Robust neural network controller for variable airflow volume system

Q. Song, W.J. Hu and T.N. Zhao

Abstract: The I-IVAC industry expects the automation and control systems to perform well throughout the year without the requirement to re-tune the system and to conserve energy. To satisfy these requirements it is necessary to provide a basic control algorithm that will respond well to the presence of nonlinear behaviour in HVAC equipment. The PID algorithm has to be enhanced to handle the highly nonlinear functionality, range of operation and robustness. The neural network is one of the best candidatcs to deal with these issues. However, it is important to address stability and disturbance properly, to obtain optimal performance of the neural control system. The design and application of a robust neural network algorithm are discussed and how it compliments the fixed proportional control algorithm to provide the desired functionality as well as the adaptation of the VAV conlrol system for a wide range of disturbances and parameter changes.

1 Introduction

Historically, in the heating, ventilating, and air-conditioning (HVAC) industry, tuning is compromised by selecting the PID gains to satisfy the worst case. This situation is further affected by the introduction ofvariable air volume (VAV) in the early 1970s and the variable water flow distribution systems in the early 1990s. In recent years there have been a few interim solutions such as variable gain controllers [I], adaptive control algorithms [2] and various ramping tech- niques that helped to improve performance. As the neural network has great capability to handle nonlinear functions, a number of neural network control algorithms attempt to address this performance problem [3, 41. However, neural network training algorithms are sensitive to disturbances and may not be able to obtain an optimal performance, or even become unstable. To the best of our knowledge, robustness and stability issues of the VAV sy:item have not been studied yet by previous researchers.

On the other hand there has been extensive research in the theoretical area of robust neural controllers [5-E]. For example, the persistent exciting condition has been removed in the presence of disturbance [7]. Variable structure and dead-zone schemes have been introduced to design a robust adaptive algorithm for a neural network control system to improve tracking performance [6, 7, 91. There are two main disadvantages of these existing algo- rithms, which require knowledge of the bound 0.T the input signal to the neural network or disturbance a priovi.

Q IEE. 2003 IEE Proceedingc online no. 20030065 DO/: 10.I049lip-cta:20030065 Paper first rcccived 24th April and in revised form 5rh Novemher 2002 Q. Sang and W.J. H u are with the School of Electrical and Electronic Engineering. Nanyang Technological Universily, Nanyang Avenue, Singapore 639798 T.N. Zhao is with the Singapoic lnstitulc of Manufactuing Technology, 71 Nanya?g Drive. Singapore 638075

I12

We propose a robust neural adaptive control scheme for the VAV control system. The stability analysis is based on sector theory [9-1 I]. A normalised learning algorithm is used to train the neural network, which eliminates the requirement of a bounded regression signal to the system. Furthermore, an adaptive dead-zone scheme is employcd to enhance adaptive capability and robustness of the VAV control system against disturbance with complete stability and a convergence proof. The selection of the dead-zone does not require knowledge of the upper bound of the disturbance, which is usually unknown for the VAV control system. Any change of the VAV system parameters or disturbance will not degrade the system performance because the size of the dead zone is updated by the adaptive algorithm accordingly.

Thc objective of the robust neural control algorithm is to provide an improved operation over a standard neural controller against the disturbance in the VAV system. The performance improvement can be described in terms of reduced controlled variable overshoot and undershoot, reduced time required to reach set-point, increased actuator life by reducing oscillations, and a fast recovely time from a process disturbance.

