Unsupervised fuzzy neural network structural active pulse
controller
Shih-Lin Hung
∗;†and C. M. Lai
Department of Civil Engineering; National Chiao Tung University; 1001 Ta Hsueh Road; Hsinchu 300; Taiwan
SUMMARY
Applying active control systems to civil engineering structures subjected to dynamic loading has re-ceived increasing interest. This study proposes an active pulse control model, termed unsupervised fuzzy neural network structural active pulse controller (UFN-SAP controller), for controlling civil engineering structures under dynamic loading. The proposed controller combines an unsupervised neural network classi cation (UNC) model, an unsupervised fuzzy neural network (UFN) reasoning model, and an active pulse control strategy. The UFN-SAP controller minimizes structural cumulative responses dur-ing earthquakes by applydur-ing active pulse control forces determined via the UFN model based on the clusters, classi ed through the UNC model, with their corresponding control forces. Herein, we assume that the e ect of the pulses on structure is delayed until just before the next sampling time so that the control force can be calculated in time, and applied. The UFN-SAP controller also averts the diculty of obtaining system parameters for a real structure for the algorithm to allow active structural control. Illustrative examples reveal signi cant reductions in cumulative structural responses, proving the feasi-bility of applying the adaptive unsupervised neural network with the fuzzy classi cation approach to control civil engineering structures under dynamic loading. Copyright?2001 John Wiley & Sons, Ltd.
KEY WORDS: active pulse control; dynamic loading
INTRODUCTION
Control of civil engineering structures subjected to dynamic loading, such as those attributed to earthquakes, heavy winds and high waves, can be classi ed into passive and active controls. A passive control system requires no external power source. On the other hand, in an active control system, external power sources control actuator(s) that apply forces to the structure in a prescribed manner. Active control devices are used in civil engineering structures to modify ∗Correspondence to: Shih-Lin Hung, Department of Civil Engineering, National Chiao Tung University, 1001 Ta
Hsueh Road, Hsinchu 300, Taiwan.
†E-mail: [email protected]
Contract=grant sponsor: National Science Council of Republic of China; contract=grant number: NSC 88-2211-E-009-001.
Received 10 December 1999 Revised 27 June 2000 Copyright 2001 John Wiley & Sons, Ltd. Accepted 19 July 2000
the structural parameters (sti ness and damping), allowing them to respond more favourably to external excitation [1].
The several analytical theories used for active control of civil engineering structures include optimal control, stochastic control, adaptive control, hybrid control and intelligent control [2] Housner et al. [3] indicated that the control strategies deemed appropriate for civil engineering structural control should be simple, but robust and fault tolerant. Additionally, such control strategies not need to be an optimal control, and must be implemented. Intelligent control is a promising approach in the application of automatic structure control systems. Several references from the special issues of Journal of Engineering Mechanics [2] give a good overview of intelligent control systems.
Fu [4] proposed the concept of intelligent control to enhance and extend the applicability of automatic control systems. Intelligent controllers can be viewed as adaptive or self-organizing systems that learn through interaction with their environment with little a priori knowledge. Two main methodologies related to intelligent control have been developed: (1) arti cial neu-ral networks (ANNs) and (2) fuzzy logic. ANNs have been used since the 1980s to identify and control structures in the context of structure control. Nikzad and Ghaboussi [5] rst ap-plied multi-layered feedforward networks (MFN) to the problem of digital vibration control of mechanical systems. Wen et al. [6] used two ANNs: one to predict the structural response subjected to the control force alone, and the other to predict ground accelerations. Yen [7] applied an indirect predictive learning control scheme for control of a large space structure. Meanwhile, Tang [8] described a simple heuristic-based control strategy capable of applying the control force to cancel the system velocity at the preceding time step for an SDOF system. Chen et al. [9] used ANNs to identify the structure system and the trained ANNs can model the structural behaviour. Furthermore, Chen et al. [10] proposed a BTTNC active structural control model consisting of two neural networks. One is used for representing the structure to be controlled; the other is trained for determining the actions taken to control the structure. Recently, Hung et al. [11] presented an active pulse control model, termed the adaptive neural structure active pulse controller (ANSAP controller), to control civil engineering structures under dynamic loading. The controller consists of two sub-networks: the neural emulator net-work (NEN) and the neural control netnet-work (NCN). The NEN netnet-work is used as a system identi cation model to obtain the structural parameters and it is trained o -line. The NCN model is then used to determine the corresponding control force for the structure represented by the NEN model according to measured structure responses. Other related investigations dealing with ANNs applied to control domains can also be presented in the special issues of the Journal of Engineering Mechanics [2].
