{"title": "Real-Time Monitoring of Complex Industrial Processes with Particle Filters", "book": "Advances in Neural Information Processing Systems", "page_first": 1457, "page_last": 1464, "abstract": "", "full_text": "Real-Time Monitoring of Complex Industrial\n\nProcesses with Particle Filters\n\nDept. of Mechatronics and Automation\n\nRub\u00b4en Morales-Men\u00b4endez\nITESM campus Monterrey\n\nMonterrey, NL M\u00b4exico\n\nrmm@itesm.mx\n\nNando de Freitas and David Poole\n\nDept. of Computer Science\n\nUniversity of British Columbia\n\nVancouver, BC, V6T 1Z4, Canada\n\u0001 nando,poole\n@cs.ubc.ca\n\nAbstract\n\nThis paper discusses the application of particle \ufb01ltering algorithms to\nfault diagnosis in complex industrial processes. We consider two ubiq-\nuitous processes: an industrial dryer and a level tank. For these appli-\ncations, we compared three particle \ufb01ltering variants: standard parti-\ncle \ufb01ltering, Rao-Blackwellised particle \ufb01ltering and a version of Rao-\nBlackwellised particle \ufb01ltering that does one-step look-ahead to select\ngood sampling regions. We show that the overhead of the extra process-\ning per particle of the more sophisticated methods is more than compen-\nsated by the decrease in error and variance.\n\n1 Introduction\n\nReal-time monitoring is important in many areas such as robot navigation or diagnosis\nof complex systems [1, 2]. This paper considers online monitoring of complex industrial\nprocesses. The processes have a number of discrete states, corresponding to different com-\nbinations of faults or regions of qualitatively different dynamics. The dynamics can be very\ndifferent based on the discrete states. Even if there are very few discrete states, exact moni-\ntoring is computationally unfeasible as the state of the system depends on the history of the\ndiscrete states. However there is a need to monitor these systems in real time to determine\nwhat faults could have occurred.\n\nThis paper investigates the feasibility of using particle \ufb01ltering (PF) for online monitoring.\nIt also proposes some powerful variants of PF. These variants involve doing more computa-\ntion per particle for each time step. We wanted to investigate whether we could do real-time\nmonitoring and whether the extra cost of the more sophisticated methods was worthwhile\nin these real-world domains.\n\n2 Classical approaches to fault diagnosis in dynamic systems\n\nMost existing model-based fault diagnosis methods use a technique called analytical redun-\ndancy [3]. Real measurements of a process variable are compared to analytically calculated\n\n\u0003 Visiting Scholar (2000-2003) at The University of British Columbia.\n\n\u0002\n\fvalues. The resulting differences, named residuals, are indicative of faults in the process.\nMany of these methods rely on simpli\ufb01cations and heuristics [4, 5, 6, 7]. Here, we propose\na principled probabilistic approach to this problem.\n\n3 Processes monitored\n\nWe analyzed two industrial processes: an industrial dryer and a level-tank. In each of these,\nwe physically inserted a sequence of faults into the system and made appropriate mea-\nsurements. The nonlinear models that we used in the stochastic simulation were obtained\nthrough open-loop step responses for each discrete state [8]. The parametric identi\ufb01cation\nprocedure was guided by the minimum squares error algorithm [9] and validated with the\n\u201cControl Station\u201d software [10]. The discrete-time state space representation was obtained\nby a standard procedure in control engineering [8].\n\n3.1 Industrial dryer\n\nAn industrial dryer is a thermal process that converts electricity to heat. As shown in\nFigure 1, we measure the temperature of the output air-\ufb02ow.