A new adaptive PI controller and its application in HVAC systems

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1 Energy Conversion and Management 48 (2007) A new adaptive PI controller and its application in HVAC systems Jianbo Bai *, Xiaosong Zhang Department of Power Engineering, Southeast University, Nanjing, Jiangsu , PR China Received 25 April 2006; accepted 29 October 2006 Available online 17 January 2007 Abstract The paper concerns the development of a new adaptive PI controller for use in HVAC systems. The process of HVAC control can be described as a first order plus dead time model. A kind of arithmetic of recursive least squares (RLS) with exponential forgetting combined with model matching of a zero frequency method is adopted to estimate the model s parameters while the system remained in closed loop. Then, a simple tuning formula for a PI controller with robustness based on the estimated parameters was used to adjust the controller s parameters automatically while under closed loop. To evaluate the effectiveness of the adaptive PI controller, the proposed method was compared with a H 1 adaptive PI controller. The simulation results show that the new adaptive PI controller has superior performance to that of the H 1 adaptive PI controller. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: HVAC system; Temperature control; Adaptive control; PI controller; System identification 1. Introduction * Corresponding author. Tel.: address: bai_jianbo@sina.com (J. Bai). Although traditional PI/PID (proportional integral/proportional integral derivative) controllers have been most commonly used in HVAC systems [1], it has sometimes been difficult to compensate fully for load disturbance and to keep controlled variables close to set point values within the prescribed range. Furthermore, finding optimal PID gains by the trial-and error process is one of the most tedious problems faced by a field operator. Even the PID controller that has been well tuned at its commissioning may perform poorly because of changes in operation conditions. In order to get good control effects, there are three common methods for adjusting the PI/PID controller in HVAC systems: manual, self tuning and adaptive control methods. With manual tuning, the operator runs different tests to determine the parameters that characterize the process. Simple empirical rules are then used to determine the controller parameters from the estimates of the process parameters. Seborg et al. reviews manual tuning methods [2]. Self tuning has been widely and successfully applied in many industrial fields [3]. A self tuning controller assumes that the process under control has initially unknown but constant parameters during normal operation, so the parameters can be estimated by a step test or relay feedback. Pinnella et al. proposed a self tuning method of an integral only controller [4]. Brandt exploited the self tuning control in HVAC systems [5]. Nesler developed a computer based self tuning controller, where the open loop step test of Z N methods was used to derive the process model and the controller [6]. Åström et al. describes a tuning method for a general digital controller based on relay feedback to HVAC plants in which the control design method is based on pole placement [7]. Bi et al. have developed a new PID design method based on step and relay experiments, which gives satisfactory performance on HVAC systems [8]. For thermal load disturbances, most HVAC systems have time varying dynamic characteristics in which the system s parameters are variable during the operation, so it is not sufficient to use a self tuning controller in /$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi: /j.enconman