2 Airflow and motor-fan dynamics for VAV control

As a first step towards the design of a robust neural VAV control system it is assumed that all the dynamics of the system, including the a i d o w volume rate Yr..('), are included in the motor-fan model [ I Z ] . This model consi- ders neither the air temperaturc incrcasc nor the sensor dynamics. The first has a much slower time response compared with airflow dynamics and can be controlled separately [13]. The following two dynamic equations give correlation among the motor voltage e,(t), motor current I ( t ) , fan speed Nfi,,,(f), a i d o w volume rate V,b.(r) and pressure rise AppiO.(t) in the VAV system [12]:

where L , is armature inductance, R, is armature resistance, Kh is back-EMF constant, J.,, is equivalent moment of inertia, Ki is torque constant. B, , is equivalent friction factor, and i ~ ~ , ~ is efficiency parameter of fan. Assume a constant mass density p and that the only source of airflow is from the fan and the final airflow rate at the building boundary is zero. Variations of these parameters are considered as disturbances in the final discrete-time model in (9). The correlation between pressure rise and the airflow volume rate is modelled by [I41

I AP( / ) = -pV'J/) with g, = l.0kg-m/N-s2 (3)

where gc is a dimensional constant. And the correlation between motor speed and pressure rise is [ I51

(4)

where C,, is the dimensionless pressure head cocfficient, n.,/nfis the ratio ofmotor to fan speed. Combining (3) and (4), the correlation between airflow volume ratc and motor speed IS

Using ( I ) , (2) and (S ) , and Euler's rule, the discrete-time 'input-output can be presented with N,,,,,,,,,(k) =y(k) as

V ( k + 2 ) = F(y(k + I) ,y(k)) + Tu(k) + d(k + 2) (6)

where ~ ( k ) = TKie,J(k)/2J,L, is the cnntrol signal, Tis the sampling time, d(k) is the missing dynamic during the

modelling procedure, like the pressure rise at the building boundary, and

F ( y @ + l ) , y (k) )

T T

(7)

The correlation between fan speed and pressure rise is highly nonlinear, and in a variable air volume the air-flow rate can change between 30 to 100% to satisfy the building intake airflow rate. This leads to the nonlinear function F(.), which is not only dependant on the system state variable vector x(kj, but also related to the airflow rate on the boundary of the building and air density p. The last two arc assumed initially to be zero and constant in the continuous-time model, respectively, and reconsidered as disturbance d(k) in the discrete-time model (6) and (7). This disturbance may cause two problems, i.e. affect the modelling accuracy and the tracking performance of the airflow control. Therefore we use a multilayered neural network with a robust training algorithm to estimate the unknown nonlinear function F(.) against the disturbance d(k) online, and incorporate the estimated value F(.) into an integrated control design to form a robust neural controller against the disturbance.

The strncture of the discrete-time VAV neural control system is shown in Fig. I . Note that the airflow pressure rise Aphn(k) is normally measured by the pressure sensor to form the feedback signal in a real system; however, we use the motor speed N,,,,,o,(k), which is a nonlinear func- tion of the pressure rise Apea,,,(kj defined in (4), as the feedback signal in the simulation for convenience. The tracking error of the control system can be defined as

44 = - Y&) (8)

113

where ydk) is the command input signal and can be converted into the airflow volume rate setpoint as V/,,,,,@). Define the control signal u(k) as

where k,,, is the gain parameter of a fixed controller, and F(x(k)) the output of a three-layered neural network, which could also be extended to other multilayered cases, is defined as

&k)) = vT(k)H(w(k), x ( k ) ) (10) with the input vector x(k) = [ y ( k - I ) y ( k - 2)]: where v(k) = [v,(k), vZ(k), . . . , vnk(k)I7 E R'Ih is the weight vector between the hidden and output layers, nh is the number of neurons in the hidden layer of the network, and H(w(k), x(k)) E R"" is the nonlinear activation function vector

H(1v(k), x(k) )

= [ A , (x-1 h , ( k ) . . . h,,(k)I7 = [ h(w:(k)x(k)) h(wT(k)x(k)) ' ' ' h(lv;<(k)x(k))]T

w(k) = [w,(k), %(k) , . . ., w,(k)l'

The adjus'able weight matrix w(k) E R'lAx2 of the hidden layer

(11)

where wj(k)=[xjI(k) wj2(k)lTeR2 with j = l , 2, . . . , n h , w,,(k) is the adjustable weight from the ith neuron of the input layer to the j th neuron of the hidden layer, and / I ( . ) is the nonlinear activation function of hidden neurons h(x) = 1/(1 +e-4") where ;. > 0 is the so called gain parameter of the threshold function. Note that we define the activation function in such a way to make max(h'(.r)) = ;. as used in the proof procedure of Lemma 1 in the Appendix (Section 8.1).

