Relay Auto Tuning Simulink
Jocelin Taylor
This paper considers frequency point identification and PID-type controller tuning through the use of relay feedback methods, based on those originally given by Åström and Hägglund in 1984. The proposed concepts and supporting theory are first explored before more recent developments are investigated. Two methods which aim to increase the accuracy of frequency point identification are studied and applied to a coupled tanks rig, where the benefits are apparent. The focus then turns to using the identified frequency points to arrive at a set of PID-type controller parameters. A number of tuning methods are considered from the simple method first proposed by Åström and Hägglund to more advanced methods for robust design. Pleasing results are seen in application to the coupled tanks rig, especially in the case of a tuning method which gives an 'iso-damping' property. Finally, recommendations are given regarding the situations in which the proposed methods may, or may not, be beneficial.
Because these rigid controller tuning rules are often not sufficient, a number of methods which are able to provide automatic and adaptive tuning to changing process characteristics have been developed. These include pattern recognition (Bristol, 1977), relay auto tuning (Åström and Hägglund, 1984a), and stepwise parameter optimisation (Radke and Isermann, 1987). These methods are based on the automatic measurement of the ultimate gain and period, and are useful where little is known about the process characteristics and there is variation in process dynamics. It is the relay auto tuning method that is considered in this paper.
In recent years, a number of automated relay tuning methods have been developed. One of the most notable is the Åström-Hägglund method as presented in 1984 (Åström and Hägglund, 1984a). There are a number of benefits in the use of relay auto-tuning, the most notable being that (a) the method does not introduce a risk of loop instability like the Ziegler-Nichols cycling method, (b) little a priori knowledge of the plant is necessary, and (c) the loop output can be kept close to the set-point throughout the test with correct selection of relay parameters.
Following the 1984 publication, Åström and Hägglund developed their initial method to include a stability-based tuning specification by placing constraints on phase or amplitude margins (Åström and Hägglund, 1984b), developments on which were also given by Dormido and Morilla (Dormido and Morilla, 2004), among many others. However, it has been shown that for a given phase or amplitude margin, very different responses can result (Åström and Hägglund, 1984c) and these stability measures are not always completely reliable, also shown by Åström and Hägglund in 1995 (Åström and Hägglund, 1995). Other methods proposed by Åström and Hägglund (Åström and Hägglund, 1995) and Dormido and Morilla (Dormido and Morilla, 2000) give specification based on sensitivity margin, which is considered to give a more reliable indication of relative stability, while Liu and Daley (Liu and Daley, 2001) and Tan et al. (Tan et al., 1996) give tuning methods concerned with a combined performance criterion. More recent developments include methods which rely on the identification of two frequency points to achieve a more robust controller tuning such as in Tan et al. (Tan et al., 1996), or arrive at a process model from which tuning can be achieved as presented by Wang and Shao (Wang and Shao, 2000; Wang and Shao, 1999). Further advanced methods rely on or multiple frequency point identification to obtain a process model as given by Wang et al. (Wang et al., 1997) or extract multiple frequency points from one relay feedback test with use of the fast Fourier transform (FFT) (Wang et al., 2003).
Another important consideration which has to be made is of the relay feedback procedure itself, which directly affects the accuracy of the amplitude and frequency which utilised used in the aforementioned tuning methods. The mathematical basis of this technique lies in describing function analysis (Atherton, 2005), which relies on a linear approximation of a non-linear element and neglects harmonics higher than the fundamental. Due to this, various modifications on the use of a simple relay have been presented in order to increase estimation accuracy by reducing the effects of higher order harmonic terms. Examples include the use of a preload relay (Tan et al., 2006) and a saturation relay (Shen et al., 1996), or the use of a twin-channel method (Friman and Waller, 1997). More recent proposals are given by Jeon et al. (Jeon et al., 2010) who use a linear combination sub-relay signals with different frequencies or gains, and Je et al. (Je et al., 2009) where an optimally combined series of 10 pulses are used to generate one period of the relay signal. In addition, methods exist which aim to counteract difficulties which can be encountered such as noise, non-linearities and load disturbances (Yu, 1999; Sung and Lee, 2006; Hang et al., 1993; Lin and Yu, 1993).
