Extensions and Modifications of Relay Autotuning Mats Friman Academic Dissertation Department of Chemical Engineering Åbo Akademi University FIN-20500 Åbo, Finland
Preface This thesis is the result of the work that was carried out in a research project about autotuning in process control, at the Process Control Laboratory, Åbo Akademi University during the period of time stretching from 1992 to 1997. A research project concerning autotuning started in 1990 with two diploma works (Kim Roos and Mats Friman) under the supervision of Professor Kurt Waller. This thesis is the continuation of that work. I am grateful to my supervisor, Professor Kurt Waller, for offering me a position as a graduate student at the laboratory and for arranging financial support. I appreciate the qualified guidance in the form of clear and logical scientific principles as well as the honest and prompt judgement of various thoughts and ideas that I suggested during the work. I would like to thank the entire personnel at the Process Control Laboratory for providing a pleasant working environment. Specially I want to thank Associate Professor Hannu Toivonen for interesting discussions and lectures and for co-ordinating the post-gradual courses, Dr. Tore Gustafsson, Dr. Kurt-Erik Häggblom, Kim Roos, and Kati Sandström for valuable comments and interesting discussions, and Jari Böling, Pekka Lehtiö, Päivi Nurmi, and Stefan Rönnblad for their assistance during the experimental runs on the distillation column. Many thanks also go to Ann-Christin Waller for correcting the language in the manuscripts. Valuable comments on the manuscripts by Professor William Luyben (Lehigh University, Bethlehem, Pennsylvania, USA) and Dr. Steve Walsh (Imperial College, London, U.K) are also appreciated. I would like to thank the various funding organizations that have helped to finance this work, including the Academy of Finland / Graduate School in Chemical Engineering, Nordisk Industrifond, Tekes, Neste OY, and Neste Foundation. Finally, I would like to thank my family and friends for their support. Åbo, July 1997 Mats Friman 2
List of Publications This thesis is based on the following four papers: I. Friman, M.; Waller, K. V. Autotuning of Multiloop Control Systems. Industrial & Engineering Chemistry Research. 1994, 33, 1708-1717. II. Friman, M.; Waller, K. V. Closed-Loop Identification by Use of Single-Valued Nonlinearities. Industrial & Engineering Chemistry Research. 1995, 34, 3052-3058. III. Friman, M.; Waller, K. V. A Two-Channel Relay for Autotuning. Accepted for publication by Industrial & Engineering Chemistry Research. 1997. IV. Friman, M. Automatic Re-Tuning of PI Controllers in Oscillating Control Loops. Accepted for publication by Industrial & Engineering Chemistry Research. 1997. 3
Abstract In this thesis extensions and modifications of Åström-Hägglund autotuning are investigated. A method where relay autotuning is extended to multi-input-multi-output control systems is suggested. The method is based on identification of n n simple two-parameter transfer functions through n identification experiments. The method is experimentally tested on a water-mixer and a pilot-plant distillation column. Two methods where the relay, used in standard autotuning, is replaced by some other nonlinearity are suggested. In the first method, the relay is replaced by a two-parameter nonlinearity so as to improve the accuracy of the identification. In the second method, the relay is replaced by two relays operating in parallel, one on the process output, and the other on the integral of the process output. With this construction, a point on the Nyquist curve at a userselected angle in the third quadrant is identified, something that has certain advantages with respect to controller tuning. Moreover, a method where the ideas of autotuning are utilized for an oscillating (unstable) PI control loop is suggested. An oscillating control loop provides a useful possibility for process identification and subsequent controller tuning. This method has proven to be useful for retuning of unstable PI control loops and for rapid start-up of feedback-controlled chemical processes. 4
Introduction Control plays a key role in the operation of chemical plants with respect to economical performance, safety and operability. To realize the role of control at a general level, consider the definition of the concept "chemical process" (Pohjola et al., 1994): "Chemical process is control of (physico-chemical) phenomena for a purpose". This definition clearly shows the necessity of control: without control there would be no functional chemical process industry. An example illustrates the assortment of possible control possibilities involved in the definition above. Say that we have a room and an electric heater, i.e. we have a phenomenon (heat transfer from heater through the room) and a purpose (to make it comfortable for people to stay in the room). If we study this process from a control point of view, we find that some of the control possibilities, ordered according to added complexity, are the following: 1. We can adjust the heating power to be constant. This is easy to implement but it would probably make the indoor temperature too high in summer or too low in winter. 2. We can adjust the heating power according to the time of the year. This would take into account temperature changes between seasons, but it would not take into account unexpected variations in the outdoor temperature. 3. We can adjust the heating power according to the measured outdoor temperature (feedforward control). This control possibility is widely used but it does not take into account all possible disturbances (like changes in the ventilation). 4. We can adjust the heating power according to the measured indoor temperature (feedback control). 