The theory of fuzzy logic theory developed by Zadeh [12] can be used to model imprecision, ambiguity and fuzziness in vague linguistic information. Adeli and Hung [13] developed a fuzzy neural network learning model by integrating an unsupervised fuzzy neural network classi cation algorithm with a genetic algorithm and an adaptive conjugate gradient neural network learning algorithm, and applied it to the domain of image recognition. Meanwhile, Hung and Jan[14] presented an unsupervised fuzzy neural network (UFN) reasoning model in structural engineering. Recently, Hung and Jan [15] integrated the UFN reasoning model with a supervised learning model, termed integrating fuzzy neural network (IFN) learning model, and applied it to the problem of structural engineering. In the framework of structural control, Casciati and Yao [16] provide an overview of neural and fuzzy techniques for the control of structures. Faravelli and Yao [17] presented guidelines for implementing a fuzzy active control
strategy for civil engineering structures. Other examples of recent research include Furuta et al. [18], Yeh et al. [19], Sun and Goto [20] and Nagarajaiah [21].
While extending the above research, this work presents a novel fuzzy-ANN active pulse control model, the unsupervised fuzzy neural network structural active pulse (UFN-SAP) con-troller, to control civil engineering structures under dynamic loading. The proposed model is based on an unsupervised neural network classi cation (UNC) model [13], an unsupervised fuzzy neural network (UFN) reasoning model [14; 15] and an active pulse control strategy [22]. The proposed control strategy attempts to reduce cumulative structural responses during earthquakes by applying the active pulse control force. The necessary pulse control forces are determined via the UFN model based on the clusters, classi ed through the UNC model, with their corresponding control forces. These classi ed clusters with their corresponding control forces are representative of structural dynamic behaviours. The control e ectiveness of the UFN-SAP controller is investigated for an SDOF and a 3DOF structure subjected to the EL Centro, Kobe, and Northridge earthquakes.
ACTIVE PULSE CONTROL ALGORITHM
Active control system in civil engineering structures is based on closed-loop control, implying that the structural response is continually monitored and this information is used to continu-ously modify the applied control forces. Consider a non-linear structural system with n degree of freedom subjected to an external excitation. The equation of motion is written in matrix notation:
M x(t) + C ˙x(t) + Kx(t) = B1u(t) + E1w(t) (1)
where constant matrices M, C and K are, respectively, the mass, damping, and sti ness ma-trices with n×n entities; x(t) is the n-dimensional displacement vector with respect to the ground; u(t) is the p-dimensional control force vector, w(t) is the q-dimensional external ex-citation vector and the n×p matrix B1 and n×q matrix E1 are location matrices that de ne the
locations of the control force and the excitation, respectively. While assuming that the external seismic force and the control force are piecewise-linear and piecewise-constant interpolation functions, respectively, the external seismic force and the control force are expressed in the following forms:
u( ) = 0; kt6 ¡[(k + 1)t−tu] (2)
u( ) = u[kt]; [(k + 1)t−tu]6 ¡(k + 1)t (3)
w( ) = −kt
t {w[(k + 1)t]−w[kt]}+ w[kt]; kt6 ¡(k + 1)t (4) where k = 0; 1; 2; : : : ; N, is an integer number, t is the time length of the sampling period and tu is the time length of applying the control force. Therefore, Equation (1) can be written
in a discrete state form as follows:
where Ap= eAt; Bp= A−1(eAtu−I); Ep1= A−1(Ap−I); Ep2= A−1(Ep1=t−Ap), and z[k + 1] = x[k + 1] ˙x[k + 1] 2n×1
is a 2n-dimensional state vector of the structure response at time t = (k + 1)t. The system matrix A and location matrices B and E can be determined by the following:
A = 0 I −M−1K −M−1C 2n×2n (6) B = 0 M−1B 1 2n×p (7) E = 0 M−1E 1 2n×q (8) The rst two terms on the right-hand side of Equation (5) represent the structural response at time k(t) after the active pulse force u[k] has been applied. With applying a pulse control force u[k], the structural response z[k] at any time k(t) should be eliminated, as z∗[k] = Apz[k] + BpBu[k] = 0. Then the control force can be obtained by
u[k] = −(BTB)−1BTB−1
p Apz[k] = Gz[k] (9)
The matrix G is called a state feedback gain matrix and equals to G = −(BTB)−1BTB−1
p Ap (10)
Such a control approach is called an active pulse control strategy, in which, the control force is applied to minimize the system response that carries over from the current time step to the next time step [11; 22]. In the active pulse control algorithm, the pulse control force applied to the structure is reduced in a tu(tu ¡ t) period. Furthermore, the e ect of
pulses is assumed to be postponed to just before the next sampling time. Consequently, there is time to calculate the control force, to prepare it during the former t−tu period and to
apply it to the structure during the later tu period of each time step t.
