of the proposed design method is verified by numerical simulation results.

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.

Least Squares Support Vector Machine-Based Multivariate Generalized Predictive Control for Parabolic Distributed Parameter Systems with Control Constraints

This manuscript addresses a new multivariate generalized predictive control strategy using the least squares support vector machine for parabolic distributed parameter systems. First, a set of proper orthogonal decomposition-based spatial basis functions constructed from a carefully selected set of data is used in a Galerkin projection for the building of an approximate low-dimensional lumped parameter systems. Then, the temporal autoregressive exogenous model obtained by the least squares support vector machine is applied in the design of a multivariate generalized predictive control strategy. Finally, the effectiveness of the proposed multivariate generalized predictive control strategy is verified through a numerical simulation study on a typical diffusion-reaction process in radical symmetry.

Dynamic H∞ Feedback Boundary Control for a Class of Parabolic Systems with a Spatially Varying Diffusivity

The dynamic H∞ feedback boundary control for a class of parabolic distributed parameter systems with a non-constant (spatially varying) diffusion rate is addressed in this paper. The observer-based controller is designed to deal with non-collocated sensors and actuators, and the H∞ performance index is employed to tackle the influence of the external disturbance and measurement noise. The resulting closed-loop system is formed by the boundary actuation with the H∞ control strategy, and the output feedback is designed from the domain-averaged and boundary-valued measurement, respectively. With the sufficient conditions of the linear matrix inequality that infer the stability of the system, the corresponding gains of observer and controller are solved. Numerical simulations are given to show the validity of the main results.

Spatiotemporal Modeling for Distributed Parameter System under Sparse Sensing

Modeling of the parabolic distributed parameter system (DPS) with the Karhunen-Loeve (KL) method under sparse sensing will become very difficult because the information from the measurements is inc...

Backstepping‐based sliding mode fault‐tolerant control for linear interconnected parabolic distributed parameter systems

This study considers the fault-tolerant control (FTC) issue for a class of linear interconnected parabolic distributed parameter systems by utilising the backstepping-based sliding mode control technique. The original interconnected system, which suffers from process/actuator faults and matched disturbances, is converted into a stable target system by using an invertible backstepping transformation. Two types of kernel functions, the distinct diffusivity and the same diffusivity, are obtained. A sliding mode FTC scheme is developed to eliminate the destabilising effects caused by process/actuator faults and matched disturbances. An industrial application example is presented to validate the applicability and the relevance of the developed methodology.