2 1044 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) HVAC systems. According to Åström and Wittenmark, a fixed gain controller should be used for systems with constant dynamics, and adaptive control methods should be used for processes with time varying dynamics [3]. So, we should use adaptive control methods in the HVAC industry. Self tuning control and adaptive control are the same and represent control systems capable of adjusting themselves to changes in their operational domain as well as changing requirements. The self tuning control system makes these adjustments in search of optimality over an initial period only and then freezes, while the adaptive controller continues to make adjustments throughout its life. In comparison with other control methods, the application of adaptive control in HVAC systems has the following benefits: (1) It can eliminate the time to retune HVAC control systems due to seasonal changes or others. (2) It can eliminate the time to tune new HVAC control systems. (3) It can be adaptive to load changes or disturbances and remain stable. (4) It can give near optimum control performance and decrease energy use. (5) It is easy to realize in current control systems. The application of adaptive control in HVAC systems can be found in the following literature. Qu et al. developed real time tuning of PI controllers in HVAC systems [9]. The HVAC process they described is a first order plus dead time model. By using the recursive least squares (RLS) method, the model s parameters were estimated while the HVAC system remained in closed loop. Then, H 1 loop shaping tuning rules were transformed to discrete time tuning rules and implemented in an adaptive PI control strategy. The main limitation of the tuning method is that the RLS method cannot estimate the model s dead time online, and the PI control strategy has slowness of response. Seem presented a new method for implementing a new pattern recognition adaptive controller (PRAC), developed through optimization, for automatically adjusting the parameters of PI controllers while under closed loop control [10]. Inputs to the PRAC are the set point, controller output and sampled values of process output. Depending on estimation of the noise level of the system, the PRAC will determine the parameters of the digital PI controller used in a HVAC system. The main limitation of the tuning method is that the PRAC can only estimated the HVAC control process s noise, which cannot represent characteristics of the HVAC system in fact. The paper presents a new adaptive PI control method in HVAC systems. In the paper, the HVAC process is described as a first order plus dead time model. By using RLS with exponential forgetting combined with model matching of zero frequency, all of the HVAC model s parameters, including the dead time, were updated while the process remained in closed loop. Then, a simple tuning formula for the PI controller acts as the adaptive control strategy. Simulation results show that this adaptive control method is very robust and has better performance than other adaptive PI control methods. The paper is organized as follows: In Section 2, the HVAC model of a temperature control system is analyzed. Section 3 describes the system s online identification method of the HVAC systems and shows the estimated results under closed loop. Section 4 designs a tuning formula for the PI controller with robustness, which can guarantee the real HVAC process with gain margin A m = 2.98 and phase margin / m = Section 5 reviews the adaptive PI controller and shows the simulation results, which mainly compare this adaptive control method with a H 1 adaptive PI controller [9]. 2. HVAC system model The HVAC system we considered in the paper is the temperature control system in the testing room of an air conditioner, enthalpy difference method testing platform. The testing-platform is a kind of facility to test the performance of air conditioners (Fig. 1). It is composed of two testing rooms, which simulate indoor environmental space and outdoor environmental space separately. When the performance of air conditioners is being tested, the testing rooms must keep the temperature and humidity invariable. The control effect and precision of the temperature and humidity in the testing room are important for running the testing platform. Every testing room has its own HVAC system including electric heater, electric humidifier and evaporator of refrigeration plant to maintain the temperature and humidity of the environmental space. To simplify the simulation, we use the heater as the only control element to execute the simulation of temperature control in the testing room. The heater is an electric heater that heats indoor air and is adjustable within the range from 0 kw to 12 kw, corresponding to the controller output from 0% to 100%. From the point of view of online control, it is necessary to reduce the computational effort required in the identification and control of the HVAC system. We describe the HVAC system as a first order plus dead time model, which was represented as in Fig. 2. In Fig. 2, u(t) is the controller output and the input of the system that represents the heater s output power ratio, y(t) is the output of the system that represents indoor temperature ( C), Q(t) is the heater s output power relative to u(t) andk s, T d, T s and n(t) represent plant gain, dead time, time constant and load disturbances of the system separately. Then, the transfer function of the HVAC system is as follows: GðSÞ ¼ Y ðsþ UðSÞ ¼ k s T s s þ 1 e T ds ð1þ

