Economic model predictive control for transport-reaction systems with target profiles
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: denotes the set of dimensional real vectors. denotes the set of real integers. denotes a Hilbert space under inner products , in which are the state vectors containing elements. denotes the associated induced norm of the Hilbert space . A function is . means that it is absolutely continuous on its domain. Set , with . , with . For a bounded linear operator , . Finally, and refers to the predicted state profile and output at instant, stemming from and , with .
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 is discussed first, given and 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 . The auxiliary function in the contractive constraint follows (12), , , and is the solution to Eq. (17), i.e. . Denote the closed-loop implementation input sequence as . We consider following two target static spatial distribution: (a) the optimal pair ,
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).
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These authors contributed equally to this work.