0-7803-9514-X/06/20.00@2006IEEE ICIEA 2006

Model Predictive Control Based on Fuzzy Linearization Technique For HVAC Systems Temperature Control

Jia Lei Lv Hongli School of control science and engineering,

Shandong University, Jinan, China, 250061,

China Email: honglitao@sohu.com

Wenjian Cai School of Electrical and Electronic Engineering,

Nanyang Technological University, Nanyang Avenue, Singapore, 619798.

Email: ewjcai@ntu.edu.sg

Abstract The Heating, Ventilating, and Air-Conditioning systems (HVAC systems) are typical nonlinear time-variable multivariate systems with disturbances and uncertainties. A new Mamdani fuzzy model predictive control strategy based on sum-min inference was proposed to control HVAC systems in this paper. The resolution relationship of two inputs and single output variables of the Mamdani fuzzy controller was obtained by its structure analysis. Then the fuzzy linearization predictive model at k+1 sampling time on base of its resolution equation was designed. And at P ahead horizon predictive models were got. The predictive control strategy based fuzzy linearization predictive model was given and the procedure to implement the control algorithm was outlined. Finally simulation test results showed that the proposed fuzzy model predictive control approach is effective in HVAC systems temperature control applications. Compared with the conventional PID control, this fuzzy model predictive control algorithm has less overshoot and shorter setting time.

1 Introduction A Heating, Ventilating, and Air-Conditioning system (for short HVAC system) consists of indoor air loop, chilled water loop, refrigerant loop, condenser water loop, and outdoor air loop. In each loop there are many dynamical variables which interact with each other. So the HVAC system is a typical nonlinear time-variable multivariate system with disturbances and uncertainties. It is very difficult to find a mathematical model to accurately describe the process over wide operating range. The design of controller for HVAC systems is a big challenge for practical engineers. Recently, some complex control strategies based on the classical control concepts have been proposed in attempt to improve the

system performance [1-3]. Among those works, classical control techniques, especially proportional-integral-derivative (PID) controllers were still widely used in practice due to easier implementation, low cost and reliable in harsh field conditions. However, the PID controller was designed based on the specific cold load while the practical load is variable, furthermore its simple structure is difficult to overcome the effect of uncertainties. As a result, it is necessary to seek more effective strategies to control HVAC systems.

Model predictive control (MPC) is a control strategy growing up from the practice industry process. Its prominent characteristic is that MPC absorbs the optimization technique of modern control theories and makes use of the predictive models and rolling optimization. So that it takes on strong robustness [4]. Since Richalet described the model predictive heuristic control based on the impulse response in 1978 [5], The model predictive control has made great progress in theories and applications [6-9].

Intelligent control techniques are applied abroad in nonlinear systems and practical industries because of no need of accurate mathematical models. In particular the fuzzy control has been used in the HVAC systems [10-13]. Ghiaus demonstrated that the nonlinearities of the heat exchange process can be well described by a rather simple fuzzy scheme and showed that the fuzzy control resulted in better performance and eliminated the retuning process required by the conventional PID controllers [13].

The common characteristic of fuzzy control and model predictive control is not need of accurate model of the controlled process. And both of them are robust to the uncertainties and disturbances in closed-loop systems. As a result, the fuzzy model based-on predictive control has came and developed from fuzzy control and model predictive. Moreover it has been more and more popular and practical in application field [6-914]. He and Cai proposed a multiple fuzzy model-

based temperature predictive control for HVAC systems [6]. Sousa and Babusla developed fuzzy predictive control applied to an air-conditioning system [7]. But the existing predictive control technology based on fuzzy model was always based on the Takagi-Sugeno fuzzy model [14]. Because the Takagi-Sugeno fuzzy model is composed of a set of local mathematical models, it is difficult to modeling and identification of complex nonlinear systems. Then the T-S fuzzy model is not easy to apply in practice.

In classical Mamdani fuzzy model, the premises and consequents of fuzzy rules are composed of fuzzy sets which can be established based on experience and knowledge of manipulators and experts. Therefore it is easier to model and identify for Mamdani fuzzy model. In addition to the advantage of predictive control, in industry the Mamdani fuzzy model based-on predictive control is convenient for application. In this paper we proposes a novel predictive control strategy based on sum-min Mamdani fuzzy model and apply the novel fuzzy MPC to the temperature control of air handling unit in HVAC system.

2 Linearization Fuzzy Predictive Model In this paper, the exact linearization predictive model can be gained by the real-time structure analysis of Mamdani fuzzy predictive model. The linear control strategies can be implemented on base of the linearization model. The frame of this novel fuzzy model predictive control for nonlinear systems is showed in figure 1. Supposed in a SISO nonlinear system

))(),(()1( kukyfky =+ (1) where )(ky )(ku are output and input variable at k sampling time, )(f is a nonlinear function. According to the universal approximate [17]. It exists a following Mamdani fuzzy model ))(),(( kukyF approaching the nonlinear system (1) infinitely:

jiR , If )(ky is iA and )(ku is jB then )1( +kym is jiU , Nji ,,2,1, L= (2)

where iA jB jiU , Nji ,,2,1, L= take on isosceles fuzzy membership functions in figure 2. Assumed there are N=2J+1 fuzzy numbers in each input variable )(ky and )(ku of fuzzy controller, they

are denoted by iA and jB , Nji ,,2,1, L= . Assumed there are 51412 += JN fuzzy numbers in the output of fuzzy controller )1( +kym , they are denoted by jiU , .

