Gene library based Resource Allocation in Time Sensitive Large Scale Networked Control Systems. Preliminary Exam Presentation Unnati Ojha

Transcription

1 Gene lbrary based Resource Allocaton n Tme Senstve Large Scale Networked Control Systems Prelmnary Exam Presentaton Unnat Ojha ADAC Lab, NCSU

2 Outlne Objectves, Motvaton System Descrpton Gene Lbrary Bologcal Background Formulaton Adaptve Bandwdth Management Formulaton Analytcal Soluton usng Lagrange Multplers Results Case Study Gene Lbrary based Bandwdth Allocaton for Intellgent Transportaton System Concluson and Future Work

3 Objectve and Motvaton Objectve: To develop a real-tme resource allocaton algorthm based on a gene lbrary for large scale networked control system such as the ntellgent transportaton system To ntroduce Artfcal Immune System based update algorthms n order to tune the gene lbrary Motvaton In large scale network control systems wth hard real-tme constrants, such as the ntellgent transportaton system, bandwdth s a lmted resource that should be allocated effcently accordng to the need of the vehcles, thus t s necessary to develop a real tme bandwdth allocaton algorthm Study on extracton of relevant features that ndcate the rsk/abnormal behavor n large scale systems can reduce the dmensonalty of the problem ADAC Lab, NCSU 3

4 System Descrpton Local Agents Envronment Sensor Agents Supervsory Agent wth Bandwdth Constrant ADAC Lab, NCSU 4

5 System Descrpton d Local Agents x ( x u ) f ( ) x u f ( ) th,, Dynamcs for of agent,,, n,,, m d = f t + = th = Dsturbance on agent T T = x x x = T th State varables of agent = u u u = T T T x = x x xn = y U ( y y ) th Input varables for agent State varables from every agent th = hx ( x, u ) = Measurement vector from agent ψ Envronmental Sensor Agents l L th :, ref, u = Local controller for agent T th j =,,,..., j j j, L = State varables of envronmental agent E E E E j j T T T T E = E, E..., E N = State varables from all envronmental agents E q th j = he ( E j ) = Measurements from j envronmental agent ADAC Lab, NCSU 5

6 System Descrpton Bandwdth constrant expressed as a samplng rate assgnment problem max Supervsory Agent F ( t)( samples / sec) = ς B ( t)( bts / sec), ς = D( samples / packet) Q ( bts / packet) total supervsory agent wll use a mult-rate samplng technque to perform smart samplng n order to conserve bandwdth T T T T T T Yk = y, k, y, k,..., y N, k, q, k, q, k..., q N, Data from local agents E k = Z = f, = Selectve samplng of avalable data τ ( τ Y ) k + k k = τ, τ,.. τ = Samplng rates for each agent E k, k, k N + N, k T max T ADAC Lab, NCSU 6

7 System Descrpton Relevant measurement for each agent Z N th ( N Z ), k, k k, k = f, = Measurement vector assocated wth agent = th Neghborhood of agent Decsons from supervsory agent ( Z L) G : ψ, τ = samplng rate assgnment τ k L = Gene Lbrary k ADAC Lab, NCSU 7

8 Outlne Objectves, Motvaton System Descrpton Gene Lbrary Bologcal Background Formulaton Optmal Adaptve Bandwdth Management Formulaton Analytcal Soluton usng Lagrange Multplers Results Case Study Gene Lbrary based Bandwdth Allocaton for Intellgent Transportaton System Concluson and Future Work 8

9 Bologcal Background Gene a locatable regon of genomc sequence, correspondng to a unt of nhertance, whch s assocated wth regulatory regons, transcrbed regons, and or other functonal sequence regons Phenotype: The set of observable characterstcs of an ndvdual resultng from the nteracton of ts genotype wth envronment. E.g. eye color, blood type, har type, Genotype: The genetc code carred by all lvng organsms, that holds the crtcal nstructons that are used and nterpreted by the cellular machnery to produce the or Phenotype of the ndvdual. ADAC Lab, NCSU 9