UNSUPERVISED NEURAL NETWORK CLASSIFICATION (UNC) MODEL This section brie y reviews the unsupervised neural network classi cation (UNC) model [13]. Assume there is a set with N training instances, X ={X1; X2; : : : ; XN}. Instance Xj is de ned
as a pair including input Xj; i and its corresponding output Xj; o. If there are M decision
vari-ables in the input and K data items in the output, then, the input Xj; i and output Xj; o of
instance Xj are represented as vectors of the decision variables and data, and are denoted as
Xj; i= [x1j; x2j; : : : ; xjM] and Xj; o= [o1j; o2j; : : : ; oKj]. The clustering process aims to classify the set
of training instances into a certain number of clusters based on predetermined features from the training data. The elements in each cluster are as similar as possible to each other and as dissimilar as possible to those in other clusters. Unsupervised classi cation is conducted using a topology and weight-change neural network. The number of input nodes is made equal to
the number of input decision variables (M) in each training instance. The number of output nodes is set to equal the number of clusters and it will be determined once the classi cation process is completed.
A neural network with M input nodes and one output node is rst generated and the input vector of the rst training instance is presented. Then, the input vector of the succeeding training instance is inputted through the network. The classi cation of a training instance into an existing cluster or a new cluster is based on the notion of maximum likelihood. The unsupervised classi cation process can be summarized as follows:
1. Calculate the degree of di erence, di (Xj; Ck), between the input vector of a training
instance Xj and each cluster Ck. The function di (Xj; Ck) is de ned as the square Euclidean
distance represented as di (Xj; Ck) = M P m=1 m(x m j −cmk)2; (11)
where m denotes prede ned weight, and is used to represent the degree of importance for
the mth decision variable in the input. The weights m are generally set as constant based
on the heuristic associated learning problem.
2. Find the smallest degree of di erence, di min= min{di (Xj; Ck); k = 1; 2; : : : ; Q}, for the
training instance and assign the cluster with the value of di min to be active. Term Q
denotes the number of classi ed clusters presently.
3. Compare the value of di min with the prede ned threshold value Ä. If di min is smaller
than Ä, the training instance belongs to the cluster with the value of di min and the weights
associated with the links are updated using a successive estimation approach [13]. If di min
greater than Ä, the training instance is classi ed as a new cluster and the topology of the neural network is altered.
Hereinafter, the set of N training instances is classi ed into a certain number homogeneous clusters, say P. These P distinct clusters represent a discrete feature set for these N training instances. The workability of these P clusters must be veri ed before they can be used as a sample base of the set. The veri cation process relies on the concept that the input of any instance Xi in the set X can be regenerated by combining these P clusters with an acceptable
error, provided these P clusters represent the important features of the set. The veri cation process is summarized in the following steps:
1. For each training instance Xj, calculate the degree of di erence, di (Xj; Ck), between the
input vector of the training instance and each cluster Ck where j = 1–N and k = 1–P.
2. Collect similar clusters for instance Xj such that the di (Xj; Ck)6Ä. Meanwhile, set the
term sk as 1−di (Xj; Ck)=Ä between 0 and 1.
