MINIMIZATION OF ACTUATOR REPOSITIONING USING NEURAL NETWORKS WITH APPLICATION IN NONLINEAR HVAC 1 SYSTEMS

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1 MINIMIZATION OF ACTUATOR REPOSITIONING USING NEURAL NETWORKS WITH APPLICATION IN NONLINEAR HVAC SYSTEMS M. J. Yazdanpanah *, E. Semsar, C. Lucas * yazdan@ut.ac.ir, semsar@chamran.ut.ac.ir, lucas@ipm.ir Cntrl & Intelligent Prcessing Center f Excellence & Department f Electrical and Cmputer Engineering, University f Tehran, P.O. Bx 495/55, Tehran, IRAN. Fax: (+98-) 609 Keywrds: Neural Netwrk, PID Cntrller, Chpper, Bilinear mdel, Radial Basis Functin (RBF). Abstract In prcess cntrl the main gal is nt nly the utput tracking but als the design f a suitable cntrl signal. It is due t the fact that it is dealt with the actuatr s perfrmance directly. This means that the undesired scillatins shuld be remved t alleviate the malfunctin f actuatrs. Mre than this, saving energy is an imprtant pint in energy-cnsuming systems such as Heating, Ventilating and Air Cnditining (HVAC) systems. In this paper, a nnlinear saturating element is designed with the help f neural netwrks. This element chps the undesired scillatins f the cntrl signal. The perfrmance f the system is guaranteed by the use f a number f PID cntrllers as central cntrllers and implementatin f a perfrmance measurement index t reduce the errr between the utputs and desired input signals. A multi-layer perceptrn netwrk is used and trained with Errr Back Prpagatin algrithm. The methd is applied t a nnlinear mdel f an HVAC system, which demnstrates the gd perfrmance f the prpsed design methd. Nmenclature hw W h fg Vhe W W C T M Q T T V s p s Enthalpy f liquid water Humidity rati f utdr air Enthalpy f water vapr Vlume f heat exchanger Humidity rati f supply air Humidity rati f thermal space Specific heat f air Temperature f utdr air Misture lad Sensible heat lad Temperature f supply air Temperature f thermal space Vlume f thermal space ρ f gpm Air mass density Vlumetric flw rate f air Flw rate f chilled water Intrductin Althugh in mst cntrl designs, utput tracking is almst the main gal f the designers, but it is imprtant t ntice that having a smth cntrl signal is as imprtant as utput tracking. The reasn is that, cntrl signal is usually implemented thrugh the actuatrs directly, s if it has a lt f vershts and scillatins, the actuatrs wuld be damaged very sn and this is nt ecnmical frm an industrial view pint. Mre than this saving energy is an imprtant pint in energy-cnsuming systems. Fr example HVAC systems cnsume abut 50% f the ttal energy used in the wrld [8]. In this way if the cntrl strategy culd gain at all the gals f tracking, energy saving and minimum actuatr repsitining with a level f tradeff between them, the benefits mentined abve can be achieved simultaneusly. Althugh these are imprtant prblems in cntrl applicatins, the authrs didn t find significant researches in this field and the nly wrks encuntered are as fllw: In [7] an ptimal cntrller is designed by the definitin f a cst functin in such a way that minimizes the cntrl signal scillatins. The H methd is used in [9,0,8] t minimize the actuatr repsitining. The Neural Netwrks are used fr this purpse in [0] and applied t the linear mdel f an HVAC system. In [6] a neural netwrk is used as a central cntrller fr the purpses f utput tracking and scillatin canceling. Internal mdel cntrl is als applied in [7] t achieve these gals. In this paper a new methd is intrduced by the use f Neural Netwrks. In spite f previus methds, which use Neural Netwrk as a central cntrller, in this paper the Central cntrllers are tw PIDs and tw Neural Netwrks are used fr ther cntrl purpses. The purpse f using such a cnfiguratin is that: In the industrial applicatins, the PID is usually the mst cmmn chice fr prcess cntrl and this chice is fr sme reasns such as: its simple tuning, easy hardware implementatin and acceptable perfrmance. But as we knw, this cntrller fails in sme situatins and s it culd nt satisfy sme desirable indices. S if sme way culd be fund t vercme the undesirable prperties Heating, Ventilating and Air Cnditining

