# PID Control

**Researchers:**Karl Johan Åström, Josefin Berner, Tore Hägglund, and Kristian Soltesz.

This project has been in progress since the beginning of the eighties and resulted in industrial products as well as several PhD theses. Three monographs on PID control that are based on experiences obtained in the project have also been published. The last is "Advanced PID Control", published in 2005. It is also translated to Spanish 2009: "Control PID avanzado". The research is currently focused on the following topics:

### Measurement noise filtering for PI and PID controllers

Measurement signals are always corrupted with noise.This will be reflected in the control signal behavior in e.g. high variance or large inter-sample jumps if considering a discrete time setting. Previous work on PI and PID controllers often focus on proportional-, integral- and derivative gains at design but the filter action is added afterwards such that a reasonable sensitivity to noise is given. However, the filter changes phase and gain of the controller and the initial tuning may not give satisfying results.

In this project, we investigate the tradeoffs between load disturbance attenuation, robustness and the undesired control activity generated by measurement noise. The goal is to find design rules that take all these aspects into account in the PID design, where the measurement noise filter is included.

A new methodology that uses a second order filter to attenuate the fluctuations of the control signal due to measurement noise, and which tuning parameter is given by the filter time constant T_f, has been derived. The main contributions are:

- Filtering design criteria for attenuation of measurement noise, which includes the Control Bandwidth, the Standard Deviation of the Control Signal (SDU), and the Noise Gain.
- An iterative method to calculate the filter time constant T_f based on the gain crossover frequency, which considers the trade-offs between performance, robustness, and measurement noise attenuation.
- Simple rules derived from the results obtained from the iterative method, which allow finding the filter time constant for common PID tuning rules based on FOTD models.
- Simple rules to find the added dynamics in the nominal FOTD model due to filter introduction, which leads to the recalculation of the controller parameters.

### Optimal Robust PID Design

A Matlab-based software tool for optimal PID design has been developed at the department. The software finds the PI or PID controller that minimizes the Integrated Absolute Error (IAE) value during a step load disturbance on the process input, with respect to robustness constraints on the sensitivity and complementary sensitivity functions. This PID design method is called SoftWare-based Optimal Robust Design (SWORD).

Varying the time constant of the low-pass filter, it is possible to find optimal or near-optimal solutions to an optimization problem extended with a noise sensitivity constraint. As the time constant of the low-pass filter increases, the PID controller will gradually transform into a PI controller and then finally an I controller. This gives a natural set of I, PI and PID controllers to choose from. The final controller can be selected based on visual feedback of the control signal activity due to measurement noise.

The optimal solutions to the extended optimization problem can also be used to compare the performance of PI and PID controllers to examine the benefit of the derivative part for different processes. Assuming continuous time white Gaussian noise with unit spectral density, it is possible to derive optimal PI and PID controllers with the same robustness and noise sensitivity constraints. The ratio of optimal PI performance divided by optimal PID performance for the case of medium noise sensitivity and high robustness is plotted in the figure. Each symbol represents a process in a batch of 134 models representative for the process industry. These have been classified with respect to their normalized time delay, τ. Processes with τ close to zero or one generally benefit less from the derivative part than processes in between. Two process types, however, benefit more from derivative action than others, namely those with two identical poles and little delay as well as second order processes with one integrating pole and little delay.

In order to use software-based optimal design methods like the one described, it is important to have better modeling tools than what is normally available in the process industry. A simple step response test has been shown insufficient to design PI and PID controllers that are close to optimal. Research has shown that process information around the phase -125° is sufficient to find first order time delayed models for optimal PI control. For optimal PID control the model needs to be accurate around a larger span of phase angles from -125° down to around -235°. With the right modeling tools, it should then be fairly easy to incorporate optimal software tuning into a new generation of autotuners that will be far better than any existing PID tuning rules.

