Design and Implementation of Predictive Controller with KHM Clustering Algorithm

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1 Journal of Computational Information Systems 9: ) Available at Design and Implementation of Predictive Controller with KHM Clustering Algorithm Jing ZENG 1,, Jun WANG 2 1 College of Information Engineering, Shenyang University of Chemical Technology, Shenyang , China 2 College of Computer Science and Technology, Shenyang University of Chemical Technology, Shenyang , China Abstract A KHM multi-model predictive control method is presented based on data-driven for a sort of unknownstructure nonlinear system based on vast history data only. A database search strategy based on K- Harmonic Means is given, hard cluster is softening. The similar cluster is selected, neighborhood and model parameters are determined on line according to the change of system operation point. This method can shorten searching time and improve precision of the data which used to get optimal local model. A multiple generalized predictive control strategy based on KHM is developed. The simulation test has stated the validity of this approach. Keywords: Clustering Algorithm; Fitting; Local Polynomial Regression; Predictive Control 1 Introduction Nonlinear system modeling and identification is very complicated, when there are a lot of observation data, determining model structure and related optimal problem will become more complex, it is difficult to get accurate global model for actual industrial process [1]. Many scholars modeling online using the present sampling data based on data driven [2-4], and it s generally believed that local polynomial fitting algorithm can achieve good control effect. Clustering analysis is one of the basic technologies of data mining, statistical analysis, machine learning and many other fields of application. In the study of clustering problem, k-means KM) clustering algorithm became one of the universal clustering algorithms because of its rapid convergence performance [5]. However, KM algorithm is very sensitive to the system initial value, different initial value can lead to different clustering results. K-Harmonic Means KHM) clustering algorithm can make up for the defect [6-9], this algorithm has the characteristic that be not sensitive to the initial value and advantages of good clustering performance. Generalized predictive control algorithm is a kind of advanced control algorithm which can overcome the system lag Corresponding author. address: zengjing0066@sohu.com Jing ZENG) / Copyright 2013 Binary Information Press DOI: /jcis6421 July 15, 2013

2 5620 J. Zeng et al. /Journal of Computational Information Systems 9: ) and deal with open loop unstable non-minimum phase system [10,11]. This paper presents a K- Harmonic Means KHM) local polynomial generalized predictive control algorithm, using KHM clustering algorithm to cluster analysis on historical database, finding out the recent class to the current working point, getting the specific working point neighborhood of input vector in this class, computing the corresponding weights and model parameters, and finally combining with generalized predictive control algorithm, designing the predictive controller. This paper gives the simulation case analysis, and the results show that the dynamic response of closed-loop system can be improved significantly. 2 Modeling Process Assume that the input and output dataset can expressed as {Y i, X i )} N i=1, it can characterize all kinds of possible basic condition of the nonlinear process, the relationship of input and output of the process can be expressed as Y i = fx i ) + ε i, i = 1,...N 1) f.) is the nonlinear mapping between input and output data, ε i is noise variable. By the Taylor formula, it is known that at the input vector x, the corresponding local function of the system f ) can be expressed by p order polynomial, fx, ξ) = ξ 0 + ξ 1 X i x) ξ r X i x) p 2) After get the similar data sample, local polynomial fitting is essentially a weighted optimal problem, its purpose is to minimize the mismatching degree between the model and data, ˆξ = arg min ξ i Ψ k x) ly i p ξ j X i x) j )w i x) 3) j=0 w i x) denote weight, Ψ k x) denote the specific working point neighborhood of input vector, it contains k sampling point. 3 K-harmonic Mean Database Searching Modeling process first need to determine the input-output history database, which should cover all kinds of working conditions, for industrial field data, removing noise filtering, smoothing and eliminating related redundant information should also be considered. Suppose the dataset which contain n group data can expressed as {x 1, x 2,..., x n }, clustering analysis the dataset, suppose k {2,..., n 1} is clustering number, V k is the kth class, k n is segmentation matrix, satisfy the following formula { uij = 1 if x j V i u ij = 0 if x j / V i

