Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients

We focus on output feedback control of distributed processes whose infinite dimensional representation in appropriate Hilbert subspaces can be decomposed to finite dimensional slow and infinite dimensional fast subsystems. The controller synthesis issue is addressed using a refined adaptive proper orthogonal decomposition (APOD) approach to recursively construct accurate low dimensional reduced order models (ROMs) based on which we subsequently construct and couple almost globally valid dynamic observers with robust controllers. The novelty lies in modifying the data ensemble revision approach within APOD to enlarge the ROM region of attraction. The proposed control approach is successfully used to regulate the Kuramoto‐Sivashinsky equation at a desired steady state profile in the absence and presence of uncertainty when the unforced process exhibits nonlinear behavior with fast transients. The original and the modified APOD approaches are compared in different conditions and the advantages of the modified approach are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4595–4611, 2013     

Output feedback control of dissipative PDE systems with partial sensor information based on adaptive model reduction

We address the problem of control of spatially distributed processes in the presence of measurement constraints. Specifically, we assume the availability of sensors that measure part of the state spatial profile. The measurements are utilized for the derivation and on‐demand update of reduced order models (ROM) based on an extension of the adaptive proper orthogonal decomposition (APOD) method using a snapshot reconstruction technique. The proposed Gappy‐APOD methodology constructs locally accurate low‐dimensional ROM thus resulting in a computationally efficient alternative to using a large‐dimensional ROM with global validity. Based on the low‐dimensional ROM and continuous measurements available from point sensors a Lyapunov‐based static output feedback controller is subsequently designed. The proposed controller design method is illustrated on an unstable process modeled by the Kuramoto‐Sivashinsky equation, when the designed controller successfully stabilizes the process even in the presence of model uncertainty. © 2012 American Institute of Chemical Engineers AIChE J, 59: 747–760, 2013     

A linear model predictive control algorithm for nonlinear large‐scale distributed parameter systems

This work provides a framework for linear model predictive control (MPC) of nonlinear distributed parameter systems (DPS), allowing the direct utilization of existing large‐scale simulators. The proposed scheme is adaptive and it is based on successive local linearizations of the nonlinear model of the system at hand around the current state and on the use of the resulting local linear models for MPC. At every timestep, not only the future control moves are updated but also the model of the system itself. A model reduction technique is integrated within this methodology to reduce the computational cost of this procedure. It follows the equation‐free approach (see Kevrekidis et al., Commun Math Sci. 2003;1:715–762; Theodoropoulos et al., Proc Natl Acad Sci USA. 2000;97:9840‐9843), according to which the equations of the model (and consequently of the simulator) need not be given explicitly to the controller. The latter forms a “wrapper” around an existing simulator using it in an input/output fashion. This algorithm is designed for dissipative DPS, dissipativity being a prerequisite for model reduction. The equation‐free approach renders the proposed algorithm appropriate for multiscale systems and enables it to handle large‐scale systems. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Dissipativity‐based distributed model predictive control with low rate communication

Distributed or networked model predictive control (MPC) can provide a computationally efficient approach that achieves high levels of performance for plantwide control, where the interactions between processes can be determined from the information exchanged among controllers. Distributed controllers may exchange information at a lower rate to reduce the communication burden. A dissipativity‐based analysis is developed to study the effects of low communication rates on plantwide control performance and stability. A distributed dissipativity‐based MPC design approach is also developed to guarantee the plantwide stability and minimum plantwide performance with low communication rates. These results are illustrated by a case study of a reactor‐distillation column network. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3288–3303, 2015     

Feedback control of dissipative PDE systems using adaptive model reduction

The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen‐Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Feedback control of hyperbolic distributed parameter systems

Hyperbolic distributed parameter systems (DPS) represent a large number of industrial processes with spatially nonuniform operating variable profiles. Research has been conducted to develop high-performance control strategies for these systems by exploiting their high-fidelity models. In this paper, a feedback control method that yields improved performance is proposed for DPS modelled by first-order hyperbolic partial differential equations (PDEs) using the method of characteristics. Simulation results show that this method can provide effective control for the systems modelled by a scalar PDE as well as a system of PDEs. Further, it can efficiently compensate the effect of model-plant mismatch and effectively reject the disturbances.     

