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Modern industrial processes demand real-time model predictive control (MPC) method within the constraints of limited computing resources. This paper introduces an accelerated MPC method to address the issue. Specifically, we employ the time series mixing model (TSMixer) to construct a predictive model capable of accurately forecasting the behavior of the system under control. Furthermore, we extend the capabilities of TSMixer by integrating a feature classification model (TSMixer with FCM). This enhanced version takes into account both future information and static variables, resulting in superior accuracy compared to the transformer model while maintaining a significantly smaller parameter count. Finally, to further enhance MPC’s responsiveness, we develop a two-dimensional block stochastic configuration network (2D-BSCN) imitative controller. This innovative network clones the behavior of rolling optimization, effectively replacing online optimization to reduce computational time. The experimental results show that the accelerated MPC has an excellent performance in numerical simulation and actual industrial processes. Notably, the algorithm achieves a smaller FLOPs and tracking time than other state-of-the-art models, well within the requirement for industrial process control.

The problem of fault estimation is addressed for one-dimensional (1-D) linear boundary control and boundary observation (BCBO) parabolic partial differential equations (PDEs) with a faulty boundary measurement. The considered plant is subjected to simultaneous unknown multiplicative faults entering the boundary input and boundary measurement. Difficulties arise due to the coupling between the sensor fault parameter and unknown boundary state appearing in the measurement. With the only boundary input and faulty boundary measurement, it is rather challenging to estimate the accurate values of faults and state simultaneously. Therefore, most existing results only consider correct and healthy measurement for PDE systems. To this end, novel adaptation laws and an adaptive observer are designed in this work to provide exponential convergent joint fault-state estimation, where we design and leverage a set of novel filters. It is first time that unknown multiplicative fault parameter in the measurement can be estimated accurately in the PDE systems.

This article investigates the adaptive neural tracking control problem for a class of hyperbolic PDE with boundary actuator dynamics described by a set of nonlinear ordinary differential equations (ODEs). Particularly, the control input appears in the ODE subsystem with unknown nonlinearities requiring to be estimated and compensated, which makes the control task rather difficult. It is the first time to consider tracking control of such a class of systems, rendering our contributions essentially different from the existing literature that merely focus on the stabilization problem. By formulating a virtual exosystem to generate a reference trajectory, we propose a novel design of the adaptive geometric controller for the considered system where neural networks (NNs) are employed to approximately estimate nonlinearities, and finite and infinite-dimensional backstepping techniques are leveraged. Moreover, rigorously theoretical proofs based on the Lyapunov theory are provided to analyze the stability of the closed-loop system. Finally, we illustrate the results through two numerical simulations.

This article develops a robust adaptive boundary output regulation approach for a class of complex anticollocated hyperbolic partial differential equations subjected to multiplicative unknown faults in both the boundary sensor and actuator. The regulator design is based on the internal model principle, which amounts to stabilize a coupled cascade system, which consists of a finite-dimensional internal model driven by a hyperbolic distributed parameter system (DPS). To this end, a systematic sliding mode equipped with a backstepping approach is developed such that the robust state feedback control can be realized. Moreover, since the available information is a faulty boundary measurement at the right side point, state estimation is required. However, due to the presence of boundary unknown faults, we need to solve an issue of joint fault-state estimation. Restrictive persistent excitation conditions are usually required to guarantee the exact estimation of faults but are unrealistic in practice. To this end, a novel concurrent learning (CL) adaptive observer is proposed so that exponential convergence is obtained. It is the first time that the spirit of CL is introduced to the field of DPSs. Consequently, the observer-based adaptive boundary fault tolerant control scheme is developed, and rigorous theoretical analysis is given such that the exponential output regulation can be achieved. Finally, the effectiveness of the proposed methodology is demonstrated via comparative simulations.

