Applied Soft Computing

Transcription

1 Applied Soft Computing () Contents lists available at ScienceDirect Applied Soft Computing journal homepage: A recurrent neuro-fuzzy system and its application in inferential sensing S. Jassar a,, Z. Liao b, L. Zhao a a Department of Electrical and Computer Engineering, Ryerson University, 35 Victoria Street, Toronto, Canada b Department of Architectural Science, Ryerson University, Canada article info abstract Article history: Received 3 September 9 Received in revised form 3 November Accepted November Available online 4 November Keywords: Adaptive systems Fuzzy logic Hybrid learning Inferential sensing Neuro-fuzzy systems Conventional neuro-fuzzy systems cannot effectively cope with dynamic processes, such as the heating systems of the buildings, due to the feed forward network structure. To overcome this problem, the existing hybrid system is incorporated with a feedback loop so that it can model the dynamical behavior of the process. As a case study, this improved hybrid system is employed to build an inferential model that estimates the average air temperature in a building served by a forced-warm-air heating system. The results show that the inferential model based on this improved hybrid system is accurate and robust. The parameter identification and tuning process is effective. Compared with conventional hybrid neuro-fuzzy system, it can significantly improve the performance of the inferential models. The neuro-fuzzy model incorporated with the feed-back loop has been tested using experimental data and the worst monthly RMSE as.6 C. This means that the inferential model can be employed to design better control schemes to improve indoor environmental quality and to save energy in buildings. Elsevier B.V. All rights reserved.. Introduction Inferential sensing is an attractive technique for modeling the dynamic behavior of the space heating systems, providing practical methods for estimating the value of critical control variables that are otherwise difficult, if not impossible, to measure using conventional physical sensors. For example a more representative measurement of average air temperature in the buildings, which is essential for optimal control of the relevant equipment, can be obtained using an inferential sensor [,]. Despite the rapidly decreasing cost and improving accuracy of most temperature sensors, it is normally impractical to use a lot of sensors to measure the average air temperature because the wiring and instrumentation can be very expensive to install and maintain. The inferential sensor trained with the available short-term measurements can be used for long-term prediction/estimation of the average air temperature. The inferential sensors can be based on empirical models including Kalman Filters, Artificial Neural Networks (ANNs), Fuzzy Logic (FL) and hybrid neuro-fuzzy methods. Over the last decade, significant advances have been made in the areas of FL and neural networks (NNs) [3]. The synergism of FL and NN has produced a functional system capable of learning, high-level of thinking and reasoning. It is an improved tool for determining the behavior of imprecisely defined complex dynamical systems. The purpose of a neuro-fuzzy system is to apply neural learning techniques to Corresponding author. address: pillo deepi@yahoo.co.in (S. Jassar). identify and tune the parameters and/or structure of neuro-fuzzy systems. Considerable work has been performed to integrate the excellent learning capability of neural networks with fuzzy inference systems (FIS) for deriving the initial rules of a fuzzy system and tuning the membership functions [4 9]. The neuro-fuzzy systems can combine the benefits of these two powerful paradigms into a single capsule. The features exhibited in these systems, such as fast and accurate learning, good generalization capabilities, excellent explanation facilities in the form of semantically meaningful fuzzy rules, and the ability to accommodate both data and existing expert knowledge, make them suitable for a wide range of engineering and scientific applications []. Hybridization technology of neuro-fuzzy systems has resulted in the development of intelligent systems through several approaches as GARIC, ANFIS, FUN, NEFCON, FALCON, SOFIN, FINEST, EFuNN, dmefunn, the evolutionary design of neuro-fuzzy systems and many others [7, 6]. Recent research demonstrates the use of Adaptive Neuro-Fuzzy Inference System (ANFIS) based modeling technique to estimate the average air temperature in the buildings from the information available to the boiler or furnace plant [7,8]. However, due to the nature of its feed forward network, the conventional hybrid neuro-fuzzy model does not allow for effectively capturing the time varying properties of the dynamic processes. Unlike recurrent networks it does not have any dynamic features. In the past few years, many studies related to recurrent neural networks (RNN) have been attempted to solve this problem using attractor dynamics and information storage ability [9 ]. However, RNNs training is found to be more difficult as compared to /$ see front matter Elsevier B.V. All rights reserved. doi:.6/j.asoc...