This method is obviously undesirable for industrial implementation due to the risk of system instability. The Åström-Hägglund relay tuning method (Åström and Hägglund, 1984a) uses the addition of a relay to the feedback loop to avoid this potential drawback. For the purposes of tuning, the relay replaces the original controller.
Fundamental to the understanding and application of the relay tuning techniques described as part of this paper is describing function analysis. This method of analysis was developed during the 1940s by Krylov and Bogolyubov (Krylov and Bogolyubov, 1947) during the search for an explanation as to why wartime gun- and antenna-tracking systems would oscillate about their set point. It was found that these oscillations arose from non-linearities in the systems such as those seen in gearing systems. An observation made was that when these oscillations occurred, the system output was often sinusoidal; hence the basis of the describing function technique is on the assumption of this case as presented by Atherton (Atherton, 2005).
There are many hundreds of tuning methods available in the literature which make use of frequency points identified by relay feedback. Differences between these methods lie in the way that the identified frequency point in the Nyquist plot is manipulated or moved in order to create the desired closed loop response in the compensated system. Tuning methods have been selected from the simple and well-known to the more advanced and robust.
The results gained from the relay feedback tests in the previous section are now used for the tuning of PID-type controllers in order to ascertain the knock-on effects of identification inaccuracy on the ultimate performance of a controller.
In order to compare the previously proposed tuning methods, controllers are designed for each, as well as a simple tuning with use of the Ziegler-Nichols rule to act as a reference. Where appropriate, identifications performed with the saturation relay in the previous section are used, due to earlier findings. Appendix A gives details of the design decisions and criterion applied in each case.
Finally, techniques such as the iso-damping method have illustrated the robust design capabilities of relay auto-tuning technique where negligible overshoot increases were seen for relatively large changes in loop gain.
In the light of these findings, it is concluded that, for the vast majority of cases, where average performance is required, a simple, fast tuning using the original methods proposed by Åström and Hägglund may suffice. However, where applications demand, methods which give superior responses are available, although much more application effort is required. With the coupled tanks rig or similar in mind, and while considering the findings made in this paper, the preload relay method for identification coupled with the iso-damping method for controller tuning would be recommended.
To aid in the analysis and simulation during this work, various Simulink-based auto-tuning models were developed relating to many of the methods studied. Initially, a basic model was designed to measure the amplitude and frequency of the limit cycle, and subsequent developments provided the ability to calculate PID parameters.
Through much development and testing, we arrive at the model faceplate shown in Figure D-1. This model incorporates a PID controller and relay auto-tuner based on the Åström PID design, and allows specification of hysteresis should this be desired. The user is able to select the times at which the tuning begins and stops. Upon completion of the tuning, the model updates the parameters of the PID controller with the ones just calculated.
Figure D-2 gives a view inside the 'Tuner and Controller' subsystem shown in Figure D-1. Here the PID, controller, simulated plant, and Åström auto-tuner subsystem can be seen. A switch can also be seen which provides a transition between the relay test and control activation. Bumpless transfer has been included in the PID block to ensure that the plant does not undergo a large disturbance during the transition between the relay test and the activation of control with the updated PID parameters provided by the 'PID Update Switch' block, which is triggered by the test stop signal in Figure D-1.
The last portion of the model is concerned with the calculation of the PID parameters to a phase margin specification, and uses an embedded Matlab function to apply the Åström calculations to the values measured previously. Options for varying the ratio Ti/Td and specifying the amount of relay hysteresis present are provided which alter the calculations accordingly. Phase margin is set outside the auto-tuner subsystem, as seen in Figure D-1.
Åström, K. J. and T. Hägglund (1984c), 'A frequency domain method for automatic tuning of simple feedback loops', in Proceedings of the 23rd IEEE conference on decision and control, Las Vegas, pp.299-304
aa06259810