3. Regenerate the input of instance Xj by combining its similar clusters as Xj; i0 = (1=PSs=1ss)
(s1C1+ s2C2+· · ·+ ssCs) and evaluate the error, j, between the desired and computed
input vectors Xj; i and Xj; i0 ; j= PMm=1(xmj −x
0m
j )2.
4. Calculate the system error E =PN
j=1j, an accumulation of the error for each instance. If
the system error is below a prede ned level, these P clusters are adequate. Otherwise, the threshold value Ä is modi ed and the classi cation process of the UNC model is initiated to once again re-classify these N training instances.
After the veri cation process is completed, the corresponding outputs for each cluster can then be obtained. These P distinct clusters with their corresponding output are then collected as a sample base for the set.
UNSUPERVISED FUZZY NEURAL NETWORK (UFN) REASONING MODEL Assume that U is an associated sample base with P samples U1; U2; : : : ; UP classi ed via
the above UNC model for the aforementioned instance base X and Y is a new instance in the same problem domain. The input Uj; i and output Uj; o of sample Uj are denoted as
Uj; i= [u1j; u2j; : : : ; uMj ] and Uj; o= [o1j; o2j; : : : ; oKj]. Similarly, the new instance Y can also be
de ned as a pair, including input Yi and unsolved output Yo, respectively. The input Yi is
a set of decision variables, Yi= [y1; y2; : : : ; yM]. The output Yo is currently a null vector
and can be solved using the unsupervised fuzzy neural network (UFN) reasoning model. [14; 15].
The UFN reasoning model is brie y reviewed as follows. Similar to other neural network learning models, the UFN model must rst be trained based on a given sample base. The learning stage in the UFN model just involves two stages: determining weights for each decision variable in the input vector and selecting appropriate working parameters for the fuzzy membership function. For the rst stage, the weights m are set based on the heuristic
associated learning problem. The second stage is more complicated and is implemented as follows. The rst step is to determine the degree of di erence between any two distinct samples in base U. Therefore, a total of T = CP
2 = P(P−1)=2 degree of di erence has to be
calculated. The prede ned square Euclidean distance function in Equation (11) is employed to measure the di erence, dij, between two inputs Ui; i and Uj; i for samples Ui and Uj as
dij= di (Ui; i; Uj; i) =PMm=1 m(umi −umj)2. After the values of dij for all samples in base U
are computed, the average of the sum of dij, denoted as dij= avg(dij) =PTt=1(dij)=T, can be
derived. The second step involves determining the fuzzy membership function. In the UFN reasoning model, a ‘similarity’ between any two samples is represented using a quasi-Z type membership function de ned as
ij= f(dij; Rmax; Rmin) = 0 if dij¿Rmax
RmaxRmin−Rmindij
(Rmax−Rmin)dij if Rmin¡dij¡Rmax
1 if dij6Rmin
(12) The working variables Rmax and Rmin are employed to de ne the upper and lower bounds of
the ‘degree of di erence’. The lower bound Rmin is set to a constant 10−5. The upper bound,
however, was set as Rmax= Á dij. The term Á is a real number between 0 and 1 and was
established through trial and error.
After the learning in the UFN model is completed, the new instance Y can be solved via the UFN reasoning model. The process of the UFN reasoning can be summarized in three steps. The rst step involves search for samples that resemble the new instance Y in the sample base U, and performed through a single-layered laterally connected network with an unsupervised competing algorithm. That is, to collect the similar samples with the degree
of di erence, dYj= di (Yi; Uj;i) =PMm=1 m(ym−umj)2, less than Rmax. The second step entails
representing the fuzzy relationships among the new instance and its similar samples using Equation (12).
The nal step is to generate the output Yo vector of instance Y by synthesizing the outputs
of similar samples Sk; o according to their associated fuzzy membership values using the center
of gravity (COG) method [14; 15]. Assume that R similar samples are collected. Then, the output Yo of instance Y yielded via the COG method is de ned as follows:
Yo= PR k=1kSk; o PR k=1k (13) where k represents the membership value for the kth similar sample.
The reasoning process of the UFN depends on determining the degree of similarity between Y and Uj. Consequently, the UFN reasoning model can generate no solution if the new instance
entirely di ers from all samples in the sample base. Herein, the output of the new instance is simply set as the output of the sample with the smallest degree of di erence.