3 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fan Fan Electric humidifier Pore plate blowing-in Pore plate blowing-in Electric humidifier Testing-platform main body Compressor Evaporator test Air-conditioner Evaporator Compressor Electric heater Electric heater To condensor From throttle mechanism Inner tesing-room Outter testing-room From throttle mechanism Fig. 1. Diagram sketch of testing-platform of air conditioner enthalpy difference method. To condensor Fig. 2. Approximate mathematical model of the HVAC system. Assuming T is the sampling time and d = T d /T, then the z transform model of the HVAC system is derived: S 1 et GðzÞ ¼z k s e ð T dsþ ¼ Bðz 1 Þ S 1 þ T s S Aðz 1 Þ z d ¼ bz 1 1 az 1 z d ð2þ where T =T s a ¼ e ð3þ b ¼ k s ð1 e T =T s Þ ð4þ According to the HVAC system s input (u(t)) and output (y(t)) values, the a, b and d parameters can be estimated by RLS with exponential forgetting combined with model matching of zero frequency method. Then, the k, T s and T d parameters of Eq. (1) can be calculated. 3. System identification 3.1. Joint identification arithmetic The RLS method is an effective approach to online identification of model parameters. In common, both the a and b parameters of the system can be estimated by the RLS method [8], but the RLS method alone cannot estimate d online. In this paper, we explored a new joint identification arithmetic, which uses RLS with exponential forgetting combined with model matching of zero frequency method to estimate a, b and d parameters all together. In order to estimate d, we deploy G(z) s numerator in Eq. (2), B(z 1 ) Æ z d,asb m (z 1 ): B m ðz 1 Þ¼b 1 z 1 þ b 2 z 2 þþb m z m m ð5þ d s In the above formula, m 1 is the biggest possible delay time of the HVAC system. Then the estimation model of the HVAC process is G m ðzþ ¼ Bðz 1 Þ Aðz 1 Þ z d ¼ B mðz 1 Þ Aðz 1 Þ b ¼ 1 z 1 þ b 2 z 2 þþb c m z m m ð6þ 1 ^az 1 and the ˆ symbol denotes an estimated parameter. The parameter estimates are compacted into a vector as ^h ¼½^a; b 1 ; b 2 ;...; b c m Š T ð7þ resulting in model output of yðkþ d ¼^h T hðkþ ð8þ where hðkþ ¼½ yðk 1Þ; uðk 1Þ; uðk 2Þ;...; uðk mþš T ð9þ For the recursive algorithm to be able to update at each sample time, it is necessary to define an error. The model prediction error is defined as eðkþ ¼yðkÞ ^h T ðk 1ÞhðkÞ ð10þ which is the difference between the actual system output and the estimation model output. This error is used to update the parameter estimates according to ^hðkþ ¼^hðk 1ÞþGðkÞeðkÞ ð11þ where the estimator gain matrix is defined as pðk 1ÞhðkÞ GðkÞ ¼ ð12þ q þ h T ðkþpðk 1ÞhðkÞ The forgetting factor, q, will be chosen by the designer as a value between 0 and 1 as described below. The covariance matrix P is updated using PðkÞ ¼ 1 q ½I GðkÞhT ðkþšpðk 1Þ ð13þ The forgetting factor in Eq. (12) allows new data to be weighted more heavily than old data when updating the

4 1046 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) parameters. Thus, a large transient in the past will be discounted as time goes by. The initial value of P(k) and h(k) are chosen as estimates and allowed to settle to their final values as the program goes through several iterations. There is no unique way to initialize the algorithm because it depends on knowledge of the process. The common suggestion is to set ^hð0þ ¼zero and P(0) = ai where a is a large scalar. After ^h is estimated, then the b and d parameters of G(Z) s numerator B(z 1 ) Æ z d can be calculated by model matching of zero frequency method. Assuming frequency x = 0, let the value of zero order and first order differential coefficient of Bðz 1 Þz d j z¼e jwt be equal to cb m ðz 1 Þj z¼e jxt Bðz 1 Þz d j z¼e jwt ¼ cb m ðz 1 Þj z¼e jxt ð14þ dbðz 1 Þz d j z¼e jwt d- ¼ d cb m ðz 1 Þj z¼e jwt d- From the above formulae, b and d can be estimated by ^b ¼ Xl bb i i¼1 X l "!, X ^d ¼ i b l i i¼1 i¼1 ð15þ ð16þ bb i # 1 ð17þ The integer value of d parameter can be calculated by ^d ¼ INTð^d þ 0:5Þ ð18þ 3.2. Simulation results The HVAC system with the following parameters was simulated. K denotes sampling time sþ1 e 5s ð0 < K 6 100Þ >< 72 GðsÞ ¼ 60sþ1 e 6s ð100 < K 6 200Þ ð19þ >: ð200 < K 6 300Þ 84 80sþ1 e 6s Assume the unit of time constant and dead time is minute, the sampling period T = 1 min, environmental temperature is 25 C and the room setting temperature is 35 C, Hence, the variations of the a, b and d parameters in Eqs. (2) (4) are as follows 8 >< a ¼ 0:98347; b ¼ 1:19; d ¼ 5 ð0 < k 6 100Þ a ¼ 0:98347; b ¼ 1:19; d ¼ 6 ð100 < k 6 200Þ >: a ¼ 0:98758; b ¼ 1:043; d ¼ 6 ð200 < k 6 300Þ ð20þ The implementation scheme for online identification of k s, T s and T d in the HVAC system under closed loop mode in Simulink is shown in Fig. 3. The parameters of the PID controller in Fig. 3 are K P = 0.2, K i = 0.02 and K d = 0.5, which is based on Z N tuning rules [11] and the HVAC system of Eq. (19) from 0 to 100 sampling time. Figs. 4 6 show the identification results of the k s, T s and T d parameters of the HVAC system under closed loop. The identification parameters converge fast and reach their final values. The identification precision is satisfactory whether the parameters are stable or unstable. From Fig. 7, we note that the prediction output according to the estimated HVAC model has almost the same shape as the actual process output. This fact indicates the joint identification arithmetic is suitable for online identification of the HVAC system. 4. A simple PI tuning strategy with robustness 4.1. PI controller According to Haines, PI control is the preferred method of control for HVAC systems because of the improvements in accuracy and energy consumption when compared to proportional control [12]. A PI controller is a kind of linear controller, as it composes the control error according to the setting value and process output and then makes the controller output value based on the linear resultant of the error s proportion and integral. For a PI controller, the control signal at time t is determined from Z t uðtþ ¼K P eðtþþk i eðtþdt þ u 0 ð21þ 0 where u(t) is the controller output, u 0 is the controller initial output, K P is the proportion parameter, K i is the integral parameter, e(t) is the error at time t defined as e(t) =y sp (t) y(t), where y sp is the set point for process output at time t and y(t) is process output at time t. The transfer function of the PI controller is as follows: G c ðsþ ¼ UðsÞ EðsÞ ¼ K P þ K i ð22þ s 4.2. Cancellation of PI controller s zero and HVAC system s pole According to the transfer function of the HVAC system and PI controller, we can get the open loop transfer function of the HVAC system that uses a PI controller G o ðsþ ¼G c ðsþgðsþ ¼ K P þ K i ks e T ds ð23þ s T s þ 1 Fig. 8 shows the Nyquist curve of Eq. (23), With PI control, it is possible to move the given point on the Nyquist curve in three directions in the complex plane, as is indicated in Fig. 8. The point A may be moved in the radial direction of G(s) by changing the proportion coefficient K P.IfK P is great than 1, the A point will move away from the origin, otherwise A will move closer to the origin. The point A may be moved in the orthogonal direction by changing the integral coefficient K i, and the degree range of the movement of point A is from 0 to 90. The time constant of the HVAC model is always large. It will reduce the performance of instantaneous response apparently. We consider that cancellation of the PI controller s zero and the HVAC system s pole in the control