Define a bounded ideal weight vector v* E R"*, which is a minimum point of the cost function E(k) = ( I/2:k2(k) (see (14)) subject to v(k), in the output layer of the neural network, the nonlinear function F(x(k)) can be approxi- mated by the neural network as

F ( x ( k ) ) = F*(k) + ~ ( k ) (12) where e(k) is the reconstructure error of the ncural network, and the ideal output of the network

F*(k) = v*'(k)H(w', x(k) ) (13) with w* ER""', which is a local minimum point of the cost function subject to w(k), is also a hounded*ideal w$ ht matrix*of the hidden layer as iv**=[1y:;, wa2,. . .,

2, . . ., nh). Then the network training error can be presented as

e(k)=F(.r(k)) - k(k) + d(k) = F*(k) - k ( k ) + ~ ( k ) + d(k)

w . , ] , !- with w j ( k ) e R 2 , w,*(k)=[nql(k) ~ j z ( k ) ] I ' ( j = I ,

(14) The closed-loop equation of the VAV control system is presented according to the discrete-time nlodei (6), (7) and (8)

r(k) = k , r ( k - l ) + F ( x ( k ) ) - k ( k ) + d ( k ) (15)

The VAV nonlinear function F(x(k)) is nonnally not measurable or unknown due to model uncertainty in the real system, that is, no desired value is providesi to teach the neural network. The available tracking error :signal r(k) IS used to update the weight of the learning algorithm,

I14

which is the difference between the set-point input and motor speed defined in (7) as shown in Fig. I . Through (12) to (15) the estimate error of the neural network e(k) in (14) can he rewritten as

e(k ) = r ( k ) - k,,r(k - 1) = ( I - z-'k,)r.(k)

= F(x(k)) - F ( k ) + d(k)

= F*(k) ~ P ( X ( k ) ) + &(k) + d(k)

= vr(k) f i (k) + ;'(k)H(w*, x(k) ) + ~ ( k ) + d ( k )

= "T(k)H(k) + e(k) (16)

where R(k)=H(y*, .r(k)) - H(w(k), I(@), k,= 1 - Tp ( p > 0), C(k) = v ~ v(k) E Rn6 is the weight error matrix of the o u t p t layer, z-' is a time delay factor and ~ ( k ) = cT (k )~( iv , x(k)) + i:(k) + d(k) is an equivalent disturbance term.

3 Robust neural network training and control algorithms

To deal with the disturbance, an adaptive dead-zone function is defined as

(17)

where A,(k) is the size of the adaptive dead-zone to be estimated. Now we can define the operator H;: e"@) --f Q'(k) , to update w(k)

where P(k) is a variable leaming rate based on the adaptive dead-zonc scheme defined asL,

with

H'(k) = diag[h;(k), . . . , hhh(k)]

= diag[12'(wi(k), x(k) ) , . . . , h'(w,(k), x(k ) ) ] E R"h.rnh

where h:.(k) is the first derivative of the threshold function in (1 I), and ha , , = min[h',(k), . . . , hlh(k)] > 0 (assume that h:(k) is not simultaneously zero) is a positive constant.

Similarly, a normalised gradient descent algorithm can he written in the matrix form

I€€ Proc.-Conirol Theory Appl., Vol. IJO. No. 2. hlnrch 2003

where r(k) is a variable learning rate based on the dead- zone scheme defined as

l o if Ile(k)ll 5 A$) The main advantage of this robust training algorithm of the neural network is the online estimation of the width A,(k) of the adaptive dead-zone in (19) and (22). We should show that the requirement of the adaptive dead-zone technology is critically important to provide guaranteed closed-loop stability and tracking performance of the VAV control system in the simulation part.

