Improved Identification of Nonlinear Dynamic Systems using Artificial Immune System

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1 Imroved Identification of Nonlinear Dnamic Sstems using Artificial Immune Sstem Satasai Jagannath Nanda, Ganaati Panda, Senior Member IEEE and Babita Majhi Deartment of Electronics and Communication Engineering, NIT, Rourkela nandasatasai@gmailcom, ganaatianda@gmailcom, babitamajhi@gmailcom Abstract-Over the recent few ears the area of Artificial Immune Sstem (AIS) has drawn attention of man researchers due to its broad alicabilit to different fields In this aer the AIS technique has been suitabl alied to develo a new model for efficient identification of nonlinear dnamic sstem Simulation stud of few benchmark identification roblems is carried out to show suerior erformance of the roosed model over the standard multilaer ercetron (MLP) aroach in terms of resonse matching, number of training samles used and convergence seed achieved Thus it is concluded that the AIS based model used is a referred candidate for identification of nonlinear dnamic sstem I INTRODUCTION The biological immune sstem (BIS) is a multilaer rotection sstem where each laer rovides different tes of defense mechanisms for detection, recognition and resonses It also resists infectious diseases and reacts to foreign substances Following the BIS rincile a new branch of comutational intelligence known as artificial immune sstem (AIS) [] has evolved which finds alications in otimization[5]-[6], comuter securit, data clustering, attern recognition, fault tolerance etc Like other evolutionar comuting algorithms it also hels to develo efficient comutational models The four forms of AIS algorithm reorted in the literature are immune network model, negative selection, clonal selection and danger theor In [] Charsto and Zuben has roosed the clonal selection rincile for otimization Identification of nonlinear dnamic lants finds extensive alication in control, communication, instrumentation, ower sstem engineering and man other fields For identifing such comlex lants, the recent trend of of research is to emlo nonlinear structures and to train their arameters b evolutionar comuting tools Man research work on this toic have been reorted but the first imortant one is using multilaer ercetron (MLP) b Narendra and Parthasarath [3] Later on in 999 [4] we had roosed a new aroach to identif such sstems roviding identical or even better erformance but emloing a low comlexit FLANN structure However, the major disadvantage of these methods is that the emlo derivative based learning rule to udate their weights which at times leads to local minima and thus incorrect estimate of the arameters To alleviate this roblem in this aer a new AIS based derivative free method using low comlexit FLANN structure is roosed for efficient identification of nonlinear dnamic sstems The aer is organized as follow In Section II we begin with a review of nonlinear dnamic sstem and their sstem identification The roosed model using FLANN structure learning b AIS algorithm is discussed in section III The basic rincile of cloning is dealt in section IV The roosed algorithm is resented in section V The simulation stud of few benchmark dnamic sstems using roosed method is resented in section VI The comarison of identification erformance obtained from simulation of MLP and the new method is resented in section VII Finall section VIII resents the concluding remarks of the aer II PRINCIPLE OF ADAPTIVE SYSTEM IDENTIFICATION u (k) Dnamic Sstem FLANN based model n(k) Adative AIS (k) Figure Block diagram of dnamic sstem identification The basic block diagram of adative sstem identification is shown in Fig in which the lant is considered as nonlinear, dnamic in nature and is associated with noise n(k) The roosed model is a nonlinear single laer functional link neural network (FLANN) with no hidden laer with its connecting weights trained b AIS based algorithm The smbols u(k), (k), (k) and e(k) reresent the inut, outut of lant, the estimated outut of the model and the error signal resectivel The objective of the identification task is to minimize the error e(k) recursivel such that (k) aroaches (k) when the same inut u(k) is alied to the lant and the model In this aer the comlex SISO lant is assumed to have one of the following three forms Model : ( k g [ u ( k n ) = i = ), u ( k α i ),, ( k u ( k Σ (k) i ) m Σ e (k) )] () /8/$5 8 IEEE Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Downloaded on Ma 8, 9 at 7:49 from IEEE Xlore Restrictions al

