# Economic Model Predictive Control with Zone Tracking

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Setup

#### 2.1. Notation

#### 2.2. System Description and Control Objective

## 3. EMPC with Zone Tracking

#### 3.1. EMPC Formulation

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Stability Analysis

**Definition**

**1.**

**Assumption**

**A1.**

**Assumption**

**A2.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

#### 3.3. Prioritized Zone Tracking

**Definition**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**4.**

**Definition**

**3.**

**Theorem**

**3.**

**Proof.**

**Remark**

**5.**

## 4. Modified Target Zone

Algorithm 1: Modified target zone. | |

1. | Choose some $M\in {\mathbb{I}}_{1}^{N}$ and $\alpha \ge 0$ |

2. | Set ${\mathbb{Z}}_{0}=({x}_{s},{u}_{s})$ |

3. | for $i=0:M-1$ |

Calculate ${\mathbb{Z}}_{i+1}$ with Equation. (20) | |

end | |

4. | The modified target zone is ${\mathbb{Z}}_{t}^{\prime}={\mathbb{Z}}_{M}$ |

**Theorem**

**4.**

- (i)
- If ${c}_{1}$ is an exact zone tracking penalty for ${\mathbb{Z}}_{t}^{\prime}$ for all $x\left(n\right)\in {\mathbb{X}}_{N}({\mathbb{Z}}_{t},{x}_{s})$, then the modified target zone is forward invariant under the closed-loop system. That is,$$(x\left(n\right),u\left(n\right))\in {\mathbb{Z}}_{t}^{\prime}\phantom{\rule{0.277778em}{0ex}}\u27f9\phantom{\rule{0.277778em}{0ex}}(x(n+1),u(n+1))\in {\mathbb{Z}}_{t}^{\prime}$$
- (ii)
- If in addition Assumptions 1 and 2 hold, the transient economic performance in the modified target zone ${\mathbb{Z}}_{t}^{\prime}$ is upper bounded such that for any time instant K where $(x\left(K\right),u\left(K\right))\in {\mathbb{Z}}_{t}^{\prime}$, the following holds:$$\sum _{n=K}^{\infty}\left(\right)open="("\; close=")">{\ell}_{e}(x\left(n\right),u\left(n\right))-{\ell}_{e}({x}_{s},{u}_{s})$$