The robust neural network VAV control system uses the adaptive dead-zone to identify the nonlinear function from system disturbance online and put into the airflow dynamics to fonn part of the control signal together with the fixed controller. The algorithm can be summarised as follows (refer to Fig. I ) .

Step I : Form the new input vector x(k) of the neural network using the motor velocity N,,,,,,>,-(k) and the acce- leration &ocoy(k); note that the pressure rise Ap&) will he used instead of the acceleration in a practical system since the prcssure sense is normally available: Step 2: Calculate the neural network output F(x(k)) using the input state x(k) and the existing or initial weights of the network in the first iteration. Step 3: The control voltage input u(k) is calculated based on (8). Step 4: The new measurement of the airilowfmotor dynamics is taken and the new tracking error signal r(k+ 1) is fed trough a fixed filter to produce the training error signal e(k) of the network according to (15). Step 5: The training error e(k) is uscd to train the neural network and calculates the new weights w(k+ 1) and v(k+ I). That improves the network performance, in tum the VAV control system, in the sense that the tracking error signal r ( k + 1) is minimised through the iteration. Step 6: Go back to step 1 to continue the iteration.

4 Robust stability and convergence of the neural control system

Stability ofthe robust neural controller is based on the sector theorem [9-1 I ] , which requires that the feedback system in Fig. 1 should he reformed as a normalised equivalent feed- back structure as shown in Fig. 2. Then neural network training algorithm can he proved to meet the certainty dissipative condition to obtain guaranteed stability of the error signal e(k) as in theorem 2, in tum, the tracking error r(k) of overall control system as in corollary 2. The main theorems are presented in this Section, while most of the proof procedures are put in the Appendix (Section 8).

Note that the activation function h(.) in ( I 1) is nonde- creasing function. According to the mean value theorem, there exists a unique positive number p(.) with i. p( . )? 0

Fig. 2

/EE Proc.-ConNol Theory Appf., ?bi. 1511. ,No. 2. Murid 2003

Normalised eyuivirlent error,feedhack system

so that the network estimation error e(k) in (14) can he rewritten as the following:

e(k) = y*(k) - k(k) + ~ ( k ) + d(k)

= v ' ( k ) ( ~ ( w * . .r(k)) - ~ ( w ( k ) , x(k))) + i ( k )

= v'(k)[h(w;'x(k)) - h(w:(k)x(k))

. . . h(WiTx(k)) - h ( ~ ; ~ ( k ) . r ( k ) ) ] ~ + Z(k)

= vr(k)[p(w;. W , ( k ) , x(k))( ,@ - wT(k))x(k)

. . . p(w;A, wnh(k), x ( k ) ) ( w g - w;6(k))x(k)]' +e@) = v'(k)p(k)i'(k)x(k) + Z(k) (24)

where G ( k ) = w* - w(k) E R""' is the weight error matrix, and p(k) t R"""" is a unique positive matrix

I@) = diag b 1 ( k L . . . . P,#)I = diag [p(w;, (k ) , x(k ) ) , . . . , dw;*. w,,jk), x(k))l

(25)

The closed-loop system can be further rewritten as a normalised equivalent error feedback system as shown in Fig. 2.

e"(k) = -H;@"(k) + 7 ( k ) (26)

with operator: H;: e"(k) + V ( k ) and ffz = I, and

p(k) = (1 + lIH'(k)v(k)Il211x(~)ll) (30)

Lemma I : The normalised operator H;: e"(k)-t @,"(k) satisfies the following equation:

P m u f . see Appendix (Section 8. I).

