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The state estimation and sensor placement for a continuous pulp digester with delayed measurements are investigated. The underlying model of interest is heat transfer in a pulp digester modeled by two coupled hyperbolic partial differential equations and an ordinary differential equation. Output measurements are considered with delay due to the possible low sampling rate. The Cayley–Tustin transformation is utilized to realize model time discretization in a late lumping manner which does not account for any type of spatial approximation or model reduction. The discrete Kalman filter is applied to estimate the system states using the delayed measurements. The selection of sensor location is addressed along with estimator design accounting for the delayed measurements and investigated by minimizing the variance of estimation error. The performance of the state estimator is evaluated, and the sensor placement is analyzed through simulation studies, which offers a planning view of sensor location in industrial applications. [ABSTRACT FROM AUTHOR]

In this manuscript, we address discrete-time state and error feedback output regulator designs for a class of linear distributed parameter systems (DPS) with bounded input and output operators. By utilising the Cayley–Tustin bilinear transform, a linear infinite-dimensional discrete-time system is obtained without model spatial approximation or model order reduction. Based on the internal model principle, discrete state and error feedback regulators are designed. In particular, discrete Sylvester regulator equations are formulated, and their solvability is proved and linked to the continuous counterparts. To ensure the stability of the closed-loop system, the design of stabilising feedback gain and its dual problem of stabilising output injection gain design are provided in the discrete-time setting. Finally, three simulation examples including a first-order hyperbolic partial differential equation model and a 1-D heat equation with considerations of step-like, ramp-like and harmonic exogenous signals are shown to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This work explores the design of a model predictive controller of the continuous pulp digester process consisting of the co‐current zone and counter‐current zone modeled by a set of nonlinear coupled hyperbolic partial differential equations (PDEs). The distributed parameter system of interest is not spectral, and slow–fast dynamic separation does not hold. To address this challenge, the nonlinear continuous‐time model is linearized and discretized in time utilizing the Cayley–Tustin discretization framework, which ensures system theoretic properties and structure preservation without spatial discretization or model reduction. The discrete model is used in the full state model predictive controller design, which is augmented by the Luenberger observer design to achieve the output constrained regulation. Finally, a numerical example is provided to demonstrate the feasibility and applicability of the proposed controller designs. [ABSTRACT FROM AUTHOR]

This manuscript develops a robust output regulator for the column froth flotation process in the presence of unknown disturbance. In particular, motivated by relevant physical modeling setting, all information (including magnitude, phase, bias, and frequency) of the disturbance is unavailable. The system considered is consisting of froth and collection zones and is governed by a set of conservative heterodirectional transport hyperbolic partial differential equations (PDEs). Moreover, the transport related time delay between the froth and the collection region is considered, and is associated with the boundary conditions of the PDEs. The velocity of the feed is considered the input and the concentration of the solid particle with air phase at the top of froth region as the output. We propose a disturbance frequency identifier and formulate a robust controller according to the internal model principle such that the output tracks the desired concentration despite the model uncertainties and unknown disturbance. [ABSTRACT FROM AUTHOR]

In this work, a discrete-time output regulator design is proposed for a class of gas pipeline networks to meet various operating requirements in energy scheduling. Based on the isothermal Euler equations, linearized continuous-time gas pipeline network models with boundary actuation and sensing in the infinite-dimensional space are established, with consideration of Rankine–Hugoniot conditions at junction joints. Cayley–Tustin bilinear transformation is applied for model time discretization without any spatial approximation, by which the continuous-time model with unbounded operators is transformed into an infinite-dimensional discrete-time system with all bounded operators and essential continuous-time properties are invariant under this transformation. Based on the internal model principle, the discrete output regulator is constructed and its solvability conditions are provided. Considering the unavailability of the full state information, observer design methods for state estimation of exogenous and pipeline systems are proposed in order to construct an output feedback regulator. Additionally, a stability analysis of the considered pipeline system is provided. Finally, two simulation examples representing a single gas pipeline and a star-shaped pipeline network are given to demonstrate the applicability of the proposed method. • A discrete-time output regulator design method for pipeline networks is proposed. • Linear discrete-time distributed pipeline models with bounded operators are obtained. • The discrete-time output regulator equations are formulated and proved. • A Lyapunov analysis is provided to assess the stability of the pipeline models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the output regulation problem is addressed for a class of linear hyperbolic infinite-dimensional systems with spatially varying coefficients modelling a large class of convection-dominated transport reaction systems. In particular, distributed parameter systems with bounded input and unbounded output operators are considered. First, we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation. Based on the assumption that the hyperbolic system satisfies the exponential stability conditions, the main manuscript contribution is the development of two novel finite-dimensional regulators, output and error feedback regulators, such that the controlled output of the plant tracks a reference signal generated by a known signal process (exosystem). In order to guarantee the feasibility of the proposed regulators, the solvability of the corresponding Sylvester equations is discussed and the solvability conditions are provided. Finally, simulations of output regulation of an axial dispersion reactor and a relevant numerical example illustrate the main results and performance of the proposed regulators synthesis. [ABSTRACT FROM AUTHOR]

