# Measurable Disturbances Compensation: Analysis and Tuning of Feedforward Techniques for Dead-Time Processes

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Feedback Plus Feedforward Controllers

#### 2.2. Extended GPC with Implicit Feedforward Compensator

- a)
- ${\rho}_{d}<0$$\begin{array}{c}{\alpha}_{p}\left(z\right)=\left(\right)open="\{"\; close>\begin{array}{cc}{\alpha}_{{p}_{1}},\hfill & (nd+{d}_{v})=1\hfill \\ {\alpha}_{{p}_{1}}+{\alpha}_{{p}_{2}}{z}^{-1}+\cdots +{\alpha}_{{p}_{(nd+{d}_{v})}}{z}^{-({d}_{v}+nd-1)},\hfill & (nd+{d}_{v})1\hfill \end{array}\end{array}{\alpha}_{f}\left(z\right)={\alpha}_{{f}_{1}}{z}^{1}+{\alpha}_{{f}_{1}}{z}^{2}+\cdots +{\alpha}_{{f}_{N-({d}_{v}-d+nd)}}{z}^{N-({d}_{v}-d+nd)}$

- b)
- ${\rho}_{d}=0$$\begin{array}{c}{\alpha}_{p\left(z\right)}=\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & (nd+{d}_{v})=0\hfill \\ {\alpha}_{{p}_{1}},\hfill & (nd+{d}_{v})=1\hfill \\ {\alpha}_{{p}_{1}}+{\alpha}_{{p}_{2}}{z}^{-1}+\cdots +{\alpha}_{{p}_{(nd+{d}_{v})}}{z}^{-({d}_{v}+nd-1)},\hfill & (nd+{d}_{v})1\hfill \end{array}\end{array}$

- c)
- ${\rho}_{d}>0$$\begin{array}{c}{\alpha}_{p}\left(z\right)=\left(\right)open="\{"\; close>\begin{array}{cc}0,\hfill & (nd+{d}_{v})=0\hfill \\ {\alpha}_{{p}_{1}},\hfill & (nd+{d}_{v})=1\hfill \\ {\alpha}_{{p}_{1}}+{\alpha}_{{p}_{2}}{z}^{-1}+\cdots +{\alpha}_{{p}_{(nd+{d}_{v})}}{z}^{-({d}_{v}+nd-1)},\hfill & (nd+{d}_{v})1\hfill \end{array}\end{array}$

## 3. Design Rules

#### 3.1. Advanced PID Plus Feedforward Tuning Rules

#### 3.1.1. Open-Loop Design

#### 3.1.2. Closed-Loop Adaption

#### 3.1.3. Tuning Guideline

- Tune the PI or PID feedback controller Equation (2) at will to fulfill the given closed-loop specifications.
- Set $\rho ={L}_{u}-{L}_{v}$, and design $\alpha >(\rho +{L}_{ff})/{T}_{v}$ accordingly:$$\alpha =\left(\right)open="\{"\; close>\begin{array}{c}{\displaystyle \frac{\rho}{2{T}_{v}\left(\right)open="("\; close=")">1-\sqrt{{e}^{-\frac{\rho}{{T}_{v}}}}}}\\ \text{aggressive}\end{array}1.7& \text{moderate}\\ 4& \text{conservative}$$
- Tune the lead-lag feedforward controller Equation (3) using the proposed rules:$$\begin{array}{c}{L}_{ff}=\text{max}(-\rho ,0)\hfill \\ {B}_{ff}={T}_{u}\hfill \\ {T}_{ff}={T}_{v}-\left(\right)open="("\; close=")">\rho +{L}_{ff}{\alpha}^{-1}\hfill \end{array}$$
- If the control signal peak is not acceptable, increase ${T}_{ff}$ as needed.
- End of design.

#### 3.2. Extended Design for GPC Implicit Feedforward Compensator

#### 3.2.1. Reference Filter

#### 3.2.2. FSP-GPC Approach

- Compute the horizons and weighting factors in order to obtain the desired set point tracking performance for the nominal plant, such as described in the previous sections.
- Estimate the uncertainties of the plant, and tune the filter ${F}_{r}\left(z\right)$ in order to obtain robust stability and the highest bandwidth for the disturbance rejection performance.