\n\nFigure 1: Industrial dryer.\n\nNormal operation corresponds to low fan speed, open air-\ufb02ow grill and clean temperature\nsensor (we denote this discrete state\nfaulty\nfan,\n\n\u0005 ). We induced 3 types of fault:\nfaulty fan and grill.\n\nfaulty grill (the grill is closed), and\n\n\u0002\u0001\u0006\u0003\b\u0007\n\n\u0003\n\t\n\n3.2 Level tank\n\n\u0002\u0001\u0004\u0003\n\n\u0003\n\u000b\n\nMany continuous industrial processes need to control the amount of accumulated material\nusing level measurement, such as evaporators, distillation columns or boilers. We worked\nwith a level-tank system that exhibits the dynamic behaviour of these important processes,\nsee Figure 2. A by-pass pipe and two manual valves (\n) where used to induce\ntypical faulty states.\n\nand\n\n\f\u000e\n\n\f\u0010\u000f\n\n4 Mathematical model\n\nWe adopted the following jump Markov linear Gaussian model:\n\n\u0001\u0012\u0011\n\n\u0013\u0015\u0014\n\n\u0001\u0017\u0016\n\u0001\u001f\u001b\n\u0001\u001f\u001b\n\n\u0001\u0019\u0018\u001a\r\u001c\u001b\n\u0001\u0019\u0018\u001a\r! #\"$\u0014\n\u0001\u000e )./\u0014\n\n\u0001\u001f\u001b&%'\u0001\u001a )(\u0015\u0014\n\n\u0001\u001f\u001b&01\u00012 )3\u0015\u0014\n\n\u0001\u001f\u001b&*+\u0001\n\u0001\u001f\u001b&*+\u000154\n\n\n\u0001\n\n\u0001\n\n\n\n\u001d\n\u0001\n\u0003\n\u001e\n\u0014\n\n\u001d\n\n\n,\n\u0001\n\u0003\n-\n\u0014\n\n\u001d\n\n\n\fFigure 2: Level-Tank\n\n\u0011\u0015\u0014\n\n\u0011\u0015\u0014\n\n\u0001\u0001\u0003\u0002\u0005\u0004\u0007\u0006\n\f\u000b\n\u0014\u0017\u0016\n4\u0019\u0018\n\u0013\u0015\u0014\n\ndenotes the measurements, \u001d\nis a known control signal,\n\u0002\u0001\nand\nfor any\n\n\u0001\b\u0003\u0002\t\u0004\u0007\n\n4\u000e\r\u000f\r\u000e\r54\u0011\u0010\u0013\u0012\n. The parameters\n\u0001\u0019\u0018\u001a\r\u0017\u001b\n\u0013\u0015\u0014\n. The initial states are \u001d \u001f\n\u0014#\"\n\u0011!\u0014\n\u001b\u001b\u001a\u001d\u001c\u001e\u0016\n, we have\n. The important thing to notice is that for each realization of\n1\u0001\nexactly using the\n\ndenotes the unknown continuous\nwhere ,\nstates,\ndenotes the unknown discrete\nstates (normal operations and faulty conditions). The noise processes are i.i.d Gaussian:\nare\n01\u0001\n%'\u0001\nidenti\ufb01ed matrices with\nand\na single linear-Gaussian model. If we knew\nKalman \ufb01lter algorithm.\nThe aim of the analysis is to compute the marginal posterior distribution 1 of the dis-\n. This distribution can be derived from the posterior distribution\ncrete states\nby standard marginalisation. The posterior density satis\ufb01es the follow-\n\u001f)&\ning recursion\n\n('4\n, we could solve for \u001d\n\n\u0014\u0017\u0016\n\u0001\u001f\u001b\u001f./\u0014\n\n4%$\n\n\u0002\u0001\n\n4\u0019\u0018\n\n\u001f'&\n\n\u001f'&\n\n\u0014\u0017(\n\n\u001f'&\n\n\u001f)&\n\n\u001f'&\n\n\u001f'&\n\n\u0001\u0017\u0016\n\n\u000154\n\n\u0001\u001f\u001b\n\n\u0001\u0019\u0018\u001a\r\nThis recursion involves intractable integrals. One, therefore, has to resort to some form of\nnumerical approximation scheme.\n\n\u0001\u0019\u00182\n\n\u0001\u0019\u00182\n\n\u0001\u0017\u0016\n\n(1)\n\n\u0001\u0019\u0018\u001a\r\u0017\u001b\n\n\u0001\u0019\u0018\u001a\n\n\u0001\u0017\u0016\n\u0001\u0019\u00182\r\u0017\u001b\n\n5 Particle \ufb01ltering\n\nIn the PF setting, we use a weighted set of samples (particles) \u0001\napproximate the posterior with the following point-mass distribution\n\n\u001d\u0005+-,/.\n\u001f'&\n\n+/,/.\n\u001f)&\n\n\u001b\u00174\n\n+-,/.\n\n\u000210\n\n,/2\n\nto\n\n\u000154\n\n\u001d4\u001f'&\n\u001f)&\n\n\u0001&\u001b\nat\n\n\u0014\u0017(\n\u001d \u001f)&\n\u0014D(\n\u0001\u00174\n+/,-.