2 936 S. Jassar et al. / Applied Soft Computing () Nomenclature X (m,p) the input value to the mth node in layer P Y (m,p) the output from the mth node in layer P T avg measured average air temperature in the built environment ( C) ˆT avg inferential model estimated average air temperature in the built environment ( C) T external temperature ( C) T ad desired room temperature ( C) T sd supply air temperature set-point ( C) Q sol solar radiation (W) Q in energy consumption, calculated on the basis of control signal fire (W) C a total thermal capacity of air in the building (J/ C) C e total thermal capacity of the internal layer of building envelope (J/ C) C e total thermal capacity of the external layer of the building envelope (J/ C) T e lumped temperature of the internal layer of the building envelope ( C) T e lumped temperature of the external layer of the envelope ( C) K total heat transfer coefficient from the internal air to the internal layer of the building envelope K 3 total heat conductance from the internal air to the outside through infiltration K 4 total heat conductance from the internal layer to the external layer of building envelope K 5 total heat transfer from the external layer of building envelope to the outside m mean for Gaussian membership function variance for Gaussian membership function constant determining the portion of the solar radiation that into the building ˇ constant determining the portion of energy released from a radiator through convection time constant of boiler (s) E overall error function for training algorithm w bc the link that connects the output of the bth node in layer 3 with the input to the cth node in layer 4. N order of the dynamic model S number of data points k step size the feed forward networks because RNNs have complex network structures [] and therefore heavier computational efforts are also required making it unsuitable for the targeted applications. Furthermore, with feed forward network structures the performance of a fuzzy neural network has been demonstrated to be better than a neural network [3]. Applied to space heating systems, the feedforward neuro-fuzzy networks showed better performance than those of the neural networks [4]. Based on these observations, researchers have attempted the construction of a recurrent fuzzy neural network composed of internal feedbacks and time-delay synapses for control and identifications systems [5 7]. Several types of fuzzy reasoning in a connectionist structure have been proposed in literature [,]. Most neuro-fuzzy inference systems can be classified into three types depending on the types of fuzzy reasoning and fuzzy if then rules employed: Mamdani type [8]; Tsukamoto type [9]; and Takagi-Sugeno type [3]. When generating the FIS structure for neuro-fuzzy systems, it is important to select appropriate parameters, including the number of membership functions (MF)s and type of MF for each input. It is also important to select proper parameters for the learning process. In this paper, initial parameter identification and further tuning for obtaining the optimal values are discussed for a neuro-fuzzy model with feedbacks. The feedback connections are introduced to capture the dynamic properties of the distributed heating system. The rest of the paper is organized as follows. In Section, the functioning of different nodes in all five layers of the hybrid neurofuzzy model is given. Initial parameter identification is discussed in Section 3. The hybrid learning algorithm for fine tuning the MFs is also presented. Section 4 discusses the application of hybrid system based inferential sensing technology for forced-warm-air space heating systems. Case study for multi-zone forced-warm-air heating systems in Canada is presented. Finally, conclusions are given and future scope is discussed.. Feedback neuro-fuzzy system The feedback neuro-fuzzy system is a multilayer neural network-based fuzzy system. In this model, the dynamic property is achieved by feeding back the output through a feedback loop. Its topology is shown in Fig.. The system has five layers with the time-delay element for feeding back the estimated output to the first layer. A model with two inputs and a single output is considered here for convenience. Accordingly, there are two nodes in layer and one node in layer 5. The input and output nodes represent the input states and output control decision signals, respectively, and in the middle layers, there are nodes functioning as MFs and rules. The input nodes in layer directly transmit input signals to the next layer. The term nodes in layer and layer 4 act as MFs to express the input output fuzzy linguistic variables. Throughout the simulation case study presented in this paper, all MFs used are Gaussian functions defined in Eq. () A (u) = Gaus sin(u; m, ) = e ((u m) / ) A Gaussian MF is determined by m and : m represents the center of the MF; and determines the width of the MF. The two fuzzy sets of the first and the second input variables consist of k and k linguistic terms, respectively. The linguistic terms, such as very high (VH), high (H), medium (M), low (L), very low (VL), are numbered in descending order in the term nodes. Hence, k + k nodes U 5 k Y k3 k U k*k Fig.. Topology of hybrid neuro-fuzzy model. LAYER 5 LAYER 4 LAYER 3 LAYER LAYER ()

