23 XXIV International Conference on Information, Communication and Automation Technologies (ICAT) October 3 November, 23, Sarajevo, Bosnia and Herzegovina Model Based Predictive in Parameter Tuning of PI Controllers E. Sahin, M. Güzelkaya 2, İ.Eksin 3 Sürmene Abdullah Kanca Meslek Yüksek Okulu Karadeniz Teknik Üniversitesi, Trabzon esahin@ktu.edu.tr 2, 3 Kontrol ve Otomasyon Mühendisliği Bölümü İstanbul Teknik Üniversitesi, 34469, Maslak, İstanbul 2 guzelkaya@itu.edu.tr, 3 eksin@itu.edu.tr Abstract The peak observer method is firstly proposed and used for PID type fuzzy logic controllers. In this study, the peak observer method is adapted and then implemented to the classical PI control structure. The basic principle of the method is to change the controller parameters of the system using the peak values of the system response in order to improve system performance. Firstly, the peak observer method is reconsidered on a simple internal model control based classical PI controller. Later, the peak observer method is further developed and a new structure called model based peak observer is proposed and the parameters of PI controller are further tuned for a much better performance. The performances of the proposed methods are tested and compared on different systems based on simulations.. Introduction It is well known that most of the industrial processes are still using the conventional PID controllers because of their simple control structure, ease of design and effectiveness for linear systems. Furthermore, many studies have been proposed to determine the parameters of these controllers. [,2,3]. In the internal model control method, the parameters of the PID controller have been reduced to selecting a single parameter value [4,5]. Various methods exist to set the parameters of the fuzzy PID controllers as for the conventional PID controllers, [6,7,8,9,]. Self-tuning peak observer method uses the peak values of the system responses in order to adjust the scaling factors of PID type fuzzy logic controller in an online manner [6]. In this method, the system response is allowed to reach its first peak and then proportional, integral and derivative control parameters are changed in such way that the system response will be more damped while the overall gain of the system remains the same or unchanged. The peak observer method that has been developed for fuzzy PID controllers in the literature has been adapted to the classical PI controller structure for the first time within this study. The parameters of the classical PI controller are designed and set via internal model control method and the adjustment of these parameters are then accomplished using peak observer idea. Next, a simple modification on the peak observer method is done. Shortly, a factor called oscillation reduction factor is introduced which is a function of overshoot values. Then, the PI controller parameters are then multiplied by the new oscillation factor adjusted overshoot values. Finally, a new method called model based dictive peak observer is proposed in order to reduce the first peak value of the system response. Simulation studies are accomplished on various systems for the proposed methods in a comparative manner to illustrate their performances. 2. Internal Model Control Method The parameters of PI and PID controllers can be adjusted by internal model control method in which simple analytical rules are used by reducing the order of the model [4]. The system responses are observed to be more robust to noise and reference changes using this method. In this method, firstly, an approximate model of first or second order with dead time is obtained from the known system by applying the model reduction rules. PI controller parameters are then determined by using this approximate model. Formulas for the time constant and dead time value of the first order approximate system are shown below. 2 () 2 h Tj (2) 2 2 i 2 i3 j In the above exssions τ, τ 2 and θ are dominant time constants and dead time of the original system respectively, τ i is the other time constants, T is the zero of the system if it exists and h is the sampling time. As an example, the following second order system without dead time is chosen. G ( s) ( s )(.2s ) (3) 978--4799-43-/3/$3. 23 IEEE
In the above system, τ =, τ 2 =.2, θ and T are equal to zero since system has no zeros and dead time as seen. By using the system reduction formulas () and (2), the approximated model is obtained as given below. g ( s) (.s ).s e (4) The open loop, unit step responses of the approximated and original systems are shown in Figure. As it is seen from the figure, the system responses are very close to each other..8.6.4.2 2 3 4 5 6 7 8 Figure : The comparison of the original and approximate model of the system. The relations for calculating parameters of the PI controller with internal model control are given below. These parameters are proportional constant k p and integral time constant τ i. k p i.5. 5 ' k (5) k min(,8 ) Original system response Approximated system resposne (6) In the above exssion k is the plant gain. The proportional constant (k p ) is calculated as 5.5 and integral time constant (τ i ) is calculated as.8 or integral constant (k i ) is 6.875 using the above relations for the sample system. The unit step response of the system with the calculated PI parameters is shown in Figure 2..4.2.8.6.4.2.5.5 2 2.5 Figure 2: System response with internal model control. As seen from Figure 2, there exists a 24% overshoot and some weak oscillations in the system response with internal model control. 3. PI Controller with Peak Observer The peak observer method is firstly used to tune the determined parameters of PID-type fuzzy logic controller in order to improve the system performance [6]. In this method, system response of the designed controller is observed until reaching its first peak. After the occurrence of the first peak, the controller parameters are tuned by multiplying them by a factor which depends on the overshoot value (δ). Thus, a correction process is initiated to the system beginning with the first peak and correction processes continues with every peak and reverse peak values.,6,4,2,8,6,4,2. Peak 2. Peak 4. Peak 3. Peak 2 3 4 5 6 7 8 9 Figure 3: The absolute value of the error calculated at the peak and reverse peak moments.