Via the Takagi–Sugeno (T-S) fuzzy model-based nonlinear control technique, nonlinear spatiotemporal feedback compensators are proposed in this article for exponential stabilization of parabolic partial differential dynamic systems with measurement outputs transmitted over a communication network. More specifically, an approximate T-S fuzzy partial differential equation (PDE) model with $C^\infty$-smooth membership functions is constructed to describe the complex spatiotemporal dynamics of the nonlinear partial differential systems, and its approximation capability is analyzed by the uniform approximation theorem on a real separable Hilbert space. A spatiotemporally asynchronous sampled-data measurement output equation is proposed to model the transmission process of networked measurement outputs. By the approximate T-S fuzzy PDE model, fuzzy-observer-based nonlinear continuous-time and sampled-data feedback compensators are constructed via the spatiotemporally asynchronous sampled-data measurement outputs. Given that sufficient conditions presented in terms of linear matrix inequalities are satisfied, the suggested fuzzy compensators can exponentially stabilize the nonlinear system in the Lyapunov sense. Simulation results are presented to show the effectiveness and merit of the suggested spatiotemporal fuzzy compensators.

This article considers the problem of adaptive output regulation of a class of 1-D anticollocated hyperbolic partial differential equations (PDEs). The model is subject to unknown scaled parameters in both the boundary condition and boundary measurement. An adaptive boundary observer, providing online estimates of the system state and parameters, is designed. Particularly, to realize that the pure boundary state at left side tracks a reference signal, a challenging problem with the estimate of the true value of unknown scaled parameter appearing in measurement (at the left side) is tackled in this article. It is shown that the proposed observer is exponentially convergent, and furthermore, the exponential output regulation and stabilization are achieved.

Joint uncertainties and state estimation of a class of linearly coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this article. For unstructured uncertainties that are completely unknown, by employing Takagi–Sugeno fuzzy logic systems to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and an infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this article.

In this article, we address the constrained output estimation of discrete-time linear distributed parameter systems in the presence of plant and measurement disturbances. Sufficient conditions are proposed for the strong stability of the proposed moving horizon estimator. We further show that the discrete-time estimation results can be linked to continuous-time infinite-dimensional systems described by partial differential equations with unbounded disturbance and output operators by using the Cayley–Tustin transformation. The theoretical results are illustrated with numerical examples on a 1-D wave equation and a 1-D diffusion equation.

The paper considers robust output regulation of a Guyer-Krumhansl (GK) heat conduction law under modeling mismatches. More specifically, we consider the differences of the GK model compared with a simple diffusion equation (also known as the Fourier heat conduction law) and treat them as modeling mismatches in view of robust output regulation. We will consider a simple internal model based controller to solve the output regulation problem and analyze its robustness with respect to the differences between the two heat conduction laws. Moreover, we will test the robustness of the considered controller in numerical simulations.

Explicit model predictive control design is carefully developed for discrete-time linear plants on Hilbert spaces, and we highlight the role of the so-called Slater condition in the reliable explicit solution of the MPC optimization. We then proceed to present an explicit MPC algorithm that accounts for the stabilization and input constraints satisfaction. We do structure preserving temporal discretization of the infinite-dimensional parabolic PDE system by application of the Cayley transformation. The salient feature of explicit MPC design is the realization of the region-free approach in explicit MPC design with identification of active constraint sets to realize optimal stabilization and constraints satisfaction. Finally, the resulting design is illustrated by the application to the PDE model given by an unstable heat equation with boundary actuation and Neumann boundary conditions. The example demonstrates simultaneous stabilization and input constraints satisfaction on the one hand, and on the ability to deal with a relatively high plant dimension and a long optimization horizon on the other hand.

In this work, we aim at addressing thermodynamic modeling and passivity properties of a class of distributed parameter systems (DPSs) with dispersion that are described by partial differential equations (PDEs). The class of distributed parameter systems account for a general class of thermodynamic models with spatial-temporal characteristics. For this class of distributed systems, various contributions on stability analysis, control and estimator designs have been made in the existing literature. On a different note, this contribution aims at providing a thermodynamic perspective on the modeling of the transport processes and passivity using flux expressions based on thermodynamical driving forces. A linearized model is derived and an invertible linear transformation is applied to further simplify the model by eliminating the transport terms. A case study on an adiabatic tubular reactor with dispersion is used as an example.