3 S. Jassar et al. / Applied Soft Computing () and k 3 nodes are included in layers and 4, respectively, to indicate the input output linguistic variables. Each node in layer 3 is a rule node and represents a single fuzzy control rule. In total, there are k k nodes in layer 3 to form a fuzzy rule base for two linguistic input variables. The semantic meaning and functions of the nodes in the proposed network are as follows. We use indices, a, b, c and d for the nodes in layers, 3, 4 and 5, respectively. Following notations are used to describe the functions of the nodes in each of the five layers. y(t) Hybrid Neuro Fuzzy Model Output Delay X (m,p) : the input value to the mth node in layer P. Y (m,p) : the output from the mth node in layer P. m (m,p), (m,p) : the mean and variance of the Gaussian functions of the mth node in layer P. w bc : the link that connects the output of the bth node in layer 3 with the input to the cth node in layer 4... Layer Nodes in layer are input nodes that represent input linguistic variables as crisp values. The nodes in this layer only transmit input values to the next layer, the membership function layer. Y (,) = U, Y (,) = U ().. Layer Nodes in this layer act as MFs to represent the terms of the respective linguistic variables. This is implemented using Gaussian MFs with two parameters, mean (m) and variance (). Initially, the connection weights in this layer are unity and the MFs are spaced equally over the weight space. { Y(,) for a =,,...,k X (a,) = (3) Y (,) for a = k +,k +,...,k + k The output function of this node is the degree to which the input belongs to the given MF. [ ) Y (a,) = exp ( ] X(a,) m (a,) for a =,,...,k + k (4) (a,) where and m are the parameters, which are adjusted through the learning process. As the values of these parameters change, the bell-shaped functions vary, thus exhibiting various forms of MFs on a linguistic label. Parameters in this layer are referred as premise parameters..3. Layer 3 Each node in layer 3 represents a possible IF-part of a fuzzy rule. Each node has two input values from layer. The weights of the links are set to unity. The nodes in this layer perform the AND operation. Thus, all the nodes in this layer form a fuzzy rule base. Hence, the functions of the layer are: X (b,3) = min(y (a,) ) (5) a A b Y (b,3) = X (b,3) for b =,,...,k k (6) where A b is the set of indices of the nodes in layer that are connected to node b in layer 3, and Y (a,) is the output of node a in layer..4. Layer 4 A node in layer 4 represents a possible THEN-part of a fuzzy rule, and each node of this layer performs the fuzzy OR operation Direct Inputs y(t N) Delay y(t ) Delay Feedback Inputs y(t ) Fig.. The feedback system for neuro-fuzzy model. to integrate the field rules leading to the same output linguistic variables. The connection weights w bc of the links connecting nodes c in layer 4 to nodes b in layer 3 represent conceptually the firing strength of the corresponding fuzzy rules when inferring fuzzy output values. The initial connection weights between layers 3 and 4 are randomly selected in the interval [,+]. The functions of this layer are expressed as k k X (c,4) = w bc Y (b,3) (7) c= Y (c,4) = min(,x (c,4) ) for c =,,...,k 3 (8).5. Layer 5 The node in this layer computes the output signal of the neurofuzzy model. The output node together with the layer 5 acts as a defuzzifier. The center of area defuzzification scheme, used in this model, can be simulated by X (,5) = k 3 c= m (c,4) (c,4) Y (c,4) (9) X (,5) y = Y (,5) = () k3 c= (c,4)y (c,4) Hence, the cth link weight in this layer is m (c,4) (c,4)..6. The feedback system In the model discussed above, the dynamic property is achieved by feeding the output back to layer through a feedback loop as shown in Fig.. Introducing this feedback connection, enables the system to remember previous states and uses both the previous and the current state to calculate the new output values. From a practical point of view, time delay embedded with the feedback is time variant which can lead to a different dynamical behavior and such behavior is vital in improving the ability to provide better and more accurate prediction capability. With the output feedback system, the output at time t as a function of current external inputs and previous inputs and outputs, is given by y(t) = f [U(t ),...,U(t M),y(t ),...,y(t N)] () where U(t ),..., U(t M) are the direct inputs and y(t ),..., y(t N) are the inputs by feeding back the output at different times. By doing so, we will be able to train the canonical hybrid neurofuzzy system as a recurrent of the order of N by providing the timedelay element at the output.