))(( kyA A-2 A-1 A0 A1 A2

-1 -0.5 0 0.5 1 )(ky Fig. 2 Isosceles fuzzy numbers membership functions

Without losing generality, at arbitrary k sampling time, for input and output variables )(ku )(ky in nonlinear system (1), it exists :, ji 1, JjiJ subject to ])1(,[)( SiiSky + , ])1(,[)( SjjSku + S=1/J. So the fuzzy membership functions of )(ky in fuzzy sets Ai, Ai+1 and )(ku in Bj, Bj+1 are acted. The member functions of )(ky and )(ku in other fuzzy sets are all zero. As a result, we only obtain four effective rules. Through sum-min fuzzy inference, center of gravity defuzzification it can be obtained:

+

++++++

+

++++++

=+

|)5.0()(|])5.0()([])5.0()([

2)1(

|)5.0()(|])5.0()([])5.0()([

2)1(

)1(

SjkuSSjkuSikySSji

SikySSjkuSikySSji

kym

),,,(

),,,(

8743

6521

CCCC

CCCC (3)

If supposed that

+

+==

|])5.0()(|[2

|])5.0()(|[2)()(

SjkuSS

SikySS

kKkK IP ),,,(

),,,(

8743

6521

CCCC

CCCC

then )()()1()1( kuKkyKSjiky uym ++++=+ (4)

It is obvious that the parameters ijS are all the constants in the certain sampling time k, so that fuzzy expression (3) have the following linearization style:

)()()()1( kCkuKkyKky uym ++=+ , SjKSiKjikC uy )]5.0()5.0()1[()( ++++= (5)

Where ))()(()1()( kuKkyKkykC uy ++= is the error between the predictive model and the nonlinear model.

3 Model Predictive Controller Design The proposed fuzzy model linearization technique can give an approximate linear time-variant model at given state vector for controlled process. Based on the linearization model, the MPC algorithm is going on. In order to define how well the predicted process tracks the set-point, the following cost function is used:

)(ky )(ku

)(kyr

Process

Fuzzy T-S model

Predictive controller

Fig. 1 predictive control based on fuzzy Mamdani model

=

++=yN

irm ikyikyJ

1

2))()(( (6)

where )( ikyr + is the output variable set-point and Tyrrrr NkykykykY )](,),2(),1([)( +++= L , yN is control horizon.

At first, if supposed )()1()( yNkukuku +==+= LL

The predictive model is )()()()( kRCkQUkPXkY ++= (7)

where Tymm NkykykykY )](,),2(),1([)( +++= L T

N ypppP ],,,[ 21 L= i

yi Kp = T

N yqqqQ ],,,[ 21 L= ui

q

qyi KKq

=

+=1

1

1

TN yrrrR ],,,[ 21 L=

=

+=1

1

1i

q

qyi Kr

)]([)( kykX = )]([)( kukU = yNi ,,2,1 L= (8)

The optimal controller output sequence can be found by minimizing the above cost function (6). A necessary condition for minimum J is

0)(

=

kuJ (9)

Differentiating the equation for J we can get the following optimal solution:

)]()([][)( 1 kRCkPXYQQQkU rTT

= (10)

Secondly, when the control horizon 1uN , generally yu NN < , supposed

)()1()( yuu NkuNkuNku +==++=+ LL , At the same time,

TmkukunkykykX )]1(,),(),1(,),([)( ++= LL T

uNkukukukU )]1(,),1(),([)( ++= L So the yN step predictive outputs change to the following vectors

)()()()( kRCkQUkPXkY ++=

where

=

+

+

)1(1

)1(111

mnNN

mn

yypp

ppP

LML

=

uyyyy

uuuuu

NNNNN

NNNNN

qqqq

qqqq

qqq

Q

LMMMMM

LMMMMM

LL

321

321

2221

11

00000

TN yrrrR ],,,[ 21 L=

Differentiating the equation for J and using the above condition, then the following predictive controller optimal solution at the k sampling time is got:

)(]001[)( kUku L= (11) where )]()([][)( 1 kRCkPXYQQQkU r

TT=

. Finally, we adopt feedback regulation to further

improve the control performance. When at k sampling time the controller variable )(ku operates on the process, the predictive output variable )1( +kym can be got by the predictive model (2), Because disturbance and uncertainties in the controlled process, It exists error between practice output )1( +ky and the predictive model output at the k+1 sampling time. That is

)1()1()1( ++=+ kykyke m (12) It can be used predictive the future error of the future predictive output variable. Then defining

)1()( ++ khekym (13) to modify the future predictive output value, where modification coefficient h regulating vector.

