Model-free PID Controller Autotuning Algorithm Based on Frequency Response Analysis

Transcription

1 Model-free PID Controller Auto Algorithm Based on Frequency Response Analysis Stanislav VRÁ A Department of Instrumentation and Control Engineering, Czech Technical University in Prague Prague, , Czech Republic and Bohumil ŠULC Department of Instrumentation and Control Engineering, Czech Technical University in Prague Prague, , Czech Republic ABSTRACT PID controllers are the most popular group of controllers in practical applications. Many theoretical techniques and procedures have been developed for PID controllers with the aim to optimize the function of the control circuits. However, these techniques are seldom applied, except in cases where there are some extreme requirements for the safety and the operational quality of the control loops. Even auto of controllers, which some producers have been offering for a quite long time, is not exploited as it might be. There are several reasons for this unsatisfactory state of affairs, one of which is the necessity to master a difficult theoretical background, often requiring perfect models of the dynamics of the controlled process. In this paper, using a laboratory set-up in model representation and also in physical representation, we will discuss some of these reasons in a confrontation with the new idea of performing controller auto without knowledge of the controlled plant model. Keywords: PID controller, oscillation, control quality indicator, harmonic oscillation, model-free 1. I TRODUCTIO Generally, control algorithms are characterized by various levels of complexity. The algorithms that they use can be divided into two groups: algorithms that contain built-in auto features, and the algorithms without auto features. The most-used controllers usually use algorithms without auto features. The simplest controllers are two- or three-state controllers, usually implemented as a two- or three-state relay. PID-type controllers are more advanced than two- and three-state controllers. In addition to standard PID controllers, some alternatives of this algorithm are offered e.g. PID controllers with two degrees of freedom, fuzzy PID controllers, and fractional PID controllers. PID-type controllers are the most used in almost all industrial branches [2]. It is easy to explain why they are favoured so much, even when there are many better algorithms in existence, and a new or improved algorithm emerges almost every month (recent improvements include control error reference course control [15] and the matrix PID controller [10]). The reason is that the PID controller is easy to implement. No deep mathematical theory is necessary to understand how the PID controller works, so everybody is able to imagine what is happening inside the controller during the control process [16]. In situations when there are high demands on control process quality, especially in cases when the complex dynamics of the controlled processes is extremely difficult to master, model-free PID controllers are usually not sufficiently powerful control tools. Then, it is necessary to implement control algorithms designed on the basis of a more detailed knowledge of the controlled plant dynamics. Control with a state variable feedback is an example of such advanced algorithms. The results of controllers based on the use of state variables are usually much better than the results with other types of controllers; this is because state controllers work as controllers in multiple feedback loops. Many other advanced control algorithms are designated for controlling processes with a time delay [23]. A special group of controllers are those based on neural networks, genetic algorithms and other artificial intelligence tools and combinations of tools [4]. The weakness of standard PID-type controllers in comparison with model-based control algorithms, i.e. the absence of auto features inside the control algorithm, can be removed by adding an external algorithm. Many algorithms have been presented [3], [5], [9], [12], but they are mostly based on a model of the controlled process/device. This is an unsatisfactory situation, because industrial practice prefers control algorithms and methods that do not need any model of the controlled process or device. As a result, the more than 50-year-old Ziegler-Nichols me-

