Deign of a Fuzzy Baed Digital PID Controller for Control of Nonlinear HVAC Sytem VAHID JOHARI MAJD, FARHAD BESHARATI and FARZAN RASHIDI Faculty of Engineering, Tarbiat Modarre Univerity Tehran Iran Abtract- Heating, Ventilating and Air Conditioning (HVAC) lant i a multivariable, nonlinear and non minimum hae ytem, which control of thi lant, i very difficult. Thi aer reent a new aroach to control of HVAC ytem. The rooed method i a hybrid of fuzzy logic and dicrete PID controller. Simulation reult how that thi control trategy i very robut, flexible and alternative erformance. To evaluate the uefulne of the rooed method, we comare the reone of thi method with dicrete PID (DPID) controller. The imulation reult how that our method ha the better control erformance than DPID controller. Keyword: HVAC Sytem, Fuzzy Logic, dicrete PID Controller, Fuzzy dicrete PID, Robut, multivariable, nonlinear. h w Enthaly of liquid water h Enthaly of water vaor fg ratio of uly air C Secific heat of air W M o Moiture load T 2 Temerature of uly air V Volume of thermal ace f Volumetric flow rate of air W o V he W T o Q o T ratio of outdoor air Volume of heat exchanger ratio of thermal ace Temerature of outdoor air Senible heat load Temerature of thermal ace ρ Air ma denity gm Flow rate of chilled water Introduction In recent year, fuzzy logic controller, eecially dicrete PID tye fuzzy controller have been widely ued in indutrial rocee owing to their heuritic nature aociated with imlicity and effectivene for both linear and nonlinear ytem. In fact, for ingle-inut ingle outut ytem, mot of fuzzy logic controller are eentially of dicrete PD tye, PI tye or PID tye with nonlinear gain. Becaue of the nonlinearity of the control gain, fuzzy dicrete PID controller oe the otential to achieve better ytem erformance over conventional dicrete PID controller rovide the nonlinearity can be uitably utilized. On the other hand, due to the exitence of nonlinearity, it i uually difficult to conduct theoretical analye to exlain why fuzzy dicrete PID controller can achieve better erformance. Conequently it i imortant, from both theoretical and ractical oint of view, to exlore the eential nonlinear control roertie of fuzzy dicrete PID controller, and find out aroriate deign method which will ait control engineer to confidently utilize the nonlinearity of fuzzy dicrete PID controller o a to imrove the cloed-loo erformance. Thi aer reent a new aroach to control of HVAC ytem. The rooed method i a hybrid of fuzzy logic and dicrete PID
controller. Simulation reult how that thi control trategy i very robut, flexible and alternative erformance. 2 Fuzzy Baed Dicrete PID Controller In thi ection, we decribe the deign of the FPIC: it mathematical derivation and technical content. We have ued a fuzzy PI control unit arrangement, called the derivative-of-outut, which i often deirable if the reference inut contain dicontinuity [6]. For thi arrangement, the derivation of the fuzzy control law i erformed in the outut of the fuzzy PID controller. It hould be mentioned that the fuzzy PID controller that we deigned i a digital controller. We firt tart with a continuou conventional PID controller and then ue the tandard bilinear tranform to convert it to the correonding digital controller. We firt decribe the deign rincile and baic tructure of the fuzzy PID controller. The outut of the conventional analog PID controller in the frequency - domain, i given by de( t) u PID = k e( t) + ki e( t) dt + kd dt and it frequency domain form i ki U PID ( ) = k + + k d E( ) () where k, k i and k d are the roortional, integral and derivative gain reectively, and E () i the tracking error ignal. Thi equation can be tranformed into the dicrete verion for comuter aided control ytem by alying the bilinear tranformation 2 z =, where T>0 i the amling T + z time, which reult in the following form: kit 2kd kit kd U PID( z) = k + + E( z) 2 T z T + z Letting k T k K = i d k 2 2, K i = kit, kd K d = T T