Economic model predictive control for transport-reaction systems with target profiles

https://doi.org/10.1016/j.conengprac.2020.104684Get rights and content

Highlights

  • Transport-reaction processes (TRP) that being described by hyperbolic/parabolic PDEs.

  • Economic MPC for TRP from both convergence and performance point of view.

  • The economic optimizations are carried out without any type of spatial approximation.

  • Typical examples illustrate the effectiveness with emphasis on enhanced performance.

Abstract

For transport-reaction processes with certain general cost functions indicating comprehensive profit, the existing linear–quadratic optimal control or model predictive control (MPC) methods may be conservative. Due to their spatial characteristics, dynamic evolutions and final target static spatial distributions under different control strategies may result in remarkable variation of operation performance. This article presents a novel economic model predictive control (EMPC) synthesis for systems described by hyperbolic/parabolic partial differential equations (PDEs) with pre-optimized static spatially distributed profile, from both convergence and performance points of view. Instead of deriving approximate ordinary differential equations (ODEs) and applying finite-dimensional control strategies, this work addresses the optimization and stabilization problem on the basis of infinite dimensional model itself, which is realized by Cayley–Tustin (CT) transformation without any type of spatial approximation. The accumulated economic cost, input/output constraints are explicitly considered in the finite horizon optimal control problem (FHOCP), with an additional energy-like contractive constraint accounting for closed-loop convergence toward the target optimal steady-state profile. By directly solving Lyapunov functions associated with PDEs models, the parameters in the FHOCP are suitably defined such that the composite EMPC system admits algorithm feasibility and closed-loop stability. Finally, the proposed method is applied to typical examples; numerical results indicates the effectiveness and remarkable performance improvement of the EMPC method compared to standard MPC results.

Introduction

Transport-reaction processes are usually mathematically described by hyperbolic/parabolic partial differential equations (PDEs), according to the underlying dominated convection/diffusion mechanism; see e.g. Curtain and Zwart (2012) and Ray (1981) for practical examples and references therein. In order to respond to stringent production demands associated with transport-reaction processes, the controller design should not only consider system constraints inclusion and stability guarantees, but also account for overall economic performance, which refers to the accumulation of indexes in time and space, such as productivity, by-product formation and/or energy consumption indexes during operation.

This operation performance is defined on the system state/input profiles, and is determined by their evolution in transient phase and final steady-state spatial distribution; for instance, the controlled fluid temperatures and the temperature of additional heat source in the heat exchanger systems (Ozorio Cassol et al., 2019); the rate of production through out the rod and the temperature of the cooling medium in the exothermic catalytic reaction system (Bendersky & Christofides, 2000); the overall heat collected by the fluid, the inlet fluid flow rate and the outlet fluid temperature in the distributed solar collector system (Farahat et al., 2009, Xu et al., 2018), etc.

The existing model-based control strategies in the past years focus on closed-loop convergence toward target steady-states and constraint satisfaction, which in general can best (and may only) be responsible to steady-state economic operation performance. The infinite-dimensional feature of the model induces complexity in the controller design and implementation. Therefore, a mainstream idea is to approximate the model into the discrete time/space model and apply finite-dimensional controllers. For diffusion dominated processes described by parabolic PDEs, the spatial approximation can be realized by Galerkin’s method with the concept of inertial manifolds, Christofides and Chow (2002). The controller formulation can subsequently be transferred into a finite horizon optimal control problem (FHOCP) and explicitly address the input and state constraints, and be realized through the finite-dimensional model predictive control (MPC) synthesis; see e.g. Dubljevic et al., 2006, Dubljevic et al., 2005, Liu et al., 2014 and Yang and Dubljevic (2013). Other works consider utilizing time–space decomposition-based methods to derive the reduced-order spatial–temporal Hammerstein model for applying efficient MPC to process with uncertainty, see e.g. Li and Qi (2010) and Li et al. (2011). For convection dominated processes described by hyperbolic PDEs, the continuous spectrum feature of the spatial differential operator prevents aforementioned Galerkin’s technique, which increases the complexity of controller design. In Dubljevic et al. (2005) the finite difference method is applied to the mechanism model; and Aggelogiannaki and Sarimveis (2009) and Hulkó et al. (2017) on the other hand, consider utilizing data-driven approaches to obtain the finite-dimensional model and thus the constrained MPC can be applied. Other notable works consider design linear–quadratic (LQ) optimal control on the basis of infinite-dimensional Hilbert space setting and the operator Riccati equation, toward the unconstrained stabilization problem of hyperbolic PDEs; see e.g. Aksikas et al. (2009) and Moghadam et al. (2013). In general, the controller design for constrained transport-reaction processes remains elusive. Utilizing approximation method enables designers to apply fruitful finite-dimensional results; however may potentially incorrectly describe the nature of PDEs, which challenges the validity of the controllers. Recently in Xu and Dubljevic (2017), an important bilinear transformation method is proposed such that the constrained MPC algorithm can be realized without any spatial approximation. The control model is obtained by introducing an important resolvent operator into the Cayley-Tustin (CT) transformation scheme, which fully captures the nature of the PDEs. In Dubljevic and Humaloja (2020), utilizing this crucial implementation, the constrained stabilization of arbitrary exponentially stabilizable regular linear systems is achieved, by specifying an additional terminal penalty function while without terminal constraint in FHOCP.

