Elsevier

Journal of Process Control

Volume 80, August 2019, Pages 103-116
Journal of Process Control

Economic model predictive control for achieving offset-free operation performance of industrial constrained systems

https://doi.org/10.1016/j.jprocont.2019.04.006Get rights and content

Highlights

  • An offset-free EMPC is proposed for systems in the presence of model mismatch.

  • The mismatch causes feasibility problem in the dynamic economic optimization.

  • The method make sure the effectiveness utilizing a dynamic target optimization stage.

  • Feasibility, input-to-state stability and offset-free property are guaranteed.

Abstract

The optimization and control of large-scale industrial systems are typically based on hierarchical structures. The real-time optimizer (RTO) solves a steady-state setpoint optimal problem in long time-scale, based on a rigorous nonlinear model. The down-layer advanced controller is designed to achieve setpoint tracking based on a linear model to get efficient computation. This paradigm on the one hand lacks consideration about dynamic economic performance, on the other hand leads to the unreachable and inconsistent issues, which cause feasibility problem in the controller and the deviation of the plant operating point, i.e., offset. In this work, a linear offset-free economic model predictive controller (EMPC) is proposed for systems in the presence of model mismatch. The method constituted by a dynamic target optimization (DTO) stage and the following EMPC stage. These two stages work in a bidirectional way. The DTO stage uses the dynamic augmented model to derive a feasible trajectory with the terminal target attainable for down-layer EMPC. The EMPC optimizes the dynamic economic performance in the presence of a contractive energy-like constraint related to the feasible trajectory. Recursive feasibility, input-to-state stability (ISS) and offset-free property are guaranteed. The proposed method is applied to a fluid catalytic cracking (FCC) process. The numerical results demonstrate the effectiveness of the proposed offset-free EMPC for improving the overall economic performance and achieving stability and offset-free property in process control.

Introduction

The optimization and control of large-scale constrained systems are inseparable from the hierarchical control structure, for its ability to separate the managements and optimizations in the time-scale point of view. The classical industrial hierarchical structure is illustrated in Fig. 1(a) [1]. The real-time optimizer (RTO) solves a steady-state economic optimal problem in long time-scale, based on scheduling and planning information delivered from the upper-layer. A rigorous steady-state model and plant constraints are used in this problem to conduct optimal setpoints. The lower-layer is also referred to as the advanced control layer. Model predictive control (MPC, see e.g. [2]) aligns well in this layer in most cases, for its ability to guide the system to track the optimal setpoints within constraints, by minimizing the deviation between predicted output and setpoint. In general, MPC regulator used in hierarchical control solves a finite horizon optimal control problem (FHOCP) in short time-scale, based on a linear local model identified at certain steady-state [3].

The classical optimization of operating performance is based on the common notion that the process economical efficiency is mainly determined by the steady-state design. It is applicable when the system is kept at the same operating point for a long time. However, the growing need of intelligent manufacturing in new industrial practice urges us to develop more efficient and nimble operation methods, accounting for dynamic market fluctuation, energy pricing, customer demand changes and variable scheduling plans. The design of control strategies which can operate the system in an overall economic cost optimal fashion when setpoint changes frequently is one of the core academic issues to be studied [4].

Two main problems need to be addressed when aforementioned paradigm is applied to new process operation. The first one is the losses of steady-state economic performance in production. This problem is caused by model mismatch among RTO, MPC and the real plant. When the setpoint changes frequently, it is difficult to identify all control models in every possible operating steady-states, which makes some RTO setpoints unattainable for down-layer MPC. The controller may fail to maintain the plant operation at the desired economic optimal setpoint, resulting in steady-state offsets [5], [6]. The second problem is the lack of consideration about dynamic economic performance in transition phase. The traditional methods for improving economic performance mainly rely on RTO. The down-layer dynamic controller is set to derive inputs with respect to a cost function accounting for the distance between predicted trajectory and setpoint, without economic consideration. However, when the optimal steady-state changes frequently, the transient phase will take a considerable part in system closed-loop trajectory. In this case the dynamic economic performance cannot be neglected [7], [8].