2 thod [22] and its derivates, such as the Chien-Hrones- Reswick method and the Cohen-Coon method, are still in use. Nowadays, the relay method [1], [8] is becoming popular. Many modifications have been made to the relay method to improve some of its properties, usually by adding a second relay [6], [11] or by adding an integrator [17] into the loop, but the basic variant is the most widely used [21]. Industrial controllers are usually equipped with this method as a pre- tool, because the method needs the control process to be stopped temporarily. Special cases of controllers equipped with auto features are those produced by Honeywell, equipped with the Honeywell Looptune Algorithm, and those produced by Foxboro, equipped with the Foxboro Exact Algorithm. However, exact details about how they work are not published - some knowledge has, however, been obtained by reverse engineering. The reasons mentioned above have motivated us to develop a new auto algorithm for PID-type controllers that does not use a model of the controlled plant and that does not need to break the control function when the controller parameters need to be changed. These positive properties are balanced by the fact that the algorithm is not suitable for rapid controller adaptation, because time is taken to achieve an optimal setting when it. This not a very disturbing consideration if the controller needs to be retuned after a certain operational time when the control process is not necessary to be stopped. The principle of the auto method presented here is based on tools provided by linear theory [7], especially on one of the tools - control quality indicators. In linear theory, the control quality indicators are connected with the Nyquist plot [18]. The two most-known control quality indicators are the Phase Margin and the Magnitude Margin. In the proposed tuner, the control quality indicators are evaluated from the frequency responses obtained by experiments in real closed control circuits. If the control circuit is excited by a harmonic excitation signal added to the control error, it is possible to evaluate from the response to this excitation in the closed control loop the magnitude and phase shift of the oscillation, as if it was obtained in the open loop. Changing the frequency of the excitation and setting the controller allows us to achieve the recommended optimal values of the openloop control quality indicators [19]. The measure in which the controlled process is disturbed can be influenced by setting the size of the amplitude of the added harmonic signal. The only condition is that the response to the added harmonic signal must remain measureable. More details about model-free frequency-based auto based on control quality indicators evaluated from frequency response analysis are published in [20]. 3. TESTI G DEVICE 2. PRI CIPLES OF THE PROPOSED AUTOTU I G For any auto method, it is important to describe somehow the changes in the control loop behaviour which are to be compensated by correcting the controller setting. Among all experimental ways to identify the current dynamic properties of the controlled plant, preference is mostly given to the following two, both of which evaluate the time responses to a defined input excitation. The first way assumes the use of step responses, while the second evaluates the responses to a harmonic signal, which are processed in dynamic characteristics known as frequency responses. The use of step responses usually focuses on obtaining the step response of a controlled plant alone, and therefore the control function of the control circuit must interrupted by disconnecting the controller. During the experiment all inputs, except the input which has changed stepwise, must be kept at constant steady values. These conditions are difficult to ensure, and for this reason an evaluation of frequency responses is preferred in auto procedures. There is no problem in obtaining the frequency response of the controlled plant just from measurements performed in the closed control loop. Auto, which is able to set the optimal parameters of the controller without breaking the control process, is of course preferred [14]. Figure 1 -Scheme of a Three-Tank Cascade A laboratory model of a Three-Tank Cascade was used for testing the algorithm. The scheme of the cascade is shown in Figure 1. Water is supplied into tank one and tank three. Each tank is equipped with a pressure sensor that is used for measuring the water level. In addition to

3 the mutual interconnection, each of the tanks in the cascade has its own outlet valve, enabling various operation modes with different dynamics to be simulated. All valves in the laboratory model are adjustable only manually. The laboratory model is controlled by a WinPAC programmable automation controller equipped with Rex control software. This software also offers RexLib library, containing all function blocks of Rex control software for the Simulink toolbox of the Matlab program. This allows us to separate the algorithm from our own control algorithm which, together with a harmonic signal generator, is executed in WinPAC. The algorithm can then be executed directly in Simulink, where both new values of the controller parameters and requests for changes of the harmonic excitation signal frequency are computed. Computer Simulink w Auto algorithm ω e sin(ωt) WinPAC Harmonic signal generator PID controller r 0, r I, r D Figure 2 - Block scheme of algorithm splitting e b u Threetank cascade y The Ziegler-Nichols method of critical oscillation Controller period r 0 = 25,8 dm 2.min -1 T I = min As the controller is set close to the stability margin, disturbance can cause dangerous response r 0 = 10,3 dm 2.min -1 T I = min Figure 3 - Ziegler-Nichols method of critical oscillation auto experiment (manipulated variable depicted after dividing the values by 10) The relay method Controller period No disturbance allowed Controller is disconnected r 0 = 10,3 dm 2.min -1 T I = min Figure 4 -The relay method auto experiment (manipulated variable depicted after dividing the values by 10)