UFN STRUCTURAL ACTIVE PULSE (UFN-SAP) CONTROLLER
For an SDOF structure with an active control device installed and subjected to an external ex-citation, the responses (dynamic behaviours) of the structure in each time step are monitored and the required control force for the structure can then be determined through the equa-tion of moequa-tion. Herein, the structural response with its corresponding control force in time step k(t) is denoted as an instance Xk. For the entire time history, say H, of excitation,
there are total N (= H=t) training instances can be obtained and collected as a training set, X ={X1; X2; : : : ; XN}.
For the entire time history of external excitation, these N instances may be identical or similar to each other. If some particular instances can be systematically collected from these N instances, the dynamic behaviours of the structure under the given external excitation can be represented by a series of these particular instances. The UNC model is a tool permitting these instances to be systematically classi ed into a certain number of homogeneous clusters, say P, according to their inputs, relative displacement and velocity data in this study. The feasibility of these P clusters is then veri ed based on the above-mentioned process. Once the veri cation process is complete, the corresponding control force for each cluster can be obtained. These P distinct clusters with their corresponding output then gathered as a sample base for the SDOF structure.
As the structure is subjected to a new excitation, the structural dynamic response of the structure is monitored at each time step. The required control force can then be obtained via the following steps. First, the monitored structural response is compared with the input of each sample in the sample base. Second, the samples with their degree of di erences below a prede ned threshold are collected as similar samples. Finally, the corresponding control force in the time step is then calculated via a fuzzy synthesis of the outputs of these similar samples. The aforementioned steps can be performed through the UFN reasoning model.
This work proposes a novel fuzzy-ANN active pulse control model, Unsupervised fuzzy neu-ral network structuneu-ral active pulse (UFN-SAP) controller, to control civil engineering structures under dynamic loading, and Figure 1 schematically illustrate the model. The proposed model
Figure 1. Model of the UFN-SAP controller.
is an integration of a UNC model [13], a UFN reasoning model [14; 15], and an active pulse control strategy [22]. The controller design process is implemented in the following stages. Figure 2 presents a owchart of these steps.
1. Run a case based on the aforementioned active pulse control algorithm to generate a set of training instances.
2. Classify the set of training instances into a certain number of homogeneous clusters using the UNC model.
3. Verify the feasibility of the classi ed clusters based on the previously mentioned veri cation steps.
4. Compute the control force for each cluster and gather it as a sample base for the structure. 5. Train the UFN reasoning model using the sample base.
These stages are o -line design of the UFN-SAP controller. However, applying control force in a structure system is an on-line process. Given a base motion, the response of the structure in each time step is rst measured, then the associated control force is determined via the UFN-SAP controller, and nally the control force is applied to the structure to destroy the response of the current time step before the next time step.
Figure 2. UFN-SAP controller design process.
THE NUMERICAL EXAMPLE Example 1: SDOF structure system
This section rst chooses an SDOF structure system, one-storey structure with an active tendon controller, to explore the control e ectiveness of the UFN-SAP controller. This structure is an approximately 1 : 4 scale model of a 1 : 2 scale model of the prototype structure. Table I lists the properties of the SDOF system. The EL-Centro earthquake recorded data (PGA = 0:348g), referred to as Base Motion One, are enlarged to 150 per cent of the original intensity to be used as the greatest external excitation imaginable (0.522g) for the 1 : 4 scale model. The sampling period t is 0.01 s. Figure 3 depicts the time history and the corresponding acceleration spectra of base motion one.
Controller design: The response of the SDOF structure is rst calculated for base motion one using Equation (5). The pulse duration tu is assumed to be half of the sampling period t,
Table I. System parameter values for the SDOF example.
Parameter Quantity Mass, m 2923.38 kg Structure sti ness, k 1391:06 kN m−1 Natural frequency, f0 3.47 Hz Damping factor, 5% Damping coecient, c 6373:74 N−s m−1
Figure 3. (a) Time history and (b) the acceleration spectrum of base motion one.