The absolute value of the error calculated at the peak and reverse peak moments may be shown by δ j (j=,2,3, ). It is obvious that the δ j values are less than one and gradually decrease in time. For the PI control case, the δ j values are directly multiplied by the controller parameters to improve the system performance. As a result, the new controller parameters applied to the system after every peak and reverse peak moments are obtained as follows: k k p ( new ) k p ( old ) (7) j i ( new ) ki ( old ) (8) j The block diagram of the control system with peak observer is given in Figure 4. 4. Enhanced with Oscillation Reduction Factor Although a satisfactory performance improvement is obtained by the peak observer method, some sub-oscillations cannot be eliminated. In this section our goal is to obtain less oscillatory system responses by a simple modification on the peak observer method. For this purpose, different systems have been examined and it has been observed that a suitable oscillation reduction factor can be obtained that will multiply by the overshoot values which are used in equations (7) and (8) to improve the controller parameters at the first peak moment. In Table, the oscillation reduction factors for different overshoot values are sented. Table : The oscillation reduction factors and corresponding overshoot values with random control parameters. Overshoot value Oscillation reduction factor,9228,756,92,833,789,288,6372,5,484,2,4723,9,3263,2 Figure 4. The block diagram of the control system with peak observer. The unit step system responses for the system (3) that is obtained using the PI controller parameters calculated by (5) and (6) are shown in Figure 5 with and without being tuned by peak observer..4,73,4 As it is seen from Table, there is an inverse relationship between reducing factor and overshoot values. Using the data given in the table, the oscillation reduction factor can be exssed as a function of overshoot. The obtained fourth order function between oscillation reduction factor and the overshoot value is given in Figure 6..2.8.6.4.2 PI Controller Response with PI Controller Response Oscillation reduction factor (ORF).4.35.3.25.2.5..5 ORF=3.*O 4-7.64*O 3 + 7.33*O 2-3.58*O +.83 2 3 4 5 6 7 8 9 Figure 5: Unit step system responses with and without peak observer method As it is seen from Figure 5, peak observer method reduces the oscillations on the system response..2.3.4.5.6.7.8.9 Overshoot (O) Figure 6: The functional relation between the oscillation reduction factor and the overshoot values. In the cases in which the integral coefficient of the PI controller is very small, multiplication with another small oscillation reduction factor may cause steady state error within the system response due to diminishing integral term in the
controller. To avoid this situation, lower and upper limits should be used for the proportional and integral coefficients. The effect of the oscillation reduction factor is shown on Figure 7. In this figure, the system (3) is controlled by the PI controller with coefficients K p =34 and K i =2..6.4.2 PI controller PI controller with peak observer PI controller with ORF.4.2.8.6.4.2 Internal model control system response Model based control system response.8.6.4.2 2 3 4 5 6 7 8 Figure 7: Unit step responses of system (3) for PI controller, PI controller with peak observer, PI controller with oscillation reduction factored peak observer As it is seen from Figure 6, less oscillated system response is obtained with the proposed method. 5. Model Based In the peak observer method, the improvement on the system performance begins after the system response reaches to its first peak value. In this section, our main goal is to intervene before the first peak is reached. For this purpose, firstly, the overshoot value that would occur if the system had been run by the PI controller parameters set a priori with internal model controller design procedure is determined. Then, when the system response reaches the reference value the controller parameters are multiplied by (/ ). The aim of this multiplication is to increase the controller parameters. After the system response passes to the reference value, the error will have a negative sign. Thus, larger controller parameter values will have an overshoot reducing effect on the system response. After this operation, the peak observer method is still kept in use and the new controller parameters are set using the equations (7) and (8) to reduce the oscillations..5.5 2 2.5 Figure 8: Unit step system response with internal model control and model based peak observer method. It seen that a system response with a reduced overshoot but still there exists certain oscillations if one does not apply peak observer method. The two related responses aer given in Figure 8. Therefore, as it has been mentioned above, the peak observer method has to be applied in order to settle down the oscillations further. The related system responses can be seen in Figure 9..4.2.8.6.4.2 Model Based Control Model Based Control with.5.5 2 2.5 Figure 9: The unit step system response with model-based peak observer method. When the controller parameters obtained by the equations (5) and (6) are applied to system (3) an overshoot of.24 has been observed. The system response that is obtained by multiplying the controller parameters (/ = /.24=4.6) is illustrated in Figure 8.