## 4. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Feedforward Compensator

## Appendix B. GPC with Disturbance Compensation

## References

- Åström, K.; Hägglund, T. Advanced PID Control; ISA Press: Research Triangle Park, Durham, NC, USA, 2006. [Google Scholar]
- Guzmán, J.L.; Hägglund, T. Simple tuning rules for feedforward compensators. J. Process Control
**2011**, 21, 92–102. [Google Scholar] [CrossRef] - Pawlowski, A.; Guzmán, J.L.; Normey-Rico, J.E.; Berenguel, M. Improving feedforward disturbance compensation capabilities in Generalized Predictive Control. J. Process Control
**2012**, 22, 527–539. [Google Scholar] [CrossRef] - Pawlowski, A.; Fernández, I.; Guzmán, J.L.; Berenguel, M.; Acién, F.G.; Normey-Rico, J.E. Event-based predictive control of pH in tubular photobioreactors. Comput. Chem. Eng.
**2014**, 65, 28–39. [Google Scholar] [CrossRef] - Veronesi, M.; Visioli, A. Automatic tuning of feedforward controllers for disturbance rejection. Ind. Eng. Chem. Res.
**2014**, 57, 2764–2770. [Google Scholar] [CrossRef] - Hast, M.; Hägglund, T. Design of optimal low-order feedforward controllers. In Proceedings of the 2nd IFAC Conference on Advances in PID Control, Brescia, Italy, 28–30 March 2012.
- Seborg, D.E.; Edgar, T.E.; Mellichamp, D.A. Process Dynamics and Control—2nd Edition; Wiley: Cambridge, MA, USA, 2004. [Google Scholar]
- Shinskey, F.G. Process Control Systems—Application, Design, and Tuning; McGraw-Hill: New York, NY, USA, 1996. [Google Scholar]
- Nisenfeld, A.; Miyasak, R. Applications of feedforward control to distillation columns. Automatica
**1973**, 9, 319–327. [Google Scholar] [CrossRef] - Coughanowr, D.R. Process Systems Analysis and Control; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Brosilow, C.; Joseph, B. Techniques of Model-Based Control; Prentice Hall: Upper Saddle River, NJ, USA, 2012. [Google Scholar]
- Isaksson, A.; Molander, M.; Modén, P.; Masko, T.; Starr, K. Low-order feedforward design optimizing the closed-loop response. In Proceedings of the Pan-Pacific Conference on Control Systems, Vancouver, BC, Canada, 16–18 June 2008.
- Adam, E.J.; Marchetti, J.L. Designing and tuning robust feedforward controllers. Comput. Chem. Eng.
**2004**, 28, 1899–1911. [Google Scholar] [CrossRef] - Rodríguez, C.; Normey-Rico, J.E.; Guzmán, J.L.; Berenguel, M. Robust design methodology for simultaneous feedforward and feedback tuning. IET Control Theory Appl.
**2016**, 10, 84–94. [Google Scholar] [CrossRef] - Vilanova, R. Feedforward control for uncertain systems. Internal model control approach. In Proceedings of the IEEE Conference on Emerging Technologies and Factory Automation, Petras, Greece, 25–28 September 2007; 2007. [Google Scholar]
- Vilanova, R.; Arrieta, O.; Ponsa, P. IMC based feedforward controller framework for disturbance attenuation in uncertain systems. ISA Trans.
**2009**, 48, 439–448. [Google Scholar] [CrossRef] [PubMed] - Elso, J.; Gil-Martínez, M.; García-Sanz, M. Quantitative feedback-feedforward control for model matching and disturbance rejection. IET Control Theory Appl.
**2013**, 7, 894–900. [Google Scholar] [CrossRef] - Rodríguez, C.; Guzmán, J.L.; Berenguel, M.; Hägglund, T. Generalized feedforward tuning rules for non-realizable delay inversion. J. Process Control
**2013**, 23, 1241–1250. [Google Scholar] [CrossRef] - Rodríguez, C.; Guzmán, J.L.; Berenguel, M.; Hägglund, T. Optimal feedforward compensators for systems with right-half plane zeros. J. Process Control
**2014**, 24, 368–374. [Google Scholar] [CrossRef] - Pawlowski, A.; Guzmán, J.L.; Rodríguez, F.; Berenguel, M.; Normey-Rico, J.E. Predictive control with disturbance forecasting for greenhouse diurnal temperature control. In Proceedings of the 18th World Congress of IFAC, Milan, Italy, 28 August–2 September 2011.
- Pawlowski, A.; Guzmán, J.L.; Rodríguez, F.; Berenguel, M.; Sánchez, J. Application of time-series methods to disturbance estimation in predictive control problems. In Proceedings of the IEEE International Symposium on Industrial Electronics, Bari, Italy, 4–7 July 2010.
- Shridhar, R.; Cooper, D.J. A tuning strategy for unconstrained SISO Model Predictive Control. Ind. Eng. Chem. Res.
**1997**, 36, 729–746. [Google Scholar] [CrossRef] - Lee, J.H.; Morari, M.; Garcia, C.E. State-space interpretation of model predictive control. Automatica
**1994**, 30, 707–717. [Google Scholar] [CrossRef] - Carrasco, D.S.; Goodwin, G.C. Feedforward model predictive control. Annu. Rev. Control
**2011**, 35, 199–206. [Google Scholar] [CrossRef] - Normey-Rico, J.E.; Camacho, E.F. Unified approach for robust dead time compensator. J. Process Control
**2009**, 19, 38–47. [Google Scholar] [CrossRef] - Yang, J.; Li, S.; Chen, X.; Li, Q. Disturbance rejection of dead-time processes using disturbance observer and model predictive control. Chem. Eng. Res. Des.
**2011**, 89, 125–135. [Google Scholar] [CrossRef] - Guzmán, J.L.; Hägglund, T.; Veronesi, M.; Visioli, A. Performance indices for feedforward control. J. Process Control
**2015**, 26, 26–34. [Google Scholar] [CrossRef] - Guzmán, J.L.; Hägglund, T.; Åström, K.J.; Dormido, S.; Berenguel, M.; Piguet, Y. Feedforward control concepts through interactive tools. In Proceedings of the 18th IFAC World Congress, Milano, Italy, 28 August–2 September 2011.
- Camacho, E.F.; Bordóns, C. Model Predictive Control; Springer-Verlag: London, UK, 2007. [Google Scholar]
- Skogestad, S. Simple analytic tuning rules for model reduction and PID controller tuning. J. Process Control
**2003**, 13, 291–309. [Google Scholar] [CrossRef] - Visioli, A.; Piazzi, A. Improving set-point following performance of industrial controllers with a fast dynamic inversion algorithm. Ind. Eng. Chem. Res.
**2003**, 42, 1357–1362. [Google Scholar] [CrossRef]