5. We can combine possibilities 3 and 4. The present work deals with feedback control of chemical processes, i.e. control possibility 4. The main motivation to use feedback control is that the control purpose usually can be fulfilled even if unmodeled (unknown or unpredictable) disturbances affect the process. However, a feedback connection involves also some disadvantages. In addition to a high degree of complexity, feedback is more difficult to handle than e.g. feed-forward control. This relates to the interaction involved, in the example above a change in the heating power affects (in contrast to feed-forward control) the measured temperature, which in turn affects the heating power, etc. Because of this interaction, there is a risk of the control loop turning unstable. In this work, feedback control is realized through proportional-integral-derivative control (PID control). In the example above, this means that a PID controller would adjust the heating power as a linear combination of the following three components: 1. The difference between room temperature and desired temperature (proportional action), 2. Time integral of the difference between room temperature and desired temperature (integral action), 3. Time derivative of temperature (derivative action). Moreover, PI control, i.e. PID control without derivative action, is studied. The vast majority of controllers in the chemical industry are of the PID type, or a reduced version (e.g. PI or P controllers). Their popularity is easy to understand they have a simple 5
structure, they are familiar to engineers, and their control capabilities have proven to be adequate for most control loops. In order for the PID controller to fulfil its assignment, the weights of proportional, integral, and derivative action must be individually chosen for each implementation. We say that we tune the controller. In a typical chemical plant there are hundreds of PID feedback loops. They are often poorly tuned because the choice of PID controller weights (hereafter called PID controller parameters) requires professional knowledge by the user. The theory of controller tuning involves sophisticated mathematical manipulations, including complex-valued functions, differential equations, and integral transforms. It is thus not a surprise that the average process engineer repeatedly tunes controllers by trial-and-error methods. Because the PID controller has three tuning parameters, controller tuning by trial and error is a search in the threedimensional space. Evidently, optimal controller parameters are seldom instantly obtained by trial and error. Many modern controllers are equipped with various adaptive techniques such as self-tuning, on-line tuning, and autotuning. These features provide easy-to-use controller tuning and have proven to be well accepted among process engineers. One of the most common approaches to tune a controller automatically is to connect a relay as a feedback controller to the process during tuning. This relay autotuner was introduced in 1984 (Åström and Hägglund, 1984) and today several commercial controllers are equipped with this device. In the present thesis, this particular controller tuning approach, commonly labelled "autotuning", is considered. Autotuning is appealing to process engineers as it provides systematic controller tuning in the easiest possible way, i.e. by pressing a button. The purpose of this work is to suggest extensions and modifications of standard relay autotuning. These extensions and modifications regularly show improvements (in terms of accuracy, speed or robustness) of the autotuning procedure compared to the standard method, albeit at the cost of some added complexity. The contents of the four papers are summarized below. I. Autotuning of Multiloop Control Systems Multivariable systems are more difficult to tune and control than single-input-single-output (SISO) systems are. A multivariable control system where the control loops are independently tuned according to recommendations given for SISO systems often results in an unstable system. The Åström-Hägglund autotuner is designed for SISO-systems and therefore cannot be directly applied to multi-input-multi-output (MIMO) systems. However, with a few modifications autotuning can be applied also to multivariable systems and in this paper one such method is suggested. The approach here suggested is to employ relay autotuning in the identification of a multivariable model of the process. When an identified process model is available, a wide assortment of model-based controller tuning designs found in the literature can be utilized. From a control point of view, it is often convenient to model chemical processes with a firstorder-plus-dead-time model, ke /( Ts + 1 ). This model has three parameters: the gain k, the Ls dead-time L and the time constant T. In relay identification where only two parameters of the process are identified (i.e. the ultimate gain and the ultimate frequency), it is not possible to 6
identify a three-parameter model in a reliable way. However, for the purpose of feedback control, accurate estimation of the time constant is often unnecessary. Considering frequencies important to feedback control, it turns out that, if the time constant is large (T>>L), the firstorder-plus-dead-time model is well approximated by an integrator-plus-dead-time model Ls ke / s. Analogically, if T is small (T<<L) a gain-plus-dead-time system ke Ls is often sufficient. Such simple two-parameter models can easily be identified through a relay experiment. The identification of simple two-parameter models is well suited for MIMO systems because off-diagonal elements can be identified in open-loop and a full process transfer function matrix consisting of n n elements can thus be identified through n experiments only. The method proposed is finally tested experimentally on a water mixer and on a pilot-plant distillation column. II. Closed-loop Identification by Use of Single-Valued Nonlinearities Standard autotuning is based on the identification of two parameters of the process, i.e. the ultimate gain and the ultimate frequency. It is interesting to notice that these parameters are