4 938 S. Jassar et al. / Applied Soft Computing () Membership Grades VL L M H VH Domain Interval 3. Hybrid learning Fig. 3. Initial MFs for u, u and y. A fuzzy model can be conveniently constructed through the following three processes: identifying initial MF parameters, finding the IF THEN rules, and optimizing the MF parameters. 3.. Initial parameter identification: number and type of MFs When generating a neuro-fuzzy structure, it is important to select proper parameters, including the number of MFs and type of MF for each individual input variable. The type of MF depends upon the type of application the model is used for. The authors have developed a scheme to choose the proper number and type of MF for given input output data pairs. The minimization of testing error for a given set of data pairs is considered as a criterion for the selection of proper number and type of MFs [3]. 3.. Initial parameters (m and ) for MFs Divide the input and output space into fuzzy regions. After the number of MFs associated with each input and output are fixed, the initial values of parameters are set in such a way that the centers of the MFs are equally spaced along the range of each input and output variable. Moreover, these MFs satisfy the condition of ε- completeness with ε =.5, which means that given a value x of one of the inputs in the operating range, we can always find a linguistic label A such that A (x) ε. In this manner, the FIS can provide smooth transitions and sufficient overlap from one linguistic label to another. One has to choose the intervals for the linguistic values of each input and output linguistic variables in such a way that they do overlap and also cover the entire space of the corresponding input output linguistic variables. For instance, it can be assumed that the domain intervals of u, u and y are [,]. Each domain interval is divided into certain number of regions that are each assigned a fuzzy MF. Fig. 3 shows an example where the domain intervals of u, u and y are divided into five regions. Obviously, other divisions of the domain regions and other shapes of MFs are possible Rule generation Fuzzy rules are generated from the training data, which consists of series of data pairs. Obtain one rule from one pair of desired input output data. Then assign a degree of certainty to each rule. To resolve a possible conflict problem, i.e. rules having the same IF-part but a different THEN-part, and to reduce the number of rules, we assign a degree to each rule generated from data pairs and accept only the rule from a conflict group that has a maximum degree. In other words, this step is performed to delete the redundant rules, and therefore obtain a concise fuzzy rule base. The effect of these rules is that each of the rules is activated to a certain degree represented by the weight value associated with that rule MF tuning After the fuzzy rules are found, the whole network structure is established. The network enters the learning phase to optimally adjust the parameters of the MFs. The hybrid learning algorithm is used to tune the MF parameters. This algorithm is a combination of the least square and back-propagation methods. The learning algorithm refines the premise fuzzy MF and consequents using the least squares estimate and back propagation. The parameter tuning through learning process minimizes the error function, E = U n (t d Y (d,5) ) () d= where n is the number of nodes in layer 5 and t d and Y (d,5) are the target and actual outputs of the node d in layer 5 for the input U. The weight factor w bc is an adjustable parameter corresponding to node c in layer 4 and node b in layer 3, the update learning formula for w bc is ( ) E w bc (t + ) = w bc (t) (3) w bc where is the learning rate and the chain rule is given as follows: E = E Y (c,4) = E Y (d,5) Y (c,4) (4) w bc Y (c,4) w bc Y (d,5) Y (c,4) w bc also, the learning rate can be further expressed as: k = ( wbc ) E w bc (5) where k is the step size. The value of k determines the speed of convergence as explained in Section 3.5. In this type of learning, the update action occurs only after the entire set of training data pairs is processed. This processing of the entire training data pairs is called an epoch Parameter selection for MF tuning For every learning algorithm, it is important to select proper parameters for the learning process. The parameters which mainly affect the model performance are given below: Initial step size, SS Step size increase rate, SS INC Step size decrease rate, SS DEC Step size is an array of step sizes. The step size is increased by a constant factor SS INC (greater than one) if the training error undergoes four consecutive reductions and decreased by a constant factor SS DEC (less than one) if the training error undergoes two consecutive combinations of one increase and one reduction. A scheme has been developed to choose the proper values of the above parameters for given input output data pairs. The minimization of testing error for a given set of data pairs is considered as a criterion for the selection of learning process parameters [3]. The learning rate used for updating the adjustable parameters is proportional to the step size as given by Eq. (5).

5 S. Jassar et al. / Applied Soft Computing () Building Td + error Thermostat Regulator Flame Level Blower Speed Furnace Control Logic Heat Heating System Tavg ^ Tavg Tavg Estimator Qin Qsol Furnace State ON/OFF Qsol Sensor T T Sensor Fig. 4. Block diagram representation of the furnace control scheme using average air temperature estimator. 4. Inferential sensing technology for multi-zone forcedwarm-air space heating systems A conventional residential forced-warm-air heating system comprises a furnace in which fuel is burned to generate warm-air, the distribution system that distributes the warm-air to the space being conditioned and returns the indoor air back to the furnace, and the air registers that diffuse warm-air to the occupied indoor spaces. In the current practice, the control of such heating systems is very problematic. The major equipment, or the furnace, is controlled by a control system that only measures the air temperature at one place in the building. Although the control system allows flexible scheduling of temperature set-point to accommodate different occupancy schemes, the building is basically treated as a single zone system. Consequently, some rooms in the building are either overheated or unnecessarily heated while not occupied, while some rooms are under-heated due to insufficient heating capacity that result from overheating in other rooms. A survey on the operation of such heating systems in residential buildings in Ontario, Canada was conducted in winter 8/9 by the authors. The survey aims to find out how serious the above mentioned problem is in the current practice and to identify potential for improvements. The survey results show that approximately % of the respondents feel dissatisfied with room temperature in the