The fuzzy controller can be viewed a a natural extenion of the conventional dicrete PID control algorithm with a fuzzy imlementation [2]. The tructure of the fuzzy dicrete PID (FDPID) controller include two block of the traditional fuzzy controller: a fuzzyfier and an inference engine. A uually, the traditional fuzzy controller work with inut ignal of the ytem error e and the change rate of error de. The ytem error i defined a the difference between the et oint r(k) and the lant outut y(k) at the te k, i.e.: e(k)=r(k)-(k) (2) The change rate of the error de at the te k i: de(k)=e(k)-e(k-) () A a third inut ignal, the FDPID can ue the accumulative error δ: δe(k)= e(i) (4) The mot ued digital PID control algorithm can be decribed with the well-known dicrete equation: u(k)=k e(k)+k i δe(k)+k d de(k) (5) where u(k) i the outut control ignal. The Sugeno fuzzy rule into the FPID can be comoed in the generalized form of ifthen tatement to decribe the control olicy and can be rereented a: R (n) : if e i E (n) i and de i de (n) (n) i and δe i δe i Then f (n) u = K (n) e(k)+ K (n) d de(k)+ K (n) i δe(k)+k 0 (6) where e, de, δe are the decribed inut variable and k, ki and kd are the ame contant a in (5). Thi way the imilarity between the equation of the conventional digital PID controller (4), (5) and the Sugeno outut function fu in the equation (6) could be found. The fuzzy imlication can be erformed by mean of the roduct comoition [2]: µ (n) u = µ (n) e µ (n) (n) de µ δe (7) where µ (n) (n) (n) e, µ de and µ δe ecify the memberhi value uon fired fuzzy et of the correonding inut ignal. For a dicrete univere with N quantization level in the controller outut, the control action u F i exreed a a weight average of the Sugeno outut function f u and their memberhi value µ u of the quantization level [5]:
N f ui i= F = N u i= µ µ ui ui HVAC Sytem (8) The conumtion of energy by heating, ventilating, and air conditioning (HVAC) equiment in commercial and indutrial building contitute % of the world energy conumtion [5]. In ite of the advancement made in comuter technology and it imact on the develoment of new control methodologie for HVAC ytem aiming at imroving their energy efficiencie, the roce of oerating HVAC equiment in commercial and indutrial building i till an low-efficient and high-energy conumtion roce [6]. Claical HVAC control technique uch a ON/OFF controller (thermotat) and roortional- integralderivative (PID) controller are till very oular becaue of their low cot. However, in the long run, thee controller are exenive becaue they oerate at very low energy efficiency and fail to conider the comlex nonlinear characteritic of the multi-inut multi-outut (MIMO) HVAC ytem and the trong couling action between them. The roblem of HVAC control can be oed from two different oint of view. In the firt, one aim at reaching an otimum conumtion of energy. In the econd, that i more common in HVAC control, the goal i keeing moiture, temerature, reure and other air condition in an accetable range. Several different control and intelligent trategie have been develoed in recent year to achieve the tated goal fully or artially. Among them, PID controller [4,4], DDC method [5,6], otimal [0,9,7], nonlinear [] and robut [,] control trategie, and neural and/or fuzzy [,6,7] aroache are to be mentioned. We have alo dealt with thi roblem and rovided novel olution in [5]. The uroe of thi aer i to ugget another control aroach, baed on fuzzy DPID controller to achieve fater reone with reduced overhoot and rie time.. HVAC Model In thi art, we give ome exlanation about the HVAC model that we have ued. For imulation of HVAC ytem, ome different model have been rooed and conidered. In [7] a linear firt order model of the ytem with a time delay i ut forward, while the nonlinearity of the HVAC ytem i conidered in [6]. In thi aer, we ued the model develoed in [4], ince it aim at controlling the temerature and humidity of the Variable Air Volume (VAV) HAVC