In this article, we propose a novel framework on the state estimation for a class of positive real infinite-dimensional systems, which suffer from unknown actuator and sensor faults and uncertain exogenous disturbance. Four cases of fault occurrence are taken into account: single multiplicative actuator fault, single multiplicative sensor fault, simultaneous actuator and sensor faults, and single actuator fault with a dead-zone nonlinearity. We develop new adaptive laws for the estimation of fault parameters and embed the proposed laws into the infinite dimensional state observer design. Moreover, by considering the actuator fault with a dead-zone nonlinearity, an infinite dimensional observer associated with coupled parameter update laws is developed to help to complete the fault accommodation. This is achieved by reconfiguring the actuator input based on the estimated information of parameters in the actuator fault map with a dead-zone nonlinearity.

The deformation of the heart tissue due to the contraction can modulate the excitation, a phenomenon referred to as mechanoelectrical feedback (MEF), via stretch-activated channels. The effects of MEF on the electrophysiology at high pacing rates are shown to be proarrhythmic in general. However, more studies need to be done to elucidate the underlying mechanism. In this work, we investigate the effects of MEF on cardiac alternans, which is an alternation in the width of the action potential that typically occurs when the heart is paced at high rates, using a biophysically detailed electromechanical model of cardiac tissue. We observe that the transition from spatially concordant alternans to spatially discordant alternans, which is more arrhythmogenic than concordant alternans, may occur in the presence of MEF and when its strength is sufficiently large. We show that this transition is due to the increase of the dispersion of conduction velocity. In addition, our results also show that the MEF effects, depending on the stretch-activated channels’ conductances and reversal potentials, can result in blocking action potential propagation.

Model-based discrete-time output regulator design is proposed for fluid flow systems using a geometric approach. More specifically, a class of vortex shedding and falling thin film phenomena modelled by complex Ginzburg–Landau equation (CGLE) and Kuramoto–Sivashinsky equation (KSE) are considered. Differently from a traditional continuous-time controller design, a novel discrete-time modelling technique is proposed in a general infinite-dimensional state-space setting, which does not pertain any spatial approximation or model reduction, and preserves model intrinsic properties (such as stability, controllability and observability). Based on the time discretized plant model (CGLE and KSE systems) by the Cayley–Tustin method, discrete regulator regulation equations are established and facilitated for an output regulator design to achieve fluid flow control and manipulation. To address model instability, a spectrum analysis is utilized in stabilizing continuous-time CGLE and KSE systems, and a link between discrete- and continuous-time closed-loop system stabilizing gains is established. Finally, the proposed methodology is demonstrated through a set of simulation cases, by which the output tracking, disturbance rejection, and model stabilization are achieved for the considered CGLE and KSE systems.

This article addresses the model predictive controller design for a jacket tubular reactor with a simple reversible exothermic reaction (A⇌ B). Using energy and mass balance laws, four nonlinear hyperbolic partial differential equations are derived to model the tubular reactor dynamics in terms of two concentrations, the reactor temperature, and the spatially varying jacket temperature. The nonlinear continuous-time model is linearized and discretized in time by the use of the Cayley–Tustin transform without spatial discretization or model reduction. Along these lines, a state-feedback model predictive controller is formulated to realize model stabilization with respect to input and output constraints. To account for the state estimation, a Luenberger observer-based model predictive control (MPC) frame is further developed, and observer gains are obtained as solutions of an operator Riccati equation. Finally, two numerical examples are provided to demonstrate the feasibility and applicability of the proposed MPC designs.

In this work, an output regulator design in a discrete-time setting is considered for a linearized Ginzburg-Landau equation (GLE) with point observation using the internal model principle. To address model instability, spectrum analysis is presented and utilized for the continuous-time GLE system. In addition, the Cayley-Tustin transform is used for model time discretization and no spatial approximation or model reduction is induced in the time discretization. As for the servo control design, the discrete-time Sylvester regulation equations are constructed and applied. By a state-feedback regulator, the output tracking and disturbance rejection are realized simultaneously for the Ginzburg-Landau equation, which is verified by a set of simulation studies.

