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Self-Tuning Fuzzy PI Controller and its Application to HVAC Systems A. K. Pal and R. K. Mudi
Abstract— In this paper, a Self-tuning Fuzzy PI controller is used for the supply air pressure control loop for Heating, Ventilation and Air-Conditioning (HVAC) system. The self-tuning Fuzzy PI controller (STFPIC) is adjusted the output scaling factor on-line by fuzzy rules according to the current trend of the controlled process. The rule-base for tuning the output-scaling factor is defined on error and change of error of the controlled variable. Ziegler-Nichols tuned PI or PID controller performs well around normal working conditions, but its tolerance to process parameter variations are severely affected. The STFPIC is used here to overcome these shortcomings. Comparing with PID and Adaptive Neuro-Fuzzy (ANF) Controllers, simulations results show that STFPIC performances are better under normal conditions as well as when the HVAC system encounters large c 2008 Yang’s Scientific Research parameter variations. Copyright ° Institute, LLC. All rights reserved. Index Terms— PID control, HVAC system, self-tuning fuzzy PI controller, adaptive neuro-fuzzy method..
I. I NTRODUCTION
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EATING, Ventilation and Air-Conditioning (HVAC) systems require control of environmental variables such as pressure, temperature, humidity etc. Like in other industrial applications, most of the controllers commissioned in HVAC systems are of Proportional-Integral-Derivative (PID) type. The conventional PID controllers are widely used in industry due to their simplicity in arithmetic, ease of using, good robustness, high reliability, stabilization and zero steady state error. But HVAC system is a non-linear and time variant system. It is difficult to achieve desired tracking control performance since tuning and self-adapting adjusting parameters on line are a scabrous problem of PID controller. Over the past few decades, several methods for determining PID controller parameters have been developed. Some employ information about open-loop step response, for example Cohen-Coon reaction curve method [1]; other methods are knowledge of the Nyquist curve, e.g., the Ziegler-Nichols frequency-response method. However, these methods use only a small amount of information about the dynamic behavior of the system, and often do not provide good tuning. Manuscript received June 17, 2007; revised December 11, 2007. A. K. Pal, Applied Electronics and Instrumentation Engineering Department, Heritage Institute of Technology, Chowbaga Road, Anandapur, Kolkata700107. R. K. Mudi, Instrumentation and Electronics Engineering Department, Jadavpur University, Kolkata. Email: arabindak
[email protected](A. K. Pal) Acknowledgements: The authors would like to thank all referees for their valuable guidance, which helped to improve the quality of this paper. Publisher Item Identifier S 1542-5908(08)10105-1/$20.00 c Copyright °2008 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on July 09, 2008 at http://www.YangSky.com/ijcc/ijcc61.htm
It has been reported that fuzzy logic controller is very suitable for non-linear system and even with unknown structure [2, 3]. In [4 - 15], various fuzzy PI / PID hybrid control schemes are discussed to improve PI / PID control performance. The tuning procedure can be a time-consuming, expensive and difficult task [16, 17]. This problem can be easily eliminated by using self-tuning scheme for fuzzy PI / PID controller. Generally a skilled human operator always tries to manipulate the process input, usually by adjusting the controller gain based on the current process states (error and change of error) to get the process “optimally” controlled. We used a simple but robust self-tuning scheme, where the controller gain is adjusted continuously with the help of fuzzy rules. Here the tuning of the output-scaling factor (SF) that is equivalent to the controller gain has been given the highest priority because of its strong influence on the performance and stability of the system [18]. The self-tuning mechanism is applied to PI type Fuzzy Logic Controller for simulation experiment. In the HVAC system, the supply air pressure is regulated by the speed of a supply air fan. Increasing the fan speed will increase supply air pressure, and vice versa. The dynamics from the fan variable speed drive to the supply air pressure can modeled as a second order plus dead time. This process is well established by Bi and Cai [19]. In real application however, both fans and dampers exhibit non-linear properties for different working points, even a well-tuned PID controller may not be able to achieve a desired performance for all set points and process variations. In the first part of this paper Self-tuning Fuzzy Logic Controller is described. The second part described the implementation of the PI type Self-tuning Fuzzy Logic Controller on a HVAC system. In the last part simulation results are presented to compare with the well-tuned PID controller and Adaptive Neuro-Fuzzy (ANF) controller. II. DEVELOPMENT OF PI-TYPE SELF-TUNING FUZZY CONTROLLER The basic function of the rule base is to represent in a structured way the control policy of an experienced process operator and/or control engineer in the form of a set of production rules such as If {process state} then {control output}. Considered a set of desired input-output data pairs: (1)
(1)
(2)
(2)
[x1 , x2 ; u(1 )], [x1 , x2 ; u(2 )]
(1)
where x1 and x2 are inputs and u is the output. Here considered error (e) as x1 and change of error (∆e) as x2. The task here is
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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 6, NO. 1, MARCH 2008
The fuzzy controller is developed using this 49 fuzzy if-then rules as shown in Table1.