10 Bologcal Background For every phenotypc trat, there s a specfc genotype that encodes t. A genotype may contan the transcrbed regon from one or more genes Genotypes FfBb Ffbb ffbb ffbb ffbb FFBb FFbb FfBB FFBB Proten Phenotypes Dark Blue Dark Blue Lght Blue Lght Blue Pale blue Brown Brown Brown Dark brown Sturm, R.A. & Frudaks, T.N. Eye colour: portals nto pgmentaton genes and ancestry. Trends n Genet., (4) ADAC Lab, NCSU

11 Bologcal Background Allele: An allele s a vable DNA (deoxyrbonuclec acd) codng that occupes a gven poston (gene) on a chromosome. It s one of two or more versons of a gene(s) *Image from ADAC Lab, NCSU

12 Bologcal Background Gene Expresson: process by whch nformaton from a gene s used n the synthess of a functonal gene product whch results n some phenotypc trat. Lbrary of genes wth actvaton nstructons Mappng from genetc code to proten Transcrpton: Process of actvaton of a gene and creaton a complmentary copy of a the DNA sequence n the transcrbed porton of the gene. Translaton: Process of decodng the transcrbed genetc code to create protens ADAC Lab, NCSU

13 What s a gene lbrary? Database, mappng Set that contans the genotypes that map the relevant varatons n a phenotypc trat Phenotypc trat system behavor of nterest (e.g. facal expresson) Genotypes Genes: lp shape, eyebrow shape 9 genotypes to represent 9 dfferent scenaros Relevant varatons Happy, sad, angry, others 3

14 Formaton of Gene Lbrary Input Space (Y): Data avalable to the system Y = {y, y, y 3,, y N } = Gene Pool Allele (λ ): The value of a th gene n the Gene Pool for an agent Genome of an ndvdual (Λ): The values of all genes for an ndvdual = [λ, λ λ N ] T Domnant genes (G d ) For specfc phenotypc trat, Φ G d = {g d,, g d,, g d,nd } G d Y Selecton of features usng PCA, kernel-pca, expert system etc Genotype (for Φ) Vector of actual values for domnant genes of Φ = [λ d,, λ d, λ d,nd ] T ADAC Lab, NCSU 4

15 Formaton of Gene Lbrary Genotype Phenotype Map (GP-Map): A -tuple that contans actvator protens to express the genotype and maps the genotype to the phenotypc trat assocated wth t ζ = ( π,m ), π = Actvator/Supressor protens T [ π, π, π,..., π ], π {, } = 3 M = mappng from transcrbed genes λ ρ d, j j N to smple protens ( M, M,..., M ), M : f ( λ, ) = = = d th An Allele of domnant gene N j j d j j j Smple proten produced by the mappng Output of the GP-Map s a complex proten that partly/fully expresses the phenotypc trat ϒ = complex proten produced by the mappng = ρ ρ j ADAC Lab, NCSU 5

16 Formaton of Gene Lbrary If a phenotype needs to be expressed usng m protens Φ = w ϒ + wϒ wmϒm, w = normalzaton weght for each proten Gene Lbrary: Collecton of GP-Maps L = { ζ }, ζ..., ζ SL ADAC Lab, NCSU 6

17 Smple Example sn( x) sn( x) cos( y) cos ( y) y ([ ],sn( gd )) ([ ],sn( gd ),cos( gd )) Let Gene Pool = (w, x, y, z), and Φ = + + ζ = ζ, =,, cos ( g,) d ζ 3 = [ ], g d, ADAC Lab, NCSU 7

18 Outlne Objectves, Motvaton System Descrpton Gene Lbrary Bologcal Background Formulaton Optmal Adaptve Bandwdth Allocaton Formulaton Analytcal Soluton usng Lagrange Multplers Results Case Study Gene Lbrary based Bandwdth Allocaton for Intellgent Transportaton System Concluson and Future Work 8