winter. The results indicate a systematic problem with the control of conventional heating and cooling systems. There exists a correlation between the comfort level and the occupant s access to the control over room temperature. Respondents with higher heating system controllability tend to feel more thermally comfortable. In the multi-zone heating systems, the survey results show that 36% and 7% of the respondents experienced overheating in the winter and overcooling in the summer. In the same house there are places with temperature higher or lower than the desired level. This is because the thermostat only sense the temperature of one zone in which it is installed. A more representative measurement of average air temperature in the building is essential for optimal control of the relevant equipment as the output of a physical sensor is unlikely to be representative for the room being measured. To deal with these problems of thermal discomfort and higher energy consumption, authors have presented the application of an inferential sensor model for estimating the average air temperature that is otherwise difficult to estimate, if not impossible, in multizone heating systems. Liao and Dexter have developed a physical model relating the average air temperature in the built environment to three easily measurable variables [8]. The average air temperature (T avg ) is estimated from three other easily measurable variables including energy used by the heating systems (Q in ), solar radiation (Q sol ) and the external temperature (T ). A set of the parameters [K K 3 K 4 K 5 C a C e C e ˇ ] were determined through the training process of the model. The estimated temperature can optimize the control of mechanical equipment such as chiller, boiler, furnace, pump and fan in the generation plant of any heating ventilating and air-conditioning (HVAC) system. Future outcome of this study may be the development of a control scheme using estimated temperature as a feedback for controlling either the flame level or the blower speed of the furnace system, as shown in Fig. 4. As can be seen from Fig. 4, the scheme consists of the three components: an estimator model for estimating the average air temperature in the building, a set-point resetting module, and the furnace control logic. The estimator model is used to estimate the average air temperature in the building (ˆT avg ). The set-point regulator is used to determine the set-point of the supply air temperature (T sd ) from the difference between the output of the estimator model (ˆT avg ) and the desired room temperature (T ad ). The furnace control logic unit is used to maintain the supply air temperature at the set-point, determined by the temperature resetting module, by controlling the flame level or blower speed for furnace firing signal control. 4.. The feedback adaptive neuro-fuzzy model for estimating average air temperature Neuro-fuzzy modeling is used for developing the average air temperature estimator Input variable selection When generating a FIS system structure, first important step is to select the relevant input and output variables. The output variable is the hard to measure variable, T avg. The inputs are easily measurable variables, as discussed above, Q in, Q sol and T. Fig. 5 shows a block diagram of the hybrid neuro-fuzzy system with a feedback loop for feeding that feedback the estimated output to the

6 94 S. Jassar et al. / Applied Soft Computing () Tavg(t) Hybrid Neuro Fuzzy Model Output Delay The external temperature is measured by two sensors. T is represented by the algebraic average of the two external temperature measurements. The third input, Q sol, was monitored by metrological station, University of Toronto, Mississauga, Ontario, Canada. This weather station is operated by the department of geography. CNR net radiometer is used for the measurement of net radiation at the earth s surface [3]. The model is trained using input output data pairs for the experimental data for the month of January 8, as shown in Figs. 6 and 7 shows the testing data for checking the performance of the model. Qin Qsol T Tavg(t ) Feedback Input Fig. 5. Block diagram of the feedback adaptive neuro-fuzzy model for average air temperature estimation. layer. The estimated value of the output variable is represented by ˆT avg Training and testing data Experimental data are collected from a residential building located in Markham, Ontario, Canada. The building is a single detached house with three levels, including basement, ground floor and second floor. Each level is divided into zones. Multiple sensors were used to monitor the air temperature in each zone and their algebraic average was treated as the representative measurement of the room temperature in the zone. It is assumed that each zone has the same floor area; the building air temperature, T avg, is represented by the algebraic average of the air temperature in all the zones in all the three levels. The sampling interval is 5 min. The time when the furnace switches between ON and OFF states is recorded. This is a discrete signal. A first order filter is used to convert this discrete signal into a continuous signal. This continuous signal is used to compute the actual energy used by the furnace system, Q in Number and type of MFs For the given set of input output data pairs, optimal number and type of MFs are selected for each individual input and output variables. Minimum testing error (MTE) is considered as a criterion for the selection [3]. Figs. 8 and 9 present the impact of number and type of MFs on the training/testing errors. Fig. 8 shows that the different combinations of number of MFs affect the training and testing errors significantly. The number of MF combination is: [4 3 4] for trn and test, [5 5 3] for trn and test, [5 5 5] for trn3 and test3 and [5 4 5] for trn4 and test4. For trn3 and test3 data set each input variable has five MFs, [5 5 5], and error curve is with minimum testing error. Fig. 9 concludes that the selection of the type of MF depends upon the shape of the training and testing data variations. The MF type combination is: [trimf trimf gaussmf] for trn and test, [gaussmf gbellmf gaussmf] for trn and test, [gbellmf gaussmf gbellmf] for trn3 and test3, and [gaussmf gaussmf gaussmf] for trn4 and test4.for data set four, trn4 and test4, the testing error curve give the best value as MTE occurs at 6 epochs and it is the minimum out of the four testing error curves. In this case, the input output training data points in the training data set are large enough as compared to the testing data, and the best combination of number of MFs is [5 5 5], i.e. each individual input variable has five MFs. MF type selected is Gaussian MF, which is the same as that used for subtractive clustering based ANFIS model [7]. The shape of the initial MFs for Q in, Q sol, T and T avg is the same as that shown in Fig x Qin (W).5 Qsol (W) T (Deg C) 5 Tavg (Deg C) Fig. 6. Training data (January 8: day ).