4 Ziegler-Nichols step response method r 0 = 14, dm 2.min -1 Period of controller No disturbance allowed Controller is disconnected T u = 0,5 min T n = 4 min r 0 = 4,1 dm 2.min -1 T I = 12 min Figure 6 -The Ziegler-Nichols step response method auto experiment (manipulated variable depicted after dividing the values by 10) The momentum method Controller period No disturbance allowed Controller is disconnected m 0 = 1,6 m 1 = 3,8 m 2 = 5,5 σ = 0,2 r 0 = 0,6 dm 2.min -1 T I = 0,5 min Figure 5 -The momentum method auto experiment (manipulated variable depicted after dividing the values by 10) 4. RESULTS OF EXPERIME TS The features of several methods were tested and compared by means of the Three-Tank Cascade laboratory setup in the configuration depicted in Figure 1. The auto features are simulated in the following sequence: first, using changes in the controller setpoint, the water level in Tank 2 is increased by h 2 = 0,1 dm, and then it is decreased back to the initial height. During the second phase, the controller is tuned, and, finally, the same experiment as in step one is repeated, but with a new controller setting after auto. Five methods were tested: the new model-free frequencybased auto, the critical setting according to the Ziegler-Nichols method, the Ziegler-Nichols step response method, the relay method, and the momentum method [13]. The methods were tested from the following viewpoints: - first, what are the differences in the achieved optimal controller parameter values, and what are their impacts on the control responses, and - second, what will happen if a disturbance occurs, or how is the control function restricted during the period of controller. Figures 3 to 7 show the results of this test. There are clear differences in the achieved controller parameter values according to the method, and their impact on control quality is evident. Only the settings obtained by the Ziegler-Nichols method of critical oscillations and by the relay method are similar to each other. This was to be expected, because the relay method is just another way to

5 The model-free frequency-based auto Period of controller r 0 = dm 2.min -1 r 0 = dm 2.min -1 T I = 1,8 min γ = 70 Disturbances do not matter is stopped during unsteady states T I = 1,8 min Figure 7 -Frequency-based auto experiment (manipulated variable depicted after dividing the values by 10) invoke critical oscillation, and then the setting follows the Ziegler-Nichols rules. All methods differ in the time needed for controller. The time needed for is a characteristic sign of the quality of a method. Even methods providing a good controller setting can be considered worse than methods that provide less good results (still better than without any ), but that achieve them in a short time. Another important attribute is the possibility to make an automatic evaluation of the response. It is easier and less memory-consuming to detect peaks in the response to a harmonic signal than to find an inflection point in a step response and to compute the parameters of the tangent at this inflection point, especially in cases when the type of step response changes with the change in operating point. Finally, it is easier to evaluate the integral of the pulse response than to evaluate a frequency response. Use of the Ziegler-Nichols method of critical oscillation and the relay method lead to oscillating responses, while use of the Ziegler-Nichols method, the momentum method, and the model-free frequency-based method lead to non-oscillating responses. Generally, oscillating responses are not totally excluded, but in this case, such a result leads to excessive wear of the actuating device. The main disadvantage of the new model-free frequencybased auto presented here, in comparison to the other methods, is that the new algorithm takes much more time to obtain the optimal controller setting. There is one important consideration that the figures do not show sufficiently clearly: new model-free frequencybased auto is the only method among those tested here that does not require disconnection of the controller or any degradation of the control function during. This means that the process is safer, and can be applied in real conditions. The momentum method, the Ziegler-Nichols step response method and the relay method perform experiments only with the controller disconnected from controlled plant, i.e. the controlled plant is totally uncontrolled during. In the Ziegler-Nichols method of critical oscillations, the controlled plant is brought into oscillations with an unpredictable amplitude size. As a result, these methods should be used preferably in which is under manual control of the operator. A further limiting factor that has not been mentioned is the restriction of the Ziegler-Nichols step response method and the momentum method to plants with non-oscillating step responses. 5. CO CLUSIO S The new model-free frequency-based PID controller method that uses control quality indicators has been presented not only in terms of its operating principles, but mainly in terms of the experience obtained during its implementation and testing. From this testing, in which the controlled plant was represented by various kinds of models (linear, nonlinear, physical), we chose testing based on the simulation model of a real physical laboratory model, reflecting all essential nonlinearities and data of the laboratory model, with emphasis on modeling all the details corresponding to the real application. This consideration is very important, because only if we reflect all real conditions under which auto starts and runs is it possible to make a serious and useful comparison of various approaches and of auto efficiency. This is what is often missing in the literature.