4.7 Theorem 7 The weight-tuning algorithm in (21) and (22) using the adaptive dead-zone scheme has the following properties:

for k = 0, 1 . . . . , (32) for k = 0, 1, . . . , (33)

+ l)il 5 ll;(k)l/

lIi, ,(k + 1)11 c ll&(k)ll Lim suple(k)l 5 AV (34) X-m

where &(k) = AV - Av(k)

Proof see Appendix (Section 8.2)

4.2 Theorem 2 conic sector stability theorem According to lemma I and theorem 1 the operator q: e"(k) + W ( k ) satisties (31), Z"(k) = [j'(k)H(&v', x(k))+ ~ ( k ) + d ( k ) l f J ( p ( k ) ) E L ~ , and K ; = H Z = 1 then the normalised error feedback system in Fig. 1 is stable and furthermore, the original estimation vector is also boundcd i.e. O(k) E L Z .

Prouf see [9-1 I ] 115

Table 1: List of model parameters

Variable Value

Jed 10.0 kg m2

Proof. see Appendix (Section 8.3)

Corollarv 2: The VAV control system tracking error signal r(k) will converge to a compact set, i.e. Ir(k)l 5 A,, where

Bed

sc C h

0.01 kg m21s 1.1 kg-m/N-sZ 1 .o

P 0.8 kg/m2 is a positive constant with A,,=min[A,., Ax,]. A 100 m2 0," 2 m Proof. see [IO]. ndnf 1

5 Simulation results !Ita" 0.5 0.5 Q

4.3 Theorem 3 The robust training algorithm in (18) and (19) using the adaptive dead-zone scheme has the following properties:

lliv(k + I ) / [ sll;V(k)ll

/I&#- + 1)ll 5llAJkJll for k = 0, I , . . . ,

for k = 0, I , . . . , ( 3 5 )

(36) (37) k-t Lim m sup/e(k)) SA",

where %(kJ = W* - w(k), and &(k) = Aw - A,(k)

The model of the VAV system used in the simulation is based on (6) with parameters listed in Table 1 1121. The simulation is carried out using both a standard back- propagation algorithm without the normalisation factor/ dead-zone and the robust training algorithm with A,(O) = A,(O)=O and random initial weights of the network. Then both neural networks, which use ten hidden-layer neurons, are applied to the online neural tracking controller with the gain parameter k,.=0.5 and the initial values

In the presence of the disturbance the minimum training criterion of the cost function was set to be e-" Oo0 for the standard back-propagation algorithm. The normally distributed random disturbances are generated at different magnitudes as shown in Figs, 3-5. It was observed that the disturbance causes the divergence of the neural network in

.x,(oJ=~x,(o)=o.

1200} 1200 \ system outpiit system output

1000

6 800

command ~ ignal

800

600 command signal 5: 600-

400 - 400

c _m

200 200.

0 -2

a a

1200 r 1200 r

' 0 200 400 600 800 low 1200 1400 1600

time. s time. s b b

Fig. 3 a Standard back-propagation algorithm b Robust algorithm

116

TracingpefornianceJor noise magnitude lld(lgll = 0.1 Fig. 4 a Standard back-propagation algorithm b Robust algorithm

Trackingperformance for noise magnirrrde lldfi)ll= 0.3

IEE Pmc.-Conrml Theory Appl.. YOI. 150, No. 2. March 2003

2 BRANDT, S. : 'Adaptive control imdementation issues'. ASHRAE i I

1200 1 1000

800

c o r k " signal'

400 6oo t 2ool

200

'0 200 400 600 800 lo00 1200 1400 1600

time. s b

Fig. 5 Trucking perfinnlmce for no& magnitude lld(kjll = O S

the sense that there is output drifting, and in turn, degrades the tracking performance of the neural controller as shown in Figs. 3-5 for the standard back-propagation algorithm. In contrast, the robust tuning algorithm provides the guaranteed convergence of the network and smaller control tracking error compared with that of the standard back- propagation algorithm, as shown in Figs. 3-5, respectively.

6 Conclusions

An adaptive robust neural network controller has been developed and applied to the VAV system to deal with model uncertainty and system disturbance due to incom- plete knowledge of the physical system. A robust adaptive training algorithm is provided to update weights of the neural network online to take care..great nonlinearly and changes of the systems. Simulations on the VAV system show the advantage of the adaptive robust neural controller over the standard neural network or tixed dead-zone scheme, which can reject disturbance online without a priori knowledge of the upper hound of system disturbance.