7 S. Jassar et al. / Applied Soft Computing () x Qin (W).8 Qsol (W) T (Deg C) 5 Tavg (Deg C) Fig. 7. Testing data (January 8: day 3) Rule base From the set of desired input output data pairs, corresponding degrees are determined for different MFs. (.,.38,.54;.5), (.8,.6,.4;.43),... (6) where the first three numbers (Q in, Q sol, T ) are inputs and the fourth one (T avg ) is the output for each data pair. All the variables are normalized to [,] range. Domain intervals of Q in, Q sol, T and T avg are divided into five regions, very low (VL), low (L), medium (M), high (H), very high (VH) as shown in Fig. 3. Q in, if applied to Fig. 3 has a degree of in VL, and a degree of. in L. Similarly, T, has a degree of.9 in L,. in M and.3 in VL. Now assign [4 3 4] [5 5 3] [5 5 5] [5 4 5].45 Testing Error 4 3 [4 3 4] [5 5 3] [5 5 5] [5 4 5] Fig. 8. Training and testing errors obtained by the neuro-fuzzy model using different numbers of MFs.

8 94 S. Jassar et al. / Applied Soft Computing () [tri tri gauss] [gauss gbell gauss] [gbell gauss gbell] [gauss gauss gauss] Testing Error [tri tri gauss] [gbell gauss gbell] [gbell gauss gbell] [gauss gauss gauss].5 Fig. 9. Training and testing errors obtained by the neuro-fuzzy model using different types of MFs. Q in,i,q sol,i,t,i and T avg,i to a region with maximum degree: Q in, is assigned to VL and T, is assigned to L. The rules are obtained from input output pairs of the experimental data shown in Figs. 6 and 7. Examples of the rules obtained from individual pairs of the desired input output data are given below: R: if Q in is VL, Q sol is L, T is L, then T avg is L R: if Q in is VH, Q sol is L, T is L, then T avg is L After assigning degree to each rule, part of the fuzzy rule base is presented in Table Learning of model parameters The model is trained using hybrid learning algorithm through which the optimal values for the parameters are determined in this phase. Figs. show the training and testing errors using different parameters for training process. Fig. shows that the initial step size does not affect the values of MTE, while it does affect the training epochs when the MTE appears. The larger the initial step size, the earlier the MTE comes. For data set trn and test, with initial step size., the MTE occurs at 6 epochs and for data set Table Part of the rule base. Rule IF THEN T avg is Degree R VL L L L.9 R M L VL VL.45 R3 H M L VL.55 R4 VH L L L.34 R5 VH L H H.3 R6 VH M M M. R7 VH L H H.65 R8 H H H M.4 R9 VH VL H M.58 R VH L H H.3 R VH L L H.9 R VH M M M.3 R3 VH M M H.6 R4 VH M M H.46 R5 VH M M H.43 trn4 and test4, with initial step size.7, MTE occurs at 6 epochs [3]. Fig. presents the effect of step size increase rate, SS INC,on the training and testing errors. A similar conclusion can be drawn as the impact of initial step size. The larger the increase rate, the faster the FIS achieves MTE. Fig. shows that no significant difference exists for FIS obtained using different step size decrease rates. For four different SS DEC values as.8 for first data set trn and test,.85 for second data set trn and test,.9 for third data set trn3 and test3 and.95 for fourth data set trn4 and test4, the testing and training error curves are approximately the same and all the four curves in each subplot are superimposing each other. Based on this analysis and choosing MTE as the deciding criterion, the training process parameters selected are: SS =., SS INC =., SS DEC =.85, number of training data pairs = 3, number of epochs = Simulation results and model validation The model performance is measured using root-mean-square error (RMSE) as a statistical indicator to provide a numerical description of the goodness of the estimates. This is calculated according to Eq. (7): S RMSE = (ˆT ) avg (i) T avg (i) (7) S i= where S is the number of data points, T avg (i) the measured data point and ˆT avg (i) the estimated data point. Before adding the feed-back loop, the model is trained. The learning process is terminated when the error is minimized. The RMSE for training data is.8 C after 59 epochs of learning. After training the final MFs assume a different form. The comparison of the measured and estimated average air temperature values for the training data set is shown in Fig. 3. The graph in Fig. 4 shows the error for each training data pair. The testing data, as shown in Fig. 7, is used for checking the performance of the trained model. The comparison between the measured and the estimated average air temperature values for

9 S. Jassar et al. / Applied Soft Computing () Testing Error Fig.. Training and testing errors obtained by the neuro-fuzzy model for different step sizes. Table Results. Test data TEST-I January (day 3) TEST-II February TEST-III March TEST-IV April TEST-V October TEST-VI November TEST-VII December TEST-VIII January TEST-IX February RMSE ( C) Hybrid neuro-fuzzy model without self feed-back Hybrid neuro-fuzzy model with self feed-back the testing data is shown in Fig. 5. The RMSE is.69 C. The sensor model used is not correctly structured and cannot be used to estimate the average air temperature in the buildings. As shown in Fig. 5, the error is as large as C and 3 C for few data pairs. This is because the dynamics of the building thermal system is not captured by the model. To resolve this problem the feedback loops are implemented using time-delay element at the output and the model performance Testing Error Fig.. Training and testing errors obtained by the neuro-fuzzy model for different step size increase rates.