**Proof.**

**Remark**

**6.**

**Remark**

**7.**

## 5. Simulation

#### 5.1. Example 1

#### 5.1.1. EMPC Tracking the Original Target Zone

#### 5.1.2. EMPC Tracking the Modified Target Zone

#### 5.2. Example 2

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Lu, J.Z. Challenging control problems and emerging technologies in enterprise optimization. Control Eng. Pract.
**2003**, 11, 847–858. [Google Scholar] [CrossRef] - Mayne, D.Q.; Rawlings, J.B.; Rao, C.V.; Scokaert, P.O.M. Constrained model predictive control: Stability and optimality. Automatica
**2000**, 36, 789–814. [Google Scholar] [CrossRef] - Rawlings, J.B.; Angeli, D.; Bates, C.N. Fundamentals of economic model predictive control. In Proceedings of the 51th IEEE Conference on Decision and Control, Grand Wailea, HI, USA, 10–13 December 2012; pp. 3851–3861. [Google Scholar]
- Grüne, L. Economic receding horizon control without terminal constraints. Automatica
**2013**, 49, 725–734. [Google Scholar] [CrossRef] - Ellis, M.; Durand, H.; Christofides, P.D. A tutorial review of economic model predictive control methods. J. Process Control
**2014**, 24, 1156–1178. [Google Scholar] [CrossRef] - Liu, S.; Liu, J. Economic model predictive control with extended horizon. Automatica
**2016**, 73, 180–192. [Google Scholar] [CrossRef] - Scokaert, P.; Rawlings, J.B. Feasibility issues in linear model predictive control. AIChE J.
**1999**, 45, 1649–1659. [Google Scholar] [CrossRef] - Kerrigan, E.C.; Maciejowski, J.M. Soft constraints and exact penalty functions in model predictive control. In Proceedings of the UKACC International Conference Control, Cambridge, UK, 4–7 October 2000. [Google Scholar]
- De Oliveira, N.M.C.; Biegler, L.T. Constraint handing and stability properties of model-predictive control. AIChE J.
**1994**, 40, 1138–1155. [Google Scholar] [CrossRef] - Zeilinger, M.N.; Morari, M.; Jones, C.N. Soft Constrained Model Predictive Control With Robust Stability Guarantees. IEEE Trans. Autom. Control
**2014**, 59, 1190–1202. [Google Scholar] [CrossRef] - Askari, M.; Moghavvemi, M.; Almurib, H.A.F.; Haidar, A.M.A. Stability of Soft-Constrained Finite Horizon Model Predictive Control. IEEE Trans. Ind. Appl.
**2017**, 53, 5883–5892. [Google Scholar] [CrossRef] - Ferramosca, A.; Limon, D.; González, A.H.; Odloak, D.; Camacho, E. MPC for tracking zone regions. J. Process Control
**2010**, 20, 506–516. [Google Scholar] [CrossRef] - Lez, A.H.G.; Marchetti, J.L.; Odloak, D. Robust model predictive control with zone control. IET Control Theory Appl.
**2009**, 3, 121–135. [Google Scholar] - González, A.H.; Odloak, D. A stable MPC with zone control. J. Process Control
**2009**, 19, 110–122. [Google Scholar] [CrossRef] - Blanchini, F. Set invariance in control. Automatica
**1999**, 35, 1747–1767. [Google Scholar] [CrossRef] - Amrit, R.; Rawlings, J.B.; Angeli, D. Economic optimization using model predictive control with a terminal cost. Annu. Rev. Control
**2011**, 35, 178–186. [Google Scholar] [CrossRef] - Diehl, M.; Amrit, R.; Rawlings, J.B. A Lyapunov function for economic optimizing model predictive control. IEEE Trans. Autom. Control
**2011**, 56, 703–707. [Google Scholar] [CrossRef] - Rao, C.V.; Rawlings, J.B. Linear programming and model predictive control. J. Process Control
**2000**, 10, 283–289. [Google Scholar] [CrossRef] - Fletcher, R. Practical Methods of Optimization; John Wiley & Sons: New York, NY, USA, 2013. [Google Scholar]
- Kerrigan, E.C.; Maciejowski, J.M. Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 12–15 December 2000; Volume 5, pp. 4951–4956. [Google Scholar]
- Keerthi, S.; Gilbert, E. Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints. IEEE Trans. Autom. Control
**1987**, 32, 432–435. [Google Scholar] [CrossRef] - Halvgaard, R.; Poulsen, N.K.; Madsen, H.; Jørgensen, J.B. Economic model predictive control for building climate control in a smart grid. In Proceedings of the IEEE Innovative Smart Grid Technologies, Washington, DC, USA, 16–20 January 2012; pp. 1–6. [Google Scholar]

**Figure 1.**Closed-loop input trajectories of EMPC of Equation (10) with ${c}_{1}={10}^{4}$ and ${c}_{2}={10}^{2}$ (solid), ${c}_{2}={10}^{3}$ (dotted), ${c}_{2}={10}^{4}$ (dashed), ${c}_{2}={10}^{5}$ (dash dotted), respectively. Shaded area depicts the input target zone. The upper and lower part correspond to initial state $x\left(0\right)=-5$ and $x\left(0\right)$ = 5, respectively.

**Figure 2.**Closed-loop economic performance of EMPC of Equation (10) with ${c}_{1}={10}^{4}$ and ${c}_{2}={10}^{2}$ (solid), ${c}_{2}={10}^{3}$ (dotted), ${c}_{2}={10}^{4}$ (dashed), ${c}_{2}={10}^{5}$ (dash dotted), respectively. The upper and lower part correspond to initial state $x\left(0\right)=5$ and $x\left(0\right)=-5$, respectively.

**Figure 3.**Closed-loop input trajectories of EMPC in Equation (10) with $x\left(0\right)=5$, ${c}_{1}={10}^{4}$ and ${c}_{2}={10}^{4}$, $N=10$ (solid), $N=20$ (dotted), $N=30$ (dashed), $N=40$ (dash dotted), respectively.

**Figure 4.**The constraint set $\mathbb{Z}$ (box), target zone ${\mathbb{Z}}_{t}$ (shaded rectangle) and modified target zone ${\mathbb{Z}}_{t}^{\prime}$ (parallelogram). The circle indicates the optimal steady state $({x}_{s},{u}_{s})$.

**Figure 5.**Closed-loop input trajectories of EMPC of Equation (10) with modified zone ${\mathbb{Z}}_{t}^{\prime}$ in Equation (24), with ${c}_{1}={10}^{4}$ and ${c}_{2}={10}^{2}$ (solid), ${c}_{2}={10}^{3}$ (dotted), ${c}_{2}={10}^{4}$ (dashed), ${c}_{2}={10}^{5}$ (dash dotted), respectively. Shaded area depicts the input target zone. The upper and lower part correspond to initial state $x\left(0\right)=-5$ and $x\left(0\right)=5$, respectively.