The present work extends known finite dimensional Luenberger observer and Kalman filter designs to the realm of linear transport-reaction systems frequently present in chemical engineering practice. A unified modelling framework for distributed parameter systems (DPS) which does not account for any type of spatial approximation or order reduction is developed. The Cayley–Tustin transformation of continuous linear distributed parameter system yields structure and properties preserving discrete distributed parameter models, amenable to observer and filter design developments. Designs presented here explore well known state reconstruction methodologies starting from least square estimation, continuous and discrete Luenberger observers and one-step predictor Kalman filter realization. Simple implementation and realization account for the appealing nature of the discrete system observers and filter designs for linear transport-reaction systems. The simulation scenarios cover the majority of representative examples found in the common engineering process control practice.

This manuscript addresses the design of a model predictive controller for the autothermal reactor with internal counter current heat exchanger (ARICHE) modeled by the system of quasilinear hyperbolic partial differential equations (PDEs). The autothermal reactor dynamical model is explored by considering parametric dynamical analysis and exhibits the multiplicity of steady states. Generally, the unstable steady state provides an optimal conversion, and selectivity is considered as steady state of interest. The linearization is utilized to obtain a linear system of coupled transport-reaction hyperbolic PDEs, and the structure preserving Caley-Tustin discretization is employed without use of any spatial approximation or order reduction. The discrete model is utilized in the model predictive controller design, and it successfully addresses the control scenarios that account for the optimality, stability, and constraints satisfaction.

In this contribution we propose dissipation conditions using the second law of thermodynamics for a class of first order hyperbolic thermodynamic systems. The entropy function is used as a storage function and the internal entropy production as the dissipation function. The supply rate is found to be proportional to the difference of the entropy at the boundaries, and the time derivative of the entropy balance allows to define a PI controller with variable parameters designed using a cascade structure, which is applied to an adiabatic plug flow reactor with mass recycle, using the recycle rate as the manipulated variable. Finally, the controller is tested with numerical simulations tracking a desired yield.

In this manuscript, a comparison between the parabolic and hyperbolic partial differential equations for heat diffusion is studied. First, numerical results and an eigenvalue analysis for these two types of equations are shown, which help to understand the difference in the system dynamics. Then, both equations are also considered in a Stefan problem for the melting of an ice block in a one-dimensional setting. The results show that the hyperbolic partial differential equation shows a finite speed of propagation of heat and can represent the system as properly as the parabolic equation.

This work addresses an optimal linear quadratic (LQ) boundary output regulator design problem for the plug flow reactor described by hyperbolic partial differential equations (PDEs), with actuation applied only at the inlet of the reactor. By applying the weak variational approach, the necessary optimality conditions are provided and then an optimal state feedback controller is presented. In particular, the time-varying state feedback gain is determined by solving Riccati-type PDEs and this work extends the linear quadratic regulator design to the class of boundary controlled hyperbolic PDE systems. Along the line of the LQ design, an optimal boundary tracking regulator is designed such that the output of the considered reaction process tracks the desired reference signal generated by an exosystem. A simulation example is included to show performance of the proposed approach.

The state feedback regulator problem for a network of countercurrent heat exchangers is considered in this paper. The system is described by two sets of hyperbolic partial differential equations (PDEs) and the model is nonlinear with respect to the control input. To deal with the nonlinearity, the equilibrium temperature profile is calculated and utilized in the linearization of the original nonlinear system. Then, based on infinite-dimensional representation, the state feedback regulator problem (in particular the tracking problem) is considered, where the target is to design a controller that, while guaranteeing the stability of the closed-loop system, drives the controlled output to track a reference signal generated by an exosystem with its spectrum on the imaginary axis. Given the explicit expression of the transfer function, we provide sufficient conditions such that the resulting linearized system is causal and stable. Given that the controlled system is stable, we propose a simple and novel method to provide the stabilization feedback gain K, such that the controlled system tracks the reference signal. Finally, a numerical simulation illustrating the results is presented.

