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# Distributed Control Systems ECE 7750

Utah State University

GPA 3.77

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This 41 page Class Notes was uploaded by Nelle Braun DDS on Wednesday October 28, 2015. The Class Notes belongs to ECE 7750 at Utah State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/230397/ece-7750-utah-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Utah State University.

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Date Created: 10/28/15

C EDIE own In sewmum ind IIMH39UJMVSIIHII Analysis of Coordination in MulitA gent Systems Through Partial Difference Equations ECE 7750 FISP 2 Larry Ballard larry ballardaggiemail usu edu Utah State University CSOIS 39 ADH 8 200870 121 CEDIB mm 1w Sanmum and Maltaquot swan Outline Introduction Mathematical Background 0 Dynamics Leaderless Case o Conclusion i ADH 8 200870 221 C EDIE Oenlnl 1w SelfOmanhinu and Intelligent 51mm Introduction 39urpose to create a new modeling framework for analysis of multiagent systems specifically showing how PdEs provide a framework in which different agent models and control laws can be analyzed with similar techniques 39 ADrll 8 200870 321 Ef gyhing and Imall39lgl l stum Introduction Motivation Partial difference equations can be recast into systems of ordinary differential equations ODEs where partial differential equations cannot Many mathematical tools for analyzing PdEs are 39 completely analogous to the ones developed for PDEs and the PdEs formalism establishes a direct link between classic functional analysis and control theory that can be fruitfully exploited for studying systems linked by a communication network April 8 2008 70 321 C EDIE Oenlnl 1w SelfOmankinu and Intelligent 51mm Introduction Motivation PdEs provide a mathematical description of the collective dynamics where spatial phenomena and temporal evolution of the agent states are kept separated and described through operators acting either on space or time 39 April 8 200870 321 C EDIE own In sewmum ind Intercannulaquot Introduction Motivation The PdEs framework leads to equations that may be reminiscent of PDEs arising in physics and this can be of great help for guessing sensible properties of the collective dynamics 39 April 8 200870 321 Introduction Control Laws Laplacian Control Elastic Control The main difference between the two controls is that the elastic control law implements a potential field for collision avoidance 39 ADHl 8 200870 321 CEQIB huh 1W senmum ind mum SlawII Mathematical Background 39 ADH 8 200870 421 E sfsacsiifi MmWWMat hemat ical Background Graph Theory A graph represents the communication topology In this case the graphs are undirected ie communication goes two ways 39 ADrii 8 200870 521 MMWMathematical Background Graph Theory i A graph represents the communication topology In this case the graphs are undirected ie communication goes two ways Let G be an undirected graph which contains the set of N nodes and a set 5 c N x N of edges The nodes Lay e N are neighbors if cy E a This means that the agents 5 and y share information about their states with each other i ADrii 8 200870 521 Theory A graph represents the communication topology In this case the graphs are undirected ie communication goes two ways Let G be an undirected graph which contains the set of N nodes and a set 5 c N x N of edges The nodes Lay e N are neighbors if cy E 5 This means that the agents 5 and y share information about their states with each other A graph G is called connected when there is at least one path of communication from any node a e N to any other node y e N d April 8 2008 70 521 atmserf WWmmMat hemat ical Background Graph Theory Let S be a nonempty connected subgraph of the connected graph G The boundry of S is defined 388839yEGSEIESg 39 ADH 8 200870 621 Mathematical Background Functional Analysis 5 Wm M Sewmum and Inclinetwinsquot Let f N H R be defined in graph G The partial derivative of f is defined as ayf i y f 39 ADHi 8 200870 721 558m Mathematical Background Functional Analysis Let f N H R be defined in graph G The partial derivative of f is defined as 8yf i M 1 Properties 39 8yfv 8fy MW 0 39 ADHi 8 200870 721 5535quot Mathematical Background Functional Analysis Let f N H R be defined in graph G The partial derivative of f is defined as 8yf i y it Properties 39 Mm 8fy 835M 0 The Laplacian of f is defined by AM i 2W 83W EM and 39 ADHi 8 200870 721 Mathematical Backgound Functional Analysis The integral is defined as fo Z Nfc The average is defined as f fo et L2GRq be the Hilbert space composed of all functions f N H R The inner product is defined as g fG ng the norm is defined as f2L2 fG f2 i ADH 8 200870 821 Mathematical Background Functional Analysis l et H1 and H3 be defined as subsets of L2 where H1GRq i f E L2GRq1ltfgt 0 H6GRq i f E L2GRq gs 0V3 E 88 Both H1 and H3 are Hilbert spaces with the norm Milli erN 2W lt9yf932 In other words if G is connected then for any f 6 L2 fH 0 iff f is constant Also let HiGRq denotes the space orthogonal to H1GR2 and is the space of constant functions in G April 8 200870 921 mm hema ca Background Triangle Operator usual letting G be a connected graph then H1 H H1 and A H3 H H1 have strictly negative igenvalues and the eigenfunctions form a basis for H1 nd H3 respectively And for f e L2Af 0 ifff e 39 ADrll 8 200870 1021 amsewgmmmmanlylathemat ical Background PdEs and Stability Let zct N x R H R be a two variable function z39at t 39 z 0 Notice that we can set F A 39 ADH18 200870 1121 yathematical Background PdEs and Stability Now analyzing the stability assume F0 0 and consider a subspace y c L2GRq The projection off 6 L2 on V is fl PVf Then the origin is stable on y for all t 2 