ytem, however SISO bilinear model of the HVAC ytem for controlling the temerature ha been given in [2]. Below, we decribe the mathematical tructure of a MIMO HVAC model ued throughout thi aer. The tate ace equation governing the model are a follow: x& = u α (x x ) u α (W x ) + 2 2 α (Q h M ) o fg o x& = u β ( x + x ) + u β 5( T x ) o u β (0.25W + 0.x W ) o 2 x& = u α (W x ) + α M 2 2 4 o y = x y = x (9) 2 2 In which the arameter are: u = f, u2 = gm, x = T, x2 = W, x = T α = / V, α = h / C V, α = / ρc V, α = / ρv, β = / V 4 2 fg 2 / Vhe, = he, w β = ρc β h / C V (0) And the numerical value are given in table. Alo, the actuator tranfer function can be conidered a: 2 z G act ( z) = () k + z In which k and τ are the actuator gain and time contant. The chematic tructure of he 2
the HVAC ytem i given in figure. The ytem ha delayed behavior which i rereented via linearized, firt order and time delay ytem. Furthermore, the model rereent a MIMO ytem in which one of the I/O channel ha a right half lane zero, meaning that it i non-minimum-hae. Table: Numerical Value for ytem arameter ρ =. 074 lb / ft C =. 24 Btu / lb. F V = 58464 ft T o = 85 F M o = 66.06 lb / hr V he =. ft W =.007 lb / lb W o =.008 lb / lb Outide Damer Exhauted Air 5 4 Heat Exchanger Filter Chiller Pum Cooling Coil 2 Thermal Sace FAN Figure. Model of the HVAC ytem Suly Air 4 Simulation Reult In thi ection, we decribe the circuit we have ued for controlling the HVAC lant. The actual lant model involve four inut and three outut rocee, of which two inut can be maniulated for achieving deired erformance level. Our initial attemt to conider an SISO roblem in which temerature et oint tracking wa the main goal roved futile, becaue the ret of the ytem could not be regarded a diturbance and unmodeled dynamic. The reone eed caued the other outut increae beyond accetable level. Next, we tried to achieve the deign goal via two earate fuzzy PID controller (Figure 2). We wihed to track temerature and humidity to their reecting et oint level of 7 F and 0.009, while maintaining the uly air temerature within the range of 40 F to 00 F. Thi roved very atifactory (Figure and 4). The erformance level achieved via the two alternative aroache are outlined in table 2. Figure 2: Control circuit with two controller 9.000 x 0- Temerature 74.5 74 7.5 7 72.5 72 7.5 7 9.0002 9.000 9 8.9999 8.9998 8.9997 Suly Air Tem 65 55.5 0 0.05 0. 0.5 0.2 0.25 0. 8.9996 0 0.05 0. 0.5 0.2 0.25 0. Figure. HVAC ytem reone with Fuzzy DPID controller 0 0.05 0. 0.5 0.2 0.25 0.
0 0.04 00 0.0 Temerature 0.02 0.0 0.0 Suly Air Tem 0.009 40 0.008 Figure 4. HVAC ytem reone with DPID controller 0 Table 2- Performance characteritic of HVAC ytem with two Fuzzy DPID and DPID controller S-SError (Tem-Humi) Rie (Tem-Humi) POS (Tem-Humi) Fuzzy DPID 0.0%-0.00% 0.00-0.0002 02.28-0.00 DPID 0.00%-0.00% 0.009-0.002 49.96-4. We examined the robutne of thee controller with reect to external diturbance. To do that, we fed the lant with time-variable heat and moiture diturbance ignal in the form given in figure 5. A oberved in the figure 5, there i ome deterioration from the nominal amount of the two external diturbance. The reone of the two Fuzzy PID controller and of the two PID controller are given in the figure 6.8 x 05 and 7. A hown figure 6 and 7, the fuzzy PID controller how the better control erformance than PID controller in term of ettling time, overhot and rie time. The outut of the ytem, with the reence of diturbance variation, how that the fuzzy PID controller can track the inut uitably. But the erformance of PID controller i too low. 220.7.6 20.5 200.4 Multure..2 Heat. 2.9 2.8 0 0.5.5 2 2.5.5 0 0.5.5 2 2.5.5 Figure 5. The heat and moiture diturbance ignal for robutne conideration 76 9.0004 x 0-9.000 9.0002 Temerature 74 7 72 7 9.000 9 8.9999 8.9998 8.9997 Suly Air Tem 65 55 0 0.5.5 2 2.5.5 8.9996 0 0.5.5 2 2.5.5 0 0.5.5 2 2.5.5 Figure 6. HVAC ytem reone of the Fuzzy DPID controller with the reence of diturbance variation.