\n\u001f'&\n\u000210\n,/2\n\u0001\u0019\u00182\n\n9C;/=\n>\u0011?\n\nwhere\n\ncles\n\n9C;/=\n>\u0011?\n\u0001\u0019\u0018\u001a\n\n\u001dF+/,/.\n\u001f)&\n\n\u0001\u0017\u0016\n\n\u001f'&\n\u0001\u001f\u001b\ndenotes\n\n+/,-.\n\u000176\u000e8:9<;-=\n>@?\nA\u001bB\n\n\u001f'&\n,-2\nthe Dirac-delta function.\n\n\u001d4\u001f'&\n\n9<;-=\n>@?\n\n\u000154\n\n\u0014\u0017(\n\n\u0001\u001f\u001b\u00174\n\nGiven\n\nparti-\n\ntime\n\n\u0005 ,\n\nGIH\n, we adopt the notation\n\napproximately distributed according to\n\n1NOTATION: For a generic vector\n\n. For simplicity, we use\n\nand discrete distributions using\n\nthe entries of this vector at time\nrealisation. Consequently, we express continuous probability distributions using\n]_^\nP#J'Na`\b[LJ'NDT\nadmit densities with respect to an underlying measure\ndensities by\nekcl[LJ'N\n\n. For example, when considering the space\nYZP\\[LJ\u000eNWTFcmfgP#J'NUTL[LJ\u000eN\n\nJLK\u001bM\nJ\u000eN\ninstead of ]_^\n\nNFOIP#J1K%Q@J\u000eR\u000fQ%S\u0019S%S%Q@J'NUTWV\nYZP\\[LJ\n\nto denote all\nto denote both the random variable and its\ninstead of\n. If these distributions\n(counting or Lebesgue), we denote these\n, we will use the Lebesgue measure,\n\nfgP#J\u000eNWT\n, so that\n\nP#J'N_cdJ\u000eNWT\n\nYbP#J'NUT\n\nhji\n\n.\n\n*\n\u0001\n\n\u0001\n\u0005\n\u0002\n\u001b\n\u001b\n\u0014\n\u001e\n4\n\"\n4\n-\n4\n.\n4\n3\n4\n\n\u0001\n\u0016\n\n\u001b\n.\n\u0014\n\n\n\u0001\n\n\u0001\n\u001f\n\u001f\n\u001b\n\n\u001f\n\u0011\n\n\u001f\n\u001b\n\u0001\n\u0013\n\u0014\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n\u0013\n\u001d\n\u0001\n4\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n*\n\u0014\n\u001d\n\u0001\n4\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n\u0003\n*\n\u0014\n\u001d\n4\n\n\u0016\n,\n\n&\n\u001b\n*\n\u0014\n,\n\u001d\n\n*\n\u0014\n\u001d\n\u0001\n4\n\n\u001d\n4\n\n*\n\u0014\n,\n,\n\n&\n\n\u0014\n\u0001\n4\n\n\u0001\n%\n\u0001\n\n3\n\u0013\n0\n\n,\n\n&\n\u0003\n0\n5\n\n%\n\u0012\nA\n\n6\n8\nA\nB\n\u0012\nA\n\nE\n\u0001\n4\n\n\nJ\nX\nN\nT\ne\n\f+-,/.\n\u001f'&\n\n\u0001\u0019\u0018\u001a\n\n\u0001\u0019\u00182\n\n+/,-.\n\u001f'&\n\u0001\u0019\u00182\n\n, PF enables us to compute\n\napproxi-\nparticles \u0001\n\u0013\u0015\u0014\u0017(\nmately distributed according to\n. Since we cannot sample from\n+/,-.\n\u001f'&\nthe posterior directly, the PF update is accomplished by introducing an appropriate impor-\ntance proposal distribution \u0001\nfrom which we can obtain samples. The basic\nalgorithm, Figure (3), consists of two steps: sequential importance sampling and selection\n(see [11] for a detailed derivation). This algorithm uses the transition priors as proposal\ndistributions; \u0001\n. For the selection step, we\n\nused a state-of-the-art minimum variance resampling algorithm [12].\n\n\u0013\u0015\u0014\u0017(\n\u001f)&\n\n, at time\n\n+/,/.\n\u001f)&\n\u001f'&\n\n\u001b\u001f\u0013\u0015\u0014\n\n\u0001\u0019\u00182\n\n\u0001\u0019\u0018\u001a\n\n\u0013\u0015\u0014\n\n+/,-.\n\u001f'&\n\n+/,-.\n\u001f'&\n\n\u001f'&\n\n\u001f)&\n\n\u000154\n\n\u0014\u0017(\n\n\u0001\u001f\u001b\n\n,-2\n\nSequential importance sampling step\n\n, sample from the transition priors\n\nand set\n\n\u0002 For\n\u0002 For\n\nc\u0005\u0004\nc\u0005\u0004\n\nQ)SCSCS\n\nQ)SCSCS\n\nQ\u0007\u0006\n\t\u000b\n\r\f\u000f\u000e\nN\u0011\u0010\n\n\f\u0016\u000e\n\n\u0019\f\u000f\u000e\nN\u001e\u001d\nQ\u0007\u0006\n\n\t\u000b\n\r\f\u000f\u000e\nN\u0014\u0013 K\n\n\f\u0016\u000e\n\nN\u0007\u0012\nO\u001f\u001b\n\nSelection step\n\n, evaluate and normalize the importance weights\n\nP\\[\n\n\u0019\f\u000f\u000e\nN\u001a\u0013\n\nN\u0018\u0012\n\n\t\u000b\n\r\f\u000f\u000e\n\n\u0019\f\u000f\u000e\n\n\f\u000f\u000e\nN#\"\n\n\u0019\f\u000f\u000e\n\nand\n\n\f\u0016\u000e\n\nN\u001a\u0013\nN\u0018\u0012\n\u001b%$\n\n\f\u000f\u000e\nN%')(\n\f\u000f*\n\n\f\u000f\u000e\n\n\f\u0016\u000e\nN\u0017\u0010\n\n\f\u000f\u000e\nK \u001d\nN\u0014\u0013\n\n\f\u0016\u000e\n\n\f\u000f\u000e\n\n.