5 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fig. 3. Simulink sketch of online identification under closed loop mode. loop will improve the control performance. So we choose the controller s zero so that it will cancel the HVAC model s pole. The HVAC model in Eq. (20) can be written in the form GðsÞ ¼ e T ds ð24þ es þ f where e ¼ T s ; f ¼ 1 ð25þ k s k s The PI controller in Eq. (22) can be written in the form G c ðsþ ¼k Fig. 4. Closed loop mode identification results (k s ). Es þ F s ð26þ Fig. 5. Closed loop mode identification results (T s ). where K P ¼ ke; K i ¼ kf ð27þ Let e = E and f = F, then, the open loop transfer function of the HVAC system is approximated by G o ðsþ ¼G c ðsþgðsþ ¼ k e T ds s ð28þ 4.3. Designation of PI controller with robustness From Eq. (28), the G o (s) s frequency characteristic can be written in the form

6 1048 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fig. 6. Closed loop mode identification results (T d ). Fig. 8. Nyquist curve of HVAC system. Fig. 7. Comparison between actual output and predictive output. Fig. 9. Nyquist curve of HVAC system after the cancellation. G o ðjxþ ¼G c ðjwþgðjwþ ¼ k sinðxt dþ x k cosðxt dþ j - ð29þ Fig. 9 shows the Nyquist curve of the open loop transfer function of the HVAC system after cancellation of the PI controller s zero and the HVAC system s pole. It can be seen that the Nyquist curve of the loop transfer function is tangent to a line parallel to the imaginary axis of the complex plane. Wang and Shao [13] defined the inverse of the distance between the line and the imaginary axis as a new specification for robustness k. 1 k ¼ max jre½g cðjwþgðjwþšj ð30þ 06x<1 From Eq. (29), we can get following equation jre½g c ðjwþgðjwþšj ¼ k sinðxt dþ x ð31þ k sinðxt d Þ x is a periodic function with amplitude attenuation (Fig. 10). The oscillation periodic time of the function is 2p/(x Æ T d ). When x >0, the function gets the maximal value k Æ T d. Hence, 1 k ¼ k T d ð32þ We can calculate the gain margin A m and phase margin (P point of Fig. 9) of Eq. (29) A m ¼ p=2 ð33þ k T d u m ¼ p ð34þ A m From Eqs. (32) (34), we get A m ¼ p 2 k ð35þ u m ¼ p 2 1 ð36þ k It is obvious that the new specification satisfies both the gain margin requirements to a degree. According to Eq. (32) and the closed loop stability criterion of Nyquist,