As a result of the inherent complexities in infinite-dimensional setting, the dynamic operation performance of transport-reaction processes was hardly considered in the existing researches, even though it takes a considerable part in the overall accumulated cost numerically due to the additional spatial scale. A possible way to address this issue is to do aforementioned approximation and apply the finite-dimensional economic model predictive control (EMPC) schemes, see, e.g. Angeli et al., 2012, Ellis and Christofides, 2014 and Ellis et al. (2014) and the references therein. In general, the control index in the FHOCP contains economic cost explicitly or implicitly, which directly responds to dynamic economic optimization. The general form (not necessarily positive and/or convex) of the control index inevitably brings stability issues to the resulting process. Additional constraints/conditions are required in the FHOCP or for the system itself. There are two common stable EMPC approaches. The first one is dissipativity/turnpike property-based EMPC schemes, which are designed specifically for ODEs satisfying dissipativity condition w.r.t. the economic cost; see Angeli et al., 2012, Diehl et al., 2011, Faulwasser et al., 2017, Ferramosca et al., 2014 and Risbeck and Rawlings (2019). These conditions mathematically allow the economic cost to be transferred into a positive definite function, and further the stability can be achieved by imposing the terminal (equality) constraints. Recently in notable work by Faulwasser and Bonvin (2015), based on an exact turnpike property of the ODE, the convergence of the closed-loop EMPC system is guaranteed without additional terminal constraints nor terminal penalties. The other mainstream idea refers to the additional constraints-based EMPC schemes, which are designed for general ODEs without dissipativity property. The so-called Lyapunov-based EMPC schemes (see, e.g. Heidarinejad et al. (2012)) achieve stability of ODEs by involving auxiliary Lyapunov feedback controllers in formulations. Along this line (Lao et al., 2014) has made the first attempt on economic optimization for Riesz-spectral parabolic PDEs, by applying Galerkin’s approximation method and finite-dimensional EMPC. Another notable works consider enforcing convergence without the assistant of auxiliary controllers, e.g. the contractive constraint-based EMPC in He et al. (2016) and its output-feedback extension in Yang et al. (2019) for systems described by ODEs.

For a more general class of transport-reaction processes described by hyperbolic/parabolic PDEs, the constrained economic optimization and control should essentially resort to the original nature of the infinite-dimensional system, which motivates the design of EMPC approaches on the basis of aforementioned CT transformation (see Dubljevic and Humaloja (2020) and Xu and Dubljevic (2017)). Different from the MPC formulations, additional conditions (including convergence constraints and/or stringent terminal constraints) are required in EMPC optimization. These demanding ingredients may result in feasibility problem when applied to infinite-dimensional cases, due to the different design settings associated with controllability, pole placement/state feedback concepts of the PDEs. To the best of our knowledge, these issues remains intractable. Thus, performance improvement of distributed parameter systems (DPS) is still an open and challenging topic to be explored.

In this paper, a constrained economic model predictive control synthesis is proposed. The economic optimizations (on both dynamic phase and final static distributions) are carried out in the infinite spatial setting that fully captures the nature of the systems. Specifically, an optimal spatially distributed profile is pre-calculated; and a constrained FHOCP toward this target profile is solved in a receding horizon framework, with the predictive model obtained by CT transformation without spatial approximation. The optimization formulation contains explicitly the accumulated economic cost in time and space to account for dynamic performance, while an additional contractive constraint is involved to achieve convergence toward the target distribution. Significant setting on the controller parameters and terminal ingredients are derived by solving the operator Lyapunov equations for different transport-reaction systems. For stable hyperbolic PDEs, the algorithm feasibility and closed-loop convergence of the infinite system is guaranteed resorting to the exact solution of the Lyapunov functions. For stable/unstable parabolic PDEs, the contractive constraint is defined on the dominant subspace resorting to the spectral projected solution of the Lyapunov functions. This allows the convergence of the closed-loop profile w.r.t. the dominant subspace and subsequently yields the stability of the parabolic PDE system; which is achieved essentially only by the contraction of energy instead of utilizing approximate ODE model in both optimization and control. The paper is organized as follows: Section 2 formulates the problem. Section 3 presents the proposed EMPC synthesis and detailed setting toward stable hyperbolic PDEs and stable/unstable parabolic PDEs, respectively. Section 4 analyzes closed-loop properties and finally, typical hyperbolic/parabolic PDE examples are provided in Section 5.