\n\n\f\u0016\u000e\n\n\f\u000f*\n\nwith respect to high/low importance\n\n\u0002 Multiply/Discard particles\nweights!\n\nto obtain\n\n\f\u0016\u000e\n\nparticles\n\nFigure 3: PF algorithm at time\n\n.\n\n, it is\nis Gaussian\n.\n\u0001&\u001b\n(2)\n\n\u001f)&\n\n\u00015\u0016\n\n6 Improved Rao-Blackwellised particle \ufb01ltering\n\nBy considering the factorisation *\n\u001f)&\n\u001f)&\npossible to design more ef\ufb01cient algorithms. The density *\n\u001f'&\n\u0001\u001f\u001b\nand can be computed analytically if we know the marginal posterior density *\nThis density satis\ufb01es the alternative recursion\n\n\u001f'&\n\n\u001f'&\n\n\u001f'&\n\n\u001f)&\n\n\u0001\u001c\u0016\n\n\u001f)&\n\n\u00015\u0016\n\n\u0001\u001f\u001b\n\n\u001f)&\n\n\u0001\u0019\u0018\u001a\n\n\u0001\u0019\u0018\u001a\r\u001c\u001b\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u0018\u001a\r14\n\n\u001f)&\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u0018\u001a\r\u0017\u001b\n\n\u0001&\u001b\n\u0001\u0019\u00182\n\nIf equation (1) does not admit a closed-form expression, then equation (2) does not ad-\nmit one either and sampling-based methods are still required. (Also note that the term\nbecause there is a depen-\ndency on past values through \u001d\n.) Now assuming that we can use a weighted set of\nsamples \u0001\n\nin equation (2) does not simplify to *\n\nto represent the marginal posterior distribution\n\n\u001f'&\n\n\u001f)&\n\n\u0001\u0017\u0016\n\n\u0001\u0017\u0016\n\n\u0001&\u001b\n\n\u0001&\u001b\n\n\u0001\u0019\u0018\u001a\r14\n+/,/.\n\u001f)&\n\n+/,-.\n\n,-2\n\n\u001f)&\n\nthe marginal density of \u001d\n\u0003,+\n\n\u001d4\u001f'&\n\n\u0001\u001c\u0016\n\n\u0001\u0019\u001b\n\n\u001f'&\n\n\u0001\u0017\u0016\n\n\u0001\u001f\u001b\n\nis a Gaussian mixture\n\n,-2\n\n+/,/.\n\n\u001f'&\n\n\u0001&\u001b\u00174\n\n9<;-=\n>@?\n\n\u001d4\u001f'&\n\n\u0001\u001c\u0016\n\n\u001f'&\n\n\u0001&\u001b@(\n\n\u0013\u0015\u0014\n\n\u001f'&\n\n\u0001\u001c\u0016\n\n\u0001\u001f\u001b\n\n+/,-.\n\n\u001d4\u001f'&\n\n\u0001\u0017\u0016\n\n+/,/.\n\u001f)&\n\n,-2\n\n\u001d\n4\n\n\u0016\n,\n\n&\n\u001b\nE\n\u001d\n\u0001\n4\n\n\u0001\n\u0002\n0\n\n\u001d\n\u0001\n4\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\nG\n\u001d\n\n\u0014\n\u001d\n\u0001\n4\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n\u0003\n\u001d\n\u0001\n\u0016\n\u001d\n4\n\n\u0001\n\n\u0001\n\u0016\n\n\u001b\n\u0003\n\b\nY\nP\n\t\nT\n\b\n\u0015\nY\n\u0015\n\u0015\nK\nQ\nN\nT\n\u001b\n\b\n\u0015\n\u001c\nM\nN\nQ\n\b\n\t\n\u001c\nM\n\b\n\u0015\nN\nQ\n\b\n\t\nN\nQ\n\u0015\n\u001c\nM\nK\nQ\n\t\n\u001c\nM\nS\n\u0003\n!\nf\n\b\n\u0015\nN\nQ\n\b\n\t\nN\n\u001d\n&\n\b\n\u0015\n\u001c\nM\nN\nQ\n\b\n\t\n\u001c\nM\nK\nN\n\u0006\n&\n\u0015\n\u001c\nM\nN\nQ\n\t\n\u001c\nM\nN\n'\n(\nK\nG\n\u0014\n\u001d\n\u0001\n4\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n\u0003\n*\n\u0014\n\u001d\n\u0001\n\u0016\n,\n\n&\n\u0001\n4\n\n\u0001\n\u001b\n*\n\u0014\n\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n\u0014\n\u001d\n,\n\n&\n\u0001\n4\n\n\u0014\n\n,\n\n&\n*\n\u0014\n\n,\n\n&\n\u0003\n*\n\u0014\n\n\u0016\n,\n\n&\n*\n\u0014\n,\n,\n\n&\n\n*\n\u0014\n\n\n*\n\u0014\n,\n\u0001\n\u0016\n,\n\n&\n\u001b\n*\n\u0014\n,\n,\n\n&\n\n\u0014\n,\n\n\u0001\n\n\u0001\n4\n%\n\u0001\n\u0002\n0\n\n3\n\u0013\n0\n\u0014\n\n,\n\n&\n\u0003\n0\n5\n\n%\n\u0001\n6\n\u0012\nA\n\u0014\n\n\u0001\n3\n*\n0\n\u0014\n,\n\n&\n*\n\u0014\n\n\u0001\n4\n,\n\n&\n\n,\n\n&\n\u0003\n0\n5\n\n%\n\u0001\n*\n\u0014\n,\n\n&\n\u0001\n4\n\n\u0001\n\u001b\n\f\u0002\u0001\n\n+/,/.