Elsevier

Automatica

Volume 73, November 2016, Pages 180-192
Automatica

Economic model predictive control with extended horizon

https://doi.org/10.1016/j.automatica.2016.06.027Get rights and content

Abstract

In this work, we consider economic model predictive control (EMPC) with extended horizon based on an auxiliary controller. The extension of the horizon is realized by employing a terminal cost which characterizes the economic performance of the auxiliary controller over a finite terminal horizon. The proposed EMPC design is easy to construct and computationally efficient. We analyze the stability and performance of the proposed EMPC design with special attention paid to the impact of the terminal horizon. It is shown that for strictly dissipative systems satisfying mild assumptions, a finite terminal horizon is sufficient to guarantee the convergence and performance of the EMPC to be approximately upper-bounded by that of the auxiliary controller.

Introduction

Model predictive control (MPC), or receding horizon control, refers to a control methodology that approximates the solution of a constrained infinite-horizon optimal control problem by solving finite-horizon optimal control problems in a receding horizon fashion. Existing efforts to achieve closed-loop stability of conventional nonlinear model predictive control (NMPC) can be divided into two categories. One is to resort to the design of MPC by employing a point-wise terminal constraint (Mayne and Michalska, 1990, Nicolao et al., 1996), terminal cost (with terminal region constraint) (Chen and Allgöwer, 1998, Jadbabaie and Hauser, 2005, Limón et al., 2006), or Lyapunov-based constraint (Mhaskar, El-Farra, & Christofides, 2006). The other one is to rely on the inherent stability of conventional MPC by adopting a sufficiently large optimization horizon (Grimm et al., 2005, Grüne and Rantzer, 2008, Primbs and Nevistić, 2000). In general, the approaches under the first category have readily provable nominal stability but tend to be conservative. Either the optimality or the size of the feasibility region has to be compromised in order to explicitly handle the nonlinearity of the system. On the other hand, while the design-free MPCs under the second category are capable of achieving near-optimal solutions, they could be computationally prohibitive with a large control horizon.

A methodology that provides an ideal trade-off between the two categories is to extend the prediction horizon of NMPC based on an auxiliary controller or control law. In this way, the prediction horizon of the optimization problem can be increased without significantly increasing the computational effort. It is shown in Alamir and Bornard (1995) that a finite prediction horizon is sufficient to guarantee stability. In Magni, De Nicolao, and Scattolini (2001), a locally optimal linear controller is utilized to extend the prediction horizon of the NMPC design. The appealing features of this approach are that enlargement of the stability region and local optimality can be achieved without relying on a large control horizon. It is also worth mentioning that the separation between the control horizon and prediction horizon is not new. The concept arises along the early versions of MPC and is well embraced in industrial MPC (Qin & Badgwell, 2003).

In recent years, a new form of MPC, which is referred to as the economic model predictive control (EMPC) has attracted considerable academic attention. In EMPC, the quadratic-type cost functions used in conventional MPC are replaced with general economic cost functions that are not necessarily positive-definite with respect to the optimal steady state. Consequently, standard stability analysis techniques to use the value function of conventional MPC as a Lyapunov function is no longer viable. In fact, steady-state operation may not even be the economically optimal operation for EMPC. It has been realized that dissipativity plays an important role in characterizing the optimality of steady-state operation as well as establishing the stability of EMPC (Angeli et al., 2012, Muller et al., 2015, Müller et al., 2015). Different EMPC designs have been proposed which stem from the conventional NMPC designs. For example, EMPC with point-wise terminal constraint (Diehl, Amrit, & Rawlings, 2011), EMPC with terminal cost (Amrit et al., 2011, Müller et al., 2014), and EMPC with Lyapunov-based constraint (Ellis and Christofides, 2014, Heidarinejad et al., 2012). These EMPC designs also suffer the shortcomings of the conventional NMPC designs under the first category—they could be overly conservative or difficult to design. In another line of research (Grüne, 2013, Grüne and Stieler, 2014), EMPC without terminal conditions is studied. This line of research reveals some intrinsic properties of EMPC. It is shown that under certain controllability and dissipativity conditions, near-optimal performance can be achieved if a sufficiently large control horizon is used. However, a large control horizon could make online implementations of the EMPC impractical.

Due to these considerations, it is natural to also consider extending the prediction horizon of EMPC based on an auxiliary controller. An attempt was made in our previous work (Liu, Zhang, & Liu, 2015). In Liu et al. (2015), a terminal cost was employed which characterizes the economic performance of the system under a stabilizing controller over a finite time window referred to as the terminal horizon. The proposed EMPC is shown to be very computationally efficient and capable of achieving near-optimal asymptotic performance. However, stability and transient performance of the proposed EMPC are not addressed in Liu et al. (2015). It is conceivable that for general nonlinear systems, the system state does not necessarily converge to the optimal steady state under the proposed EMPC scheme with a finite terminal horizon. In the present work, we systematically discuss the stability and performance of EMPC with extended horizon based on an auxiliary controller. We show that for systems strictly dissipative with respect to the stage cost, a finite terminal horizon is sufficient to guarantee the convergence and performance of the EMPC to be approximately upper-bounded by that of the auxiliary controller. The results explain the computational efficiency of the EMPC framework and provide insights into the terminal cost design of EMPC.