the same as we identify by an old, but still widely used, method for PID controller tuning, the Ziegler-Nichols ultimate sensitivity experiment (Ziegler-Nichols, 1942). Unfortunately, the parameter identification with relay autotuning is based on approximations and therefore inexact. The purpose of this study is to investigate the magnitude of the errors in relay identification and to propose different nonlinearities to replace the relay so as to give better identifications. The advantages with automatic tuning are, however, retained. An investigation in the literature reveals that there is a difference in theory and common practice in the interpretation of the output amplitude during identification. It is common to measure the output amplitude based on peak-to-peak values of one period of oscillation, even though we, according to the theory, should measure the first harmonic amplitude of the oscillations. With a peak-to-peak measurement and relay identification, the error can be as large as 30% for a first-order-plus-dead-time process. The approach here proposed is to replace the relay used in standard autotuning by another suitable non-linearity. This improves the identification. As an example, consider the nonlinearity given by a P-controller with saturation limits on the manipulated variable. With this non-linearity in the control loop we get smoother oscillations and therefore a better identification. The results show that the identification frequently can be improved if the output amplitude is interpreted correctly or if the relay is replaced by another suitable non-linearity during identification. III. A Two-Channel Relay for Autotuning Relay autotuning is appealing to process engineers as it provides systematic controller tuning the easiest possible way, i.e. by pressing a button. Conventional PI controller autotuning based on Ziegler-Nichols (1942) recommendations means, however, that overall stability cannot be guaranteed even for systems whose gain and phase are monotonically decreasing functions of frequency (low-pass systems). This robustness problem is not a weakness of the Z-N recommendations, it holds for all PI controller settings based on the ultimate gain and the ultimate frequency. 7
In this paper we suggest a method for autotuning of PI controllers with guaranteed closed-loop stability for low-pass systems. We propose a setup with two relays connected in parallel, one operating on the process output and the other on the integral of the process output. In contrast to conventional autotuning, which identifies the point on the Nyquist curve that intersects with the negative real axis, the proposed setup identifies a point on the Nyquist curve at a userselected angle in the third quadrant. For control of low-pass systems, the proposed identification enables PI controller tuning with minimum specifications on both amplitude margin and phase margin. Moreover, the suggested approach is useful also for tuning PID controllers. Basic process knowledge, often available, can be utilized in controller tuning as well as in the selection between PI control and PID control. The innovation presented here is considered as a step towards trouble-free autotuning of a wide range of processes. IV. Automatic Re-Tuning of PI-Controllers in Oscillating Control Loops Even though relay autotuning is easy to use and appeals to process engineers, there are situations where it may be time consuming. One such example is when relay autotuning is utilized during the process start-up. Because it is required that the process is at rest before identification, the operator can choose between e.g. the following two procedures during process start-up. 1) He can manually control the output to the set-point before connecting the autotuner, or 2) he can put the controller on manual, wait for the process to stabilize, and identify the process at the observed operating point (which can be far from the desired setpoint). These procedures are sometimes time consuming. Moreover, the control loop is seldom at rest when controller tuning is desired. For example, if a control loop turns unstable due to e.g. changes in the operating conditions, it is self-evident that the operator must re-tune the controller, but relay autotuning cannot instantly be utilized because the process is not at rest. However, the method here proposed provides fast process identification and controller tuning of oscillating control loops. It makes the recovery procedure faster because controller parameters can instantly be determined and implemented without a time-consuming identification experiment. With the method proposed, a conventional PI controller loop is used for identification. The control loop is driven to instability through aggressive controller settings but the magnitudes of oscillations are bounded by saturation levels on the input variable. Input saturation is present in all real processes but saturation levels can also be user-modified. This method has a number of advantages compared to other closed-loop identification methods. 1) Because the same PI controller is employed for both control and identification, the identification experiment has a simple and well-known structure. 2) The system does not have to be brought to the operating point prior to identification and tuning (i.e. it is well suited for rapid start-up). 3) Moreover, the process is identified close to the desired set-point, regardless of initial conditions or possible load disturbances affecting the control loop during identification. In the method proposed, a standard PI-controller is tuned such that the controlled system turns unstable. The subsequent oscillations are then used for process identification and consequent controller tuning. This method is particularly useful in cases where a control loop turns unstable due to e.g. changes in the operating conditions. In that case, no time-consuming identification experiment is needed, but the controller can instantly be re-tuned. Moreover, the method is useful for rapid start-up because the operator does not have to bring the process to the operating point prior to identification. 8