0 if V6 gt 0 36 gt 0 2VL2 g 6 gt tL2 g e If there also exists k gt 077 gt 0 st V2 6 L2 and Vt 2 0 it holds that zl tL2 g ke quott2lL2 then the origin is globally asymptotically stable ADrll 8 200870 1121 CSOIE huh W sunmum and Malta emuu Dynamics I ADH 8 2008 7D 1221 asammmmm Dynamics System The communication between agents is modeldy by the connected graph G mm is the position of the agent vct is the velocity ur t denotes the input eat is the error Where r e L2v e L2u e L2 and e 6 L2 39 ADrii 8 200870 1321 CEOIB cum In senmum and Imalhaut sum Dynamics System hen we can set up the following dynamics 721 r0f L2 vu e v017 L2 ae e0 eL2 Where 6 y 0 and a gt 0 are constants The state of the system is z rT UT eTT 39 ADH 8 200870 1321 CEDIE WW 39039 Selfmm m MINER Suqu Dynamics Control The desired controller will cause the agents to Align There must be a time invariant velocity 11 e R st vc t a 11 as t e 00 for all agents 6 E N Avoid collision with safety distances my gt 0 39 ADrll 8 200870 1421 553mm and Italian Slimquot Dynamics Control Two different controllers are considered Laplacian uat uLct Avct 39 Elastic uct uEct i Avt Ur t Where U7quot at i Vrtf7 ayrhw W95 ayryeg i Z Wit y llama tll2 Note that the Laplacian controller cannot guarantee collision avoidance because it is independent of position 39 ADrll 8 200870 1421 Dynamics Control m y C is an elastic potential energy function with the ollowing properities 39 VrL7C Z 0 V7 y Uy7 7 39 VC y C a 00 as C gt rig where my 2 0 represent given safety distances 0 For all 331 6 N Vcy C attains its unique minimum when C fig where my gt my are given desired distances Vltayltgteooaslteoo 39 ADrii 8 200870 1521 CSOIE huh W SQFWIIIIW ind Malian Sim Leaderless Case ADH 8 2008 7D 1621 Ef gyhing and Intelligent stuu Leaderess Case Main Results The main results are the following 1 Let u be the Laplacian control Then the origin of the system is globally asymptotically stable That is uL guarantees exponentially stable alignment 2 Let u be the Elastic Control and assume that the initial conditions satisfy the collision avoidance conditions Then alignment and collision avoidance are guaranteed i ADrll 8 200870 1721 Leaderless Case Main Results fl39he proof uses the fact that the L2 space can be ecomposed into H1 x This decomposition allows s to cancel or ignore the Hi because it is a constant sing this decomposition makes it possible to find a Lyopunov function that shows that both uL and uE are exponentially asymptotically stable d April 8 200870 1821 sols Cenhar for SelfOmani and Malian SW Leaderess Case Matab Simulations Laplacian This graph is connected as required for convergence ie there is at least one path connecting each node From now on this graph will be referred to as G1 April 8 2008 D 1921 CSDIS tank for SelfMill and Intellilant swam Leaderess Case Matab Simulations Laplacian 5 m V 39A This graph is fully connected ie every thing is connected to everything else From now on this graph will be referred to as GZ April 8 2008 D 1921 CEDIE WW 39039 SelfWm m MINER Suqu Leaderless Case Matab Simulations Laplacian mmmmmmmmmmmmmmm s Using G1 the Laplacian control guaran tees the velocities to converge in simu lation as predicted by the paper ADHi 8 200870 1921 C EDIE own In SellWm and lmuuauunuu Leaderless Case Matab Simulations Laplacian As noted previously with the Lapla cian control there is no convergence ofthe positions and there is no collision avoidance ADHi 8 200870 1921 Leaderless Case Matab Simulations Laplacian C EDIE own In sewmum and lmll39uautsutuu Switching to G2 topology for the Lapla 39 cian controller there isn t much change other than slightly faster convergence of the velocities 39 ADrll 8 200870 1921 afamnwmwm Leaderless Case Matab 7 Simulations Elastic This plot shows the elastic potential 1 2439 2 function V 5K lt 7 gt that I came up with to test the uE 39 ADH 8 200870 2021 cams Leaderless Case Matab huh M SQW WWIIMI ind Imhml Shun Simulations Elastic This plot shows the elastic force func 39 i 2 7f3ygt C272C7jmy4 iay tlon U i K lt ltme Cy2 39 ADH 8 200870 2021 Leaderless Case Matab Simulations Elastic CEOIS cam In senmum and laminatuna With the connection graph G1 the elastic control guarantees that the ve locities converges 39 ADH 8 200870 2021 C EDIE emu 1w SelfOmanhinu and Intelligent Sumquot Leaderless Case Matlab Simulations Elastic s The positions are supposed avoid each other but because the agents do not implement any sort of consensus algo rithm the collision avoidance will only work with a fully connected graph such as G2 April 8 200870 2021 afamhwwwm Leaderless Case Matab Simulations Elastic Using G2 the agents avoid collision and converge to the desired distances from each other W g g a a u i 3 39 ADHi 8 200870 2021 C EDIE cam 1w SelfOmankinu and Imelincu 51mm Leaderless Case Matab Simulations Elastic This plot of the velocities shows that the velocities do not actually converge to a constant value but a linearly in creasing or decreasing value This seems to be a result of the fact that the agents are never able to reach their de sired distance from every other agent ADrll 8 200870 2021 afg nmki g and Intelligent 516m C o n cl usi o n o PdEs allow one to unify and generalize many results on the analysis of the collective dynamics scattered in the control literature I PdEs provide a useful mathematical framework even when dealing with 1 complex agent models accounting for effects of various perturbations 2 complex control laws that can include obstacle avoidance 3 timevarying communication links 0 There are many similarities between PdEs and PDEs describing physical phenomena that can i inspire new decentralized control schemes April 8 200870 2121

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