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1 Some System Identification Challenges and Approaches Brett Ninness School of Electrical Engineering & Computer Science The University of Newcastle, Australia Many basic scientific problems are now routinely solved by simulation: a fancy random walk is performed on the system of interest. Averages computed from the walk give useful answers to formerly intractable problems Persi Diaconis, 2008 Slide 2 2 System Identification - a rich history 1700s: Bernoulli, Euler, Lagrange - probability concepts 1763: Bayes - conditional probability 1795: Gauss, Legendre - least squares 1800-1850: Gauss, Legendre, Cauchy - prob. distributions 1879: Stokes - periodogram of time series 1890: Galton, Pearson - regression and correlation 1922: Fisher - Maximum Likelihood (ML) 1921: Yule - AR and MA time series 1933: Kolmogorov - Axiomatic probability theory 1930s: Khinchin, Kolmogorov, Cramr - stationary processes Slide 3 3 System Identification - a rich history 1941-1949: Wiener, Kolmogorov - prediction theory 1960: Kalman - Kalman Filter 1965: Kalman & Ho - Realisation theory 1965: strm & Bohlin - ML methods for dynamic systems 1970: Box & Jenkins - a unified and complete presentation 1970s: Experiment design, PE formulation with underpinning theory, analysis of recursive methods 1980s: Bias & Variance quantification, tradeoff and design 1990s: Subspace methods, control relevant identification, robust estimation methods. Slide 4 4 Recent & Current Activity Slide 5 5 This talk Slide 6 6 Acknowledgements Results here rest heavily on the work of colleagues: Dr. Adrian Wills (Newcastle University) Dr. Thomas Schn (Linkping University) Dr. Stuart Gibson (Nomura Bank) Soren Henriksen (Newcastle University) and on learning from experts: Hkan Hjalmarsson, Tomas McKelvey, Fredrik Gustafsson, Michel Gevers, Graham Goodwin. Slide 7 7 Challenge 1 - General Nonlinear ID Effective solutions available for specific nonlinear structures NARX, Hammerstein-Wiener, Bilinear..... Extension to more general forms? Example: Slide 8 8 Challenge 1 - General Nonlinear ID Obstacle 1: How do we compute a cost function? Prediction error (PE) cost: Maximum Likelihood (ML) cost: Slide 9 9 Computing Turn to general measurement and time update equations: Time Update Measureme nt Update Problem - closed form solutions only for special cases: Linear, Gaussian (Kalman Filter), Discrete state HMM More generally: Need to compute solution numerically Multi-dimensional integrals the main challenge Slide 10 10 Computing Slide 11 11 SEQUENTIAL IMPORTANCE RESAMPLING SIR - More commonly known as particle filtering Key idea - use the strong law of large numbers (SLLN) Suppose a vector random number generator gives realisations from a given target density Then by the SLLN, with probability one: Suggests approximate quantification How to build the necessary random number generator? Slide 12 12 Recursive solution (Particle filter) Measurement Update Time Update Resampling Slide 13 13 Example vs. Slide 14 14 History Handschin & Mayne, Intl J. Control, 1969 Resampling Approach: Gordon, Salmond & Smith, IEE Proc. Radar & Signal Processing, 1993. (1136 citations) Now widely used in signal processing, target tracking, computer vision, econometrics, robotics and statistics, control.... Some applications in system identification. Bulk of work has involved considering parameters as state variables. Slide 15 15 Back to Nonlinear System Identification Prediction error cost: General(ish) model structure Max. Likelihood cost: Slide 16 16 Nonlinear System Identification How to compute the necessary gradients? Strategies: Differencing to compute derivatives? Direct search methods: Nelder-Mead, simulated annealing? Obstacle 2: How do we compute an estimate? Gradient based search is standard practice: Slide 17 17 Expectation-Maximisation (EM) ALG. Slide 18 18 Expectation-Maximisation (EM) ALG. Example - linear system: Estimate by regression? Need state - use estimate? E.g. Kalman smoother Suggests iteration: Use estimates of A,B,C,D to estimate state ; Use estimates of state to estimate A,B,C,D; Return and do again. Slide 19 19 Expectation-Maximisation (EM) ALG. Key idea - complete and incomplete data Actual observations: Wished for (incomplete) obervations: Form estimate of wished for likelihood: E Step: Calculate M Step: Compute Slide 20 20 KEY EM Algorithm Property Bayes rule: Take conditional expectation of both sides: Increasing implies increased likelihood: Slide 21 21 Expectation-Maximisation (EM) ALG. Slide 22 22 Expectation-Maximisation (EM) ALG. Slide 23 23 Expectation-Maximisation (EM) ALG. History Generally attributed to Baum: Ann. Math. Stat. 1970; Generalised by Dempster et al: JRSS B, 1977 (9858 cites) Widely used in image processing, statistics, radar... Slide 24 24 Nonlinear system estimation Example: N=100 data points, M=100 particles, 100 experiments Slide 25 25 Evolution of Look at b parameter only - others fixed at true values: Slide 26 26 Gradient Based Search Revisited Fishers Identity Slide 27 27 EM vs Gradient search iterates Slide 28 28 Challenge 2: Application Relevant ID Traditional practice - note asymptotic results Quality of an estimate must be quantified for it to be useful Assume convergence effectively occurred for finite N Slide 29 29 Assessment & Design Often, a function of the parameters is of more interest Again - classical approach - use linear approximation: Couple with approximate Gaussianity of Slide 30 30 One perspective Need to combine prior knowledge, assumptions and data : Measure of the evidence supporting an underlying system property - parameter value, frequency response, achieved gain/phase margin...... Slide 31 31 Computing Posteriors In principle, posterior computation straightforward: Likelihood Example: Bayes Rule prior knowledge Combine: Slide 32 32 Using Posteriors Marginal on ith parameter: Other measures? Now the difficulty - using the posterior Evaluation on -dim. grid, evaluations of model order Simpsons rule - evaluation error: Slide 33 33 Markov Chain Monte Carlo (MCMC) Slide 34 34 A randomised approach Use the Strong Law of Large Numbers (SLLN) again. Build a (vector) random number generator giving realisations: Suggests the approximation: Then by the SLLN, with probability one: One view - numerical integration with intelligently chosen grid points. Slide 35 35 The Metropolis Algorithm 1. Initialise: Choose and set The required vector random number generator: Z.y=y; Z.u=u; M.A=4; g1=est(Z,M); theta=g1.theta; Slide 36 36 The Metropolis Algorithm 2. Draw a proposal value xi = theta + 0.1*randn(size(theta)); g2 = theta2m(xi,g1); Slide 37 37 The Metropolis Algorithm 3. Compute an acceptance probability: cold = validate(Z,g1); cnew = validate(Z,g2); prat = exp((-0.5/var)*(cold-cnew)*N); alpha = min(1,prat); Slide 38 38 The Metropolis Algorithm 4. Set with probability if (rand 43 Sample Histograms of There is strong evidence that the proposed controller will achieve a gain margin > 3.8 and phase margin > 95 o Slide 44 44 Conclusions Many thanks for your attention; Collective thanks to the SYSID2009 Organisation Team! Deep thanks to the Uni. Newcastle Signal Processing Micro-electonics group (sigpromu.org) Steve Weller, Chris Kellett, Tharaka Dissanayake, Peter Schreier, Sarah Johnson, Geoff Knagge, Bjrn Rffer, Adrian Wills, Lawrence Ong, Dale Bates, Ian Griffiths, David Hayes, Soren Henriksen, Adam Mills, Alan Murray who endured multiple road-test versions of this talk, that were even worse than this one.