Notation: Rm denotes the set of m dimensional real vectors. I denotes the set of real integers. L2(0,1)n denotes a Hilbert space under inner products z1(ζ),z2(ζ)01z1(ζ)Tz2(ζ)¯dζ, in which z1(ζ),z2(ζ) are the state vectors containing n elements. Z denotes the associated induced norm of the Hilbert space Z. A function z(ζ) is a.c. means that it is absolutely continuous on its domain. Set Sa{xS;xa}, with aI. S[a,b]{xS;axb}, with a,bI. For a bounded linear operator T():Z1Z2, TsupzZ1,z0TzZ2zZ1. Finally, z(ζ,k+i|k) and yk+i|k refers to the predicted state profile and output at k instant, stemming from z(ζ,k|k) and vk+i|k, with k,iI0.

Section snippets

System description

Transport-reaction processes (TRP) are typical distributed parameter systems that exhibit prominent spatial and temporal characteristics, due to the convection/diffusion dominated physical phenomena in operation.

As is presented in Fig. 1, the control synthesis development for TRP, should not only respond to the (spatially dependent) constraints on the system state (e.g., the outlet fluid temperature) and the manipulated input (e.g. the jacket temperature), but also consider operation

Controller synthesis and implementation

The main contribution of this work is introduced in this section. The design framework is illustrated in Fig. 4. The synthesis is established on the infinite Hilbert state-space setting and its discrete-time property-preserving representation. Depend on different spectrum characteristics of TRP, the stabilizability conditions are suitably defined to ensure algorithm feasibility and closed-loop stability. The dynamic economic optimization and control of PDEs are realized in an unified

Closed-loop properties

In this section, we investigate the closed-loop PDE systems under proposed EMPC synthesis, from both convergence and performance perspectives. Our results remain essentially the basic concepts in the existing (economic) model predictive control theory, with novel extension to PDE-relevant chemical process systems. In order to establish feasibility and stability results, the invariant property of terminal set zT(ζ) is discussed first, given Q(ζ) and Q̄(ζ) satisfying the continuous Lyapunov

Motivation example revisited

In this section, the proposed EMPC scheme is applied to the steam-jacketed tubular heat exchanger system presented in Section 2.2. Prediction horizon N=10. The auxiliary function in the contractive constraint follows (12), γ=0.8, Q(ζ)=5[0,1](ζ), R=0.4 and Q̄(ζ) is the solution to Eq. (17), i.e. Q̄(ζ)=3.1250e1.6+1.6ζ+3.1250. Denote the closed-loop implementation input sequence as UEMPC. We consider following two target static spatial distribution: (a) the optimal pair us,1=65.5070, xs,1=30e0.8

Conclusion

In this work, a novel economic model predictive control synthesis is proposed, for constrained transport-reaction systems described by hyperbolic PDEs and Riesz-spectral parabolic PDEs. The optimization and control are carried out in the infinite spatial setting derived from accurate Cayley-Tustin transformation. The method is able to achieve constrained stabilization w.r.t. optimal target steady-state, and in the same time improve overall operation performance, which has been demonstrated in

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank the founding from the National Key Research and Development Program of China (2018AAA0101700 and 2018YFB1701101), National Natural Science Foundation of China (NSFC) (61590924 and 61833012).

References (37)

Cited by (8)

  • Frequency response-based decoupling tuning for feedforward compensation ADRC of distributed parameter systems

    2022, Control Engineering Practice
    Citation Excerpt :

    As the theory of functional analysis and PDEs developed, more advanced control methods and findings gradually improved. Model predictive control is applies to thermal control (Tilli et al., 2022), transport-reaction systems (Yang et al., 2021), fractional order systems (Ntouskas et al., 2018), solar plants (Silva et al., 2003) and drying stage (Daraoui et al., 2010). Extremum seeking is used for Laser pulse shaping (Ren et al., 2012).

  • Distributed economic model predictive control for an industrial fluid catalytic cracking unit ensuring safe operation

    2022, Control Engineering Practice
    Citation Excerpt :

    A two-layer EMPC structure was proposed by Sildir et al. (2014), where the upper EMPC calculates an optimal trajectory of the economy and the lower regulation MPC tracks the reference trajectory. A linear offset-free economic model predictive controller was applied to the FCC unit to cope with the feasibility problem in the controller and the deviation of the plant operating point (Yang et al., 2021). As for the one-layer EMPC framework, a nonlinear process system model of the FCC unit is developed to describe the dynamic performance of the unit as well as the predictive model in the EMPC optimization problem.

  • Self-Optimizing Control Strategy for Distributed Parameter Systems

    2023, Industrial and Engineering Chemistry Research
  • Managing logistics systems: Planning and analysis for a successful supply chain

    2022, Managing Logistics Systems: Planning and Analysis for a Successful Supply Chain
View all citing articles on Scopus
1

These authors contributed equally to this work.

View full text