10 944 S. Jassar et al. / Applied Soft Computing () Testing Error Fig.. Training and testing errors obtained by the neuro-fuzzy model for different step size decrease rates. Average Air Temperature (Deg C) Measured Temperature Estimated Temperature Fig. 3. Training results for the static model with no feed-back. is tested using the same testing data sets. As shown in Fig. 5, the previous value of the output variable is fed back, resulting into N =. So, the recurrent neuro-fuzzy structure is first order. Fig. 6 compares the measured and estimated average air temperatures for the.8.6 Fig. 5. Testing results for the static model with no feed-back. same testing data set of Fig. 7. It shows the testing error for testing data pairs is reduced to.35 Cto.75 C. Nine different sections of the experimental data obtained from the same experimental set up have been tested. Table gives the values of all three performance indices for these testing data sets Fig. 4. Training error for the static model with no feed-back. Average Air Temperature (Deg C) Measured Temperature Estimated Temperature Fig. 6. Testing results for the dynamic model with feed-back system.

11 S. Jassar et al. / Applied Soft Computing () Testing data sets I VI are for year 8 and testing data sets VII IX are for year Conclusion This paper has presented an average air temperature estimator model based on feedback neuro-fuzzy system for capturing the dynamic properties of the space heating system. The method for parameter identification and training for the neuro-fuzzy model is also presented. The recurrent neuro-fuzzy system is established through a number of processes, including selecting the input variables and MFs, establishing the rules based on experimental data, training the structure by a hybrid learning algorithm. The initial step size, step size increase rate, step size decrease rate and number of epochs are determined to minimize the testing error. The developed model can accurately estimate the average air temperature based on the information available to the heating equipment once it is trained using short-term monitoring data. The results show that the dynamical behavior of the systems can be better represented using feedback systems. The inferential model performance, measured by RMSE, is significantly reduced with the implementation of the feedback loop. The positive and negative peak for error is also reduced from C to.35 C and from 3 Cto.75 C, respectively. The estimation results show that the model is correctly structured and trained and is suitable for the enhancement of conventional control systems in the buildings. The future work will focus on the application of the estimator model to develop the predictive control schemes for furnace operation. The control scheme may control either the blower speed or flame level for improving the energy efficiency of the heating systems. Through the implementation of these control schemes, the scope for improvement of energy efficiency and thermal comfort in the built environment will be investigated. The applicability of the developed model for cooling systems will also be a part of the further studies. Further research will be conducted on the simplification of the training process by augmenting the model with priori-knowledge. Acknowledgements The work presented in this paper is partially funded by National Sciences and Engineering Research Council of Canada (NSERC) with research project reference numbers as: and and is carried out at Ryerson University, Toronto, Canada. The support of this organization is gratefully acknowledged. References [] Z. Liao, A.L. Dexter, An inferential control scheme for optimizing the operation of boilers in multi-zone heating systems, Building Services Engineering Research and Technology 4 (4) (3) [] Z. Liao, A.L. Dexter, An experimental study on an inferential control scheme for optimising the control of boilers in multi-zone heating systems, Energy and Buildings 37 () (5) [3] L.A. Zadeh, Fuzzy sets, Information and Control 8 (3) (965) [4] S. Horikawa, T. Furuhashi, Y. Uchikawa, On fuzzy modeling using fuzzy neural networks with the back-propagation algorithm, IEEE Transactions on Neural Networks 3 (5) (99) [5] C.C. Hung, Building a