This work focuses on the receding horizon optimal control for a solar‐thermal district heating (STDH) system containing lumped parameter and distributed parameter subsystems. A common STDH system includes solar collector system, a short‐term energy storage tank and a district heating loop system with a secondary gas boiler system. The inclusion or exclusion of these components leads to different operational and working modes. Detailed system description and mathematical models are provided, and three working modes are introduced and in each mode several operations are demonstrated and addressed. Single‐objective and multiobjective problems are formulated. Moreover, in the mode where gas boiler system is included to help addressing the district heating demand, the internal model based boundary servo‐control approach is proposed and applied to obtain desired boiler water temperatures such that the expected district heating demand can be satisfied. Moreover, a boundary state observer is designed for the considered solar collector system. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1217–1233, 2018 [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Highlights: [•] Boundary model predictive control of thin film thickness. [•] Kuramoto–Sivashinsky (K–S) equation. [•] Model modal predictive controller (MPC). [•] Input and PDE states constraints satisfaction. [Copyright &y& Elsevier]

The thermal regulation problem for a lithium ion (Li-ion) battery with boundary control actuation is considered. The model of the transient temperature dynamics of the battery is given by a nonhomogeneous parabolic partial differential equation (PDE) on a two-dimensional spatial domain which accounts for the time-varying heat generation during the battery discharge cycle. The spatial domain is given as a disk with radial and angular coordinates which captures the nonradially symmetric heat-transfer phenomena due to the application of the control input along a portion of the spatial domain boundary. The Li-ion battery model is formulated within an appropriately defined infinite-dimensional function space setting which is suitable for spectral controller synthesis. The key challenges in the output feedback model-based controller design addressed in this work are: the dependence of the state on time-varying system parameters, the restriction of the input along a portion of the battery domain boundary, the observer-based optimal boundary control design where the separation principle is utilized to demonstrate the stability of the closed loop system, and the realization of the outback feedback control problem based on state measurement and interpolation of the temperature field. Numerical results for simulation case studies are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3782-3796, 2013 [ABSTRACT FROM AUTHOR]

Abstract: In this work an asymptotic stabilization of highly dissipative Kuramoto–Sivashinsky equation (KSE) by means of boundary model modal predictive control (MMPC) in the presence of input and state constraints is demonstrated. The KS equation is initially defined in an appropriate functional space setting and an exact transformation is used to reformulate the original boundary control problem as an abstract boundary control problem of the KSE partial differential equation (PDE). An appropriate discrete infinite-dimensional representation of the abstract boundary control problem is used for synthesis of low dimensional model modal predictive controller (MMPC) incorporating both the pointwise enforced KSE state constraints and input constraints. The proposed control problem formulation and the performance of the closed-loop system in the full state feedback controller realization have been evaluated through simulations. [ABSTRACT FROM AUTHOR]

Advanced pipeline leak detection and localization techniques are needed to reduce greenhouse gas emissions from hydrocarbon transportation pipelines. Developing effective leak detection and localization methods is challenging due to the spatiotemporal dynamics of process variables, the presence of process/measurement disturbances and constraints, and the limited measurement data. To address this issue, this manuscript proposes a novel moving horizon estimation design for pipeline leak detection, constrained estimation of leak size and location by using an infinite-dimensional pipeline hydraulic model. Based on the mass and momentum balance laws and the Cayley–Tustin time-discretization method, an infinite-dimensional discrete-time pipeline hydraulic model is proposed considering (unknown but bounded) disturbance and leak. By introducing a coordinate transformation, we decouple the leak size and location estimation problems. The implementable discrete-time moving horizon estimator and observer are designed for constrained leak size and location estimation. The effectiveness of the proposed designs is validated via simulation examples. • An infinite-dimensional discrete-time pipe leak model with disturbance is proposed. • A moving horizon estimator (MHE) is designed for constrained leak size estimation. • Discrete-time observer and MHE are designed for leak localization. • Numerical simulation studies are performed to verify the proposed methods. [ABSTRACT FROM AUTHOR]