In the past several years, many researchers have developed alternative control methods to improve the operating performance of process systems. In order to address the steady-state offset problem, a mainstream idea is to implement offset-free MPC in the advanced control layer [9], [10], [11], [12]. This algorithm consists of two stages: the steady-state target optimization (SSTO) stage and the MPC stage, as shown in Fig. 1(b). An integrating disturbance model is used in both stages, with state and integrating term estimated by an observer at each sampling instant. Specifically, the SSTO stage is designed to response for the unattainable issue of the setpoint, which uses the steady-state version of the augmented model to generate new attainable target for the down-layer MPC. The MPC regulator eliminates the effect of estimated disturbances and tracks the target at each execution time. Fruitful works on the SSTO-MPC have been done in the last decades. In [10], [13], observability conditions on the augmented system are discussed for achieving offset-free tracking property in different cases. Combined designs of the disturbance model and the observer to improve the closed-loop performance are discussed in [14], [15], [16]. In the existing literatures, there are only a few offset-free MPC methods considering the feasibility and stability issues because establishing these results is difficult under model mismatch among RTO, MPC and the plant. In [17], a robust offset-free MPC algorithm and the convergence results are presented. The method is suitable for linear systems subject to bounded disturbances with the full state measurable. In [18], an output feedback offset-free MPC is proposed by means of a multi-model approach, the resulting fuzzy MPC system is shown to be input-to-state stable.

As for the second consideration, nonlinear economic model predictive control (EMPC) is a prevalent way to account for this issue in the recent years. Researches on EMPC the rearrange the problem in one layer, that is to combine the economic optimization and dynamic control in one FHOCP, with the model accurate enough in the regulator [19], [20]. This setup has significantly improved the economic performance, especially for systems with slow dynamic and frequent setpoint changes. Theoretical discussions on stability conditions have been developed in many works, including dissipativity assumption based EMPC [19], [20], [21], [22], and Lyapunov constraint based EMPC [23], [24], [25]. In the former, a Lyapunov function can be constructed by transferring the economic cost to a “rotated cost” based on dissipativity assumption and terminal equality constraint [20]. This assumption is a condition on the nominal system with respect to the economic criteria. In [23], [24], a dual-mode nonlinear EMPC is proposed with the explicit (auxiliary) Lyapunov controller used in constraints to guarantee stability. The method is suitable for systems with bounded disturbances. In [25], a contractive constraint based EMPC is proposed for nominal nonlinear systems. The asymptotic stability is established only using contraction of the constraint. Although EMPC has made a lot of progress in theory, how to design an industrially applicable EMPC in the presence of model mismatch is still an open topic. In recent works [26], [27], a nonlinear offset-free EMPC constituted by the economic steady-state target optimization stage and the EMPC stage is proposed, with modifier-adaption strategy used in the augmented model to ensure the reached steady-state is the optimal equilibrium of the real plant. The improved algorithm in [27] shows better performance than conventional EMPC and offset-free MPC. This controller is formulated for systems without the need for control hierarchy. To further enrich the practical implementations for processes in the presence of time-scale separation, a multi-model offset-free EMPC is proposed in [8]. This method shows better robustness for systems with large operating regimes. Feasibility and stability are discussed under the dissipativity assumption. However, in many cases the plant uncertainty may destroy the dissipativity property of the optimal setpoint. More works for general cases need to be developed.

The goal of this paper is to develop an offset-free EMPC algorithm in the advanced control layer for large-scale systems in the presence of model mismatch. There are some difficulties when EMPC is combined with the offset-free design. In order to achieve offset-free control, integrating term estimated and corrected by observer needs to be added in the linear predictive model. However, different from conventional tracking MPC, additional stability constraints are required due to the general form of the cost function in EMPC. Such stability conditions usually require the assistance of terminal constraints. The existence of the feasible dynamic trajectory initialized at the integrating states may not be guaranteed. The effect of the estimation error needs to be treated carefully on the feasibility and stability conditions of EMPC design. The synchronous convergence of the observer and the controller needs to be discussed for ensuring the composite system stability, which has not been well investigated yet.