The port-Hamiltonian representation of undamped wave equation system captures the essential energy and dynamical features in an unified framework. In this paper, an undamped wave equation system is given in the port-Hamiltonian formulation. The important issue of obtaining discrete version of the port-Hamiltonian system representation is explored. The exact discretization method is applied to the port-Hamiltonian system which transforms the system from a continuous to a discrete state space setting. The development of the discretized port Hamiltonian system formulation is necessary foundation for the discrete and computer based regulator and model realizations. The results are illustrated by numerical simulations.

In this work, we consider the output tracking and disturbance rejection problems of a mono-tubular heat exchanger process and a novel finite-dimensional output feedback regulator is developed. In the proposed output feedback regulator design, measurements available for the regulator do not belong to the set of controlled outputs. In other words, design emphasizes that other than controlled output is used as input signal to the regulator. The proposed output feedback regulator with only plant measurement ym(t) can realize the exosystem state estimation, disturbance rejection and reference signal tracking, simultaneously. Finally, the results are demonstrated by a mono-tubular heat exchanger process with specific parameters via the computer simulation.

This paper considers the optimal strategies for constrained linear state estimation. Prior information for estimating state variables is often available in the form of inequality constraints on states. In the latest developments of optimal state estimation theory consideration of state constraints has been often neglected since constraints do not fit easily in the structure of the optimal filter, for example, the issue of state constraints being present has to be addressed adequately, for example nonnegativity of concentration. In order to address this issue and to extend previous developments on the accuracy of state estimation, this work develops the constrained optimal state estimation for finite-dimensional systems. Finally, a numerical example illustrating the proposed method is presented.

Considering disturbances within domain and at the boundary, a backstepping-based output boundary regulator design is developed for a class of first-order linear hyperbolic partial integro-differential equation (PIDE) in the presence of state and sensor delays. The delays are represented by two transport PDEs, which results in an extended spatial domain where the hyperbolic PIDE, transport PDEs and ordinary differential equation (ODE) are in cascade. The ODE is a finite-dimensional signal model describing exogenous signals. First, a state feedback regulator is realized to achieve a finite time stability by applying an affine Volterra integral transformation. Then, an output regulator is developed on the basis of the nominal plant transfer behavior, which results in an exponential stability. Numerical examples illustrate the performance of the proposed regulators.

In this work, an observer based controller synthesis is proposed based on the linear matrix inequality (LMI) framework to achieve stabilization of the parabolic PDE in the presence of input constraints. A novel feature of the proposed synthesis is to construct the LMI formulation within the modal space by converting the original parabolic PDE into an infinite-dimensional abstract state space setting. The state feedback controller and the Luenberger observer are developed by accounting for the entire infinite number of modal states, thereby stabilizing the system and reconstructing the state rigorously. The input constraints naturally existing in realistic applications are considered in the design framework. Finally, since initial modal states are hardly known in advance, the LMI formulation, based on the augmented modal state space, including unstable modal states, its estimation error and fast modal output, is developed to maximize the region of attraction (ROA) for the real process state, such that the controller and the observer are robust enough to the error in the initial state guess. By considering a numerical example of an unstable parabolic PDE, we demonstrate that if feasible, the state feedback controller and the observer, generated by the LMI formulation, have capability to drive the process state to the equilibrium steady state.

This contribution addresses the development of a Linear Quadratic Regulator (LQR) for a set of time-varying hyperbolic PDEs coupled with a set of time-varying ODEs through the boundary. The approach is based on an infinite-dimensional Hilbert state-space description of the system and the well-known operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of equivalent differential and algebraic matrix Riccati equations. The feedback operator can then be found by solving the resulting matrix Riccati equations. The performance of the designed control policy is assessed by applying it to a system of interconnecting continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) through a numerical simulation.

This work focuses on predictive output feedback control of linear parabolic partial differential equation (PDE) systems with state and control constraints. Under the assumption that a finite, yet sufficiently large, number of output measurements is available, two predictive output feedback controllers are constructed and sufficient conditions are derived under which these controllers enforce stability and constraint satisfaction in the infinite-dimensional closed-loop system, respectively. The controllers are applied to an example of a linear parabolic PDE with Dirichlet boundary conditions subject to state and control constraints, and the numerical simulations demonstrate their ability to enforce closed-loop system stability and constraint satisfaction.