19 Optmal Adaptve Samplng N Total number of vehcles τ τ mn τ max Samplng tme for vehcle,τ mn τ τ max Mnmum allowable samplng tme. Maxmum allowable samplng tme. Data from each vehcle should be sampled at least once wthn τ max α L BW BW T τ w αn The actual rsk measure for vehcle Length of the data packet sent from the vehcle to the supervsor Bandwdth allocated for vehcle Total bandwdth avalable for the network Samplng wndow: The samplng rates are updated every seconds The number of samples of vehcle n the update nterval The average rsk of vehcle over samplng wndow ADAC Lab, NCSU 9

20 Objectve To selectvely sample data from the vehcles under the bandwdth constrant such that the changes n the rsk s accurately captured n order to reconstruct the rsk sgnal ADAC Lab, NCSU

21 Objectve Mathematcally, Prortzaton by rsk Bandwdth Constrant = ( k + ) τ J = α α ( t) α ( kτ ) dt s. t. N N n = k = kτ BW where, BW BW T L τ w =, n = τ τ Error between the reconstructed and orgnal sgnal Samplng tme to bandwdth converson Select the samplng tmes τ to mnmze the cost J ADAC Lab, NCSU

22 Objectve Functon Smplfcaton by usng Taylor Seres Expanson N n ( k + ) τ J = α α( kτ ) + α( kτ )( t kτ ) + H. OT. α( kτ ) dt = k = kτ N n ( k + ) τ N n ( k + ) τ J α α( kτ )( t kτ ) dt = α α( kτ ) ( t kτ ) dt = k = kτ = k = kτ N n ( k + ) τ N n 3 τ α ( α( τ )) ( τ ) α ( α( τ )) = k = kττ 3 = k = J k t k dt = k J k = J 3 N n 3 α τ ( α( τ )) approx = k = dα = dα =6 dα = 3 Cost (J approx ) Cost (J approx ) Cost (J approx ) Rsk (α) samplng tme (τ) (s) Rsk (α).5.5 samplng tme (τ) (s).5 Rsk (α).5.5 samplng tme (τ) (s) ADAC Lab, NCSU

23 Solvng Optmzaton usng Lagrange Multplers n By replacng ( ) wth and notcng that s the n α ( kτ ) ( n n ( k ) ) k = n α τ ( ( k ) ) k = n α τ k = average of α ( t) over τ w : Consderng n τ w, τ J N 3 approx = α τ n α 3 = Thus we can represent our optmzaton problem as: J τ α τ α N approx = w 3 = mn J s. t. N L w = = BW BW BW, BW = BW mn T = where, L BW =, w = α α N τ ADAC Lab, NCSU 3

24 Usng Lagrange Multplers Changng nequaltes to equaltes where, N N L w ' 3 =,.. ' = T mn = ( BW ) = mn J s t BW BW N BW BW = BW + BW mn ' The lagrangan thus becomes To fnd the optmal pont, we have L BW L w ' ' 3 ( BW ) = + λ = L = = N ' ( BW ) ( BWT N BWmn ) λ = λ = L N 3 ( BW N BW ) Thus, optmal bandwdth s gven as 3 * ( BW ) = BWmn + ( BWT N BWmn ) N 3 w = where w = α α w T = = N N L w λ ' = ( BW ) = ' L = + BW ( BWT N BWmn ) 3 mn 3 w ' BW = ( BWT N BWmn ) N 3 w w α BW (porton of total bandwdth) 5 5 dα 5 α α dα dα BW = - BW 3 4 ADAC Lab, NCSU 4

25 Smple Case Study: Results 8 6 Adaptve Bandwdth Allocaton: Vehcle Scenaro 8 6 Reconstructed Sgnal Agent Agent 4 α α Rsk Tme (s) 5-5 dα dα Comparson between Constant Samplng and Adaptve Samplng BW BW Cost (J approx ) Usng Optmal Adaptve Samplng Usng Constant Samplng Tme (s) ADAC Lab, NCSU 5