\n\n+/,-.\n\nand then propagate the mean\n\n. In particular, we sample\nwith a Kalman \ufb01lter:\n\nthat can be computed ef\ufb01ciently with a stochastic bank of Kalman \ufb01lters. That is, we use\nand exact computations (Kalman \ufb01lter) to estimate the\nPF to estimate the distribution of\nmean and variance of\n+/,-.\nand covariance\nof \u001d\n\n\f\u0016\u000e\nN\u0001\nN\u001a\u0013\n\n\f\u0016\u000e\nN\u001a\u0013\nN\u0001\n\n\f\u0016\u000e\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013 K\n\n\f\u0016\u000e\n\n\f\u0016\u000e\n\u001b\u001d\u001c\n\nwhere\n\u001b\u001d\u001c\nThis is the basis of the RBPF algorithm that was adopted in [13]. Here, we introduce an\nextra improvement. Let us expand the expression for the importance weights:\n\n\f\u000f\u000e\nN\u0014\u0013\nK\u0004\u0003\u0006\u0005\n\n\f\u0016\u000e\n\n\f\u0016\u000e\nN\u0014\u0013\n\n\f\u0016\u000e\n\r\u001dP\nN\u0001\nN\u0014\u0013\n\n\f\u0016\u000e\nN\u0001\nN\u001a\u0013\n\u0003\u0011\u0010\n\n\f\u0016\u000e\n\r\u001dP\nN\u001a\u0013\nN\u0001\n\n\f\u0016\u000e\n\r\u001dP\nN\u0001\nN\u0014\u0013\n\u0016\u0019\u0017\nand\n\n\f\u000f\u000e\nT\u0001\u0007\nT\n\t\n\u0003\u0006\u000b\n\n\f\u000f\u000e\nT\n\t\n\u0003\u000f\u000e\n\n\f\u000f\u000e\nT\u0001\u0007\n\n\u0019\f\u000f\u000e\nT\n\t\n\n\f\u000f\u000e\nT\n\t\n\n\u0019\f\u000f\u000e\n\n\u0019\f\u000f\u000e\n\n\f\u0016\u000e\n\r\u001dP\n\n\f\u0016\u000e\n\r\u001dP\n\n\f\u000f\u000e\nN\u0014\u0013 K\nN\u0001\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013 K\n\n\f\u0016\u000e\n\n\f\u0016\u000e\n\n\f\u000f\u000e\n\n\f\u0016\u000e\n\u0001\u0015\u0014\n\nN\u0013\u0012\n\r\u001dP\n\u0001\u0019\u0018\u001a\r\n\u0001\u0019\u0018\u001a\r\u001c\u001b\n\nT\n\t\n\n\u0019\f\u000f\u000e\nT\n\t\n\n\f\u000f\u000e\nN\u0014\u0013\nN\u0001\n\n\u0013 K\n\u0013 K\n, ,\n\u0001\u0017\u0016\n\n\f\u0016\u000e\n\u0016\u001a\u0017\n.\n\n\u0016\u0018\u0017\n,\n$!\u0001\n\n\u0001\u0019\u0018\u001a\r\n\u0001\u0019\u00182\r\u0017\u001b\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013\n\n\u0001\u0019\u00182\r\n\u0001\u001c\u0016\n\n\f\u0016\u000e\n\n\u0001\u0019\u0018\u001a\n\n\u0001\u0019\u0018\u001a\n\n\u001b\u001d\u001c\n\nTDe\n\nTDe\n\n\u0001\u0015\u0014\n\n\u0001\u0015\u0014\n\n\u0001\u001c\u0016\n\n\u0001&\u001b\n\n,\n\n,\n\n\u001f'&\n\u00015\u0016\n\u001f'&\n\u0001\u0017\u0016\n\u0001\u0017\u0016\n\n\u0001&\u001b\n\u0001\u001f\u001b\n\u0001\u0019\u00182\r\u00024\n\n\u001f'&\n\n\u0001\u0019\u001b\n\n\u0001\u001f\u001b\n\u0001\u0019\u0018\u001a\r\u001c\u001b\n\u0001\u0019\u0018\u001a\r\u00024\n\n\u001f'&\n\n\u001f)&\n\n\u001f'&\n\u0001\u0019\u00182\r\n\u0001\u0019\u0018\u001a\r\n\u0001\u0017\u0016\n\n\u0001\u0019\u0018\u001a\r\n\u001f'&\n\u0001\u0019\u00182\r\u00024\n\n\u001f)&\n\n\u00015\u0016\n\u0001\u0019\u00182\r\u0017\u001b\n\u001f'&\n\n\u001f)&\n\u001f)&\n\n\u0001\u0019\u00182\r\u00024\n\u0001\u0019\u00182\n\n\u0001\u001f\u001b\n\u0001\u001f\u001b\n\n(3)\n\n(4)\n\nThe proposal choice, \u001f\n\n, states that we are\nnot sampling past trajectories. Sampling past trajectories requires solving an intractable\nintegral [14].\n\n\u0001\u0019\u00182\r\u0017\u001b\n\n\u0001\u0019\u00182\n\n\u001f)&\n\n\u0001\u0017\u0016\n\n\u0001\u0017\u0016\n\n\u0001\u001f\u001b\n\n\u0001\u001f\u001b\n\nWe could use the transition prior as proposal distribution:\n\n\u001f)&\n\n\u0001\u0019\u0018\u001a\r14\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u0018\u001a\r5\u001b\n\nweights simplify to the predictive density\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u0018\u001a\r\u0017\u001b\n\n. Then, according to equation (4), the importance\n\n\u001f'&\n\n\u0001\u0019\u00182\n\n\u001f'&\n\u0001&\u001b\nposal distribution corresponds to the choice \u001f\n\ndistribution satis\ufb01es Bayes rule:\n\n\u0001\u0019\u00182\r\u00024\n\n%'\u0001\n\n\u00015\u0016\n\n\u0014\"!