2000 instances are obtained for base motion one and the rst 1500 instances are taken as training instances for the UNC model. Setting the threshold value, Ä, as 0.0012 via trial and error, 127 clusters are classi ed after the clustering process in the UNC model is completed. The feasibility of these 127 clusters is then veri ed. The inputs of these 1500 training instances are regenerated using these 127 classi ed clusters. The average error for each instance equals 0.008 per cent and it is tolerable. Hence, these 127 clusters are feasible for the SDOF structure. Subsequently, the corresponding control force for each cluster is calculated. Figure 4 depicts the distribution of these clusters and the relationship among the classi ed clusters and the 1500 training instances. This gure reveals that each training instance is circled to one or more clusters where the radius of each circle (cluster) exactly equals the square root of threshold value Ä. Figure 5 also shows the range and distribution of the control forces. According to this gure, these control forces are linearly distributed in the maximum and minimum range.
Figure 4. Distribution of the classi ed clusters and training instances.
Figure 5. Distribution of the control forces for the clusters.
The UFN reasoning model is trained based on the sample base. A total of 900 instances are used as training instances, including 500 untrained and 400 trained instances randomly selected from the 1500 training instances. The value dij is calculated based on these 900
instances. Setting the term Á as 0.8, the value of Rmax for the fuzzy membership function in
the UFN reasoning model is 0.003. Based on these working parameters, the corresponding control force for each training instance is generated via the UFN reasoning model. For the
Table II. Results for some veri cation instances: Similar samples associated with the veri cation instances.
No. of No. of similar Input decision variables Desired Computa- Absolute Relative veri cation samples output tional error error (%) instance Displace- Velocity output
ment 12 3,89,90,92 0.47644 0.56195 0.43529 0.43592 0.00063 0.15 50 1,5,6 0.51481 0.46148 0.54028 0.54041 0.00014 0.03 61 1,2,3,89,92,97 0.45976 0.52995 0.46198 0.45619 0.00579 1.25 100 1,4,5 0.54624 0.49857 0.51217 0.51236 0.00020 0.04 105 1,5,6 0.51463 0.44449 0.55647 0.56244 0.00597 1.07 150 1,3,4,92,98 0.49626 0.54101 0.45993 0.45377 0.00616 1.34 200 1,2,3,91,92 0.47652 0.50555 0.48921 0.49180 0.00259 0.53 300 1,2,3,5,91 0.49330 0.49309 0.50504 0.50345 0.00159 0.31 400 1,2,3,4,5,92 0.49475 0.50859 0.49057 0.49707 0.00651 1.33 450 1,3,4,5 0.52043 0.49968 0.50508 0.50233 0.00275 0.55 509 25,75,76 0.63945 0.30467 0.71926 0.71912 0.00014 0.02 525 68,69 0.40942 0.76716 0.22350 0.23157 0.00807 3.61 539 25,26,76 0.64301 0.22831 0.79308 0.80088 0.00780 0.98 565 74,75 0.67932 0.38123 0.65539 0.64859 0.00680 1.04 570 57,58 0.44587 0.22480 0.75040 0.74223 0.00817 1.09 589 73,74 0.70365 0.42766 0.61670 0.60582 0.01088 1.76 650 5,6 0.54475 0.43925 0.56852 0.56050 0.00801 1.41 750 1,2,5,91 0.48585 0.48320 0.51275 0.50563 0.00712 1.39 789 10,11 0.36103 0.61772 0.35503 0.35447 0.00056 0.16 794 13,14,15,81,100 0.54810 0.72367 0.29745 0.29917 0.00172 0.58 900 1,2,5,6,91 0.49931 0.46027 0.53782 0.53022 0.00760 1.41
900 training instances, the average absolute error equals 0.0047. The error is tolerable and hence the learning process in the UFN reasoning model is terminated. Table II lists similar samples associated with randomly selected instances from the 900 training instances. Two to six similar samples are found for each instance. For example, instances 50 and 105 share the same similar samples: 1, 5 and 6. However, the computed outputs di er, and equal 0.54041 and 0.56244. This is attributed to the fact that the associated fuzzy membership values for these three similar samples di er from each other in instances 50 and 105. Remarkably, the average computing time in the UFN reasoning process for each instance is less than 0.001 s. This nding implies that the required control force in each time step can be calculated in time (t−tu= 0:005 s) and applied to the structure.
