The Problem 2(47) Nonlinear System Identification: A Palette from Off-white to Pit-black. A Common Frame 4(47) This Presentation... 3(47) ˆθ = arg min

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1 y y y y y Tunn: measurement; Tjock: simulation [s] 6 OUTPUT # 6 - INPUT # OUTPUT # INPUT # - 6 The Problem (7) : A Palette from Off-white to Pit-black Missile Dynamics: Lennart Ljung Automatic Control, ISY, Linköpings Universitet Pulp Buffer Vessel: Forest Crane: Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 This Presentation... (7) A Common Frame (7) The world of nonlinear models is very diverse. A common framework: Discrete time observations of inputs and outputs:... aims at a display of the essence of the problem of non-linear identification a color-coded overview of typical parametric approaches Z t = [u(), u(),..., u(t), y(), y(),..., y(t)] A model is a parameterized predictor of the next output y(t) made at time t : ŷ(t t, θ) = ŷ(t θ) = h(z t, θ) The parameters can be estimated using the prediction error method: ˆθ = arg min θ (could be Maximum Likelihood) y(t) h(z t, θ) t Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

2 y y What s Special with Nonlinear Models? (7) The Model Surface 6(7) ŷ(t θ) = h(z t, θ) is a nonlinear function of Z. What makes the nonlinear problem much more difficult and rich than the linear problem? Two major reasons: The richness of the model surface Propagation of noise signals to the output not immediate Let us take Z t = [u(t ), u(t )] and a scalar output y(t). A model is then a surface in the space spanned by [y(t), u(t ), u(t )] and the estimation task is to estimate this surface u(t ) u(t ) Linear: Nonlinear: ŷ(t) = a u(t ) + a u(t ) ŷ(t) = h(u(t ), u(t )) 6 x u(t ) u(t ) The observations Z t are points in this space. Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Propagation of Noise Signals 7(7) The Palette of Nonlinear Models 8(7) In linear systems that are cascaded we can always propagate the noise signals to the output: y = Gu + He, where H picks up the coloring obtained by propagating the noise through a linear system. For nonlinear systems, this in generally not possible. Example: A linear system + noise, z = Gu + w is followed by a static nonlinearity f (z). At the output we have y(t) =f (Gu + w) = f (Gu) + w w =f (Gu + w) f (Gu) Here, w is not really a noise : It is clearly contaminated with the input u which will create bias-problems when minimizing the output error. Indicates that the calculation of the true predictor could be challenging. White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures universal approximators Pit-black: Non-Parametric Smoothing Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

3 Off-white Models: Physical Modeling 9(7) Example: Missile (7) Perform physical modeling (e.g. in MODELICA) and denote unknown physical parameters by θ. Collect the model equations as ẋ(t)= f (x(t), u(t), θ) y(t)= h(x(t), u(t), θ) (or in DAE, Differential Algebraic Equations, form.) For each parameter θ this defines a simulated output ŷ(t θ) which is the parameterized function from sampled data ŷ(t θ) = h(z t, θ) (Z t = u t ) in somewhat implicit form. To be a correct predictor this really assumes white measurement noise. Then the estimate is the θ that minimizes the output error fit t y(t) ŷ(t θ) Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Nonlinear System inputs, Identification outputs, 6 unknown parameters. The Equations (7) Initial Fit between Model and Data (7) function [dx, y] = missile(t, x, p, u); MISSILE A non-linear missile system. Output equation. y = [x();... x();... x();... -p(8)u()(p()x()+p()u())/p();... -p(8)u()(p()x()+p()u())/p()... ]; State equations. dx = [/p(9)(p(7)p(8)(p()x()+.p(6)p(7)x()/u()+... % Angular velocity around x-axis. p(7)u())u()-(p()-p())x()x())+... p()(u(6)-x());... /p()(p(7)p(8)(p(8)x()+.p(9)p(7)x()/u() p()-p() unknown parameters u, y : measured inputs and outputs y y y y Tunn: measurement; Tjock: simulation y Lennart Ljung Brussels Workshop, April, [s] Lennart Ljung Brussels Workshop, April, 7

4 Adjusted Fit between Model and Data (7) Off-white Models with Noise Models (7) y y y y y Tunn: measurement; Tjock: simulation [s] Lennart Ljung Brussels Workshop, April, 7 The (output error, off-white) approach is conceptually simple, but could be very demanding in practice. A main shortcoming is the use of the output error criterion, which really assumes white measurement noise. Noise signals in nonlinear models cannot really be propagated to the output. If the size of the noise is non-trivial, more careful noise modeling should be done: ẋ(t)= f (x(t), u(t), w(t), θ) y(t)= h(x(t), u(t), θ) + e(t) where w and e are white noises.to find correctly predicted outputs ŷ(t Z t, θ) = E(y(t) Z t, θ) is then the well-known intractable problem of nonlinear filtering. Often one has to resort to some simplistic observer. Lennart Ljung Brussels Workshop, April, 7 Probabilistic Learning (7) The Palette of Nonlinear Models 6(7) Recently however, with the increasing computing power, new computing intensive simulation based methods have been developed for nonlinear filtering problem, and hence for applying the Maximum Likelihood method to non-linear state space models. Particle filtering, Markov Chain Monte Carlo, MCMC, Sequential Monte Carlo... Loosely, and briefly, these are based on simulation of the noisy state-space model, and evaluating the state probabilities, focusing on paths that give the measured output sequence. It is a central current research area, Probabilistic Learning, to make these calculations as efficient as possible. See e.g. Thomas B. Schön, Andreas Svensson, Lawrence Murray, and Fredrik Lindsten: Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo, ArXiv White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures universal approximators Pit-black: Non-Parametric Smoothing Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