Figure : Unit step system responses with the model-based peak observer method and other control methods. In Figure, a comparative illustration of the responses due to various mentioned methods sented in this study is given and it is obvious that the settling time is shortened and oscillations are almost eliminated upon using model based peak observer method..4.2.8.6.4.2 2 3 4 5.2.8.6.4.2 Internal Model Control Model Based Control Model Based wtih Peak observer method Internal model control Model based peak observer method 2 3 4 5 Figure : Illustration of the disturbance effect on the system responses for the proposed methods. introduced which is a function of overshoot values and the PI controller parameters are then multiplied by the new oscillation factor adjusted overshoot values. Finally, the peak observer method is further developed and a new structure called model based peak observer is proposed and the parameters of PI controller are further tuned for a much better performance. The performances of the proposed methods are tested and compared on different systems based on simulations. In all of these simulations, it has been seen that the oscillations and overshoots occurring within the system has been decreased or diminished to a very reasonable degree. References [] Äström, K.J. and Hägglund, T., 995: PID Controllers: Theory, Design and Tuning, Instrument Society of America, Research Triangle Park, North Carolina. [2] O Dwyer, A., 2: A Summary of PI and PID Controller Tuning Rules For Processes With Time Delay. Part 2: PID Controller Tuning Rules. IFAC Digital Control: Past, Present and Future od PID Control,Teressa,Spain. 2-26. [3] Mudi, R.K. and Dey, C., 29: An Improved auto-tuning scheme for PID controllers, ISA Tras, 36-49. [4] Skogestad S.,22: ''Simple analytic rules for model reduction and PID controller tuning'' Journal Process Control 3, 29-39 [5] Riveria, D.E., Morari, M. & Skogestad, S., ''Internal model control. PID controller design'' Ind.En.Chem.Res., 25 :252-265, 986. [6] Qiao, W.Z., Mizumato M., 996: ''PID type fuzzy controller and parameters adaptive method,'' Fuzzy Sets and Systems 78, 23-35. [7] Mudi, R. K., & Pal, N. R. 999: A robust self-tuning scheme for PI- and PD-type fuzzy controllers. IEEE Transactions on Fuzzy Systems, 7(), 2 6. [8] Güzelkaya M., Eksin İ., Yesil E., 23: Self-tuning of PID-type fuzzy logic controller coefficients via relative rate observer, Engineering Applications of Artificial Intelligence, 227 236. [9] Ahn, K. K., Truong, D. Q., 29: Online tuning fuzzy PID controller using robust extended Kalman filter. Journal of Process Control, 9, -23. [] Yesil E., Guzelkaya M., Eksin I., 24: Self-tuning fuzzy PID-type load-frequency controller. Electrical Conversion and Management. 45/3, 377-39. The responses of the proposed methods due to a disturbance are illustrated in Figure. At the fifth second a unit step load disturbance of duration of. seconds is applied to the system and the proposed methods are observed to be also robust to disturbances. 6. Conclusion In this study, the peak observer method that has been developed for tuning of the fuzzy PID scaling factor has been adapted and then implemented to the classical PI control structure for the first time in literature. The basic principle of the peak observer method is to change the controller parameters of the system using the peak values of the system response in order to improve system performance. Next, a simple modification on the peak observer method is done. Shortly, a factor called oscillation reduction factor is