Download the latest SWORD PID Design Tool version for Matlab

Download the previous PID Design Tool version for Matlab

### PID design by convex optimization

Convex optimization has grown to become a mature and powerful tool in a vast number of research fields. Design of PID controllers subject to robustness constraint is not a convex optimization problem, however, it fits well into the framework of the convex-concave procedure. Using that procedure, tuning algorithms for both SISO and MIMO PID controllers have been developed. Although globally optimal controllers cannot be guaranteed, the method produces robust controllers with good performance. The work is done in collaboration with Stephen Boyd, Stanford University.

### Optimization-Based Robust PID Design in Matlab

Compact and relatively efficient implementations of software for solving the PID design problems introduced above under *Optimal Robust PID Design *and* PID design by convex optimization* are available through the Matlab code PIDopt. The software also allows for co-design of PID controllers and measurement filters.

Download latest version of PIDopt for Matlab

Or clone directly from git@gitlab.control.lth.se:kristian/PIDopt.git.

### Criteria and Trade-offs in PID Design

Control design is a rich problem which requires that many issues such as load disturbances and set-point tracking, model uncertainty, and measurement noise are taken into account. In this work, we introduce trade-off plots for PI and PID controllers, which give insight into the design methods, criteria and design compromises.

The trade-off plot above is drawn for PI control of a second order delay-dominant process. The right plot is a magnification of the lower-left part of the left plot.The blue level curves show constant values of Integrated Absolute Error (IAE), equal to 1/ki (kp is proportional gain and ki is integral gain), during a unit step load disturbance on the process input. The red level curves show constant values of max(Ms,Mt), where Ms is the max norm of the sensitivity function and Mt is the max norm of complementary sensitivity. The green, dash-dotted, line shows the loci of IAE optimal controllers for different values of max(Ms,Mt) and the green dot shows the absolute minimum. Five different tuning methods are shown in the plot, namely: Ziegler-Nichols step response method (Z-N), Lambda tuning, Skogestad's two SIMC methods (S and SM) and AMIGO tuning. The black line marked with triangles is a parametrization of the optimal controllers.

### Interactive learning modules for PID control

We are also developing interactive learning modules for PID control. The modules are designed to speed up learning and to enhance understanding of the behavior of loops with PID controllers. The modules are implemented in SysQuake, and the work is done in collaboration with professor Sebastián Dormido at UNED, Madrid, and José Luis Guzmán at Universidad de Almería.

The tools can be downloaded from Universidad de Almería.

### Publications

**Josefin Berner, Kristian Soltesz:**
"Short and Robust Experiments in Relay Autotuners".
In: *22nd IEEE International Conference on Emerging Technologies and Factory Automation; ETFA2017* 2017.

**Josefin Berner, Kristian Soltesz, Karl Johan Åström, Tore Hägglund:**
"Practical Evaluation of a Novel Multivariable Relay Autotuner with Short and Efficient Excitation".
In: *IEEE Conference on Control Technology and Applications, CCTA 2017* 2017.

**Josefin Berner, Kristian Soltesz, Tore Hägglund, Karl Johan Åström:**
"Autotuner identification of TITO systems using a single relay feedback experiment".
In: *20th IFAC World Congress, 2017* 2017.

**Kristian Soltesz, Pedro Mercader, Alfonso Baños:**
"An automatic tuner with short experiment and probabilistic plant parameterization".
*Int. Journal of Robust and Nonlinear Control,*
**
27:**11,
pp. 1857–1873, 2017.

**Kristian Soltesz, Chriss Grimholt, Sigurd Skogestad:**
"Simultaneous design of proportional–integral–derivative controller and measurement filter by optimisation".
*IET Control Theory and Applications,*
**
11:**3,
pp. 341–348, 2016.

**Pedro Mercader, Kristian Soltesz, Alfonso Baños:**
"Autotuning of an In-Line pH Control System".
In: *21st IEEE International Conference on Emerging Technologies and Factory Automation, ETFA 2016* 2016.