2 W =.008 lb / lb C p = 4 Btu / lb. F W s.007 lb / lb =. ρ =. 074 lb / ft V s = ft V he = ft T = 85 F M = lb / hr τ =. 008 hr, k = 5 Q = Table : Numerical values fr system parameters withut changing the structure f the cntrller then the benefits f this cntrller can be maintained and at the same time a gd cntrl perfrmance can be achieved. The prpsed methd in this paper has the aim t vercme the scillatry respnse f PID cntrllers by the insight t gd tracking. This methd is based upn design f a nnlinear element, which chp the undesired scillatins. Accrdingly, a perfrmance index is intrduced t achieve an acceptable utput signal and minimize the cntrl signal s scillatins. This index is used fr the training f the netwrk t, such that when the netwrk is trained, bth gals f utput tracking and scillatin cancellatin will be achieved. This element is placed in feed-frward path and the training is dne n-line. Because f the prperty f Neural Netwrks, that prcess infrmatin in cntext and nt singly [4], they can receive at a better respnse in presence f the plant s uncertainties and disturbances. As mentined abve tw PIDs are used as the central cntrllers and their gains are tuned by cnventinal methds. Neural Netwrks are trained n-line by Errr Back- Prpagatin (EBP) algrithm. The rganizatin f the paper is as fllws: In sectin the analytical mdel f the system is brught. Sectin declares the cntrl prblem and the blck diagram f the system. In sectins 4 and 5, design f PID and nnlinear element are explained. Sectin 6 discusses the Neural Netwrk structure and in sectin 7, advantages f the new methd and especially the prperty f reductin f cntrl signal s energy is inspected. Sectin 8 presents the simulatins fr an HVAC system. Finally the cnclusins are given in sectin 9. HVAC Mdel Several mdels have been cnsidered fr HVAC systems in different references. Fr example in [9,6] a linear first rder mdel f the system with a time delay is cnsidered which is a cmmn simplified mdel f the system. In [,,4] a nnlinear SISO mdel is used fr simulatins. In sme ther wrks a bilinear mdel is cnsidered fr cntrlling either the temperature [] r bth the humidity and temperature [8] which accrdingly results in a SISO r a MIMO cntrl prblem. In this paper the last case is cnsidered, as it seems t present mre practical mdel f a single-zne Variable Air Vlume (VAV) HVAC system. This system is an inherent time-delayed system, which is mstly linearized as first rder with a time delay system. Mre than this, it is a MIMO system in which ne f the I/O channel has a right half plane zer, which means it is a nn-min-phase system. The HVAC system mathematical relatins are as fllw [8]: x& = uα 60( x x ) uα 60( Ws x) + α( Q h fgm ) x& = uα 60( Ws x) + α4m x& = uβ60( x + x) uβ5( T x ) uβ 60(.5W +.75x Ws ) 6000u y = x, y = x () Lets define sme parameters as bellw: u = f, u = gpm, x = T, x = W x = T, α = / Vs, α = hfg / CpVs, α = / ρcpvs, α4 β = / V, β = / ρc V, β = h / C V he p he w p he = / ρv, In which the variables are defined in nmenclature and the numerical values are as in table. In the mdel brught in [8], the actuatr s transfer functin is nt cnsidered but here in simulatins it is used as well. In [] it is shwn that the actuatr s transfer functin, here a valve, can be assumed as: G act ( s) = k /( + τs) () In which k and τ are the actuatr s gain and time cnstant and their values are as in table. Prblem Statement Fig. shws the blck diagram f the cntrl system in the prblem f minimizing the cntrl signal s scillatin. As seen, the system cnsists f a central cntrller which itself cnsists f tw PIDs and their gains will be defined in sectin 4 by cnventinal tuning methds. Then tw blcks f H- LIMITTER can be seen. H-LIMITTER is a nnlinear element and will be described in sectin 5. This element has tw parameters, R, θ, which are the slpe and regin f saturatin functin fr chpping undesired scillatins respectively. If the value f the input t this element exceed a desired value R + b, b is a cnstant, the nnlinear element chps the input signal, and in this way, a smth utput signal can be achieved by the use f this element in feed-frward path. Figure : Blck diagram f the MIMO Cntrl System s