\n\u001f)&\n\n\u0001$#\n\u0001\u0019\u00182\n\n1\u0001\n\nHowever, we can do better by noticing that according to equation (3), the optimal pro-\n. This\n\n\u0001\u0015\u0014\n\n\u0001\u0019\u0018\u001a\n\n\u0001\u0015%\n\n\u001f)&\n\n\u0001\u0019\u0018\u001a\n\n(5)\n\n\u0001\u0019\u00182\r\nand, hence, the importance weights simplify to\n\n\u0001\u0019\u00182\n\n\u001f'&\n\n\u001f'&\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u00182\r\u00024\n\n\u001f'&\n\u0001\u0019\u0018\u001a\r\n\u0001\u0019\u0018\u001a\r5\u001b\n\n\u001f'&\n\n\u0001\u0019\u00182\n\n%'\u0001\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u00182\r\u00024\n\n\u001f'&\n\n\u0001\u0019\u0018\u001a\r5\u001b\n\n\u0004'&\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u00182\r\u00024\n\n\u001f'&\n\n\u0001\u0019\u0018\u001a\r\u00024\n\n\u0001\u0019\u001b\n\n\u0001\u0017\u0016\n\n\u0001\u0019\u0018\u001a\r\u001c\u001b\n\n(6)\n\n(7)\n\nWhen the number of discrete states is small, say 10 or 100, we can compute the distributions\nIn addition to Rao-Blackwellisation, this leads to\nin equations (6) and (7) analytically.\nsubstantial improvements over standard particle \ufb01lters. Yet, a further improvement can still\nbe attained.\n\nEven when using the optimal importance distribution, there is a discrepancy arising from\nin equation (3). This discrepancy is what causes\nthe ratio *\nthe well known problem of sample impoverishment in all particle \ufb01lters [11, 15]. To cir-\ncumvent it to a signi\ufb01cant extent, we note that the importance weights do not depend on\n\n\u0001\u0019\u00182\n\n\u0001\u0019\u00182\n\n\u0001\u0019\u0018\u001a\n\n\u001b)(\n\n\u001f'&\n\n\u001f'&\n\n\n\u0001\n\n\u0001\n\"\n\u0001\n$\n\u0001\n\u0001\ne\nK\nc\n\u0002\nP\n\t\nN\nP\n\t\nN\nN\n\b\nK\nc\n\u0002\nP\n\t\nN\nT\n\b\nK\n\u0002\nP\n\t\nN\nP\n\t\nN\nT\n\u000b\nP\n\t\nN\n\f\nN\nc\n\t\nN\nT\n\b\nK\n\t\nN\nP\n\t\nN\nT\n\u000e\nP\n\t\nN\n$\nc\n\t\nN\nK\nP\n\t\nN\nN\ne\nN\nc\ne\n\u0003\n\b\nK\n\t\nN\n\f\nN\nP\n$\n$\nK\nT\n\b\nN\nc\n\b\n\u0012\n\b\nK\n\t\nN\n\f\nN\n\t\nN\nT\n\b\nK\nQ\n\"\n\u0014\n\u001d\n\u0001\n\u0016\n,\n\n&\n\u001b\n\"\n\u0001\n\u0014\n\u001d\n\u0001\n\u0016\n,\n\n&\n\u0001\n\u001b\n\u0014\n,\n\u0001\n\u0016\n,\n\n&\n\u001b\n$\n\u0016\n0\n\u0014\n\u001d\n,\n\n&\n\u0016\n0\n\u0014\n\u001d\n,\n\n&\n\u001e\n\u0001\n\u0016\n0\n\u0014\n,\n,\n\n&\n%\n\u0001\n\u0003\n*\n\u0014\n\n,\n\n&\n\u001f\n\u0014\n\n,\n\n&\n\u0003\n*\n\u0014\n\n\u0016\n,\n\n&\n*\n\u0014\n\n\u0016\n,\n\n&\n*\n\u0014\n\n\u0001\n\u0016\n\n,\n\n&\n\u001f\n\u0014\n\n\n4\n,\n\n&\n \n*\n\u0014\n,\n,\n\n&\n\n*\n\u0014\n\n\n,\n\n&\n\u001f\n\u0014\n\n\u0001\n\u0016\n\n4\n,\n\n&\n\u0001\n\u001b\n\n\u0014\n\n,\n\n&\n\u0003\n\u001f\n\u0014\n\n\n,\n\n&\n*\n\u0014\n\n\u0016\n,\n\n&\n\u001f\n\u0014\n\n\u0001\n\u0016\n\n4\n,\n\n&\n\u0001\n\u001b\n\u0003\n*\n\u0014\n\n\n,\n\n&\n\u0003\n*\n\u0014\n\n\n \n*\n\u0014\n,\n,\n\n&\n\n\u0003\n,\n,\n4\n\u001e\n\n\u0014\n\u0016\n\n4\n,\n\n&\n\u0001\n\u001b\n\u0003\n*\n\u0014\n\n\u0001\n\u0016\n\n4\n,\n\n&\n\u0001\n\u001b\n*\n\u0014\n\n\u0001\n\u0016\n\n4\n,\n\n&\n\u0001\n\u001b\n\u0003\n*\n\u0014\n,\n\u0001\n\u0016\n,\n\n&\n4\n\n\u0001\n\u001b\n*\n\u0014\n\n\u0001\n\u0016\n\n4\n,\n\n&\n\u001b\n*\n\u0014\n,\n,\n\n&\n\n \n*\n\u0014\n,\n,\n\n&\n\n\u0003\n5\n\u0012\nA\n2\n\n*\n\u0014\n,\n,\n\n&\n\n\n*\n\u0014\n\n\n\u0014\n\n\u0016\n,\n\n&\n\u0001\n*\n\u0014\n\n\u0016\n,\n\n&\n\u001b\n\f(we are marginalising over this variable). It is therefore possible to select particles be-\nfore the sampling step. That is, one chooses the \ufb01ttest particles at time\n\u0005 using the\n. This observation leads to an ef\ufb01cient algorithm (look-ahead RBPF),\ninformation at time\nwhose pseudocode is shown in Figure 4. Note that for standard PF, Figure 3, the impor-\ntance weights depend on the sample\n, thus not permitting selection before sampling.