To this end, the offset-free EMPC is designed in a double-layered structure, which consists of a dynamic target optimizer (DTO), and a down-layer EMPC controller, as illustrated in Fig. 1(c). The DTO-EMPC works in a bidirectional way. At every sampling instant of the advanced control layer, the DTO stage uses the dynamic augmented model to derive a feasible trajectory with the terminal target attainable for down-layer EMPC. The trajectory is initialized at current estimated integrating states, and the terminal output is optimized as close as possible to the RTO setpoint. The down-layer EMPC is designed to derive the input sequence with respect to a cost function accounting for dynamic open-loop performance, instead of tracking the DTO target directly. A contractive constraint related to the feasible trajectory is designed in order to guarantee the convergence of the closed-loop system. The paper is organized as follows: Section 2 formulates the problem and preliminaries setup. Section 3 presents the main contribution of this paper, including the DTO stage, the EMPC stage and the implementation strategy. Section 4 presents the convergence results and offset-free property of the resulting closed-loop system. Section 5 presents the DTO-EMPC simulation and analysis on the FCC system. Section 6 gives the conclusion of the paper.

Notation: and I denote the set of real and integer number. Set Sa ≔ {x ∈ S ; x ≥ a}, with aI. S[a,b] ≔ {x ∈ S ; a ≤ x ≤ b}, with a,bI. Norm || · || denotes the Euclidean norm. For positive definite matrix A, xA2 denotes xTAx. A continuous function α(·):00 is a class-κ function if it is strictly increasing and α(0) = 0; it is a class-κ function if it is a class-κ function and α(s)→ ∞, as s→ ∞. (xk+i|k, yk+i|k) denotes the predicted state and output pair at k + i based on (xk|k, yk|k), with k,iI0.

Section snippets

System description

In this paper, we consider a general form of (approximate) nonlinear discrete-time model that describes the input and output relationship of the system dynamic:xk+1m=f(xkm,uk)yk=g(xkm)where f:nx×nunx and g:nxny are continuous functions; xkmnx denotes the system state at time kI0; uknu and ykny denote, respectively, the manipulated input and measurable output. Eq. (1) is also referred to as the approximated model because in industrial practice, identifying the nominal plant model

Offset-free EMPC formulation

The main contribution of this work is introduced in this section. As depicted in Fig. 2, the offset-free EMPC design is incorporated in the two-stage structure, which consists of the dynamic target optimizer (DTO) and the down-layer EMPC controller. The DTO-EMPC works in a bidirectional way.

At each sampling instant k, the observer estimates state xˆk and integrating term dˆk by the prediction error of the output. The augmented system (5) initialized at (xˆk,dˆk) is used as the predictive model

Stability analysis

In this section we analyze the closed-loop property of the DTO-EMPC algorithm. The effect of the estimation error in prediction is specified first and then the feasibility of DTO Problem (15), Problem (17) and EMPC Problem (21) at each instant are verified. Finally, by deriving a candidate Lyapunov function, we show that under mild assumptions in Section 3, the resulting system is input-to-state stable and achieves output offset-free performance w.r.t. the DTO reset target.

Lemma 1

For the system in

Application to a fluid catalytic cracking (FCC) process example

The FCC unit plays a very important role in the plant-wide economic optimization of the refinery industry, for its function of converting high-molecular-weight hydrocarbons to more valuable lower-molecular-weight products [31]. Fig. 3 shows a typical schematic of the FCC process. The preheated fresh feed is sent to the riser reactor. After mixing with the regenerated catalyst, the feed vapors flow up the riser. Meanwhile the catalytic cracking reaction occurs. The hot regenerated catalyst not

Conclusion

In this work, an offset-free EMPC based on the dynamic target optimizer is proposed, for improving the overall operation performance of large-scale constrained systems. The DTO-EMPC works in a bidirectional way. At each sampling instant, the DTO stage derives a feasible trajectory attainable for the EMPC stage. The down-layer EMPC is designed not to track the DTO reset target directly, but to optimize the dynamic open-loop economic performance in the presence of a contractive constraint related

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 61590924, 61673273, 61833012). The authors thank the editor and reviewers for their valuable and constructive comments.

References (34)

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