Structure control with the UFN-SAP controller: Herein, 2000 training instances of the base motion one are used to explore the control e ectiveness of the UFN-SAP controller. Figure 6 displays the controlled and uncontrolled relative displacements under base motion one. This gure reveals that the applied control force destroys the gradual rhythmic buildup of the structural responses. Figure 6 also indicates that the controlled structural responses using the UFN-SAP controller and those obtained via the numerical formulas in Equations (5) and (9) are almost identical. Signi cantly, the results reveal not only the feasibility of the sample base in representing the dynamic behaviour of the structure but also the high accuracy of the process of fuzzy synthesizing in the UFN reasoning model.
Figure 6. Controlled and uncontrolled relative displacements of the SDOF structure input external excitation: base motion one.
Uncertainty in the input external excitations: Two earthquake recorded data, the Kobe (PGA = 0:834g) and Northridge (PGA = 1:779g) earthquakes, referred to as base motion two and base motion three, serve as the uncertainty input data, and are scaled to 25 per cent of their original intensity. Figures 7 and 8 display the time history and the corresponding response spectra of base motion two and three, respectively. Herein, base motion two and three are used as external excitation to the SDOF structure.
For the SDOF structure system, base motion two and three are employed as unknown input data. The corresponding control forces are yielded by the UFN-SAP controller based on the 127 samples derived from base motion one. Figure 9 portrays the distribution among the 127 clusters and the instances for base motion two and three. The gure clearly reveals that some instances do not belong to any cluster. In this work, the control forces for these instances approximately equal that of the most similar sample. The results reveal that the responses of the structure are signi cantly reduced even through the controller design is based on base motion one and there are many di erences between the peak, shape, and amplitude of the spectra of base motion one and base motions two and three.
Example 2: 3DOF structure system
A 3DOF structure system is also used to investigate the control e ectiveness of the UFN-SAP controller. Notably, the parameters for each oor are identical and assumed to be the same as those of the SDOF structure used in Example 1. Figure 10 depicts the structure with three active tendon controllers installed, respectively, on each oor. This example also uses base motion one to generate the instance set that is then used to train and verify the perfor-mance of the UFN-SAP controller. The pulse duration tu is also set as half of the sampling
Figure 7. (a) Time history and (b) the acceleration spectrum of base motion two.
Figure 9. Distribution of the classi ed clusters and the instances for base motions two and three.
Figure 10. Example of the 3DOF structural system.
Controller design and structure control with the UFN-SAP controller: This example uses the same procedures for designing UFN-SAP controller described in Example 1. However, the response of the structure is the relative displacements and velocities of each oor. That is, a training instance is a pair of inputs (structural responses) and outputs (required control forces), and it is denoted as Xk=b(x1[k]; ˙x1[k]; x2[k]; ˙x2[k]; x3[k]; ˙x3[k]); (u1[k]; u2[k]; u3[k])c,
Figure 11. Controlled and uncontrolled relative displacements of the 3DOF structure input external excitation: base motion one.
where the subscripts, 1–3, denote the number of oors. The pair indicates that, at any time step, the required control force in each oor is determined based on the response of each oor at the current time step. Likewise, a total of 2000 instances are obtained based on base motion one and 1500 instances are randomly selected as training instances. The working parameter, Ä, for the UNC model is set as 0.0025 through trial and error. The upper bound (Rmax) of the
quasi-Z type membership function in the UFN reasoning model equals 0.006 as the term Á is set to 0.8.
Based on these parameters, 269 clusters are rst classi ed through the 1500 training in-stances using the UNC model. Next, the suitability of these 269 clusters is veri ed. The average absolute error equals 0.0099 per cent for each training instance and hence is tolera-ble. Meanwhile, the UFN reasoning model is also trained with the aforementioned parameters. With six input decision variables in each instance, the computing time in the UFN reasoning model for 1500 training instances is about 1 s. That is, the control force can be calculated within the period of (t−tu) and then applied to the structure in on-line control. Finally,
Figure 12. Controlled and uncontrolled relative displacements of the 3DOF structure input external excitation: base motion two.