The procedure for designing and implementing the proposed fuzzy MPC can be described in four steps: Construct the sum-min inference Mamdani fuzzy

model (2) based on the experience and expert knowledge; At k sampling time, obtain the on-line linearization

model (5) of the fuzzy model through the analytical expression of structure analysis; Based on the set-point )( ikyr + and the fuzzy

predictive model parameters, calculate the matrixes RQP , then get the optimal controller )(ku through the equations (10) or (11); Examine the actual output value )1( +ky , through

the equation (12)calculate the output errorget the k+1 sampling time predictive output value through (13)then return to step .

4 Simulation Example 4.1 Verification by Simulations Considering the HVAC systems, An air-handling unite is composed of cooling coil, air dampers, fans, chilled water pump and valves etc. The schematic diagram of

ExhaustAir

ControlDamper

OutsideAir

ChilledWater

Return Fan

FilterCooling Coil

Supply Fan(VSD)

ControlValve

MixedAir

Return Air

Supply Air

Fig. 3 Air Handling Unit

AHU in HVAC systems is shown in Figure 3. The dry-bulb temperature, web-bulb temperature and airflow rate of the on-coil air are aiT , aiwbT and am& respectively. The on-coil temperature of the chilled water is assumed as a constant, and the airflow rate am& varies in corresponding to cooling load demand of conditioned space, these two variables are considered as disturbances to the process. Thus the output aoT can be described as

( , , , )ao chw a ai chwiT f m m T T= & & (14) At first, before the real process testing, the Toolbox in Matlab7.0 software is applied to simulate the control of AHU systems in order to validate the performance of the proposed fuzzy linearization predictive control strategy. Because the complexity dynamical characteristic of AHU system, it is difficult to obtain its accurate mathematical model. However, according to the operation experiences and expert knowledge it is comparatively easy to construct the fuzzy model for it. The linear fuzzy inference rules are showed on the table 1. In membership functions of input and output invariables we supposed 2=J , 512 =+= JN . When

1=T , 20=yN , 10=uN and the set point is 1, the proposed fuzzy model predictive controller is used to control the off-coil dry-bulb temperature of AHU system. The simulation result is given in figure 4.

Table 1 The rule table of fuzzy predictive model y(k)

)1( +kym NB NS ZO PS PB

NB NT NB NM NS ZO NS NB NM NS ZO PS ZO NM NS ZO PS PM PS NS ZO PS PM PB

u(k)

PB ZO PS PM PB PT

The simulation result in figure 4 showed that it is available to apply the proposed Mamdani fuzzy model predictive control strategy to control off-coil dry-bulb temperature aoT of AHU systems. From the original

condition to the set point it only need about 4 minutes. Compared with the conventional PID controller, the novel FMPC technology has advantageous dynamical performance of less overshoot and shorter setting time etc. Furthermore it has better robust.

4.2 Real Testing In HVAC System Labs A pilot centralized HVAC systems is showed as in figure 5. The system has three chillers, three zones with three AHUs, three cooling towers and flexible partitions up to twelve rooms. All motors (fans, pumps and compressors) are equipped with VSDs. The system is made very flexible to configure these three units to form different schemes. The cooling coils for system are two rows with the dimension of 382525 cm . The measurement signals for the experiment are the water and airflow rates, on-coil air dry-bulb/web-bulb temperature, CCU inlet and outlet water temperature. The experiment is conducted under the following conditions: The chilled water supply temperature is fixed; the cooling load variation is achieved through the air and water flow rates. Through lots of tests we got the satisfied results. Figure 6 showed it need 480 seconds to reach the setting temperature 19.5. The result showed that the system nonlinearity can be well captured by the proposed fuzzy MPC strategy.

Fig. 5 HVAC systems in labs

Fig. 6 AHU modeling result by the proposed fuzzy model predictive control approach

0 5 1 0 1 50

0 .5

1

1 .5

2

2 .5

3

time (min)

Output tem

perature

Fig. 4 The simulation result of AHU system (------conventional PID output proposed FMPC output )

5 Conclusion Aiming at HVAC system control, a novel predictive control strategy based sum-min Mamdani fuzzy model was proposed in this paper. At first the resolution relationship of two inputs and single output variables of fuzzy predictive model was obtained by its structure analysis. Then the conventional model predictive control algorithm was designed based on the linearization fuzzy resolution model. Finally the simulation in Matlab toolbox and Real process testing in HVAC system labs were to validate its correction. The test results showed that the proposed fuzzy model predictive control approach is effective in HVAC systems control applications. Compared with the conventional PID controllers, this fuzzy MPC approach has the similar dynamical performances but also has the advantages of less overshoot and shorter setting time.

The proposed fuzzy linearization model based-on predictive control is successfully applied to the off-coil dry-bulb temperature control of air handling unit in HVAC systems, which showed the practicability of the novel fuzzy model predictive control strategy algorithm. The author will continue to study the predictive control algorithms based on the multi-input multi-output fuzzy model of HVAC systems.

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