6 A significant advantage of a controller equipped with the proposed model-free frequency-based auto method is that the controller is fully serviceable during. In contrast to the other methods, where the operator cannot influence the controller setting obtained by these methods, the model-free frequency-based method also allows the operator to change the criteria and their optimal values at any time, if the response does not satisfy the expected course. This research has been supported by grant MSM Development of ecological decentralized energetics, and by Czech Science Foundation Grant o. GAČR 101/07/ REFERE CES [1] K. J. Åstrom, T. Hägglund, Automatic Tuning of Simple regulators with Specifications on Phase and Amplitude Margins, Automatica, 20, 1984, pp [2] K. J. Åstrom, T. Hägglund, The Future of PID Control, Control Engineering Practice, Volume 9, Issue 11, ovember 2001, pp [3] K. J. Åstrom, T. Hägglund, Advanced PID control, ISA-The Instrumentation, System, and Automation Society, Research Triangle Park, NC 27709, 2005 [4] L. Blahová, J. Dvoran, Neural-Fuzzy Control of Chemical Technological Processes, Proceedings of the 17th International Conference on Process Control 09, Bratislava: Slovak University of Technology in Bratislava, 2009, pp [5] V. Bobál, J. Böhm, J. Fessl, J. Macháček, Digital Self- Controllers: Algorithms, Implementation and Applications, Springer, Berlin [6] M. Friman, Automatic Re of PI Controllers in Oscillating Control Loops, Ind. Eng. Chem. Res., 36, 1997, pp [7] I. M. Horowitz. Synthesis of feedback systems. Academic Press [8] J. Macháček, Plant identification through the use of nonlinearity in the feedback Automatizace č. 9. (in Czech) Praha [9] A. O'Dwyer, Handbook of PI and PID Controller Tuning Rules. London: Imperial College Press, ISBN X. [10] A. Rojas-Moreno, A. Parra-Quispe. Design and Implementation of a Water Tank Control System Employing a MIMO PID Controller, IMETI 2008: International Multi-Conference on Engineering and Technological Innovation Proceedings, vol II. Orlando: IIIS International Institute of Informatics and Systemics, 2008, pp ISBN [11] S. H. Shen, J. S. Wu, C. C. Yu, Autotune Identification under Load Disturbance, Ind. Eng. Chem. Res., 35, 1996, pp [12] G. J. Silva, A. Datta, S. P. Battacharrya, PID Controllers for Time-Delay Systems, Birkhäuser, Boston [13] J. Sobota, M. Schlegel, LTI systems identification using generalized method of moments, Proceedings of the 7th International Conference on Process Control Pardubice: University of Pardubice, 2006, pp ISBN [14] B. Šulc, "Assessment of Excited Oscillation in Controller Parameter Setting", WSEAS Transactioms on Systems and Control. 2006, vol. 1, no. 2, pp ISSN [15] B. Šulc, "Autotuned PI Level Control in a Two- Tank-Cascade Model with Sliding Control Error Reference Course", WSEAS Transactions on Circuits and Systems. 2005, vol. 4, no. 9, pp ISSN [16] B. Šulc, S. Vrána, Some Observations on Development and Testing of a Simple Auto Algorithm for PID Controllers, WSEAS Transactions on Systems and Control, October 2009, Volume 4, Issue 10, pp ISSN [17] M. Vítečková, A. Víteček, Experimental Plant Identification by Relay Method, Proceedings of XXX. Seminary ASR '05 Instruments and Control. Ostrava: VŠB Technická univerzita Ostrava, 2005, pp ISBN [18] S. Vrána, B. Šulc, Control Loop Oscillation Based PID Tuning, Proceedings of the 7 th International Carpathian Control Conference ICCC Ostrava: VŠB - Technická univerzita Ostrava, Faculty of Mechanical Engineering, Department of Control Systems and Instrumentation, 2006, pp ISBN [19] S. Vrána, B. Šulc, Control Quality Indicators Determination from Frequency Responses, Technical Computing Prague (in Czech) Praha: Humusoft, 2006, pp ISBN [20] S. Vrána, B. Šulc, Control Quality Indicators in PID Controller Auto, The 4th International Conference on Cybernetics and Information Technologies, Systems and Applications: CITSA 2007 Jointly with The 5th International Conference on Computing, Communications and Control Technologies: CCCT 2007 Proceedings, Volume II. Orlando: IIIS International Institute of Informatics and Systemics, 2007, pp ISBN [21] C. C. Yu, Auto of PID Controllers, London: Springer Verlag, London Limited ISBN [22] J. G. Ziegler, N. B. Nichols, Optimum settings for automatic controllers, Trans. ASME 64, 1942 pp [23] P. Zítek, V. Kučera, Algebraic Design of Anisochronic Controllers for Time Delay Systems. Int. J. Cont. Vol 76, No.16, pp