**Figure 6.**Closed-loop economic performance of EMPC of Equation (10) with modified zone ${\mathbb{Z}}_{t}^{\prime}$ in Equation (24), with ${c}_{1}={10}^{4}$ and ${c}_{2}={10}^{2}$ (solid), ${c}_{2}={10}^{3}$ (dotted), ${c}_{2}={10}^{4}$ (dashed), ${c}_{2}={10}^{5}$ (dash dotted), respectively. The upper and lower part correspond to initial state $x\left(0\right)=5$ and $x\left(0\right)$ = $-5$, respectively.

**Figure 8.**Phase space plot of different control schemes. The polyhedron depicts the modified target zone ${\mathbb{Z}}_{t}^{\prime}$.

**Figure 9.**Room temperature and heat input profiles of EMPC with modified target zone ${\mathbb{Z}}_{t}^{\prime}$ by Algorithm 1 with $\alpha =2000$ and $M=1,6,12,18$.

**Table 1.**Transient economic performance $\sum _{n=0}^{50}}{\ell}_{e}(x\left(n\right),u\left(n\right))$ of EMPC of Equation (10).

${\mathit{c}}_{2}={10}^{2}$ | ${\mathit{c}}_{2}={10}^{3}$ | ${\mathit{c}}_{2}={10}^{4}$ | ${\mathit{c}}_{2}={10}^{5}$ | |
---|---|---|---|---|

$x\left(0\right)=-5$ | 2.0195 | 2.0225 | 1.2560 | 1.2465 |

$x\left(0\right)=5$ | 76.1218 | 79.5542 | 86.5742 | 103.0781 |

**Table 2.**Comparison of the transient economic performance $\sum _{n=0}^{50}}{\ell}_{e}(x\left(n\right),u\left(n\right))$ of EMPC tracking the target zone ${\mathbb{Z}}_{t}$ and EMPC tracking the modified zone ${\mathbb{Z}}_{t}^{\prime}$.

Tracking ${\mathbb{Z}}_{\mathit{t}}$ | ${\mathit{c}}_{2}={10}^{2}$ | ${\mathit{c}}_{2}={10}^{3}$ | ${\mathit{c}}_{2}={10}^{4}$ | ${\mathit{c}}_{2}={10}^{5}$ |
---|---|---|---|---|

$x\left(0\right)=-5$ | 2.0195 | 2.0225 | 1.2560 | 1.2465 |

$x\left(0\right)=5$ | 76.1218 | 79.5542 | 86.5742 | 103.0781 |

Tracking${\mathbb{Z}}_{\mathbf{t}}^{\prime}$ | ${\mathbf{c}}_{\mathbf{2}}={\mathbf{10}}^{\mathbf{2}}$ | ${\mathbf{c}}_{\mathbf{2}}={\mathbf{10}}^{\mathbf{3}}$ | ${\mathbf{c}}_{\mathbf{2}}={\mathbf{10}}^{\mathbf{4}}$ | ${\mathbf{c}}_{\mathbf{2}}={\mathbf{10}}^{\mathbf{5}}$ |

$x\left(0\right)=-5$ | 2.0195 | 2.0225 | 1.2560 | 1.2465 |

$x\left(0\right)=5$ | 57.4483 | 52.7305 | 54.9608 | 64.2366 |

Variable | Unit | Description |
---|---|---|

${T}_{r}$ | °C | Room air temperature |

${T}_{f}$ | °C | Floor temperature |

${T}_{w}$ | °C | Water temperature in floor heating pipes |

${W}_{c}$ | W | Heat pump compressor input power |

${T}_{a}$ | °C | Ambient temperature |

${\varphi}_{s}$ | W | Solar radiation power |

MPC Tracking $({\mathit{x}}_{\mathit{s}},{\mathit{u}}_{\mathit{s}})$ | EMPC Tracking ${\mathbb{Z}}_{\mathit{t}}$ | EMPC Tracking ${\mathbb{Z}}_{\mathit{t}}^{\prime}$ | |
---|---|---|---|

Additional electricity cost (USD) | 363.3 | 411.1 | 369.8 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, S.; Liu, J.
Economic Model Predictive Control with Zone Tracking. *Mathematics* **2018**, *6*, 65.
https://doi.org/10.3390/math6050065

**AMA Style**

Liu S, Liu J.
Economic Model Predictive Control with Zone Tracking. *Mathematics*. 2018; 6(5):65.
https://doi.org/10.3390/math6050065

**Chicago/Turabian Style**

Liu, Su, and Jinfeng Liu.
2018. "Economic Model Predictive Control with Zone Tracking" *Mathematics* 6, no. 5: 65.
https://doi.org/10.3390/math6050065