to generate a set of fuzzy rules from the desired input- output pairs of equation (1) through following steps [20]: Divide the input and output spaces into fuzzy regions. + Assumed the domain intervals of x1 , x2 and u are [x− 1 , x1 ], − + − + [x2 , x2 ] and [u , u ] respectively. Fig.1 shows each domain interval divided into 7 equal regions, denoted by NB (negative big), NM (negative medium), NS (negative small), ZE (zero), PS (positive small), PM (positive medium) and PB (positive big) and assigns each region a fuzzy membership function. The shape of each membership function is triangular.
Similar like fuzzy controller, using symmetrical triangle calculate membership functions of (i) e, ∆e, u (as shown in Fig.1) and (ii) gain updating factor (β) (as shown in Fig.2) for self-tuning mechanism. An additional logic for addition at the output of controller is incorporated for PI controller. Because the discrete-time version equation of PI controller is ∆u(k) = Kp ∆e(k) + KI e(k); ∆u(k) = u(k) − u(k − 1); or u(k) = ∆u(k) + u(k − 1),
The term sets of e, ∆e and u contains the same linguistic expressions for the magnitude part of the linguistic values, i.e.,
where ∆u(k) is the change of control output and u(k) is the total control output. Fig.3 shows that the output scaling-factor (SF) of the fuzzy controller is modified by a self-tuning mechanism, which is marked by bold rectangular portion in the figure. Then based on the knowledge of process control or by trial and error method choose suitable SF’s for inputs and output. The relationship as follows for PI type self-tuning fuzzy controller scheme.
LE = L∆E = LU = {N B, N M, N S, ZE, P S, P M, P B} as shown in Fig.1 and represents the rule base in the table format as shown in Table1. The cell defined by the intersection of the first row and the first column represents a rule such as, If e(k) is NM and ∆e(k) is PS then u(k) is NS.
eN = Ne e, ∆eN = N∆e ∆e and ∆u = (βNu )∆uN Where Ne and N∆e are input scaling factor of error and change of error respectively and Nu is output scaling factor. Thereafter apart from fuzzy PI controller rule determination,
PAL & MUDI, SELF-TUNING FUZZY PI CONTROLLER AND ITS APPLICATION TO HVAC SYSTEMS
also determines the rule base for gain updating factor, in the similar way like if e is E and ∆e is ∆E then β is β. A structure of which is shown in Table 2, though it may vary. Further modification of the rule base for β may be required, depending on the type of response the control system designer wishes to achieve. As shown in Fig.3, when this β is multiplied with the fuzzy controller gain Nu , gives the overall gain of STFPIC. It is very important to note that the rule base for computation of β will always be dependent on the choice of the rule base for the controller. Choice of Scaling Factor (gain): The scaling factors also known as gains, which describe the particular input normalization and output denormalization, plays an important role similar to that of the gain coefficients in a conventional controller. For example, a fuzzy controller can be represented as Nu ∗ u(k) = F (Ne ∗ e(k), N∆e ∗ ∆e(k)), where Ne , N∆e and Nu are the scaling factors for e, ∆e and u respectively, and F is a nonlinear function representing the fuzzy controller. Same gain principle is used in the design of self-tuning fuzzy controller. III. SIMULATION RESULTS In this section, we made the performance of Self-Tuning Fuzzy PI Controller for HVAC system. Different performance parameters such as rising time (tr ), settling time (ts ), % peak overshoot (Mp ), integral absolute error (IAE) and integral of the time multiplied absolute error (ITAE) are studied to analyze the performance of different controllers [21]. The two integral criterion IAE and ITAE are considered because mere visual observations of response curves are not always enough to make a good comparison. Large errors contribute heavily to
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IAE; on the other hand ITAE penalizes heavy errors that occur late in time. Thus IAE and ITAE show true characteristics of the control system. A typical cooling only HVAC system is shown in Fig.8. In the system, the outside air is mixed with the building return air. Then the mixed air (supply air) is sucked through the cooling coil via a filter by a supply air fan. The cooled air is then supplied to different zones as shown in the figure. In this HVAC system, the supply air pressure is regulated by the speed of a supply air fan. Increasing the fan speed will increase the supply air pressure, and vice versa. The dynamics of the control signal feeding to the fan Variable Speed Drive to the supply air pressure can be modeled as a second order plus dead time plant. A. Performance Analysis of the STFPIC Study as well as analysis is made if the performance of STFPIC is applied under normal condition and changing of HVAC process model. Under Normal Condition: The transfer function of the supply air pressure loop under normal condition is obtained as G(s) = 0.81e−2s /(0.97s + 1)(0.1s + 1) where gain (K) = 0.81, τ1 = 0.97, τ2 =0.1 and dead time (δ) = 2 sec. For this process scaling factors are set at Ne = 0.9, N∆e = 5 and Nu = 2.5. Under HVAC Process Parameters Variation: 1) When gain (K) = 0.81, τ1 =0.2, τ2 =2 and dead time (δ) = 2 sec., then the transfer function of the supply air pressure loop is obtained as G(s) = 0.81e−2s /(0.2s + 1)(2s + 1). For this process scaling factors are set at Ne = 0.9, N∆e = 15 and Nu = 0.3.