This contribution develops the model predictive control for an unstable chemostat reactor. Initially, we analyze the system's model — a nonlinear first-order hyperbolic partial integro-differential equation (PIDE) — and carry the model linearization around an unstable operating condition. Employing the structure-preserving Cayley–Tustin transformation, we obtain a discrete-time model representation of the continuous model. Subsequently, we solve the operator Ricatti equations in the discrete-time setting to derive a full state feedback controller that stabilizes the closed-loop and design a Luenberger observer for state reconstruction given the system output measures. Finally, we formulate a dual-mode MPC ensuring constraint satisfaction and optimality, integrating the gain-based unconstrained full-state feedback optimal control obtained from the Ricatti equation. This dual-mode strategy describes an optimization problem where the predictive controller acts only if constraints become active within the control horizon. Simulation studies validate the controller performance, where the MPC only takes action if the constraints are predicted to be active within the control horizon while also guaranteeing closed-loop stabilization under only output feedback. This type of controller can be easily implemented with other control strategies and significantly decreases the computational costs of solving the optimal control problems when compared to other MPC approaches. • Modeling and Optimal Constrained Control of a Chemostat Reactor as a Distributed Parameter System. • Analysis of the Stability and possibility of multiple Steady-states of the open-loop system. • The observer design for the system allows the use of only output feedback. • The dual-mode controller used guarantees closed-loop stability and constraints satisfaction. [ABSTRACT FROM AUTHOR]

The boundary feedback regulator design for heat exchangers with delayed feedback is developed. Counter‐flow/parallel‐flow heat exchanger systems described by a pair of coupled transport hyperbolic partial differential equations (PDEs) with delayed boundary feedback loop modeled by the boundary time lag are considered. The coupled transport hyperbolic PDEs and boundary delay by application of boundary transformation are transformed in the corresponding linear infinite‐dimensional system utilized in the regulator design. The regulator design initially addresses a full state feedback controller realization augmented by the observer design to achieve simultaneously output exponential stabilization as well as tracking and disturbance rejection of polynomial and/or harmonic type of reference signals. The simulations studies demonstrate the proposed design for counter‐flow and parallel‐flow heat exchangers, two common configurations present in industrial practice. [ABSTRACT FROM AUTHOR]

As the optimal linear filter and estimator, the Kalman filter has been extensively utilized for state estimation and prediction in the realm of lumped parameter systems. However, the dynamics of complex industrial systems often vary in both spatial and temporal domains, which take the forms of partial differential equations (PDEs) and/or delay equations. State estimation for these systems is quite challenging due to the mathematical complexity. This work addresses discrete-time Kalman filter design and realization for linear distributed parameter systems. In particular, the structural- and energy-preserving Crank–Nicolson framework is applied for model time discretization without spatial approximation or model order reduction. In order to ensure the time instance consistency in Kalman filter design, a new discrete model configuration is derived. To verify the feasibility of the proposed design, two widely-used PDEs models are considered, i.e., a pipeline hydraulic model and a 1D boundary damped wave equation. [ABSTRACT FROM AUTHOR]

A realistic pipeline modeled by a nonlinear coupled first‐order hyperbolic partial differential equations (PDEs) system is studied for the long transportation pipeline leak detection and localization. Based on the so‐called water hammer equation, a linear distributed parameter system is obtained by linearization. The structure and energy preserving time discretization scheme (Cayley–Tustin) is used to realize a discrete infinite‐dimensional hyperbolic PDEs system without spatial approximation or model order reduction. In order to reconstruct pressure and mass flow velocity evolution with limited measurements, a discrete‐time Luenberger observer is designed by solving the operator Riccati equation. Based on this distributed observer system, data on different normal and leakage conditions (various leak amounts and positions) are generated and fed to train a support vector machine model for leak detection, amount, and position estimation. Finally, the leak detection, amount estimation, and localization effectiveness of the developed method are proved by a set of simulations. © 2019 American Institute of Chemical Engineers AIChE J, 65: e16532 2019 [ABSTRACT FROM AUTHOR]