Figure 11 displays the controlled and uncontrolled displacements for each oor, respectively, of the 3DOF structure system. The results reveal that the structural responses of the structure are signi cantly reduced.
Uncertainty in the input external excitations: Base motions two and three are also used as the uncertainty input data in this example. The UFN-SAP controller, designed based on the recorded data of base motion one, is used to control the 3DOF structure system under two unknown external excitations. Figures 12 and 13 depict the time history of relative displace-ments, uncontrolled and controlled, for each oor of the 3DOF structure under base motions two and three, respectively. The gures reveal that the response of the structure is signi cantly reduced even though the UFN-SAP controller is trained based on the data of base motion one. Table III summarizes the peak values of displacements, uncontrolled and controlled, of the 3DOF structure under three di erent base motions. The results surely con rm the ability of the UFN-SAP controller to handle uncertain information in the input of external excitations.
Figure 13. Controlled and uncontrolled relative displacements of the 3DOF structure input external excitation: base motion three.
Table III. Peak values of uncontrolled and controlled relative displacements for di erent input base motions.
Base motion Peak values of uncontrolled Peak values of controlled relative disp. (cm) relative disp. (cm)
1st oor 2nd oor 3rd oor 1st oor 2nd oor 3rd oor
One 2.311 4.109 5.282 0.694 1.025 1.163
Two 3.538 6.455 8.137 0.720 1.103 1.274
Three 1.022 1.825 2.238 0.400 0.635 0.757
CONCLUSIONS
This work presented a novel active structure control model, unsupervised fuzzy neural network structure active pulse controller, termed the UFN-SAP Controller, to control structures under
external excitations. The UFN-SAP controller integrates an unsupervised neural network clas-si cation (UNC) model [13], an unsupervised fuzzy neural network (UFN) reasoning model [14; 15], and an active pulse control strategy [22]. The pulse control force in the UFN-SAP controller aims to eliminate the cumulative structural responses. For a structure under a given base motion, rst, the training instances are obtained using the active pulse control algorithm. Then, these instances are systematically classi ed into certain number of clusters according to the response of the structure using the UNC model. These clusters with their corresponding control forces are gathered as a sample base. With any unknown base motion, the correspond-ing control force at each time step for the structure is then derived accordcorrespond-ing the response in the current time step and samples in the sample base using the UFN reasoning model.
Two structure systems, an SDOF and a 3DOF, under three di erent base motions are used to explore the control e ectiveness and capability of handling uncertain input information for the UFN-SAP controller. Examination of the SDOF system reveals that the active control algorithm destroys the gradual rhythmic buildup of the structural response and signi cantly reduces the structural responses at the peak of the relative displacements. Additionally, the response of the structure are signi cantly reduced for base motions two and three even the UFN-SAP controller is designed based on the recorder data of base motion one. Remarkably, there are many di erences in the peak, shape, and amplitude of the spectra of base motion one and base motions two and three. Meanwhile, a 3DOF structure system with active control device installed in each oor, respectively, demonstrates the control e ectiveness and capability of handling unknown input data of the UFN-SAP controller. The results also reveal that the structural responses for each oor are signi cantly reduced throughout the time history. The UFN-SAP controller is thus clearly useful and extendable to a system with much greater freedoms and non-linear systems.
The control performance of the UFN-SAP controller is heavily dependent on the selection of the working parameter, Ä. The smaller Ä is, the more clusters can be classi ed in the instance base, but also the greater the computing time. On the other hand, the larger Ä is, the fewer clusters can be derived in the instance base, and these clusters may be insucient to represent the dynamic behaviour of the structure. This work adopts a heuristic approach to assist the user to obtain an appropriate parameter Ä with a minimum of trail-and-error iterations. That is, the distribution of the corresponding control forces for classi ed clusters will be well-distributed within the range of maximum and minimum values. Meanwhile, the results for the veri cation process in the UNC model can provide further information for selecting an appropriate parameter Ä. A more systematic approach for selecting working parameters is currently being investigated to enhance the e ectiveness of the UFN-SAP controller.
ACKNOWLEDGEMENTS
The authors would like to thank the National Science Council of the Republic of China for nancially supporting this research under Contract No. NSC 88-2211-E-009-001.
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