\nSelecting particles before sampling results in a richer sample set at the end of each time\nstep.\n\n+/,-.\n\nKalman prediction step\n\nSelection step\n\nSequential importance sampling step\n\nK\u001bM\n\nTFc\n\nT\u001bT-f\n\ncmf\n\n\u0019\f\u000f\u000e\n\ncompute\n\n\f\u000f\u000e\n\n\f\u0016\u000e\n\nQ%S%S\u0019S)Q\u0001\u0003\u0002\n\n\f\u0016\u000e\nN\u0001\nN\u001a\u0013\n\nN_c\u0005\u0004\n\n\f\u0016\u000e\nN\u0014\u0013\n\n\u0002 For i=1, . . . , N, and for\t\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013 K\n\u0002 For i=1 , . . . , N , evaluate and normalize the importance weights\nN\u0014\u0013\n\n\f\u0016\u000e\n\u0002 Multiply/Discard particles\nN\u0014\u0013\n\n\f\u0016\u000e\nimportance weights!\n\u0002 Kalman prediction. For i=1, . . . , N, and for\t\n\u0002 For\t\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013 K\n\n\f\u0016\u000e\nN\n\nN\u0014\u0013\n\n\f\u0016\u000e\nN\u0014\u0013\nN_c\u0005\u0004\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013 K\n\ninformation, re-compute\n\nNWT\ncompute\n\nQ\u0019S%S)S%Q\u0001\u0003\u0002\n\n\f\u0016\u000e\nN\u0001\nN\u001a\u0013\n\n\u0019\f\u000f\u000e\nN\u001a\u0013\n\nQ)S%S\u0019S%Q\u0001\n\nto obtain\n\nparticles\n\n\f\u0016\u000e\nN\u0014\u0013\n\n\u0019\f\u000f\u000e\nN\u001a\u0013\n\n\u0019\f\u000f\u000e\nN\u001a\u0013\n\n\f\u000f\u000e\n\n\f\u0016*\n\nNWT\n\nNUT\n\nc\u0005\u0004\n\nSampling step\n\nNUT\n\n\f\u0016\u000e\nN\u0014\u0013\n\n.\n\n\f\u000f*\n\nwith respect to high/low\n\nusing the resampled\n\n\f\u0016\u000e\nN\u0014\u0013 K\nN\u0001\n\n\f\u0016\u000e\nN\u0014\u0013 K\n\nN\u0018\u0012\n\n\f\u000f\u000e\n\n\f\u0016\u000e\n\nK\u001bM\n\nNUT\n\n\f\u0016\u000e\n\nUpdating step\n\ncient statistics\n\n\u0002 For i=1 , . . . , N, use one step of the Kalman recursion to compute the suf\ufb01-\n\ngiven\n\nNWT\n\nNUT\n\nK\u001bM\n\nN\u0018\u0012\n\nNDT\u001bT-f\n\n\f\u0016\u000e\n\n\f\u0016\u000e\nN\u0014\u0013\n\n\f\u0016\u000e\nN\u0001\nN\u0014\u0013 K\n\n\f\u0016\u000e\nN\u0018\u0012\nN\u0014\u0013 K\n\n\u0019\f\u000f\u000e\nN\u0014\u0013\n\u0005 using the information at time\n\nN\u0014\u0013 K\n\n\f\u0016\u000e\n\nN\u0001\n\nN\u0001\n\n.\n\n. The algorithm uses an optimal proposal\nG\u0005H\n\n.\n\nFigure 4: look-ahead RBPF algorithm at time\ndistribution. It also selects particles from time\n\n7 Results\n\nThe results are shown in Figures 5 and 6. The left graphs show error detection versus\ncomputing time per time-step (the signal sampling time was 1 second). The right graphs\nshow the error detection versus number of particles. The error detection represents how\n\n\n\u0001\nG\nH\nG\n\n\u0001\n\b\ne\nP\n\t\nN\nT\nQ\n\b\n\b\nK\nP\n\t\nN\nT\nQ\n\b\n$\nP\n\t\nN\nT\nQ\n\b\n\f\nN\nP\n\t\nN\nT\n!\nN\nP\n$\nN\n\u0012\n$\nK\nQ\n\t\n\u001c\nM\nK\ni\n&\n\u0004\n\u0002\nA\n*\nK\n\u0005\nP\n\b\n$\nK\nP\n\t\nN\nT\nQ\n\b\n\f\nN\nP\n\t\nN\nP\n\t\nN\n\u0012\n\t\nK\nT\n&\n\b\ne\nK\nQ\n\b\n\b\nK\nQ\n\b\n\t\n\u001c\nM\nK\n'\n(\nK\nN\n\u0006\n&\ne\nK\nQ\n\b\nK\nQ\n\t\n\u001c\nM\nK\n'\n(\nK\n\b\ne\nP\n\t\nQ\n\b\n\b\nK\nP\n\t\nQ\n\b\n$\nP\n\t\nQ\n\b\n\f\nN\nP\n\t\nN\n\u0002\nf\nP\n\t\n\t\n\u001c\nM\nQ\n$\n\"\n\u0005\nP\n\b\n$\nP\n\t\nQ\n\b\n\f\nN\nP\n\t\nP\n\t\n\t\nK\nT\n\u0002\n\t\nN\n\u0010\nf\nP\n\t\n\t\n\u001c\nM\nQ\n$\n&\ne\nN\nQ\n\b\nN\n'\n&\n\b\ne\nK\nP\n\t\nN\nT\nQ\n\b\n\b\nP\n\t\nN\nT\n'\nG\nG\n\fmany discrete states were not identi\ufb01ed properly, and was calculated for 25 independent\nruns (1,000 time steps each). The graphs show that look-ahead RBPF works signi\ufb01cantly\nbetter (low error rate and very low variance). This is essential for real-time operation with\nlow error rates.