2 Model : ( k ) = m i = β i ƒ[ u ( k i) ( k ), Model 3 : ( k ) = ƒ[ ( k ), ( k ),, u ( k ), u ( k ), u ( k m )] ( k ),, )] where [u (k), (k)] reresents the inut, outut air of SISO lant at time instant k and m n The terms ƒ[] and g[] denote nonlinear functions and α i and β i reresent constant values III THE FLANN STRUCTURE OF THE Pao originall roosed FLANN as a novel single laer ANN structure caable of forming arbitraril comlex decision regions b generating nonlinear decision boundaries [9] In this method the initial reresentation of a attern is enhanced b using nonlinear function and thus the attern dimension sace is increased The functional link acts on an element of a attern or entire attern itself b generating a set of linearl indeendent function and then evaluates these functions with the attern as the argument Hence searation of the atterns becomes ossible in the enhanced sace The use of FLANN not onl increases the learning rate but also has less comutational comlexit [9] and [] A FLANN structure with one inut is shown in Fig The inut signal x(k) is functionall exanded to a number of nonlinear values to feed to an adative linear combiner whose weights are altered according to an iterative learning rule Usuall the least mean square (LMS) algorithm is used for training the weights The tes of exansion suggested in the literature are either ower series or trigonometric exansion But the later exansion has been recommended for most alications as it offers better erformance Therefore in the roosed model we select trigonometric based linear exansion matrix given b Where i =,,,M / and X(k) reresents the inut at the k th instant Total exanded values including an unit bias inut become Q=MLet the corresonding weight vector is reresented as w(k) having Q elements The estimated outut of the model is then given as The weights are udated b clonal selection algorithm ( k n () ( k n ); (3) for i = x ( k ) for i = (4) s i ( k ) = sin( iπ x (k )) for i =,4 M cos( iπx(k) ) for i = 3,5 M Q ( i i i= k ) = s (k )w (k ) (5) x(k) Exansion s (k) s (k) s m (k) w (k) w (k) w (k) w m (k) Figure Structure of FLANN model IV PRINCIPLE OF CLONAL SELECTION ALGORITHM e(k) Immunit refers to a condition in which an organism can resist disease The cells and molecules resonsible for immunit constitute biological immune sstem (BIS)AIS is develoed b following the rinciles of BIS Bersini first used immune algorithms to solve roblems The books [], [] rovide the details about the various rinciles and algorithms of AIS The clonal selection rincile of AIS describes how the immune cells react to athogens (foreign cells also known as antigens) and is simle but efficient evolutionar comuting tool for achieving otimum solution LNde Castro and F J Von Zuben have dealt the clonal selection in [] When a athogen invades the organism; a number of immune cells that recognize these athogens survives Among these cells some become effecter cells, while others are maintained as memor cells The effecter cells secrete antibodies and memor cells having longer san of life so as to act faster or more effectivel in future when the organism is exosed to same or similar athogen During the cellular reroduction, the somatic cells reroduce in an asexual form, ie there is no crossover of genetic material during cell mitosis The new cells are coies of their arents as shown in Fig3During this rocess the under go an mutation mechanism which is known as somatic hermutation as described in [] and [6] The affinit of ever cell with each other is a measure of similarit between them It is calculated b the distance between the two cells The antibodies resent in a memor resonse have on average a higher affinit than those of earl rimar resonse This henomenon is referred to as maturation of immune resonse During the mutation rocess the fitness as well as the affinit of the antibodies gets changed So in each iteration after cloning and mutation those antibodies which have higher fitness and higher affinit are allowed to enter the ool of memor cell Those cells with low affinit or self-reactive recetors must be efficientl eliminated The clonal selection algorithm has several interesting features such as oulation size is dnamicall adjustable, exloration of the search sace, location of multile otima, caabilit of maintaining local otima solutions and defined stoing criteria Adative (k) (k) ^ Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Downloaded on Ma 8, 9 at 7:49 from IEEE Xlore Restrictions al