3 Other elements in Fig. are Neural Netwrks (NN, NN), which have 5 inputs and tw utputs and are used in tw places fr chpping the cntrl signals used as inputs t the system. The inputs t these netwrks are tracking errr ei (, ei ( t ) and its time integral e i ( dt, cntrl signal u i, which is applied t the plant directly and a cnstant that can be either 0 r -. The utputs f the Neural Netwrks are R,θ. Finally the plant is seen in the figure which was described in sectin. In training the Neural Netwrk, a cst functin J, is used which is a measure f the system perfrmance and is described as belw: J i = [ kei + k ( ui ) ] ei = ydi yi, ui = ui ( k) ui ( k ) () In which i =, is the number f inputs, k and k are design parameters and the gal is t minimize J i. As it is bvius, using a nnlinear element in cntrl lp wuld make the system s respnse undesirable. T slve this prblem, a term i e is cnsidered in cst functin t achieve utput tracking and the term u i reduces the cntrl signal s scillatin. In this way by reductin f e i, the utput respnse wuld be cntrlled and by minimizatin f ui, the rate f variatin f u i wuld be decreased. In ther wrds, by design f nnlinear elements in such a way which minimize this cst functin, the initial gals f the prblem are achieved 4 Design f Central Cntrller As said befre the HVAC plant can be cnsidered as a first rder with a time delay system, s by usual methds such as Prcess-Reactin Curve [], the parameters f such an apprximatin can be calculated accrding t the step respnse f each I/O channel. When the system is identified the PID cntrller can be tuned with cnventinal methds described in [6,,] and are mentined in the fllwing sectins. It is imprtant t ntice that plant cnsidered here is MIMO, but it has sme prperties, which can be used fr designing the individual cntrllers fr tw I/O channels. The linear mdel f the system, calculated by Matlab sftware, results in 4 I/O transfer functins described by g, i, j =, and it can be seen that: DC gain fr g =. 006 DC gain fr g = DC gain fr g = 970 DC gain fr g = 0 As it is bvius the secnd utput is affected nly by the first input because g = 0 and the first utput is affected by bth inputs. But the first input has a very small dc-gain. S it s effect is s much weaker than the secnd utput ( g / g 0e 7) and it is acceptable t use the first input fr cntrlling the secnd utput and vice versa. In this way the central cntrller can be written as: 0 c C = (4) c 0 and c is the cntrller between the ith set pint errr and jth utput [5]. The relatinship fr each PID is as fllws: u = k p e( + ki e( dt + kde& ( (5) In this paper, the gains are designed by Chen-Cn methd, which results in almst the best respnse. Here three different methds are mentined fr PID gain tuning. 4. Ziegler-Nichls Methd This is ne f the mst cmmn methds fr the design f PID gains used in recent years, but the mst prblem with this methd is the undesired vershts, appearing in the utput respnse. In [6,,] the tables fr the gains calculatin are available. 4. Chen-Cn Methd This methd, similar t Ziegler-Nichls methd, causes vershts in the utput respnse. Fr mre details refer t [,]. 4. Internal Mdel Cntrl Methd This methd was intrduced in abut 980 and is almst a new methd. The respnse s speed is increased in this methd [5]. The ther tuning methds that can be mentined are Haalman and Ple Placement []. 5 Design f nnlinear chpper The input-utput characteristic f nnlinear element is seen in Fig.. The input t this element is the cntrl signal, which is prduced by PID cntrller. Its utput is the cntrl signal, which is applied t the plant directly. R + b and θ + a are the values f saturatin pint and slpe fr the nnlinear element t be designed by the Neural Netwrk. Implementatin f this element results in chpping the undesired scillatins f the cntrl signal. The input-utput relatin f this element is defined as belw, in which a, b are cnstants and R, θ are t be designed n line: u( tan( θ + a) u( < R + b y( = (6) ( R + b) tan( θ + a) u( > R + b Figure : Nnlinear Element Characteristic

4 6 Design f Neural Netwrk The Neural Netwrk cnsidered here, is a three-layer perceptrn with a hidden layer, which has tw neurns in hidden and utput layers and 5 neurns in the input layer. Activatin functins fr the first and secnd layers ( f ( x ), f ( x) respectively) are radial basis functins fr NN and sigmid fr NN and their relatins are as in Equatins (7-a) and (7-b): exp( ) f( x) = m0 exp( cx ) r ( z cx f x) = m0 (7-a) + z exp( cx) f ( x) = n0 exp( cx ) r f n0 ( x) = + z exp( cx) (7-b) In Equatins (7-a) and (7-b), n0 and m0 are parameters t be designed, which can be varied t achieve an ptimum respnse. Fr initial weighting, the fllwing -stage scalled Nguyen-Widrw algrithm is used []: Step : β parameter is defined as fllws: n β =.7( p) (8) In which n and p are the number f neurns f the input and utput layers. Step : Fr each hidden layer i, the weights between input and hidden layer W ( j =,..., p) are chsen values between -.5 and.5 and by calculatin f W j the updated weights can be calculated as fllws: W ( ld) W = β (9) W ( ld) j Step : The weights between hidden and utput layers are randm values chsen between β and β. 6. Errr Back-Prpagatin algrithm The learning algrithm used here, is EBP and the ptimizatin idea f steepest descent is used []. If input and utput weights are cnsidered as W and V jk then the fllwing equatins fr updating the weights are valid in which u is discrete derivative f cntrl signal, η is the rate f learning, X j and z k are utputs f the first and secnd layers respectively. Ii stands fr the i th input t the netwrk, f, f.are the derivatives f first and secnd layer activatin functins and u is the derivative f cntrl signal with z k respect t the netwrk utputs. This term can be cnsidered accrding t the nnlinear element characteristic r nly by its sign. n, m, h are the number f neurns in the input, hidden and utput layers crrespndingly. W W J = η W = h k= = η W f ( j) f ( k) V n i= jk I y ( ke + k u), i z X = f ( W I ), z = f ( V X ) j i k k m jk j= y V jk = η X j f ( k)( ke + k u) (0) z 7 Prperties f the Prpsed Cntrller This cntrller has a gd and almst fast respnse t the input because f the pwer f Neural Netwrks in case f the system s uncertainties. It means that when PID can t shw a gd perfrmance because f the uncertainties and lack f knwledge f the system s dynamics, the Neural Netwrk can help it t achieve a gd perfrmance. The mst imprtant prperty f this cntrller is the cntrl signal s scillatins canceling, which was mentined in previus sectins. Mrever, the methd reduces the ttal energy f the cntrl signal, which will be discussed in next sectin. 7. Energy Reductin By the use f this methd as well as achieving the abve cntrl gals, the cntrl signal s energy is reduced t. This can be reasnable, since by canceling the undesired scillatins f the cntrl signal, the energy can be used in a useful way and its lsses can be reduced, s thugh using less energy, better perfrmance can be gained. Related numerical results are brught in simulatins and tw indices are prpsed fr cmparing the cntrl signal energy between PID and the intrduced methd: The numerical values fr indices and shw the prperty f energy saving f the intrduced methd t ) 0 a. E = ( u u eq dt () in which u t eq is the cntrl signal value in steady state. t b. E = ( u) dt = ( u( u( t )) dt () 0 8 Simulatins 0 Simulatins presented here, are the results f applying the prpsed methd n a nnlinear mdel f an HVAC system. In these systems, cntrl signal is applied t the actuatrs directly. As an example, cnsider the valve fr cntrl f the ht r cld water, r the vlumetric flw rate f air. The main prblem with these actuatrs, are their damages because f the undesired scillatins applied t them [9]. In this way, using this new methd n an HVAC system, is a gd measure fr examining the quality f the new methd. In figures,4,5 and 6 a cmparisn is made between the utput and cntrl signals result in by cnventinal cntrl designs, which nly use the PID cntrllers and the prpsed methd k j

5 in this paper. The trajectry t be tracked is a quintic trajectry which get at it s final value in.5 time base ( It can be called a sft step with.5 (t.b.) time cnstan. The set pints fr tw utputs are as 75 F and. 009 lb / lb fr the space temperature and misture percentage. As seen in Fig., the transient scillatins f the utput respnses have been reduced significantly. Mre than this cntrl signal in Fig. 6 has reduced number f scillatins in cmparisn with PID. In figures 4 and 5 the secnd utput and its crrespndence input signal are seen. In this case as the riginal signals didn t cntain many scillatins s the limitter desn t need t chp them, but the H-Limitter increases the respnse speed and reduce the value f the respnse peak. In figures 7 and 8 the respnse and cntrl signal f the system t a square wavefrm fr the first utput is seen. As it is bvius after ne perid the number and peak value f vershts and undershts reduces. In figures 9-0 the tw indices intrduced fr cntrl signal energy are brught fr secnd input. As seen besides the gd perfrmance f the prpsed methd, the energy f the cntrl signal is reduced significantly in cmparisn with PID cntrl. The PID gains fr an HVAC system are brught in tables,. These values are fr a linear system f the differential equatins described in Equatin (), designed by Ziegler-Nichls and Chen- Cn methds [6]. The values f a and b in nnlinear elements are chsen t be zer fr b in tw limitter blcks and.4,.046 fr a. Als the final value fr R, θ are as.75,.74 fr first limitter and.5,.5 fr secnd limitter. The design parameters f Neural Netwrks are as η =, k =., k =., n =.75, m =, z =, c.8 and. 0 0 =. 