Column flotation is an efficient method commonly used in the mineral industry to separate useful minerals from ores of low grade and complex mineral composition. Its main purpose is to achieve maximum recovery while ensuring desired product grade. This work addresses a model predictive control design for a mineral column flotation process modeled by a set of nonlinear coupled heterodirectional hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs), which accounts for the interconnection of well-stirred regions represented by continuous stirred tank reactors (CSTRs) and transport systems given by heterodirectional hyperbolic PDEs, with these two regions combined through the PDEs’ boundaries. The model predictive control considers both optimality of the process operations and naturally present input and state/output constraints. For the discrete controller design, spatially varying steady-state profiles are obtained by linearizing the coupled ODE–PDE model, and then the discrete system is obtained by using the Cayley–Tustin time discretization transformation without any spatial discretization and/or without model reduction. The model predictive controller is designed by solving an optimization problem with input and state/output constraints as well as input disturbance to minimize the objective function, which leads to an online-solvable finite constrained quadratic regulator problem. Finally, the controller performance to keep the output at the steady state within the constraint range is demonstrated by simulation studies, and it is concluded that the optimal control scheme presented in this work makes this flotation process more efficient. [ABSTRACT FROM AUTHOR]

This manuscript proposes moving horizon control and state/parameter estimation designs for pipeline networks modelled by partial differential equations (PDEs) with boundary actuation. The spatial–temporal pressure and velocity dynamics within the pipelines are described by a system of six coupled one-dimensional first-order nonlinear hyperbolic PDEs. To address the discrete-time modelling challenge and preserve the infinite-dimensional nature of the pipeline system, the Cayley–Tustin transformation is deployed for model time discretization without any spatial discretization or model reduction. Considering the lack of full state information across the entire pipeline manifold, unknown states and uncertain parameters are estimated using moving horizon estimation (MHE). Based on the estimated states and parameters, a tracking model predictive control (MPC) strategy for the discrete-time infinite-dimensional pipeline system is proposed, which enables specific operation while ensuring physical constraint satisfaction. The effectiveness of the proposed controller and estimator designs is demonstrated via numerical examples. • A branched distributed parameter pipeline system modelled by PDEs is proposed. • A moving horizon estimator for state and parameter estimation is developed. • A tracking MPC is proposed to steer the controlled output to the desired set-points. [ABSTRACT FROM AUTHOR]

Infinite-dimensional systems are essential for describing complex phenomena that exhibit continuous spatial and temporal variations. This article introduces a robust model predictive control (RMPC) design to regulate constrained multi-model infinite-dimensional systems governed by a class of hyperbolic/parabolic partial differential equations (PDEs). Model uncertainty stems from system parameters that are imprecisely determined, but can be quantitatively characterized within a certain range. The RMPC algorithm is designed in a discrete-time infinite-dimensional setting, achieved through the structure-preserving Cayley–Tustin transformation without model reduction nor spatial approximation. Robustness of the controller is ensured via constraining the future cost for each model dynamics accounting for uncertainty description. Properties of the closed-loop system are discussed, including feasibility, convergence, and asymptotic stability. The proposed controller is implemented by considering three typical infinite-dimensional distributed parameter process models, with simulation demonstrating the effectiveness and enhanced performance of the RMPC over the nominal model predictive controller. • Propose a general modeling framework for DPS with multi-model uncertainty. • Develop an RMPC for multi-model infinite-dimensional systems. • Properties of the closed-loop system are analyzed. • Three DPS examples demonstrate the efficacy of the proposed RMPC. [ABSTRACT FROM AUTHOR]

This work addresses moving horizon estimation for switching conservative linear infinite-dimensional systems described by partial differential equations (PDE), where the plant and measurement equations are corrupted with bounded disturbances, and the system mode is regarded as an unknown and unpredictable discrete state to be estimated. To address the issues associated with unbounded operators (induced by boundary or point observation and disturbance) and facilitate discrete-time moving horizon estimator design, the Cayley–Tustin transformation is deployed for model time-discretization without any spatial discretization or model reduction while preserving model essential properties. A series of observability concepts along with corresponding properties are proposed and analyzed for the switching linear infinite-dimensional discrete-time systems. A moving horizon estimation algorithm that accounts for state/output and mode estimation and constraint handling is proposed. Based on the proposed observability properties, we prove the stability of the proposed moving horizon estimator. The derived results are demonstrated by an undamped Schrödinger equation with switching group-velocity dispersion. [ABSTRACT FROM AUTHOR]