3 The objective is to minimize the fitness function of (7) b clonal selection rincile 6 Selection: To select the weight vector (corresonding cells) for which MSE is minimum 7 Clone: The weight vector (corresonding cells) which ields best fitness value (minimum MSE) is dulicated 8 Mutation: Mutation oeration introduces variations into the immune cells Probabilit of mutation P m is a smaller value which indicates that the oeration occurs occasionall Total number of bits to mutate is the roduct of total number of cells, number of bits in each cell and robabilit of mutation of each cell Among the cloned cells the cell to be mutated is chosen randoml A random osition of the cell is chosen first and then its bit value is altered 9 Stoing Criteria: The weight vector which rovides the desired solution (minimum MSE) and corresonding cells are known as memor cells Until a redefined MSE is obtained stes 4-8 are reeated Figure3 The Clonal Selection Princile V WEIGHT UPDATE OF FLANN STRUCTURE USING CLONAL SELECTION ALGORITHM Determination of outut of lant: The inut is a random signal drawn from a uniform distribution in the interval [-, ]Let k be the numbers of inut samles taken The inut samle is then assed through the lant to roduce lant outut (k) exansion of inut: Same inut samles are assed through the model consisting FLANN structure Each inut samle under go trigonometric exansion as reresented in (4) 3 Initialization of a grou of cells: As it is an evolutionar algorithm we begin with a grou of solutions Here a grou of weight vector of FLANN is taken A weight vector consist of (M) no of elements Each element of weight vector is reresented b a cell which is basicall a binar string of definite length So a set of binar strings is initialized to reresent a weight vector and n number of such weight vectors is taken each of which reresent robable solution 4 Calculation of outut of model: Initiall weight vector is taken random value The outut of model is comuted using exanded values of u(k) and weight vector as described in (5) This is reeated for n times 5 Fitness Evaluation: The outut of the model (k,n) due to k th samle and n th vector, is comared with the lant outut to roduce error signal given b e(k, n) = (k, n) (k, n) (6) For each n th weight vector the mean square error (MSE) is determined and is used as fitness function given b K e ( k, n ) k = MSE ( n ) = (7) K VI SYSTEM SIMULATION In this stud the dnamic lants are reresented in form of difference equation described in ()The inut to the sstem is a uniforml distributed random signal over the interval [-, ] The testing is carried out b the sinusoidal inut given b π n sin for n 5 5 (8) un () = πn πn 8sin sin for n > To validate the erformance of the roosed method simulation stud using MATLAB is carried out using some standard dnamic lants Different examles simulated are Examle : This dnamic sstem is assumed to be of second order and is given b the difference equation (9) ( k ) = 3 (k ) 6 (k ) g[u (k )] where function g is given b For identification of the lant the model used is of the form where N[u(k)] reresents either the roosed model or the MLP structure {} For MLP both convergence factor and momentum term is taken as The number of iteration and inut samle for training used are 5 The FLANN-AIS based structure in Fig4 is used In FLANN structure each inut value is exanded to 5 trigonometric terms Initial oulation of cell is taken as The weights are trained for iterations 3 (u) = 5sin ( πu) cos(4πu) 5 () u g 3 ( k ) = 3 (k) 6 (k ) N[u(k )] () Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Downloaded on Ma 8, 9 at 7:49 from IEEE Xlore Restrictions al

4 u(k) Exansion g ( ) S Z Z (k) ^ (k) Z - e(k) equation which reresents the sstem is given b ( k ) = ƒ[ (k), (k-)] u(k) () The function ƒ is given b ƒ ( ( 5)(, ) = ) The model used for identification task is reresented b ( k ) = N[ (k), (k )] u(k) (3) (4) Here N reresents the structure of either MLP {} or the new model structure of Fig5 For MLP the convergence factor is taken as 5 and momentum term is The no of iteration and inut samle for training is taken 5 In FLANN structure the inuts are exanded to 8 terms (each inut 4 terms) Weights are trained for iterations Initial oulation of cells is taken as 6 Clonal Figure4 Proosed FLANN-AIS based structure for identification of dnamic sstem of Examle Examle : The dnamic sstem to be identified is described b the difference equation of Model The second order difference u(k) Z - f ( ) Exansion Exansion Z - (k) ^ (k) Z - e(k) Examle 3: The examle of the lant taken here is described b the difference equation of Model 3 is reresented b (k ) = ƒ[(k),(k -), (k - ), u(k),u(k -)] (5) where the functions ƒ is given b ƒ ( a a a a ( a a, a, a, a, a ) = a a 3 The model for identification of lant is of the form ( k ) = N[ (k), (k -), - ), For MLP structure of N [] is taken as {5}The convergence factor and momentum term each is taken as No of iteration and inut samle for training is taken 5 The FLANN-AIS based identification structure used is shown in Fig6 In FLANN for N [] the inuts are exanded to terms (each inut 4 terms) The number of inut samle taken is 6 Weights are trained for 5 iterations Initial oulation of cells is taken as VII RESULTS ) The erformance of MLP structure trained b BP algorithm and FLANN structure trained b clonal algorithm is comared in terms of estimated outut and the error lot as shown in Figures 7-9 Table also reveals that the roosed models offer smaller CPU time, lesser inut samles and minimum sum squared errors comared to its MLP counter art (k a u(k), u(k -)] 4 (6) (7) Clonal Figure5 Proosed FLANN-AIS based structure for identification of dnamic sstem of Examle Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Downloaded on Ma 8, 9 at 7:49 from IEEE Xlore Restrictions al

5 6 4 Z - u(k) Z - f ( ) Z - (k) outut and error - Z lant model error no of iteration Exansion 6 Figure7(a) 4 Z - Exansion outut and errors - -4 Exansion no of iterations Figure7(b) Figure7 Identification of Examle : (a) using MLP (b) using roosed FLANN-AIS Exansion 5 Exansion outut and error Clonal Z - Z no of iteration Figure8(a) Figure6 Proosed FLANN-AIS based structure for identification of dnamic sstem of Examle3 Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Downloaded on Ma 8, 9 at 7:49 from IEEE Xlore Restrictions al

6 5 TABLE I COMPARATIVE RESULTS OF DYNAMIC SYSTEMS OBTAINED THROUGH SIMULATION STUDY outut and error no of iteration Sstem CPU Time During training (in Sec) MLP FLA NN - AIS Sum of Square Errors (SSE) No of Samle For testing MLP FLA NN - AIS No of inut samle used in training MLP FLAN N -AIS Ex Ex Ex Figure8(b) Figure8 Identification of Examle : (a) using MLP (b) using roosed FLANN-AIS Outut and Error Oututs and Error No of Iteration Figure9(a) No of iteration Figure9(b) Figure9 Identification of Examle 3 : (a) using MLP (b) using roosed FLANN-AIS VIII CONCLUSION The resent aer rooses a novel alication of the AIS to identification of nonlinear dnamic sstem The simulation stud of the roosed model is carried out using standard examles to demonstrate its erformance The comuted results show its suerior erformance comared to the MLP based methods in terms of CPU time required for training of structure, sum of square errors and number of inut samles required for training The close agreement of outut resonse of the lants and the model exhibit that the AIS is a otential learning tool for accurate identification of comlex dnamic nonlinear sstems REFERENCES [] D Dasguta, Advances in Artificial immune Sstems, IEEE Comutational Intelligence Magazine, vol, issue 4, 4 49, Nov 6 [] L N de Charsto and J V Zuben, Learning and Otimization using Clonal Selection Princile, IEEE Trans on Evolutionar Comutation,Secial issue on Artificial Immune Sstems, vol 6,issue 3, 39-5, [3] KSNarendra and K Parthasarath, Identification and Control of dnamical sstems using neural networks, IEEE Trans Neural Networks, vol, 4-7, Mar 99 [4] J C Patra,R N Pal, B N Chtterji,G Panda, Identification of nonlinear dnamic sstems using Link Artificial Neural Networks, IEEE Trans Sst,,Man, Cbern B, vol 9, issue, 54 6, Ar 999 [5] Zhuhong Zhang, Tu Xin, Immune with Adative Samleing in Nois Environments and Its Alication to Stochastic otimization Problems, IEEE Comutational Intelligence Magazine, 9-4, Nov7 [6] L N de Charsto and J Timmis, An Artificial Immune Network for Multimodal Function Otimization, IEEE Congress on Evolutionar Comutation (CEC ), vol, ,ma,hawaii, [7] B Widrow and S D Stearns, Adative Signal Processing, chater-6, 99-6, Second Edition, Pearson, [8] M THagan, HBDemuth, MBeale, Neural Network Design, chater- 4 Thomson Asia Pte Ltd,Singaore, [9] YH Pao, S M Phillis and D J Sobajic, Neural-net comuting and intelligent control sstems, Int J Conr, vol 56, no, 63-89, 99 [] YH Pao, GH Park and DJ Sobjic, Learning and Generalization Characteristics of the Random Vector function, Neuro Comutation, vol6, 63-8, 994 [] DDasguta, Artificial Immune Sstems and their Alications, Sringer-Verlag, 999 [] LN de Castro, J Timmis,An Introduction to Artificial Immune Sstems: A New Comutational Intelligence Paradigm,Sringer- Verlag, Authorized licensed use limited to: NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA Downloaded on Ma 8, 9 at 7:49 from IEEE Xlore Restrictions al