8, k = k =., n0 =, m0 = 9, z =, =.0 η = c fr tw netwrks respectively. The values f tw indices intrduced fr energy are reduced by 45%, % respectively by the help f new intrduced methd. 9 Cnclusins In this paper, a nnlinear element was designed in rder t chp the undesirable scillatins f the cntrl signal. A perceptrn Neural Netwrk, EBP training algrithm and RBF activatin functins were used fr this purpse. This design methd was applied t a nnlinear mdel f an HVAC system and reached t the pre-specified gals f design such as utput tracking and reductin f cntrl signal s scillatins, which are apparent in simulatins. Mrever, the ttal energy f the cntrl signal was cnsiderably reduced by this methd. References [] K. Astrm, T. Hagglund. PID cntrllers: thery, design and tuning, ISA, (995). [] L. Fausett. Fundamentals f neural netwrks, Prentice Hall, (994). [] S. Haykin. Neural netwrks: a cmprehensive fundatin, Macmillan Cllege Publishing, (999). [4] M.B. Menhaj. Fundamentals f neural netwrks, (in Farsi), Prf. Hessabi Publicatin, (998). [5] M. Mrari, E. Zafiriu. Rbust prcess cntrl, Prentice Hall, (989). [6] M.R. Omidi. Minimizatin f cntrl signal scillatins in HVAC systems, (in Farsi), M.S. Thesis, University f Tehran, (000). [7] M.R. Omidi, M.J. Yazdanpanah. Minimizatin f actuatr repsitining using Internal Mdel Cntrl, Furth Prtugues Autmatic Cntrl Cnf., (000). [8] B.A. Serran, M. Velez-Reyes. Nnlinear cntrl f a Heating, Ventilating and Air Cnditining system with thermal lad estimatin, IEEE Trans. n Cntrl Systems Tech., Vlume 7, N., (999). [9] M. Tavanaii. Achieving ptimal and rbust perfrmance in HVAC systems, (in Farsi), M.S. Thesis, University f Tehran, (000). [0] M. Tavanaii, M.J. Yazdanpanah. A new methd fr cntrl f finite-energy systems with H cntrllers, (in Farsi), 9th Iranian cnf. On Electrical Engineering prceedings, (00). [] T. Tigrek, S. Dasgupta, T.F. Smith. Nnlinear ptimal cntrl f HVAC systems, Prceedings Of IFAC Wrld Cngress, (00). [] C.P. Underwd. HVAC cntrl systems: mdeling, analysis and design, E & FN SPON pub., (999). [] C.P. Underwd. Rbust cntrl f HVAC plant I: mdeling, Prc. CIBSE, Vlume, N., (000). [4] C.P. Underwd. Rbust cntrl f HVAC plant II: cntrller design, Prc. CIBSE, Vlume, N., (000). [5] Q.G. Wang, C.C. Hang, Y. Zhang, Q. Bi. Multivariable cntrller aut-tuning with its applicatin in HVAC systems, Prceedings Of ACC, (999). [6] M.J. Yazdanpanah. Tward minimizatin f actuatr repsitining, Internal Reprt, Dept. f ECE, Cncrdia University, (997). [7] M.J. Yazdanpanah, M.R. Omidi. Cntrl signal smthing by ptimal cntrllers, (in Farsi), 8 th Iranian Cnf. On Electrical Engineering, (000). [8] M.J. Yazdanpanah, M. Tavanaii. Minimizatin f cntrl signal scillatins in time-delayed systems by H methds, (in Farsi), 8 th Iranian cnf. On Electrical Engineering, (000). [9] M.J. Yazdanpanah, M. Zaheeruddin and R.V. Patel, Feasibility study f adaptive cntrl algrithms, Technical Reprt, Dept. f ECE, Cncrdia university, (997). [0] M.J. Yazdanpanah, E. Semsar, C. Lucas, Minimizatin f Actuatr Repsitining in Delayed Prcesses Using Flexible Neural Netwrks, Cnference n cntrl applicatin Prc., vlume, pp. -5, (00) PID Kp Ki Kd Z.N -e C.C -.e Table : PID gain tuning fr first utput PID Kp Ki Kd Z.N e-5 C.C e-5 Table : PID gain tuning fr secnd utput

6 Figure : First utput using PID and PID+ H-limitter Figure 7: st utput using PID+ H-limitter: a square wavefrm Figure 4: nd utput using PID and PID+ H-limitter Figure 8: nd input using PID+ H-limitter: a square wavefrm Figure 5: First input using PID and PID+ H-limitter Figure 9: First index using PID and PID+ H-limitter fr cntrl signal energy Figure 6: Secnd input using PID and PID+ H-limitter fr Quintic trajectry Figure 0: Secnd index using PID and PID+ H-limitter fr cntrl signal energy