**Josefin Berner, Tore Hägglund, Karl Johan Åström:**
"Asymmetric relay autotuning - Practical features for industrial use".
*Control Engineering Practice,*
**
54**,
pp. 231–245, 2016.

**Josefin Berner, Tore Hägglund, Karl Johan Åström:**
"Improved Relay Autotuning using Normalized Time Delay".
In: *American Control Conference, 2016* 2016.

**Tore Hägglund:**
"Autotuning".
In: *Encyclopedia of Systems and Control*,
Springer, 2015.

**Alfred Theorin, Tore Hägglund:**
"Derivative backoff: The other saturation problem for PID controllers".
*Journal of Process Control,*
**
33**,
pp. 155–160, 2015.

**Alfred Theorin, Josefin Berner:**
"Implementation of an Asymmetric Relay Autotuner in a Sequential Control Language".
In: *IEEE International Conference on Automation Science and Engineering (IEEE CASE 2015)* 2015.

**Olof Garpinger:**
"Optimal PI and PID Parameters for a Batch of Benchmark Process Models Representative for the Process Industry".
Technical Report
Department of Automatic Control, Lund University, Sweden, August 2015.

**Stephen Boyd, Martin Hast, Karl Johan Åström:**
"MIMO PID Tuning via Iterated LMI Restriction".
*International Journal of Robust and Nonlinear Control,*
**
26:**8,
pp. 1718–1731, 2015.

**Martin Hast, Tore Hägglund:**
"Optimal proportional–integral–derivative set-point weighting and tuning rules for proportional set-point weights".
*IET Control Theory & Applications,*
**
9:**15,
pp. 2266–2272, 2015.

**Josefin Berner:**
"Automatic Tuning of PID Controllers based on Asymmetric Relay Feedback".
Licentiate Thesis
Department of Automatic Control, Lund University, Sweden, June 2015.

**José Luis Guzmán, Tore Hägglund, Max Veronesi, Antonio Visioli:**
"Performance indices for feedforward control".
*Journal of Process Control,*
**
26**,
pp. 26–34, 2015.

**José Luis Guzmán, Tore Hägglund, Karl Johan Åström, Sebastián Dormido, Manolo Berenguel, Yves Piquet:**
"Understanding PID design through interactive tools".
In: *19th IFAC World Congress, 2014* 2014.

**Vanessa Romero Segovia, Tore Hägglund, Karl Johan Åström:**
"Design of measurement noise filters for PID control".
In: *19th IFAC World Congress, 2014* 2014.

**Vanessa Romero Segovia, Tore Hägglund, Karl Johan Åström:**
"Measurement noise filtering for common PID tuning rules".
*Control Engineering Practice,*
**
32**,
pp. 43–63, 2014.

**Olof Garpinger, Tore Hägglund, Karl Johan Åström:**
"Performance and robustness trade-offs in PID control".
*Journal of Process Control,*
**
24:**5,
pp. 568–577, 2014.

**Vanessa Romero Segovia, Tore Hägglund, Karl Johan Åström:**
"Measurement noise filtering for PID controllers".
*Journal of Process Control,*
**
24:**4,
pp. 299–313, 2014.

**Carlos Rodríguez, José Luis Guzmán, Manuel Berenguel, Tore Hägglund:**
"Optimal feedforward compensators for systems with right-half plane zeros".
*Journal of Process Control,*
**
24:**4,
pp. 368–374, 2014.

**Josefin Berner, Karl Johan Åström, Tore Hägglund:**
"Towards a New Generation of Relay Autotuners".
In: *19th IFAC World Congress, 2014* 2014.

**Martin Hast, Karl Johan Åström, Bo Bernhardsson, Stephen P. Boyd:**
"PID Design by Convex-Concave Optimization".
In: *European Control Conference, 2013* 2013.