5 Smoke-grey: Semi-physical Models 7(7) Buffer Vessel Dynamics 8(7) OUTPUT # 8 6 Apply non-linear transformations to the measured data, so that the transformed data stand a better chance to describe the system in a linear relationship. Rules: Only high-school physics and max minutes Toy Example: Immersion heater: Input: voltage to the heater. Output: temperature of the fluid Square the voltage! Sense morale: No excuse for not thinking over the basic physical facts! Another example:... 6 INPUT # 6 κ-number of outflow, κ-number of inflow, 6 6 flow volume Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Linear Model Based on Raw Data 9(7) Now it s time to (7) 6 Measured Output and Simulated Model Output Measured Output mraw Fit:.% Think:... No mixing ( Plug flow ): The vessel is then just a pure time delay for the pulp flow: Delay time: Vessel Volume/Pulp Flow (dimension time.) y Dashed line: κ-number after the vessel, actual measurements. Solid line: Simulated κ-number using the input only and a process model estimated using the first data points. G(s) = s e 8s x Perfect mixing in tank: A text-book first order system with gain= and time constant = Volume/Flow So if Volume and Flow are changing, we have a non-linear system! The natural time variable is really Volume/Flow, (which we have measured). Let us re-sample the observed data according to this natural time variable. Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

6 y 6 6 Measured Output and Simulated Model Output Measured Output mraw Fit:.% Re-sample Data (7) Semi-physical Model (7) z = [y,u]; pf = flow./level; t = :length(z) newt = interp([cumsum(pf),t],[pf():sum(pf)] ); newz = interp([t,z], newt); κ number of Inflow 6 G(s) = s e 69.8s Measured Output and Simulated Model Output Measured Output mves Fit: 6.9% x Recall Linear model κ number of Outflow y 6 The semi-physical model gives a sufficiently good description of the buffer, to allow proper time-marking of the pulp before and after. 8 Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 The Palette of Nonlinear Models (7) White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures universal approximators Pit-black: Non-Parametric Smoothing Steel-Grey: Composite Local Models (7) Non-linear systems are often handled by linearization around a working point. The idea behind Composite Local (Local Linear) Models is to deal with the nonlinearities by selecting or averaging over relevant linearized models. Example: Tank with inflow u and free outflow y and level h: (Bernoulli s) equations: ḣ = h + u; y = h Linearize around level h with corresponding flows u = y = h : ḣ = h (h h ) + (u u ); y = y + h (h h ) Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

7 Tank Example, ctd (7) Data and Linear Model 6(7) Sampled data model around level h (Sampling time T s ): Measured data: Linear Model (d = ) y(t) = γ(h ) + α(h )y(t T s ) + β(h )u(t T s ) = θ T (h )ϕ(t) An ARX-model with level-dependent parameters. Now compute linearized model for d different levels, h, h,..., h d. Total model: select or average over these local models ŷ(t) = d w k (h(t), h k )θ T (h k )ϕ(t) k= Inflöde u Nivå h Choices of weights w k :.... Thick line: Model. Thin: Measured. Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Local Linear Models 7(7) Composite Local Models: General Comments 8(7).. Two levels (models) (d=) Five levels (models) (d = ).. Let the measured working point variable (tank level in example) be denoted by ρ(t) (sometimes called regime variable or scheduling variable). If the regime variable is partitioned into d values ρ k, and model output according to value ρ k is ŷ (k) (t) the predicted output will be ŷ(t) = d w k (ρ(t), ρ k )ŷ (k) (t) k= If the prediction ŷ (k) (t) corresponding to ρ k is linear in the parameters, ŷ (k) (t) = ϕ T (t)θ (k), and the weights w are fixed, the whole model will be a linear regression. Important connections to active research areas LPV (Linear Parameter-Varying) Models Hybrid Models ( w(, ) is estimated too.) Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

8 LPV State Space Models 9(7) The Palette of Nonlinear Models (7) x(t + ) = A(ρ)x(t) + B(ρ)u(t) y(t) = C(ρ)x(t) is a linear model for each fixed ρ. If ρ Ω = {ρ,..., ρ d } it is a set of local linear models.if ρ = ρ(t) is time varying, we have a Linear Parameter Varying model. A basic difficulty is to find a common state basis from input/output observations and to manage the time variable in ρ(t) in relation to the observations. White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models Black: Flexible structures universal approximators Pit-black: Non-Parametric Smoothing Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Slate Grey: Block-oriented Models (7) Common Models (7) Building Blocks: Wiener Linear Dynamic System G(s) Hammerstein Nonlinear static function f (u) Hammerstein- Wiener Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

