Evolving PID tuning rules

A brief history, starting with the earliest PID controllers to the most recent developments. There is more continuity than you might expect.
By Willy K. Wojsznis, Terry Blevins March 13, 2013

Looking into the distant past of automation history, one can see the underlying concept of a PID controller exhibited in a nineteenth-century steam engine governor design, with the first real PID-type controller developed by Elmer Sperry in 1911. The first theoretical analysis of a PID controller was published by Nicolas Minorsky in 1922. His observations grew out of efforts to design automatic steering systems for the U.S. Navy. He realized that a helmsman controlled the ship based not only on the current error, but also on past error and current rate of change. Proportional control could provide stability against small disturbances, but it was insufficient for dealing with a steady disturbance, which required adding the integral term. Adding the derivative term further improved control. Currently, the basic principle is applied in many variations, but for our discussion we assume commonly used basic standard ISA PID controller in discrete form:

 (1)

where

OUT(k) – Controller output

Kp – Controller proportional gain

SP  – Controlled parameter setpoint

PV – Process value, controlled parameter measurement

e(k) – Control error, e(k)=SP-PV

Ti – Controller integral time (reset) in seconds

Td  – Controller derivative time (rate) in seconds

Tuning a control loop is the adjustment of its control parameters (proportional gain Kp, integral time/reset Ti, derivative time/rate Td) to the optimum values for the desired control response. Stability (bounded oscillation) is a basic requirement, but beyond that, different systems have different behavior, different applications have different requirements, and requirements may conflict with one another. In the following sections we briefly review the basic tuning techniques, focusing on objectives of the tuning and how they were achieved.

Model-free tuning

Model-free tuning techniques don’t use a process model for PID controller tuning in a direct way. They are based on the observation of a process which is under control, and became among the first initially applied for controller tuning. The most well-known techniques from this group are manual tuning and Ziegler Nichols tuning. In manual tuning the tuned loop remains in automatic mode. At first integral and derivative actions are removed by setting Ti to infinity (or very high) and Td to zero. Then, controller proportional gain Kp is increased until the loop oscillates with constant amplitude. After that, the proportional gain Kp should be set to approximately half of that value for a quarter amplitude decay type response. Ti is adjusted until any offset is corrected in sufficient time for the loop operation. It should be noted that too low a value for Ti may cause instability. Finally, Td is increased until the loop is acceptably quick to reach its reference after a load disturbance, without excessive overshoot.  

Ziegler Nichols tuning also requires setting proportional-only gain Kp high enough to achieve loop oscillations with constant amplitude, similar to the manual tuning method. The difference is that the period of the oscillation Tu should be measured. Controller gain at the test is named the ultimate gain, Ku. The controller parameters are defined then from the formulas with no need for further loop testing like in the manual tuning shown in Table 1. 

Table 1: Ziegler-Nichols calculations of P, PI, and PID controller parameters for a standard-form PID.

[Table 1: Ziegler-Nichols calculations of P, PI, and PID controller parameters for a standard-form PID.]

There were some obvious drawbacks to these techniques, at least initially. Getting the loop to cycle continuously was a time-consuming process, and there was a risk that the oscillations would grow beyond stability since there was no deterministic way to specify or limit the oscillation magnitude.

The approach became significantly more attractive after introducing relay-oscillation auto-tuning, as described in Figure 1. Relay-oscillation auto-tuning delivers loop oscillation with amplitude limited by the relay step size, shown in Figure 2.

Figure 1. Relay Oscillation Tuning Diagram

[Figure 1. Relay Oscillation Tuning Diagram] 

Figure 2. Trend plots of relay output and process output during active tuning

[Figure 2. Trend plots of relay output and process output during active tuning]

The Ziegler Nichols method identifies the ultimate gain and ultimate period, so controller settings may be determined. The original Ziegler-Nichols tuning rules were designed to provide a quarter amplitude damped response to a load disturbance. Once considered ideal, the underdamped and oscillatory nature of Ziegler-Nichols tuning has been criticized for destabilizing control loops, actually increasing variability instead of reducing it.

There were several options for modifying original Ziegler-Nichols tuning. One way was to calculate a complete first order plus dead time model from the oscillation test and use model based tuning rules. Another method was to reduce the aggressiveness of the original Ziegler-Nichols tuning rules, which tend to provide the most oscillatory response when dead time is small. Therefore, an improvement can be achieved by making controller gain smaller and integral time larger (i.e., integral gain smaller) for smaller dead times. Dead time, in addition to the ultimate gain and ultimate period, can be easily defined during relay oscillation test shown in Figure 2. An example of such nonlinear adjustment of tuning parameters is given in Advanced Control Foundation (see additional reading list).