7 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) In this section, we explore a new adaptive PI controller for the HVAC system. Fig. 11 shows the diagram sketch of the control system. The function of the Joint identification arithmetic is to estimate the parameters of the HVAC system online. The function of the Robust PI tuning strategy is to adjust the PI controller s parameters according to Eq. (38). Finally, comparisons between the adaptive PI control and a H 1 adaptive PI controllers will be made Adaptive PI controller implementation The new adaptive PI controller that is recommended by the paper has the following procedure k ¼ p=2 T d is the critical value of the HVAC control loop, where k = It means that the HVAC control loop will become unstable when k < The optimal selected value of k will satisfy the requirements of good temporal response and robustness. After a large amount of simulation tests, we choose the robust specification value k = 1.9, where the gain margin A m = 2.98 and the phase margin / m = Hence, 1 k ¼ ¼ 0:526 ð37þ 1:9 T d T d With Eqs. (24) (27) and (2) (4), we form the PI controller as K P K i " # ¼ 0:526 T s k s T 1 d k s Fig. 10. Function of k sinðxt dþ. ¼ 0:526 " # a 1 bdln a 1 a bdt x ð38þ (1) Preset initial condition ^hð0þ ¼e (e is a fully small real vector) P(0) =a 2 I (a is a fully large number) K P [0] = 0.1 K i [0] = Sampling period = 1 min (2) Compose h(k) according to Eq. (9) based on the u(k) and y(k) of the HVAC system s input and output data; (3) Compose G(k) and P(k) of Eqs. (12) and (13) according to a new group of data y(k) and h(k); (4) According to Eq. (11), estimate a new parameter vector ^h k that includes a k,b 1k,b 2k,... and b mk; 5. Adaptive PI controller Fig. 12. Response of three kinds of a adaptive PI controller for time variable HVAC system. Load disturbance R e(t) u(t) PI Controller +_ Heater + + Air conditioning Room space y(t) Robust PI tuning strategy Identified HVAC system Joint identification arithmetic Fig. 11. Diagram sketch of the new adaptive PI controller for HVAC system.

8 1050 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fig. 13. Variation of K P and K i of recommended adaptive PI controller for time variable HVAC system. (5) Calculate b k and d k according to Eqs. (16) (18); (6) Obtain K P and K i from Eq. (38); (7) At k = k + 1, go back to step 2 and continue iteration Simulation results and comparison Fig. 14. Recommended PI controller output for time variable HVAC system Adaptive PI control comparison for time variable HVAC system In this simulation, assume the time variable HVAC system is as Eq. (19). Fig. 12 shows the control results of three kinds of adaptive PI controller. It can be seen that the adaptive PI controller based on Z N tuning strategy has great overshoot and oscillation. The adaptive PI controller based on the H 1 loop shaping tuning rules or the recommended adaptive PI controller has little overshoot and oscillation, but the control results of the suggested adaptive PI control based on this paper is more stable than that of Fig. 15. Variation of K P and K i of H 1 adaptive PI controller for time variable HVAC system.

9 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fig. 16. H 1 system. adaptive PI controller output for time variable HVAC Fig. 19. Recommended adaptive PI controller output for setpoint changes in HVAC system. the H 1 loop shaping tuning rules. Because the Z N tuning rules controller has poor control performance, we only compare the control results of the adaptive PI control suggested in the paper with the H 1 loop shaping tuning rules control hereinafter. Fig. 13 shows the variation of K P and K i in the suggested adaptive PI controller, and Fig. 14 shows the controller output. Fig. 15 shows the variation of K P and K i of the H 1 loop shaping tuning rules controller, and then, Fig. 16 shows the controller output. Fig. 17. Response of adaptive PI controller for setpoint changes in HVAC system Adaptive PI control comparison for setting value variable in HVAC system Assume the HVAC system is GðsÞ ¼ 72 60s þ 1 e 5s ; from 0 to 150 sampling time and the setting value is 35 C. From 151 to 300 sampling time, the setting value is 37 C. Fig. 17 shows the control response of the new adaptive PI controller and the adaptive PI controller based on the H 1 Fig. 18. Variation of K P and K i of recommended adaptive PI controller for setpoint changes in HVAC system.