Off-line model reduction for on-line linear MPC of nonlinear large-scale distributed systems

Model predictive control (MPC) is an efficient method for the controller design of a large number of processes. However, linear MPC is often inappropriate for controlling nonlinear large-scale systems, while non-linear MPC can be computationally costly. The resulting optimization-based procedure can lead to local minima due to the, non-convexities that non-linear systems can exhibit. To overcome the excessive computational cost of MPC application for large-scale nonlinear systems, model reduction methodology in conjunction with efficient system linearizations have been exploited to enable the efficient application of linear MPC for nonlinear distributed parameter systems (DPS). An off-line model reduction technique, the proper orthogonal decomposition (POD) method, combined with a finite element Galerkin projection is first used to extract accurate non-linear low-order models from the large-scale ones. Trajectory Piecewise-Linear (TPWL) methodologies are subsequently developed to construct a piecewise linear representation of the reduced nonlinear model, both in a static and in a dynamic fashion. Linear MPC, based on quadratic programming, can then be efficiently performed on the resulting low-order, piece-wise affine system. Our combined methodology is readily applicable in combination with advanced MPC methodologies such as multi-parametric MPC (MP-MPC) (Pistikopoulos, 2009). The stabilisation of the oscillatory behaviour of a tubular reactor with recycle is used as an illustrative example to demonstrate our methodology.     

基于PDE降阶模型的最优控制

针对带有约束条件的偏微分方程（PDE）模型最优控制的实时性要求和巨大的内存开销问题,提出了基于降阶模型的输入/状态约束的最优实时控制方法。采用特征正交分解和奇异值分解以及Galerkin投影方法导出了动态PDE具有高精度离散形式的状态空间低阶模型,提出了一定输入/状态约束条件下的基于二次规划单步滚动最优控制并与基于线性二次调节器的极值验证最优控制策略相互验证。通过对流-扩散-反应过程的控制仿真结果证明了所提方法的高效性和准确性。     

Sequential and iterative architectures for distributed model predictive control of nonlinear process systems

In this work, we focus on distributed model predictive control of large scale nonlinear process systems in which several distinct sets of manipulated inputs are used to regulate the process. For each set of manipulated inputs, a different model predictive controller is used to compute the control actions, which is able to communicate with the rest of the controllers in making its decisions. Under the assumption that feedback of the state of the process is available to all the distributed controllers at each sampling time and a model of the plant is available, we propose two different distributed model predictive control architectures. In the first architecture, the distributed controllers use a one‐directional communication strategy, are evaluated in sequence and each controller is evaluated only once at each sampling time; in the second architecture, the distributed controllers utilize a bi‐directional communication strategy, are evaluated in parallel and iterate to improve closed‐loop performance. In the design of the distributed model predictive controllers, Lyapunov‐based model predictive control techniques are used. To ensure the stability of the closed‐loop system, each model predictive controller in both architectures incorporates a stability constraint which is based on a suitable Lyapunov‐based controller. We prove that the proposed distributed model predictive control architectures enforce practical stability in the closed‐loop system and optimal performance. The theoretical results are illustrated through a catalytic alkylation of benzene process example. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013     

Distributed lyapunov‐based model predictive control with neighbor‐to‐neighbor communication

This work considers distributed predictive control of large‐scale nonlinear systems with neighbor‐to‐neighbor communication. It fulfills the gap between the existing centralized Lyapunov‐based model predictive control (LMPC) and the cooperative distributed LMPC and provides a balanced solution in terms of implementation complexity and achievable performance. This work focuses on a class of nonlinear systems with subsystems interacting with each other via their states. For each subsystem, an LMPC is designed based on the subsystem model and the LMPC only communicates with its neighbors. At a sampling time, a subsystem LMPC optimizes its future control input trajectory assuming that the states of its upstream neighbors remain the same as (or close to) their predicted state trajectories obtained at the previous sampling time. Both noniterative and iterative implementation algorithms are considered. The performance of the proposed designs is illustrated via a chemical process example. © 2014 American Institute of Chemical Engineers AIChE J 60: 4124–4133, 2014     

An aggregation model reduction method for one‐dimensional distributed systems

A method for deriving reduced dynamic models of one‐dimensional distributed systems is presented. It inherits the concepts of the aggregated modeling method of Lévine and Rouchon originally derived for simple staged distillation models and can be applied to both spatially discrete and continuous systems. The method is based on partitioning the system into intervals of steady‐state systems, which are connected by dynamic aggregation elements. By presolving and substituting the steady‐state systems, a discrete low‐order dynamic model is obtained. A characteristic property of the aggregation method is that the original and the reduced model assume identical steady states. For spatially continuous systems, the method is an alternative to discretization methods like finite‐difference and finite‐element methods. Implementation details of the method are discussed, and the principle is illustrated on three example systems, namely a distillation column, a heat exchanger, and a fixed‐bed reactor. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Design of multi‐actor distributed processing systems: A game‐theoretical approach