\n\nn\no\ni\nt\nc\ne\nt\ne\nD\n\n \nr\no\nr\nr\n\n \n\nE\n%\n\n100\n\n90\n\n80\n\n70\n\n60\n\n50\n\n40\n\n30\n\n20\n\n10\n\n0\n10\u22123\n\nReal time\n\nRBPF\n\nPF\n\nla\u2212RBPF\n\n10\u22122\n100\nComputing time per timestep (=) sec\n\n10\u22121\n\n101\n\nn\no\n\ni\nt\nc\ne\n\nt\n\ne\nD\n\n \nr\no\nr\nr\n\n \n\nE\n%\n\n100\n\n90\n\n80\n\n70\n\n60\n\n50\n\n40\n\n30\n\n20\n\n10\n\n0\n\nRBPF\n\nPF\n\nla\u2212RBPF\n\n100\n\n101\nNumber of particles\n\n102\n\n103\n\nFigure 5: Industrial dryer: error detection vs computing time and number of particles.\n\nThe graphs also show that even for 1 particle, look-ahead RBPF is able to track the discrete\nstate. The reason for this is that the sensors are very accurate (variance=0.01). Conse-\nquently, the distributions are very peaked and we are simply tracking the mode. Note that\nlook-ahead RBPF is the only \ufb01lter that uses the most recent information in the proposal\ndistribution. Since the measurements are very accurate, it \ufb01nds the mode easily. We re-\npeated the level-tank experiments with noisier sensors (variance=0.08) and obtained the\nresults shown in Figure 7. Noisier sensors, as expected, reduce the accuracy of look-ahead\nRBPF with a small number of particles. However, it is still possible to achieve low error\nrates in real-time. Since modern industrial and robotic sensors tend to be very accurate, we\nconclude that look-ahead RBPF has great potential.\n\nAcknowledgments\n\nRuben Morales-Men\u00b4endez was partly supported by the Government of Canada (ICCS) and\nUBC CS department. David Poole and Nando de Freitas are supported by NSERC\n\nReferences\n[1] J Chen and J Howell. A self-validating control system based approach to plant fault detection\n\nand diagnosis. 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Control\n\nengineering practice, 5(5):639\u2013652, 1997.\n\n\fn\no\n\ni\nt\nc\ne\ne\nD\n\nt\n\n \nr\no\nr\nr\n\n \n\nE\n%\n\n100\n\n90\n\n80\n\n70\n\n60\n\n50\n\n40\n\n30\n\n20\n\n10\n\n0\n10\u22123\n\nRBPF\n\nla\u2212RBPF\n\nPF\n\nReal time\n\nn\no\n\ni\nt\nc\ne\n\nt\n\ne\nD\n\n \nr\no\nr\nr\n\n \n\nE\n%\n\n10\u22122\n100\nComputing time per timestep (=) sec\n\n10\u22121\n\n101\n\n100\n\n90\n\n80\n\n70\n\n60\n\n50\n\n40\n\n30\n\n20\n\n10\n\n0\n\nRBPF\n\nPF\n\nla\u2212RBPF\n\n100\n\n101\n\nNumber of particles\n\n102\n\nFigure 6: Level-tank (accurate sensors): error detection vs computing time and number of\nparticles.\n\n100\n\n90\n\n80\n\n70\n\n60\n\n50\n\n40\n\n30\n\n20\n\n10\n\n0\n\nn\no\n\ni\nt\nc\ne\n\nt\n\ne\nD\n\n \nr\no\nr\nr\n\n \n\nE\n%\n\n10\u22123\n\nReal time\n\nPF\n\nRBPF\n\nla\u2212RBPF\n\n10\u22122\nComputing time per timestep (=) sec\n\n10\u22121\n\n100\n\n100\n\n90\n\n80\n\n70\n\n60\n\n50\n\n40\n\n30\n\n20\n\n10\n\n0\n\nn\no\n\ni\nt\nc\ne\n\nt\n\ne\nD\n\n \nr\no\nr\nr\n\n \n\nE\n%\n\nRBPF\n\nPF\n\nla\u2212RBPF\n\n101\n\n100\n\n101\n\n102\n\nNumber of particles\n\n103\n\nFigure 7: Level-tank (noisy sensors): error detection vs computing time and number of\nparticles.\n\n[8] K Ogata. 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Filtering via simulation: auxiliary particle \ufb01lters.\n\nAmerican statistical association, 94(446):590\u2013599, 1999.\n\nJournal of the\n\n\f", "award": [], "sourceid": 2162, "authors": [{"given_name": "Rub\u00e9n", "family_name": "Morales-Men\u00e9ndez", "institution": null}, {"given_name": "Nando", "family_name": "de Freitas", "institution": null}, {"given_name": "David", "family_name": "Poole", "institution": null}]}