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2) When gain (K) = 1.2, τ1 = 0.97, τ2 =0.1 and dead time (δ) = 3 sec., then the transfer function of the supply air pressure loop is obtained as G(s) = 1.2e−3s /(0.97s + 1)(0.1s + 1). For this process scaling factors are set at Ne =0.9, N∆e =3 and Nu =1. 3) When gain (K) = 1.2, τ1 = 0.97, τ2 =0.1 and dead time (δ) = 4 sec., then the transfer function of the supply air pressure loop is obtained as G(s) = 1.2e−4s /(0.97s + 1)(0.1s + 1). For this process scaling factors are set at Ne = 0.9, N∆e = 3 and Nu = 1. The Fig. 4, Fig.5, Fig.6, Fig.7 and Table 3 are shown that the supply air pressure loop of HVAC works satisfactorily both
under normal and as well as under model variations. Table 3 refers that both the rise time and settling time is very much satisfactory. Peak overshoots are also shown negligible when STFPIC is used. B. Comparison of Practical Performance with Existing Methods In order to demonstrate the effectiveness and robustness, the performance of the proposed STFPIC has been compared with those of existing methods, the Bi, Cai’s PID controller and Jian, Cai’s ANF controller [22] for supply air pressure loop control. The comparison has been done under changing process model. The results are provided in Table 4. For the application of STFPIC, substantial improvements have been observed in settling time and also in peak overshoot for all the transfer function of the air supply model compare to ANF and
PAL & MUDI, SELF-TUNING FUZZY PI CONTROLLER AND ITS APPLICATION TO HVAC SYSTEMS
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PID controller. Furthermore, it is more important that when the process encounters large parameter variations, the method provided presents much robustness as shown in Table 4. IV. CONCLUSIONS The STFPIC method used here is rather simple to understand by the control engineer. The results show that the STFPI controller does not hamper the HVAC process performance. This scheme differs from others as it attempts to implement the operator’s strategy while running a plant. The operators / control engineers can design the fuzzy rule base for fuzzy controller and as well as the fuzzy rule base for gain updating factor according to their knowledge. It can be expected that to exploit in other loops of HVAC system and other application areas. R EFERENCES [1] Z. R. Radakovic, V. M. Milosevic, S. B. Radakovic, “Application of temperature fuzzy controller in an indirect resistance furnace,” Applied Energy. 73 (2002) 167-182. [2] H.R.Benerji, Fuzzy Logic Controllers, in: R. R. Yager, L. A. Zadeh (Eds.), “An introduction to Fuzzy logic application in intelligent systems,00 Kluwer, Boston, MA, 1992. [3] H. J. Zimmermann, “Fuzzy sets theory and its applications.” Kluwer, Nijhoa, Boston, Dordrecht, Lancaster, 1984. Singapore: World Scientific, 1993. [4] Kim J-H, Kim K-C, Chong EKP, “Fuzzy precompensated PID controllers,” IEEE Trans. Con. Syst. Technology, 1994: 2(4). [5] Cho Hyun-Joon, Cho Kwang-Bo, Wang Bo- Hyeun, “Fuzzy-PID hybrid control: automatic rule generation using genetic algorithm,” Fuzzy sets and systems, 1997, 92(3), 305-316. [6] Wu Zhi Qiao, Masaharu Mizumoto, Fuzzy sets and systems 78 (1996), 23-35. [7] M. Sugeno, “Industrial applications of Fuzzy Control,”Amsterdam, Netherlands: Elsevier,1985. [8] M. Sugeno and K. Tanaka, “Successive identifica- tion of a fuzzy model and its application to predi- ction of a complex system,” Fuzzy Sets Syst., vol. 42, pp. 315 – 334, 1991. [8] M. Sugeno and T. Yasukawa, “ A fuzzy–logic- based approach to qualitative modeling,” IEEE Trans. Fuzzy Syst., vol. 1, pp. 7 – 31, Feb. 1993. [9] R. M. Tong, “The construction and evaluation of fuzzy models,” in advances in Fuzzy Set Theory and applications ed. M. M. Gupta et. al., North Holland, 1979. [10] R. Palm, “Sliding mode fuzzy control”, in Proc. Fuzz IEEE, San Diego, CA, 1992, pp. 519-526.
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