The rest of this work is organized as follows: The system and EMPC formulation are set up in Section  2. Section  3 addresses the stability and convergence of the EMPC design. Practical stability is established for strictly dissipative systems satisfying mild assumptions. Under stronger conditions, the shrinkage of the region which the system state is eventually driven into is shown to be exponential with respect to the increase of the terminal horizon. For a special type of systems satisfying a further condition on the storage function (including conventional MPC with quadratic cost), exponential stability can be achieved. Interestingly, the same result for this type of systems may not be achieved by an EMPC without terminal condition. Section  4 discusses the asymptotic and transient performance of the EMPC design. Results on the asymptotic performance for general nonlinear systems are provided first. Stronger results on the transient performance for strictly dissipative systems are derived subsequently, based on different stability conditions from Section  3. In Section  5, two numerical examples are used to verify our analysis. Finally, we conclude our results in Section  6.

Section snippets

Notation

Throughout this work, the operator || denotes the Euclidean norm of a scalar or a vector. The symbol ‘’ denotes set subtraction such that AB{xA,xB}. The symbol Br(xs) denotes the open ball centered at xs with radius r such that Br(xs){x:|xxs|<r}. A continuous function α:[0,a)[0,) is said to belong to class K if it is strictly increasing and satisfies α(0)=0. A class K function α is called a class K function if α is unbounded. A continuous function σ:[0,)[0,a) is said to belong to

Stability and convergence

We restrict our attention to systems that are strictly dissipative with respect to the economic cost functions. Systems of this type are optimally operated at steady state (Angeli et al., 2012, Muller et al., 2015, Müller et al., 2015). Since the optimal steady state is not necessarily a minimizer of the economic stage cost, EMPC with a finite horizon in general cannot stabilize the optimal steady state. We will first establish practical stability of the EMPC design with respect to the terminal

Asymptotic and transient performance

In this section, we characterize upper bounds on the performance of the EMPC design. First, we consider the asymptotic performance of the EMPC for systems that are not necessarily dissipative. Then we show that for strictly dissipative systems satisfying different stability conditions as discussed in Section  3, stronger results on the transient performance can be obtained. In our analysis, the transient performance of the EMPC is compared against a benchmark system h which is an augmented

Numerical examples

Example 5.1

This example is a linearized continuously stirred tank reactor model taken from Diehl et al. (2011) and Grüne (2013). Consider the control system: x(k+1)=(0.835300.10650.9418)x(k)+(0.004570.00457)u(k)+(0.55590.5033) with the stage cost l(x,u)=|x|2+0.05u2. The state and input constraints are: X=[100,100]2 and U=[10,10]. The optimal steady state that solves the steady-state optimization problem of Eq. (2) is xs[3.5463,14.6531]T and us6.1637. We choose the auxiliary controller to be the

Conclusions

In this work, we provided a general framework to analyze the stability and performance of EMPC with extended terminal horizon. While a finite terminal horizon is in general not sufficient to ensure stability of the optimal steady state, it is sufficient to achieve practical stability for strictly dissipative systems under mild assumptions. Further conditions to ensure the exponential shrinkage of the practical stability region are provided. For a special case including conventional MPC with

Su Liu was born in Wuxi, China, in 1987. He received his B.S. degree in Process Control from Beijing Institute of Technology in 2010, the M.S. degree in Control Science and Engineering from Zhejiang University in 2013. He is currently a Ph.D. candidate in Chemical Engineering at the University of Alberta, Canada. His research interests include model predictive control, distributed predictive control and control performance assessment.

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Su Liu was born in Wuxi, China, in 1987. He received his B.S. degree in Process Control from Beijing Institute of Technology in 2010, the M.S. degree in Control Science and Engineering from Zhejiang University in 2013. He is currently a Ph.D. candidate in Chemical Engineering at the University of Alberta, Canada. His research interests include model predictive control, distributed predictive control and control performance assessment.

Jinfeng Liu was born in Wuhan, China, in 1982. He received the B.S. and M.S. degrees in Control Science and Engineering in 2003 and 2006, respectively, both from Zhejiang University, and the Ph.D. degree in Chemical Engineering from the University of California, Los Angeles in 2011. Since 2012 he has been with the University of Alberta, where he is currently an Assistant Professor in the Department of Chemical and Materials Engineering. His research interests are in the general areas of process control theory and practice with emphasis on model predictive control, networked and distributed state estimation and control, and fault-tolerant process control. A more detailed description of his research interests and a list of his publications can be found at http://www.ualberta.ca/~jinfeng/index.html.

The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Riccardo Scattolini under the direction of Editor Ian R. Petersen.

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