**Figure 1.**Generalized Predictive Control (GPC) resulting scheme for feedback and feedforward controllers.

**Figure 2.**Time-domain response of the open-loop disturbance compensation for a step change in v at $t=1$ and the following process: ${P}_{u}\left(s\right)=\frac{1}{1+s}{e}^{-s},\phantom{\rule{14.22636pt}{0ex}}{P}_{v}\left(s\right)=\frac{0.5}{1+0.8s}{e}^{-0.5s}$.

**Figure 3.**Comparison of compensation performance for the PID (Proportional - Integral - Derivative) and GPC (Generalized Predictive Control) approaches for realizable feedforward dynamics.

**Figure 4.**Comparison of compensation performance for PID (Proportional - Integral - Derivative) and GPC (Generalized Predictive Control) approaches for non-realizable feedforward dynamics.

**Figure 5.**Control results for non-realizable feedforward compensator with constraints on the control signal.

**Table 1.**Performance indexes for different configurations of PID (Proportional - Integral - Derivative) and GPC (Generalized Predictive Control) feedforward control schemes. $IAE$, Integrated Absolute Error; $IAC$, Integrated Absolute Control; $CSE$, Control System Effort.

PID + CFF | GPC + IFF ($\mathbf{\lambda}\mathbf{=}\mathbf{10}$) | GPC + IFF ($\mathbf{\lambda}\mathbf{=}\mathbf{0}$) | |
---|---|---|---|

$IAE$ | 0 | 0.276 | 0 |

$IAC$ | 8.31 | 13.13 | 13.29 |

$CSE$ | 0.15 | 0.12 | 0.14 |

**Table 2.**Performance indexes for different configurations of PID and GPC feedforward control schemes.

PID + CFF | PID + R1FF | PID + R2FF | PID + R3FF | GPC + IFF ($\mathbf{\lambda}\mathbf{=}\mathbf{10}$) | GPC + IFF ($\mathbf{\lambda}\mathbf{=}\mathbf{0}$) + F | |
---|---|---|---|---|---|---|

$IAE$ | 0.39 | 0.24 | 0.22 | 0.28 | 0.10 | 0 |

$IAC$ | 5.79 | 5.88 | 5.89 | 5.99 | 6.26 | 5.75 |

$CSE$ | 0.09 | 0.21 | 0.13 | 0.09 | 0.82 | 0.33 |

**Table 3.**Performance indexes for different configurations of PID and GPC feedforward control schemes.

PID + CFF | PID + R1FF | PID + R2FF | PID + R3FF | GPC + IFF ($\mathbf{\lambda}\mathbf{=}\mathbf{10}$) | GPC + IFF ($\mathbf{\lambda}\mathbf{=}\mathbf{0}$) + F | |
---|---|---|---|---|---|---|

$IAE$ | 7.15 | 4.62 | 5.50 | 6.72 | 10.61 | 3.76 |

$IAC$ | 47.36 | 47.85 | 47.38 | 46.89 | 45.31 | 43.98 |

$CSE$ | 0.97 | 1.02 | 0.98 | 0.95 | 1.54 | 1.55 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pawlowski, A.; Rodríguez, C.; Guzmán, J.L.; Berenguel, M.; Dormido, S.
Measurable Disturbances Compensation: Analysis and Tuning of Feedforward Techniques for Dead-Time Processes. *Processes* **2016**, *4*, 12.
https://doi.org/10.3390/pr4020012

**AMA Style**

Pawlowski A, Rodríguez C, Guzmán JL, Berenguel M, Dormido S.
Measurable Disturbances Compensation: Analysis and Tuning of Feedforward Techniques for Dead-Time Processes. *Processes*. 2016; 4(2):12.
https://doi.org/10.3390/pr4020012

**Chicago/Turabian Style**

Pawlowski, Andrzej, Carlos Rodríguez, José Luis Guzmán, Manuel Berenguel, and Sebastián Dormido.
2016. "Measurable Disturbances Compensation: Analysis and Tuning of Feedforward Techniques for Dead-Time Processes" *Processes* 4, no. 2: 12.
https://doi.org/10.3390/pr4020012