Search

This article studies the problem of prescribed‐time constrained tracking control for a class of 2×2$$ 2\times 2 $$ hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation. [ABSTRACT FROM AUTHOR]

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]

The model predictive controller (MPC) design is developed for a tubular chemical reactor, considering a second‐order hyperbolic partial differential equation as the model of the transport‐reaction process with boundary actuation. Without loss of generality, closed–closed boundary conditions and relaxed total flux are assumed. At the same time, the model is discretized in time by the Cayley–Tustin method, and, under the assumption that only the reactor's output is measurable, the observer design for the state reconstruction is addressed and integrated with the MPC design. The Luenberger observer gain is obtained by solving the operator Ricatti equation in the discrete‐time setting, while the MPC accounts for constrained and optimal control. The simulations show that the output‐based MPC design stabilizes the system under the input and output constraints satisfaction. In addition, to address the models' disparities, the results for both parabolic and hyperbolic equations are presented and discussed. [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]

The present manuscript considers the observer design problem for a class of scalar semi-linear hyperbolic partial differential equation (PDE) systems with a recycle loop through the boundary point. The design method of Kazantzis and Kravaris [(1998). Nonlinear observer design using Lyapunov's auxiliary theorem. Systems & Control Letters, 34(5), 241–247] developed for the nonlinear finite dimensional systems observer design is extended to semi-linear hyperbolic PDE systems. The observer design problem is tackled through a first-order associated PDE. Due to the existence of spatial partial derivative operator, Lyapunov's Auxiliary Theorem originated from finite-dimensional systems can be no longer applied to seek conditions ensuring solvability of the associated PDE. In this manuscript, a new theorem is formulated and proved to ensure the solvability. The solution for the associated PDE is locally analytic nonlinear coordinate transformation which provides foundation for observer realization and a series solution approach is developed. Simulation examples are presented to study the performance of the proposed observer. [ABSTRACT FROM AUTHOR]

In this manuscript, common chemical engineering transport‐reaction systems are modeled by taking into consideration the inertia in the transport phenomena, leading to hyperbolic partial differential equations (PDEs), which have finite speed of propagation. The findings provide the derivation of the hyperbolic transport‐reaction PDEs and a direct comparison with their corresponding dissipative parabolic PDEs. The stability analysis of the dynamical systems is considered and complemented by numerical simulations for typical processes, such as the heat diffusion in a slab, transport‐reaction inside a chemical reactor, and a phase change problem. The results demonstrate the effects associated with the modeling and/or limits of the approximation used in the derivation and application of the hyperbolic PDE for transport‐reaction systems, and the difference between the dynamics of this type of equation and the generally used parabolic PDEs. [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]

This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation. [ABSTRACT FROM AUTHOR]

The new adaptive fault estimation scheme is proposed for a class of hyperbolic partial differential equations in this paper. The multiplicative actuator and sensor faults are considered. There are two cases that require special consideration: (1). only one type of fault (actuator or sensor) occurs; (2). two types of faults occurred simultaneously. To solve the problem of fault estimation, three challenges need to be solved: (1). No prior information of fault type is known; (2). Unknown faults are always coupled with state and input; (3). Only one boundary measurement is available. The original plant is converted to Observer canonical form. Two filters are proposed and novel adaptive laws are developed to estimate unknown fault parameters. With the help of the proposed update laws, the true state of the faulty plant can be estimated by the proposed observers composed of two filters. By selecting a suitable Lyapunov function, it is proved that under unknown external disturbance, the estimation errors of state parameters and fault parameters decay to arbitrarily small value. Finally, the validity of the proposed observer and adaptive laws is verified by numerical simulation. [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]

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]

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]

Abstract: This contribution addresses the development of a linear quadratic (LQ) regulator in order to control the concentration profiles along a catalytic distillation column, which is modelled by a set of coupled hyperbolic partial differential and algebraic equations (PDAEs). The proposed method is based on an infinite-dimensional state-space representation of the PDAE system which is generated by a transport operator. The presence of the algebraic equations, makes the velocity matrix in the transport operator, spatially varying, non-diagonal, and not necessarily negative through of the domain. The optimal control problem is treated using operator Riccati equation (ORE) approach. The existence and uniqueness of the non-negative solution to the ORE are shown and the ORE is converted into a matrix Riccati differential equation which allows the use of a numerical scheme to solve the control problem. The result is then extended to design an optimal proportional plus integral controller which can reject the effect of load losses. The performance of the designed control policy is assessed through a numerical study. [Copyright &y& Elsevier]

The paper develops an adaptive constrained tracking control technique for a class of 2 × 2 hyperbolic partial differential equation (PDE) systems with boundary actuator dynamics, which are described by a set of ordinary differential equations (ODEs) in the presence of unknown parametric nonlinearities. Since the control input only appears in the uncertain ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite difficult and the existing direct boundary control approaches are ineffective. Moreover, in this paper, a more challenging problem is considered such that the controlled output and the states of ODE actuators are constrained. To this end, by utilizing finite and infinite dimensional backstepping techniques, barrier Lyapunov functions (BLFs) and adaptive methods, a novel adaptive tracking control approach is proposed. It is the first time that such a constrained tracking control problem is addressed for the PDE-ODE coupled systems considered in this paper. On the basis of the presented method, the rigorous theoretical proof is provided to show that the PDE controlled output and all the states of the ODE actuator stay within the predefined compact sets. Finally, the results are illustrated via a comparative numerical simulation. [Display omitted] • For a linear 2 × 2 hyperbolic PDE systems, the high-order unknown nonlinear actuator dynamics described by a set of ODEs are considered in the boundary point of the PDE. A tracking controller with adaptive dynamic compensation mechanism is developed such that the tracking control problem is addressed for the PDE-ODE coupled system considered in this paper. • For practical requirements for state constraints such as safe operations, physical limitations and so on, the Barrier Lyapunov function is employed to constrain the spatially distributed output of infinite-dimensional system and all the actuator states. [ABSTRACT FROM AUTHOR]