3 J. Zeng et al. /Journal of Computational Information Systems 9: ) and k i=1 u ij = 1, n j=1 u ij > 0 i = 1,..., k; j = 1,..., n). v i Ris defined as every kind of clustering centers, V = {v 1,..., v k } is clustering center set. The distance between data point x j and clustering center v i is defined as distance measure d ij, and define differ matrix by d d 1n D =..... d k1 d kn D can be calculated by X and V using appropriate norm, using the Euclidean norm to describe as follows d ij = x j v i )x j v i ) T Using the harmonic mean distance between data point and clustering center instead of the minimum distance between data point and clustering center k k i=1 1 d 2 ij min { d 2 ij i = 1,..., k } Objective function is expressed as J KHM V, X) = n j=1 k k i=1 1 d 2 ij 4) Make J / V =0 and then get the clustering center update methods as formula 5) shows. According to initial value continuously iteration, make formula 4) continuously reduce until stability. v i = n 1 ) 2 x j j=1 k d 4 1 ij d l=1 2 l,j n 1 j=1 k d 4 1 ij d l=1 2 l,j ) 2 5) Relative to the KM algorithm, KHM using the harmonic mean distance between data point and clustering center instead of the minimum distance between data point and clustering center. The conditional probability from clustering center to data point and dynamic weight of every iteration process are introduced. Algorithm procedure is as follows, Step 1 k samples from the historical input-output database {Y i, X i )} N i=1 are selected randomly as the initial clustering center v 1, v 2,..., v k. Step 2 According to formula 4), {Y i, X i )} N i=1 are divided into these K clusters according to the minimum harmonic mean principle between the data and clustering center. Step 3 The new clustering centers v 1 *, v 2 *,... v k * are computed by formula 5).

4 5622 J. Zeng et al. /Journal of Computational Information Systems 9: ) Step 4 If v l *= v l l = 1, 2,...,k ), jump into step 5, or turn to step 2. Step 5 Find out the recent class to the current working point, compute the specific working point neighborhood of input vector x in the specific class and compute weight w i x), compute model parameters according to formula 3). 4 Controller Design Consider the following NARX non-linear auto-regressive with exogenous inputs) system, yk) = fφk)) + ek), K = 1, M 6) f ) is an unknown nonlinear curve, e ) is an error random variables for example a random variable with mean zero and variance σk 2 ), φt) is regression vector, φt) = [yt 1)...yt n a )ut n k )...ut n b n k )] T n a, n b and n k are on behalf of the previous output, input and model hysteresis uantity respectively. Supposing formula 3) using uadratic form for l.), lε) = ε 2, and model structure is local linear, f φ k), ζ) = ζ 0 + ζ T 1 φ k) φ t)) 7) If ˆβ 0 and ˆβ 1 represent the parameter estimator, then the estimator of yt) can be expressed as: ŷt) = fφt), ˆζ) = ˆζ 0 8) By formula 7 can learn, a linear input-output form is available in the local neighborhood of φt), A 1 )yt) = B 1 )ut 1) + α 9) A 1 ) = 1 + a a n n B 1 ) = b 0 + b b nb n b A 1 ) and B 1 ) are polynomials about the recession operator 1, which can be obtained by ˆζ 1. α = ˆζ 0 ζ T 1 φt) 10) α is a deviation item. Multi-step prediction strategy is introduced at here, and in order to find out the optimal output predictive value after j step, Diophantine formula is introduced as follows 1 = E j 1 ) A 1) + j F j 1 ) 11) E j 1 ) = e j,0 + e j, e j,j 1 j+1 F j 1 ) = f j,0 + f j, f j,na n a By formula 9) 11) can learn y t + j) = ) Ḡj 1 u t + j + 1) + F ) j 1 y t) + E ) j 1 ξ t + j)

5 J. Zeng et al. /Journal of Computational Information Systems 9: ) Ḡ j 1 ) = E j 1 ) B 1) = B 1 ) 1 j F 1 )) A 1 ) = ḡ j,0 + ḡ j, The optimal predictive value at t + j can be expressed as follows ŷtj) = Ḡj 1 ) ut + j 1) + F j 1 )yt) 12) Another Diophantine formula is introduced which can decompose Ḡj 1 ) u t + j + 1) into two parts: known input item and unknown input item E j 1 ) B 1) = G j 1 ) + j H j 1 ) 13) G j 1 ) = g 0 + g g j 1 j+1 H j 1 ) = h j,0 + h j, h j,nb 1 n b+1 Then formula 12) can be expressed as vector form Ŷ = G U + f 14) Ŷ = [ŷ t + 1), ŷ t + 2), ŷt + 3)...ŷ t + N 2 )] T U = [ u t), u t + 1), ut + 2)... u t + N u )] T f t + 1) F 1 f t + 2) F 1 f = = H u t 1) + y t).. f t + N 2 ) F N2 G 1 1 ) g 1 [G 2 1 ) g 2 1 g 1 ] H = 2 [G 3 1 ) g 3 2 g 2 1 g 1 ]. [ ] N 2 1 G N2 1 ) g N2 N1+1 g 1 Then the output of the controlled object can be predicted by the known input, output information and the future input value. By formula 14), the performance index function can be expressed as follows { } J = E G U + f ω) T Q y G U + f ω) + U T Q u U 15) [ ω = ω t + 1) ω t + 2) ϖ t + N 2 ) ] T