neuro-fuzzy learning control system, AI Expert 8 () (993) [6] H. Ishibuchi, H. Tanaka, H. Okada, Interpolation of fuzzy if then rules by neural networks, International Journal of Approximate Reasoning () (994) 3 7. [7] J.S.R. Jang, ANFIS: adaptive-network-based fuzzy inference systems, IEEE Transactions on Systems, Man and Cybernetics 3 (3) (993) [8] C.T. Lin, C.S.G. Lee, Neural-networks-based fuzzy logic control and decision system, IEEE Transactions on Computers 4 () (99) [9] J.J. Shann, H.C. Fu, A fuzzy neural network for rule acquiring on fuzzy control system, Fuzzy Sets and Systems 7 (3) (995) [] V.A. Constantin, Fuzzy Logic and Neuro-fuzzy Applications Explained, Prentice Hall, Eaglewood Cliffs, NJ, 995. [] J.S.R. Jang, C.T. Sun, E. Mitzutani, Neuro-fuzzy and soft computing: a computational approach to learning and machine intelligence, Prentice Hall, Upper Saddle River, NJ, 997. [] Y.C. Lee, C.H. Hwang, Y.P. Shih, A combined approach to fuzzy model identification, IEEE Transactions on Systems, Man and Cybernetics 3 () (994) [3] H.R. Berenji, P. Khedkar, Learning and tuning fuzzy logic controllers through reinforcements, IEEE Transactions on Neural Networks 3 (5) (99) [4] C.F. Juang, C.T. Lin, An online self constructing neural fuzzy inference network and its applications, IEEE Transactions on Fuzzy Sets 6 () (998) 3. [5] D. Nauck, F. Klawonn, R. Kruse, Foundations of Neuro-fuzzy Systems, Wiley, New York, 997. [6] S. Tano, T. Oyama, T. Arnould, Deep combination of fuzzy inference and neural network in fuzzy inference, Fuzzy Sets Systems 8 () (996) 5 6. [7] S. Jassar, Z. Liao, L. Zhao, Adaptive neuro-fuzzy based inferential sensor model for estimating the average air temperature in space heating systems, Building and Environment 44 (9) [8] Z. Liao, A.L. Dexter, A simplified physical model for estimating the average air temperature in multi-zone heating systems, Building and Environment 39 (9) (4) 9 8. [9] X. Li, W. Yu, Dynamic system identification via recurrent multilayer perceptrons, Information Science 47 ( 4) () [] W. Yu, Nonlinear system identification using discrete-time recurrent neural networks with stable learning algorithms, Information Sciences 58 (4) [] F.J. Lin, H.J. Shieh, P.H. Shieh, P.H. Shen, An adaptive recurrent-neural-network motion controller for X Y table in CNC machine, IEEE transactions on Systems, Man and Cybernetics-Part B: Cybernetics 36 () (6) [] T.W.S. Chow, Y. Fang, A recurrent neural-network-based real-time learning control strategy applying to nonlinear systems with unknown dynamics, IEEE Transactions on Industrial Electron 45 () (998) 5 6. [3] L.X. Wang, J.M. Mendel, Generating fuzzy rules by learning from examples, IEEE Transactions on Systems, Man and Cybernetics (99) [4] S. Jassar, T. Behan, L. Zhao, Z. Liao, The comparison of neural network and hybrid neuro-fuzzy based inferential sensor models for space heating systems, in: IEEE International Conference on Systems Man and Cybernetics, 9, pp [5] C.F. Juang, A TSK-type recurrent fuzzy network for dynamic systems processing by neural network and genetic algorithms, IEEE Transactions on Fuzzy Systems () () [6] C.H. Lee, C.C. Teng, Identification and control of dynamic systems using recurrent fuzzy neural networks, IEEE Transactions on Fuzzy Systems 8 (6) () [7] P.A. Mastorocostas, J.B. Theocharis, A recurrent fuzzy-neural model for dynamic system identification, IEEE Transactions on Systems Man and Cybernetics-Part B: Cybernetics 3 () () [8] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man Machine Studies 7 () (975) 3. [9] Y. Tsukamoto, An approach to fuzzy reasoning method, in: M.M. Gupta, R.K. Ragade, R.R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North- Holland, Amsterdam, 979, pp [3] T. Takagi, M. Sugeno, Derivation of fuzzy control rules from human operator s control actions, in: Proceedings of the IFAC Symposium on Fuzzy Information Knowledge Representation and Decision Analysis, 979, pp [3] S. Jassar, Z. Liao, L. Zhao, K.L.R. Ng, Parameter selection for training process of neuro-fuzzy systems for average air temperature estimation, in: IEEE International Conference on Mechatronics and Automation, 9, pp. 8. [3] CNR Net Radiometer Manual, Campbell Scientific Corp., 8, pp. 3.