7 References

1 PINELLA, M.. HITTLE, D.C., WHECHSELBERGER, E., and PEDERSEN, C.: 'Self-tuning digital integral control', ASHRAE Truns., 1986, 92, pt. 2B, pp. 16&168

IEE Pmc.-Conrrol Theom Appl.. RI. 150, No. 2, Morch 2UU3

Tmns., 1986,92,pt. 2B,pp. 211-219,' 3 MHMED, O., MITCHELL, J.W., and KLEIN, S.A.: 'Feedfonvard-

fcedback controller using general regression neural network for labora~ tory HVAC system: Pan I-pressure control', ASIIRAE Dons., 1998,

111-, pL. >

5 H E W M.: 'Error-minimizing dead m n c for basis function networks'. IEEE Trans. Neural New.. 1996, 7, (b), pp. 39-50

6 CHEN, F.C., and KHALIL, H.K.: 'Adaptive control of a class of nonlinear discretc~time svstems using neural networks'. IEEE Trunr. Autom. Contml, 1995,40 (S), p p . 79j-XOI

7 JAGANNATHAN. S., and LEWIS, EL.: 'Discrete-time neural net controller for a class of nonlinear dynamic systems'. IEEE Trans. Aurum. Control, 1996,41, (11). pp. 1693-1699

8 SONG, Q., XIAO, 1.. and SOH, Y.C.: 'Robust back-propagation training algaithm for multilayered neural tracking controller', IEEE Trans. NeuralNew., 1999, lO,(S),pp. 1133-1141

9 ORTEGA, R., PRALY, L., and LANDAU, I.D.: 'Robustness ofdiscrete- time direct adaptive controllers'. IEEE Trans. Autom. Control, 1985.30, (12),pp. 1179-1187

1U SONG, Q., KATEBI, R., and GRIMBLE. M.J.: 'Robust multivanable implicit adaptive control'. /MA J Math. Contml Info., 1993, IO. pp. 49-70

11 CLUETT, VR., SHAH, L., and FISHER, G.: 'Robustness analysis of

multizone VAV system for control analysis', ASHRAE Trans., 1994, 94,

13 MATSUBA, 7, KASAHARA, M., MURASAWA, I., and HASHIMOTO. Y.: 'Stabilihi limit of room air temperature of a VAV

pt. 2, pp. 219-228

15 ASHRAE: 'Handbook-systems equipment' (American Society of Heating, Refrigerating and Air-conditioning Engineers, Atlanta, 1992) Chap. 18

8 Appendix

Only the main proof procedure is presented here (refer to [lo] for more details) dealing with the fixed dead-zone and has similarity with adaptive dead-zone case.

8.1 Proof of lemma 7 Define the Lyapunov function VJk) = tr{ iv7(kjiv(k)}, and consider the property of the matrix trace tr{M(k)v(k)e(k) x7(k)ivr(k)} 2 (hkgn/?.) tr{p7(k)v7(k)p(k)iv(k)x(k)} [ I O ] we have

8.2 Proof of theorem 1 Define a Lyapunov function

VJr) = ll;(k)Il* + At(k)

= tr(C'(k)C(k)) + Zq(k)' i , (k) (42)

and use ( I 9) to give

le(k)i2 - Ie(k)tIrJk)I 5 trIe(k)i'(k)H(iv(k). x(k))~ (43) where Ir,(k) = ~ * ~ & k ) + ~ ( k ) + d(k)l 5 AV According to the definition of the adaptive leaming rate in (23), (42) can be rewritten as

Then Vv(k) is bounded below by zero and is not increasing. The limit exists and the last term in (44) will go to zero, Since H(w(k), x(k)) is a bounded signal as defined follow- ing (9), we have

This completes the proof

8.3 Proof of theorem 3 According to [IO]

Furthermore, as the input vector @ ( k ) is bounded as proved in theorem 2 , then (34) follows.

This completes the proof.