10 1052 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fig. 20. Variation of K P and K i of H 1 adaptive PI controller for setpoint changes in HVAC system. Fig. 21. H 1 adaptive PI controller output for setpoint changes in HVAC system. loop shaping tuning rules. It can be seen that the new adaptive PI controller has less transition time than the latter. Fig. 18 shows the variation of K P and K i of the new adaptive PI controller, while Fig. 19 shows the controller s output. Fig. 20 shows the variation of K P and K i of the adaptive PI controller based on the H 1 loop shaping tuning rules, while Fig. 21 shows the controller s output Adaptive PI control comparison under load disturbances in HVAC system Assume the HVAC system is GðsÞ ¼ 72 60s þ 1 e 5s ; the load disturbance s period is 100 min; the amplitude is 1 and the duty cycle is 2%. Fig. 22 shows the control results of the two adaptive PI control methods. The new adaptive PI controller method is more robust than the H 1 adaptive PI controller. Fig. 23 shows the variation of K P and K i of the new adaptive PI controller, while Fig. 24 shows the Fig. 22. Response of adaptive PI controller under load disturbances in HVAC system. controller output. Fig. 25 shows the variation of K P and K i of the H 1 adaptive PI controller, while Fig. 26 shows the controller s output. 6. Conclusions We have proposed a new adaptive PI control method for temperature control of the testing platform of the air conditioner enthalpy difference method. The tuning methodology can estimate the parameters of the HVAC system accurately while the control system remains in the closed loop, and then, a new PI tuning strategy is adopted to adjust the PI controller s parameters in real time. Simulation results show that the new adaptive PI control method has faster response, smaller overshoot and higher accuracy than the H 1 adaptive PI control method for a time variable HVAC system, and it has stronger robustness and better stability than the latter under load disturbances or

11 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) Fig. 23. Variation of K P and K i of recommended PI controller under load disturbances in HVAC system. Fig. 24. Recommended adaptive PI controller output under load disturbances in HVAC system. Fig. 26. H 1 adaptive PI controller output under load disturbances in HVAC system. Fig. 25. Variation of K P and K i of H 1 adaptive PI controller under load disturbances in HVAC system.

12 1054 J. Bai, X. Zhang / Energy Conversion and Management 48 (2007) setting value changes. The new adaptive PI controller can be widely used in the HVAC industry. Acknowledgement The research work presented in this paper was financially supported by the Natural Science Foundation of China, No References [1] Underwood CP. HVAC control systems: modeling, analysis and design. New York: Rout ledge; p [2] Seborg DE, Edgar TF, Mellichamp DA. Process dynamics and control. New York: John Wiley Sons; [3] Astrom KJ, Wittenmark B. Adaptive control. New York: Addison- Wesley; p [4] Pinnella MJ, Wechselberger E, Hittle DC. Self-tuning digital integral control. ASHRAE Trans 1986;92(2B): [5] Brandt SG. Adaptive control implementation issues. ASHRAE Trans 1986;92(2B): [6] Nesler CG. Automated controller tuning for HVAC application. ASHRAE Trans 1986;92(2B): [7] Åström KJ, Hagglund T, Wallenborg A. Automatic tuning of digital controllers with application to HVAC systems. Automatic 1993;29(5): [8] Bi Q, Cai WJ, Wang QG, Hang CC, Lee EL, Sun Y, Liu KD, Zhang Y, Zou B. Advanced controller auto-tuning and its application in HVAC systems. Control Eng Pract 2000;8(6): [9] Qu Guang, Zaheeruddin M. Real-time tuning of PI controllers in HVAC systems. Int J Energy Res 2004;28: [10] Seem JE. Implementation of a new pattern recognition adaptive controller developed through optimization. ASHRAE Trans 1997;103(1): [11] Ziegler JG, Nichols NB. Optimum settings for automatic controllers. Trans ASME 1942;64(8): [12] Haines RW. HVAC system design handbook, Blue Ridge Summit, PA, Tab Professional and Reference Books, [13] Wang YG, Shao HH. Optimal tuning for PI control 2000;36(1):