6 5624 J. Zeng et al. /Journal of Computational Information Systems 9: ) Therefore, the future input which minimize the control performance index in the unconstrained conditions can be expressed U = G T Q y G + Q u I ) 1 G T Q y w f) 16) Then the real-time optimal control incremental is taken for the first element u t) = d T w f) 17) d T is the first line of the matrix G T Q y G + Q u I ) 1 G T Q y. d T = 1 0 0) G T Q y G + Q u I ) 1 G T Q y The first element of U in formula 16) is u t), then the real-time input u t) can be expressed by u t) = u t 1) + d T w f) 18) In the industrial controlled process, when there are input constraints u min ut + k) u max 19) u min ut + k) u max 20) and because the controlled variables can be written as vector form U = T U + Ū 21) T Ū ut 1) ut 1). ut 1) Then the constraints can be expressed as [ ] C U T U T U T U T c 22) C I I T T u max c u min u max Ū u min + Ū It is a uadratic programming problem as follows: min U T G T Q y G + Q u I) U 2ω f)q y G U U [ ] s.t. C U T U T U T U T c

7 J. Zeng et al. /Journal of Computational Information Systems 9: ) Simulation Study Consider the following strong nonlinear system y t)1 + y t) ) = ut) This system gain decreases along with the increase of the output signal amplitude. Because the open loop system is unstable, we generate 2500 input-output data sets in this paper by using closed loop proportional controller. The sampling time is selected as T s =0.1. The controller is designed by the data sets and the method in this paper, the simulation results are compared with the conventional generalized predictive control method. Suppose the system output is step signal, at t=1, the output is jumped form 0 to 1. In the simulation process, the predictive time domain is set as N=8, the controlled time domain is set as N u =6, the weight of output variable is Qy=1, the increment weight of the controlled signal is Q U = The output results using conventional generalized predictive control algorithm is shown in figure 1, the results by the method in this paper is shown in figure 2, the real line represent system output, dotted line represent reference in figure a), the controlled input signal is shown in figure b). Form the diagram we can see that compared with routine predictive control algorithm, the system output overshoot by the controller of our method is decrease obviously, and system response time is shortened, the input signal freuency jump is avoided. The superiority of the algorithm in this paper is seen clearly. a) Fig. 1: Step response curve of conventional predictive controller b) a) Fig. 2: Step response curve of our algorithm b)

8 5626 J. Zeng et al. /Journal of Computational Information Systems 9: ) Conclusion A K-harmonic mean clustering algorithm which is not sensitive to initial value selection is introduced in this paper, the harmonic mean distance between data point and clustering center instead of the minimum distance between data point and clustering center. The conditional probability from clustering center to data point and dynamic weight of every iteration process are also introduced. The recent class to current working point is selected by tracking real time condition changes, the specific working point neighborhood and model parameters are computed in the class, the appropriate control increment is calculated too. The simulation results show the effectiveness of the method. Acknowledgement This work is supported by the Scientific Research Project of Liaoning Province, China L , L ). References [1] N.N. Nandola and S. Bhartiya. Hybrid system identification using a structural approach and its model based control: An experimental validation. Nonlinear Analysis: Hybrid systems, 32): , [2] Yunan S, Maozai T. Adaptive local linear uantile regression. Acta Mathematicae Applicata Sinica, 273): , [3] Seuenz H, Schreiber A, Isermann R. Identification of nonlinear static processes with local polynomial regression and subset selection. IFAC Symposium on System Identification, , Saint- Malo, France, July [4] Hang Yue; Jones E G, Revesz P. Local polynomial regression models for average traffic speed estimation and forecasting in linear constraint databases. 17th International symposium on temporal representation and reasoning, [5] Kumar A, Sinha R and Bhattacherjee V. Modeling using K-means clustering algorithm st International Conference on Rencent Advances in Information Technology, , [6] B. Zhang, M. Hsu, and U. Dayal. K-harmonic means a data clustering algorithm. Technical Report HPL , HP Laboratories Palo Alto, [7] A. Inler and Z. Gungor. Applying k-harmonic means clustering to the part-machine classification problem. Expert Systems with Applications, 362): , [8] Jing Zeng, Hui Jin. Local Polynomial Regression with K-harmonic Means and Subset Selection. Journal of Computational Information Systems, 2012, 822): [9] K. Thangavel and N. Karthikeyani Visalakshi. Ensemble based distributed k-harmonic means clustering. International Journal of Recent Trends in Engineering, 21): , November [10] Wei Wang. Theory and application of generalized predictive control. Science press, [11] N.N. Nandola and D.E Rivera. A novel model predictive control formulation for hybrid systems with application to adaptive behavioral interventions. ACC, , Baltimore, Maryland, USA, June, 2010.