Model-based tuning

Note that the so-called model-free tuning just discussed is in fact partial or indirect model-based tuning. This is because the ultimate gain directly relates to the inverse of the process gain and ultimate period relates to the process dead time and lag. Significant progress in process model identification with commonly available identification tools makes it possible and easy to develop a process model and apply process-model parameters directly for model-based tuning. The first-order-lag-plus-dead-time model is the most common approximation for self-regulating processes (see Figure 3), and linear-integrator-with-gain-and-dead-time is used for integrating processes (see Figure 4). 

Figure 3 First Order Plus Dead Time Self-Regulating Process Response

[Figure 3. First Order Plus Dead Time Self-Regulating Process Response]

Figure 4 Integrating Process Response

[Figure 4. Integrating Process Response] 

There are many model-based tuning techniques; the most popular are Internal Model Control (IMC), Lambda tuning, and recently developed SIMple Control (SIMC) rules.

The most important feature of model-based tuning is its ability to shape control loop performance and robustness by using a tuning parameter. The tuning parameter relating to the speed of response is used to vary the trade-off between performance and robustness, coordinate response among loops, and achieve process control objectives (averaging level, tight control, etc.). In principle for self-regulating processes, the methods adjust the PID controller reset (or reset and rate) to match process dynamics and then adjust the controller gain to achieve the desired closed loop response. IMC and Lambda tuning have become popular because oscillation and overshoot are avoided, controllers are less sensitive to noise, and control performance can be specified in an intuitive way through the closed-loop time constant. However, load disturbance rejection is typically worse than in quarter-amplitude decay tuning. The SIMC rules were developed to improve model-based tuning performance, primarily for disturbance rejection when desired. SIMC rules provide a higher integral gain (smaller reset time) for the processes with a small dead time than Lambda or IMC tuning rules, by applying this formula: 

 

As it follows from the formula, for the processes with a small dead time and large time constant with a properly selected λ to satisfy the condition τ > 4 (τd + λ), reset time is set as Ti = 4(τd + λ) , instead of Ti = τ, as in Lambda or IMC tuning.

Controller proportional gain Kp is calculated in the same way as for the Lambda or IMC tuning:

 

For the integrating process controller, parameters are:

It is interesting to notice that optimum tuning rules geared toward minimum integrated absolute error (IAE) advanced by F. Greg Shinskey are only a particular case of SIMC tuning rules for the integrating process:

 

In fact, such formulas are very close to what is obtained when using  λ= 0. This results in the following gain and reset time:

Formulas which do not apply filter λ are therefore for a maximum performance with no designed robustness margin and no possibility of setting a desired loop performance. Therefore, using such formulas is particularly undesirable when process parameters may change causing loop instability. Instead, simple formulas provide the ability to design loop performance and robustness in a required way.

Which brings us back to…

Historically, PID controller tuning started from observing a loop with proportional action on the verge of stability, and then decreasing proportional gain to get stable operation and calculating integral and derivative terms from the loop oscillation period. In fact, all above indicators are related in some way to the process model parameters. Therefore, if all process model parameters are explicitly known, it is possible to satisfy tuning requirements in the best way. There are several model-based tuning rules which give a simple and intuitively understandable method to set a desired loop performance and robustness for a given process.

Willy K. Wojsznis is a senior technologist, and Terry Blevins is principal technologist, future architecture, for Emerson Process Management.

Additional reading:

Bennett, Stuart, “A history of control engineering, 1930-1955.” IET, p. 48. ISBN 978-0-86341-299-8, 1993.

Minorsky, Nicolas (1922). "Directional stability of automatically steered bodies." Journal of the American Society of Naval Engineers, 34 (2): 280–309.

J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic controllers,” Transactions of the ASME, Vol. 64, Nov. 1942.

J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic controllers,” Transactions of the ASME, Vol. 115, June 1993.

K. J. Astrom and T. Hagglund, “Automatic tuning of PID controllers,” ISA 1988, Research Triangle Park, NC, USA.

K. J. Astrom and T. Hagglund, “A frequency domain method for automatic tuning of simple feedback loops”, IEEE 23rd Conference on Decision and Control, Las Vegas, Dec. 1984.

W.L. Bialkowski and B. Haggman, “Quarter-amplitude damping method is no longer the industry standard,” American Papermaker, March 1992.

T. Blevins, W. Wojsznis, and M. Nixon, “Advanced Control Foundation,” ISA, 2012.

Skogestad, S. “Simple analytic rules for model reduction and pid controller tuning,” Journal of Process Control 13, 2003. 

Key concepts

  • PID controllers are virtually everywhere, yet effective tuning remains a challenge
  • Conceptually, there is more similarity among various methods than one might expect
  • Ultimately, a strategy needs to reflect the needs of the process, and selection depends on understanding those needs  

Go online

For more information, visit:

www.advancedcontrolfoundation.com 

www.emersonprocess.com

Read more on control strategy:

Fixing PID, Nov. 2012

Feedback controllers do their best, Oct. 2012

Disturbance-rejection vs. setpoint-tracking controllers, Sept. 2011

Understanding derivative in PID control, Feb. 2010

Three faces of PID, Mar. 2007