**C. Rodríguez, José Luis Guzmán, Manuel Berenguel, Tore Hägglund, J. E. Normey-Rico:**
"Diseño de controladores por Adelanto para Inversión de Retardo no realizable".
In: *XXXIV Jornadas de Automática* 2013.

**C. Rodríguez, Josè Luis Guzman, Manuel Berenguel, Tore Hägglund:**
"Generalized feedforward tuning rules for non-realizable delay inversion".
*Journal of Process Control,*
**
23:**9,
pp. 1241–1250, 2013.

**Tore Hägglund:**
"A Unified Discussion on Signal Filtering in PID Control".
*Control Engineering Practice,*
**
21:**8,
pp. 994–1006, 2013.

**José Luis Guzmán, Tore Hägglund, Antonio Visioli:**
"Feedforward compensation for PID control loops".
In
Ramon Vilanova, Antonio Visioli (Eds.): *PID control in the third millenium*,
Springer, 2012.

**Olof Garpinger, Tore Hägglund, Lars Cederqvist:**
"Software for PID design: benefits and pitfalls".
In: *IFAC Conference on Advances in PID Control* 2012.

**Olof Garpinger, Tore Hägglund, Karl Johan Åström:**
"Criteria and Trade-offs in PID Design".
In: *IFAC Conference on Advances in PID Control* 2012.

**Tore Hägglund:**
"Signal Filtering in PID Control".
In: *IFAC Conference on Advances in PID Control* 2012.

**Martin Hast, Tore Hägglund:**
"Design of Optimal Low-Order Feedforward Controllers for Disturbance Rejection".
In: *17th Nordic Process Control Workshop, 2012* 2012.

**Martin Hast, Tore Hägglund:**
"Design of Optimal Low-Order Feedforward Controllers".
In: *IFAC Conference on Advances in PID Control* 2012.

**Per-Ola Larsson, Tore Hägglund:**
"Comparison Between Robust PID and Predictive PI Controllers with Constrained Control Signal Noise Sensitivity".
In: *IFAC Conference on Advances in PID Control* 2012.

**Kristian Soltesz:**
"On Automation of the PID Tuning Procedure".
Licentiate Thesis
Department of Automatic Control, Lund University, Sweden, January 2012.

**Lars Cederqvist, Olof Garpinger, Tore Hägglund, Anders Robertsson:**
"Cascade control of the friction stir welding process to seal canisters for spent nuclear fuel".
*Control Engineering Practice,*
**
20:**1,
pp. 35–48, 2012.

**Juan Garrido, Francisco Vázquez, Fernando Morilla, Tore Hägglund:**
"Practical advantages of inverted decoupling".
*Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control,*
**
225:**7,
pp. 977–992, 2011.

**José Luis Guzmán, Tore Hägglund, Karl Johan Åström, Sebastián Dormido, Manuel Berenguel, Yves Piguet:**
"Feedforward Control Concepts through Interactive Tools".
In: *18th IFAC World Congress, 2011* 2011.

**Per-Ola Larsson, Tore Hägglund:**
"Control Signal Constraints and Filter Order Selection for PI and PID Controllers".
In: *American Control Conference, 2011 * 2011.

**Per-Ola Larsson:**
*Optimization of Low-Level Controllers and High-Level Polymer Grade Changes*.
PhD Thesis
TFRT-1088, Department of Automatic Control, Lund University, Sweden, October 2011.

**Tore Hägglund:**
"A shape-analysis approach for diagnosis of stiction in control valves".
*Control Engineering Practice,*
**
19:**8,
pp. 782–789, 2011.

**Per-Ola Larsson, Tore Hägglund:**
"Robustness Margins Separating Process Dynamics Uncertainties".
In: *European Control Conference, 2009 * 2009.

**Per-Ola Larsson, Tore Hägglund:**
"Relations Between Control Signal Properties and Robustness Measures".
In: *17th IFAC World Congress, 2008* 2008.

2017-09-11