The manufacturing of a final product could be the result of a value chain involving several processing plants distributed across several distinct owners; a feature that may prevent the application of classical process design approaches that depend on a centralized treatment of the complete processing network. In this article we propose and develop a game‐theoretical framework and specific methodologies, which allow the optimal design of distributed processing systems, through the decentralized strategies of independent actors. The resulting process design corresponds to a Nash Equilibrium point among the interacting actors. Its optimality and the justification of the independent strategies that led to it, are theoretically based on (and constrained by) the properties of the 2‐level Lagrangian approach. The article also discusses the use of penalty‐term approaches, which can extend the applicability of the proposed framework and design methodologies to problems for which the underlying convexity assumptions of the 2‐level Lagrangian approach may not be possible to ascertain. A series of case studies illustrate the application of the proposed ideas to distributed processing networks of various structures. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3369–3391, 2016     

Robust distributed economic model predictive control based on differential dissipativity

In this article, a robust distributed economic model predictive control (DEMPC) approach is developed for plant-wide chemical processes. The proposed approach achieves arbitrary feasible setpoints that may vary frequently, attenuates the plant-wide effects of unknown disturbances and minimizes a plant-wide economic cost. In this approach, a plant-wide process is represented as a network of process units and each process unit is controlled by an individual controller which shares a plant-wide optimization economic objective and stability conditions through the network. To ensure the convergence of process variables to arbitrary setpoints, a contraction condition is developed for the DEMPC, based on the contraction theory. To deal with the effects of interactions among process units, the concept of dissipativity is adopted. Using sum-separable control contraction metrics, a reference-independent robust stability condition is developed to ensure the plant-wide disturbance effects (under interactions among process units) to be attenuated in terms of differential ℒ₂ gain and represented by a plant-wide differential dissipativity condition, which is converted into the differential dissipativity conditions that individual controllers need to satisfy. This approach facilitates the optimization of plant-wide economic costs with global constraints in a distributed way, allowing efficient implementation of alternating direction method of multipliers (ADMM). The proposed approach is illustrated using a reactor-separator process.     

Real‐time economic model predictive control of nonlinear process systems

Closed‐loop stability of nonlinear systems under real‐time Lyapunov‐based economic model predictive control (LEMPC) with potentially unknown and time‐varying computational delay is considered. To address guaranteed closed‐loop stability (in the sense of boundedness of the closed‐loop state in a compact state‐space set), an implementation strategy is proposed which features a triggered evaluation of the LEMPC optimization problem to compute an input trajectory over a finite‐time prediction horizon in advance. At each sampling period, stability conditions must be satisfied for the precomputed LEMPC control action to be applied to the closed‐loop system. If the stability conditions are not satisfied, a backup explicit stabilizing controller is applied over the sampling period. Closed‐loop stability under the real‐time LEMPC strategy is analyzed and specific stability conditions are derived. The real‐time LEMPC scheme is applied to a chemical process network example to demonstrate closed‐loop stability and closed‐loop economic performance improvement over that achieved for operation at the economically optimal steady state. © 2014 American Institute of Chemical Engineers AIChE J, 61: 555–571, 2015     

Explicit distributed model predictive control design for chemical processes under constraints and uncertainty

In multivariable industrial processes, the common distributed model predictive control strategy is usually unable to deal with complex large-scale systems efficiently, especially under system constraints and high control performance requirements. Based on this situation, we use the distributed idea to divide the large-scale system into multiple subsystems and transform them into the state space form. Combined with the output tracking error term, we build an extended non-minimal state space model that includes output error and measured output and input. When dealing with system constraints, the new constraint matrix is divided into range and kernel space by using the explicit model predictive control algorithm, which reduces the difficulty of solving constraints in the extended system and further improves the overall control performance of the system. Finally, taking the coke furnace pressure control system as an example, the proposed algorithm is compared with the conventional distributed model predictive control algorithm using non-minimal state space, and the simulation results show the feasibility and superiority of this method.     

Error‐triggered on‐line model identification for model‐based feedback control

In industry, it may be difficult in many applications to obtain a first‐principles model of the process, in which case a linear empirical model constructed using process data may be used in the design of a feedback controller. However, linear empirical models may not capture the nonlinear dynamics over a wide region of state‐space and may also perform poorly when significant plant variations and disturbances occur. In the present work, an error‐triggered on‐line model identification approach is introduced for closed‐loop systems under model‐based feedback control strategies. The linear models are re‐identified on‐line when significant prediction errors occur. A moving horizon error detector is used to quantify the model accuracy and to trigger the model re‐identification on‐line when necessary. The proposed approach is demonstrated through two chemical process examples using a model‐based feedback control strategy termed Lyapunov‐based economic model predictive control (LEMPC). The chemical process examples illustrate that the proposed error‐triggered on‐line model identification strategy can be used to obtain more accurate state predictions to improve process economics while maintaining closed‐loop stability of the process under LEMPC. © 2016 American Institute of Chemical Engineers AIChE J, 63: 949–966, 2017