26 Connectng the Dots Wthn the Supervsory Agent Overall structure ADAC Lab, NCSU 6

27 Outlne Objectves, Motvaton System Descrpton Gene Lbrary Adaptve Bandwdth Management Formulaton Analytcal Soluton usng Lagrange Multplers Results Case Study Gene Lbrary based Bandwdth Allocaton for Intellgent Transportaton System Gene Lbrary for Rollover Resource Allocaton Concluson and Future Work 7

28 Gene Lbrary n ITS ADAC Lab, NCSU 8

29 Gene Lbrary for Rollover Rsk RI = a y = h a cosϕ + h snϕ R y R L g cosϕ w θ n π.3739 v sn ( ) ( 4L ) w I + mh ϕ = ma h cosϕ + mgh snϕ xx R y R R kls cosϕ snϕ cls ( cos ϕ ) ϕ m, mass (kg) 5 I xx (kg m ) 55 h R, Heght of c.g. (m).7 L w, Wheel base (m).75 k, Sprng Stffness (N/m) c, Dampng coeffcent (Ns/m) 39 l s (m).5 R. Rajaman, et al., "Real-tme estmaton of roll angle and CG heght for actve rollover preventon applcatons," n Amercan Control Conference, 9. ACC '9., 9, pp Y. Jangyeol and Y. Kyongsu, "A rollover mtgaton control scheme based on rollover ndex," n Amercan Control Conference, 6, 6, p. 6 pp. ADAC Lab, NCSU 9

30 Step : Domnant genes Phenotype, Φ = Rollover Probablty Rollover Index (RI) can be expressed as hra y hr snϕ RI = a + b, a =, b = L g L g cosϕ Canddate Complex protens, ϒ = a, ϒ w w = b, such that Φ = w ϒ + w ϒ Velocty (m/s) Steerng Angle (deg) [ 43] ( mph) [-7 7] ADAC Lab, NCSU 3

31 Step : Domnant genes NC STATE UNIVERSITY RI RI a Rollover Probablty vs. a (The relatonshp s almost lnear) b.5 b Rollover Probablty vs. b RI - b / RI RI - a /RI a Error when usng a to represent rollover phenotype Error when usng b to represent the rollover phenotype ADAC Lab, NCSU a has almost lnear relatonshp to Φ Error s small at low values of a If we use a to represent rollover rsk, then we can save computaton tme 3

32 Step : Create GP-Map G = { h, a, L } d R y w g, d ζ = [ ], gd,,, 9.8 g d,3 ADAC Lab, NCSU 3

33 Evaluatng Gene Lbrary Case : Step velocty nput (from v = 5m/s to v =3m/s)..5 x -3 Error when usng GL.9 35 Sgnal Rollover Probablty Actual Estmated usng GL Tme (s) Tme (s) Case : Step theta nput (from θ =. to θ = 5. rad).4.35 Error when usng GL Sgnal Rollover Probablty.8.6 Actual Estmated usng GL Tme (s) Tme (s) ADAC Lab, NCSU 33

34 Evaluatng Gene Lbrary Case 3: Ramp Velocty nput (-4m/s n 5 secs) 45 4 Sgnal Actual Estmated usng GL 3.5 x -3 Error when usng GL Rollover Probablty Case 4: Sne wth Dwell Tme (s) Tme (s) 3 SD.6.5 Actual Estmated usng GL 6 x -3 Error when usng GL Tme (sec) Rollover Probablty Tme (s) Tme (s) ADAC Lab, NCSU 34

35 Improvng Performance NC STATE UNIVERSITY Assgnng weght for a.35 Φ = w ϒ + w ϒ wm ϒ m,.3 RI - a /RI w = normalzaton weght w =.385 (usng lnear regresson) Φ = w a Prevously a Actual Usng Gene Lbrary Error RI a ADAC Lab, NCSU a

