Predictive control of parabolic PDEs with state and control constraints (original) (raw)
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Chemical Engineering Science, 2006
This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with boundary control actuation subject to input and state constraints. Under the assumption that measurements of the PDE state are available, various finite-dimensional and infinitedimensional predictive control formulations are presented and their ability to enforce stability and constraint satisfaction in the infinitedimensional closed-loop system is analyzed. A numerical example of a linear parabolic PDE with unstable steady state and flux boundary control subject to state and control constraints is used to demonstrate the implementation and effectiveness of the predictive controllers.
Boundary predictive control of parabolic PDEs
2006 American Control Conference, 2006
This work focuses on boundary model predictive control of linear parabolic partial differential equations (PDEs) with input and state constraints. Various predictive control formulations are presented and their ability to enforce stability and constraint satisfaction in the infinite-dimensional closed-loop system is analyzed. A numerical example of a linear parabolic PDE with unstable steady state and flux boundary control subject to state and control constraints is used to demonstrate the implementation and effectiveness of the predictive controllers.
Model predictive control formulation for a class of time-varying linear parabolic PDEs
Proceedings of the 2011 American Control Conference, 2011
This paper considers the model predictive control (MPC) formulation for a class of discrete time-varying linear state-space model representations of parabolic partial differential equations (PDEs) with time-dependent parameters. The time-dependence of the parameters are due to the changes in physical properties or operating conditions of the system such as phase transformation, reactor catalyst fouling, and/or domain deformations which arise in many industrial processes. The MPC formulation is constructed for the low dimensional discrete finite-dimensional state space representation of the PDE system and constraints on input and infinite-dimensional state evolution are incorporated in the convex optimization algorithm. The underlying MPC synthesis is utilizing the appropriately defined model representation of the PDE and yields convex quadratic optimization problem which includes input and PDE state constraints. Using the illustrative example of a crystal growth process in which the time-varying property is associated with the evolution of grown crystal, the proposed time-varying MPC formulation is implemented for the optimal crystal temperature regulation problem under the presence of input and state constraints.
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IEEE Transactions on Automatic Control
Model predictive controllers leverage system dynamics models to solve constrained optimal control problems. However, computational requirements for real-time control have limited their use to systems with low-dimensional models. Nevertheless many systems naturally produce high-dimensional models, such as those modeled by partial differential equations that when discretized can result in models with thousands to millions of dimensions. In such cases the use of reduced order models (ROMs) can significantly reduce computational requirements, but model approximation error must be considered to guarantee controller performance. In this work a reduced order model predictive control (ROMPC) scheme is proposed to solve robust, output feedback, constrained optimal control problems for highdimensional linear systems. Computational efficiency is obtained by leveraging ROMs obtained via projection-based techniques, and guarantees on robust constraint satisfaction and stability are provided. Performance of the approach is demonstrated in simulation for several examples, including an aircraft control problem leveraging an inviscid computational fluid dynamics model with dimension 998,930.
SIAM Journal on Control and Optimization, 2009
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Feedback control of hyperbolic PDE systems
AIChE Journal, 1996
This article deals with distributed parameter systems described by first-order hyperbolic partial differential equations (PDEs), for which the manipulated input, the controlled output, and the measured output are distributed in space. For these systems, a general output-feedback control methodology is developed employing a combination of theory of PDEs and concepts ffom geometric control. A concept of characteristic index is introduced and used for the synthesis of distributed state-feedback laws that guarantee output tracking in the closed-loop system. Analytical formulas of distributed outputfeedback controllers are derived through combination of appropriate distributed state observers with the developed state-feedback controllers. Theoretical analogies between our approach and available results on stabilization of linear hyperbolic PDEs are also identified. The developed control methodology is implemented on a nonisothermal plug-flow reactor and its performance is evaluated through simulations. P. D. Christofides is presently at the approach limits the controller performance, and may Iead to unacceptable control quality.
Finite-Dimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds
Journal of Mathematical Analysis and Applications, 1997
This paper introduces a methodology for the synthesis of nonlinear finite-dimensional output feedback controllers for systems of quasi-linear parabolic partial Ž . differential equations PDEs , for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinitedimensional stable fast complement. Combination of Galerkin's method with a novel procedure for the construction of approximate inertial manifolds for the Ž . PDE system is employed for the derivation of ordinary differential equation ODE Ž . systems whose dimension is equal to the number of slow modes that yield solutions which are close, up to a desired accuracy, to the ones of the PDE system, for almost all times. These ODE systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow up to a desired accuracy, a prespecified response for almost all times. ᮊ 1997 Academic Press 398
Linear model predictive control for transport-reaction processes
AIChE Journal, 2017
The article deals with systematic development of linear model predictive control algorithms for linear transport-reaction models emerging from chemical engineering practice. The finite-horizon constrained optimal control problems are addressed for the systems varying from the convection dominated models described by hyperbolic partial differential equations (PDEs) to the diffusion models described by parabolic PDEs. The novelty of the design procedure lies in the fact that spatial discretization and/or any other type of spatial approximation of the process model plant is not considered and the system is completely captured with the proposed Cayley-Tustin transformation, which maps a plant model from a continuous to a discrete state space setting. The issues of optimality and constrained stabilization are addressed within the controller design setting leading to the finite constrained quadratic regulator problem, which is easily realized and is no more computationally intensive than the existing algorithms. The methodology is demonstrated for examples of hyperbolic/parabolic PDEs.
Asian Journal of Control, 2020
In this paper, we propose a novel method to solve the model predictive control (MPC) problem for linear time-invariant (LTI) systems with input and output constraints. We establish an algebraic control rule to solve the MPC problem to overcome the computational time of online optimization methods. For this purpose, we express system constraints as a continuous function through the tangent-hyperbolic function, hence the optimization problem is reformulated. There are two steps for the solution of the optimization problem. In the first step, the optimal control signal is determined by the use of the necessary condition for optimality, assuming that there is only input constraint. In the latter, the solution obtained in the first step is revised to keep the system states in a feasible region. It is shown that the solution is suboptimal. The proposed solution method is simulated for three different sample systems, and the results are compared with the classical MPC, which show that the new algebraic method dramatically reduces the computational time of MPC.