To use a PI-controller instead of a relay in the feedback loop during identification means that the process is identified in the third quadrant. This has certain advantages for controller tuning. In contrast to ultimate-gain-and-ultimate-frequency based tuning, the third-quadrant identification allows PI controller tuning with minimum specifications on both the amplitude margin and the phase margin. Summary and Conclusions The Åström-Hägglund autotuner is an outstanding innovation, and provides many opportunities in process control. Relay autotuning has proven to be of great assistance to control engineers in the chemical process industry, as it provides systematic controller tuning the easiest possible way, i.e. by pressing a button. A large number of experimental implementations on a wide range of processes have confirmed its possibilities and usefulness. The extensions and modifications presented in this thesis are not meant to replace standard relay autotuning, but they provide solutions in some special cases when relay autotuning would not provide an acceptable solution. Some of the situations where relay autotuning can be inappropriate, and when a method considered in this thesis is believed to be of assistance, are listed below. 1. If autotuning is used for the tuning of controllers in a multivariable environment, it is often advantageous to take into account the interactions between control loops. Otherwise problems, in terms of poor control quality or unstability, will usually arise. 2. Relay identification is based on approximations and therefore provides an identification that sometimes is not accurate enough. 3. Relay autotuning is based on the identification of the ultimate gain and the ultimate frequency, an identification that does not always give the best information about the process with respect to controller tuning. For instance, PI control of an integrating process, a combination not rare in chemical process industry, is an example that often results in poor control quality if universal, Ziegler-Nichols-like, ultimate-gain-and-ultimate-frequencybased controller settings are utilized. 4. It is usually required that the process is at rest, and close to the operating point, before relay autotuning is applied. If this is not the case, a time-consuming wait-for-the-processto-stabilize step is required before the autotuning experiment. Situations where the process is not at rest before the autotuning experiment are not rare; one example is when autotuning is applied during process start-up, another when autotuning is employed for the re-tuning of an oscillating control loop. For the tuning of multivariable systems, a method based on identification of simple twoparameter transfer functions has been suggested. This method is plain and robust and thus expected to work well in practice. Off-diagonal elements are identified in open-loop and a full n n transfer function matrix can thus be identified through n identification experiments only. The approximations involved are crude but well motivated as they simplify the identification procedure, but they do not seem to affect the feedback performance in any significant way. The approximations involved in standard autotuning are seldom large enough to affect the control quality to a high degree. There are, however, situations when an accurate identification of the ultimate gain and the ultimate frequency is favorable. One such example is when the relay experiment is utilized for identification of transfer functions. In that case it is common to connect a known, variable, dynamic element in series with the process during identification, so 9
as to identify multiple points on the Nyquist curve. If these identified points are inexact, the errors are often enlarged during back-calculation of transfer function parameters. Moreover, it might be added that it is usually not known if the identification is erroneous or not, and it is usually not possible to estimate the magnitudes of the errors in the identification. The two-parameter relay suggested in the third paper is believed to be useful as it provides a non-iterative method for identification of a point on the Nyquist curve at a user-selected angle in the third quadrant. For PI controller tuning, extensive simulations of a wide range of processes, including processes with dead times, integrators, first-, second- and higher order processes as well as processes with non-minimum-phase dynamics, showed that good average robustness and performance were obtained when controller settings were based on the information about the process at a phase lag of 150, which can be compared to standard autotuning which identifies the process at a phase lag of 180. Moreover, the proposed identification is believed to be useful as it enables PI controller tuning with minimum specifications on both the amplitude margin and the phase margin for control of low-pass systems. This innovation is considered to be a step towards trouble-free autotuning of a wide range of processes. The method suggested for controller tuning based on unstable PI control loops is also believed to be useful in some special circumstances. Many processes are time-variant, which, in combination with constant controller parameters, may lead to unstable control loops. The automatic re-tuning procedure suggested here makes the recovery procedure faster as it provides instant process identification and controller tuning of oscillating control loops. The method suggested has also proven to speed up the tuning procedure during the start-up of chemical processes. References Åström, K. J.; Hägglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645-651. Pohjola, V. J.; Alha, M. K.; Ainassaari, J. Methodology of Process Design. Comp. Chem. Engng. 18, 1994, p. S307-S311. Ziegler J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 65, 433-444. 10