36 Improvng Performance - Results Rollover Probablty Actual Estmated usng GL Tme (s).5. Error when usng GL Rollover Probablty Tme (s).4 x -3 Error when usng GL. Actual Estmated usng GL Rollover Probablty Actual Estmated usng GL Tme (s) x -3 Error when usng GL Rollover Probablty Actual Estmated usng GL Tme (s) Error when usng GL Tme (s).35 Error when usng GL Tme (s).5 x -3 Error when usng GL Tme (s) 3.5 x -3 Error when usng GL Tme (s) 6 x -3 Error when usng GL Tme (s) Tme (s) Tme (s) Tme (s) ADAC Lab, NCSU 36

37 Resource Allocaton usng Gene Lbrary Recall Total Bandwdth (bytes/sec) Data Packet Sze (bytes) 5 3 * w BW max 9% = mn + ( T mn ), N BW mn % 3 w τ mn (secs).5 = τ w.5s BW BW BW N BW where, w = α α α = Φ (calculated usng Gene Lbrary).5 Bandwdth Allocated to each vehcle, BWmax = bytes/s Vehcle Vehcle Vehcle Vehcle v = m/s, Delayed Sne wth dwell (s) v = m/s, Delayed Sne wth dwell (s) Estmated and Actual RI for Vehcle.5 RI RI hat Estmated and Actual RI for Vehcle.5 RI RI hat.8.6 Samplng Tme for each Vehcle, ts mn =.5s Vehcle Vehcle ADAC Lab, NCSU 37

38 Resource Allocaton - Comparson Vehcle Vehcle v = m/s, Delayed Sne wth dwell (s) v = m/s, Delayed Sne wth dwell (s) Estmated and Actual RI for Vehcle.5 RI RI hat Estmated and Actual RI for Vehcle.5 RI RI hat Estmated and Actual RI for Vehcle.5 RI RI hat Estmated and Actual RI for Vehcle.5 RI RI hat MSE =.38e-5.4 MSE =.756e-5.3 MSE =9.975e-6.3 MSE =9.7395e-6 Approxmaton Error tme Approxmaton Error tme Approxmaton Error.. 5 tme Approxmaton Error.. 5 tme Usng Constant Bandwdth Usng Adaptve Bandwdth ADAC Lab, NCSU 38

39 Resource Allocaton - Comparson BW T (bts/s) Constant Bandwdth Adaptve Bandwdth Percent Improvement Vehcle Vehcle Vehcle Vehcle Vehcle Vehcle 5 3.e-5.56e-5.98e-5.66e e-5.76e e e e e e-6 6.8e e e e e e e-6.76e e e-6.95e-6.37e-6.5e ADAC Lab, NCSU 39

40 Concluson A formulaton for gene lbrary was ntroduced. The gene lbrary store the genotype-phenotype maps. An optmal adaptve bandwdth scheme was presented that mnmzed the error n sgnal reconstructon Prelmnary results on developng a gene lbrary for ntellgent transportaton system were shown. The dmensonalty and the computaton tme to calculate the rollover rsk was reduced by usng Gene Lbrary. The Gene Lbrary was used for resource allocaton n an ntellgent transportaton montorng system. The mprovement over the statc (equal) bandwdth allocaton scheme was as hgh as 4% ADAC Lab, NCSU 4

41 What Next? Formulaton of Gene Lbrary Formulaton of Optmal Adaptve Resource Allocaton Gene Lbrary based Resource Allocaton Senstvty Analyss of Gene Lbrary n Network Control Systems (NCS) Adaptve Gene Lbrares Senstvty Analyss: Determne performance degradaton after ntroducng delays, packet loss n the communcaton process of the NCS and ntroduce methods to allevate the performance loss Adaptve Gene Lbrary: Use Artfcal Immune Systems to adaptvely update gene lbrary ADAC Lab, NCSU 4

42 Thank You Questons? ADAC Lab, NCSU 4