0 0.0 00 0.025 0.02 0.05 Temerature 0.0 0.005 0.0 Suly Air Tem 40 0.0095 0.009 0 0.0085 20 Figure 7. HVAC ytem reone of the DPID controller with the reence of diturbance variation. 5 Concluion In thi aer, we howed the alicability of fuzzy DPID controller to the fulfillment of comlex tak of adative et oint tracking and diturbance rejection of a HVAC ytem. The control of the non-minimum hae, multivariable, nonlinear and nonlinearizable lant with contraint on it uly air temerature i indeed a demanding tak from control theoretic viewoint. The controller reented in thi aer oeed excellent tracking eed and robutne roertie. The comarion with a DPID controller i only meant to ignify the extent of the goal overfulfillment and hould by no mean imly that no other intelligent and adative controller can erform uitably. Reference [] Alber, T. P., Chan, W. L., Chow, T. T., A Neural Network-Baed Identifier/Controller for Modern HVAC Control ASHRAE Tranaction, Volume 00, 994 [2] G. Chen, Conventional and fuzzy PID controller: An overview, Int. J. Intell. Control Syt., vol.,. 25 246, 996. [] Curti, P. S., Kreider, J. F., Branenuehl, M. J., Local and Global Control of Commercial Building HVAC Sytem Uing Artificial Neural Network, ACC, 994 [4] Fargu, R. S., Chaman, C., Imlementation of Commercial PI-Neural Controller for Building Service IEE, Savoy Place, 999 [5] Geng, G., Geary, G. M., On Performance and Tuning of PID Controller in HVAC Sytem Conference on Control Alication, 99 [6] Hartman, T. B., Direct Digital Control for HVAC Sytem MC Graw-Hill, 99 [7] Heworth, S. J., Dexter, A. L., Neural Control of Nonlinear HVAC Plant Conference on Control Alication, 994 [8] Jian, W., Wenjian, C., Develoment of an Adative Neuro-Fuzzy Method for Suly Air Preure Control in HVAC Sytem IEEE, 2000 [9] Miller, R. C., Seem, J. E., Comarion of Artificial Neural Network with Traditional Method of Predicting Return from Night or Weekend Setback ASHRAE Tranaction, Volume 97, 99 [0] Newman, H. M., Direct Digital Control for Building Sytem John Wiley, 994 [] Oman, A., Mitchell, J. W., Klein, S. A., Alication of General Regreion Neural Network in HVAC Proce Identification and Control ASHRAE Tranaction, Volume 02, [2] Petrov, M., I.Ganchev and I.Dragotinov. Deign Aect Of Fuzzy PID Controller, 5th International Conference on Soft Comuting Mendel 99, June 9-2 999, Brno, Czech Reublic,.277-28. [] Rahidi. F., Luca. C., Alying CBRL (Context Baed Reinforcement Learning) to Multivariable Control of a HVAC Sytem, Tran. Of WSEAS, 200 [4] Serrano, B. A., Velez-Reye, M., 999, Nonlinear Control of A Heating, Ventilation and Air Conditioning Sytem with Thermal Load Etimation IEEE Tranaction on Control Sytem Technology, Volume 7, () [5] T. Takagi and M. Sugeno, Fuzzy identification of ytem and it alication to modeling and control, IEEE Tran. Syt., Man, Cybernetic., vol. 5,. 6 2, 985. [6] Tigrek, T., Daguta, S., Smith, T. F., Nonlinear Otimal Control of HVAC Sytem Proceeding of IFAC, 2002