9 Other Combinations (7) Active Research Field: Example: Hydraulic Crane Data (7) These are data from a forest harvest machine: OUTPUT # With the linear blocks parameterized as a linear dynamic system and the static blocks parameterized as a function ( curve ), this gives a parameterization of the output as ŷ(t θ) = h(z t, θ) and the general approach of model fitting can be applied. However, in this contexts many algorithmic variants have been suggested. Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, INPUT # 6 Input: Hydraulic Pressure. Output: Tip Position Linear Model (7) Hammerstein Model of the Hydraulic Crane 6(7) Black: Measured Output Blue: Model Simulated Output Linear model: Fit.7 % Hammerstein model: Fit 7.6 % The Hammerstein Model gives a good fit. The extra flexibility offered by the input nonlinearity is quite useful, (even though no direct physical explanation is obvious.) 6 8 Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

10 Noise Effects in Hammerstein-Wiener? 7(7) Output Error Method for the HW Model 8(7) There is frequently reason to assume that some noise enters before the output nonlinearity g. wt vt et Bode for linear system Input NLs Output NLs Input nonlinearity Saturation Input nonlinearity Deadzone Output nonlinearity Deadzone Output nonlinearity Satu G G Hw(q, µ) Hv(q, η) z w t z v t..... ut νt z g t xt f(, α) G(q, ϑ) g(, β) yt What happens if we propagate that noise to the output and apply an Output Error criterion to the above input output system? G. G OE Method OE ML NI Method Method OE ML NI ML EM Method Method Method ML NI OE ML EM Method Method Method Blue curve: Plots for the true system Red curves: Median and standard deviations for estimated systems over 8 Monte Carlo runs Number of observed data:.. ML NI ML EM Method Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Maximum Likelihood (EM) for HW Models 9(7) Results for the ML-EM Method for HW Models (7) It is clear that more effort must be paid to the noise structure. We turn to the Maximum Likelihood method for the HW model structure. It is a complication that the ML criterion cannot easily be formed. But if the unmeasured noise z w t were known it is easy to compute the ML criterion. So, treat is as incomplete data X and apply the EM algorithm, which iterates between estimating X and estimating the model for this X. Ref: Adrian Wills, Thomas B. Schön, Lennart Ljung, Brett Ninness: Identification of Hammerstein-Wiener models, Automatica Bode for linear system Input NLs Output NLs G.. G OE Method.. Input nonlinearity Saturation Input nonlinearity DeadzoneOutput nonlinearity DeadzoneOutput nonlinearity Saturation G OE G ML NI Method Method OE ML NI Method ML EM Method Method OE ML NI Method ML EM Method Method ML NI ML EM Method Method ML EM Method Blue curve: Plots for the true system Red curves: Median and standard deviations for estimated systems over 8 Monte Carlo runs Number of observed data:.... Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

11 The Palette of Nonlinear Models (7) Black-box Models (7) A general way to generate very flexible mappings from Z t to ŷ is White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures universal approximators Pit-black: Non-Parametric Smoothing x(t + ) = f (x(t), u(t), y(t), θ) ŷ(t θ) = h(x(t), θ) where f and/or h are flexible functions e.g. in terms of basis function expansions. Working with both f and h may be too general, and a very common special case is the NLARX model: x(t) = ϕ(t) = [y(t ),..., y(t na), u(t ),... u(t nb)] T ŷ(t θ) = h(ϕ(t), θ) = d α k κ k (ϕ(t)) k= for some basis functions κ k Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Basis Functions (7) It is natural to think of Taylor expansions: κ k (ϕ) = ϕ k. If na = (NLFIR), this becomes the classical Volterra series expansion. But note that if dimϕ = r, then ϕ k has r k components! A more common choice is to form all the basis functions κ k from one mother function κ and scale and position the argument differently: κ k (ϕ) = κ(β k (ϕ γ k )) ŷ(t θ) = d α k κ(β k (ϕ γ k )), θ = {α k, β k, γ k } k= Intuitive picture: Think of a scalar ϕ and let κ(z) be a unit pulse for z. Then κ(β(ϕ γ)) is a pulse of width /β starting in ϕ = γ. The sum above is then a piecewise constant function, capable of approximation "any" function arbitrary well for large enough d. ANN, LS-SVM etc (Sjöberg et al, Automatica 99) Focusing on f (7) Basis functions for f, h: x(t + ) = f (x(t), u(t), θ) ŷ(t θ) = h(x(t), θ) Polynomial expansion (Paduart et al, Automatica, ) Gaussian Process (GP) model for f (Rasmussen, inverted pendulum experiments; x measured); [Basis expansion in terms of the eigenfunctions associated with the kernel (covariance function for the GP)] Sine basis; If process noise affects x, particle filtering must be applied to find the predictor (Svensson and Schön, Automatica 7) Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7

12 y The Palette of Nonlinear Models (7) Pit-black Models: Non-Parametric Smoothing Methods 6(7) x White: Known model Off-white: Careful Physical Modeling w or w/o noise models Smoke-grey: Semi-physical modeling (Could be used more!) Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. Black: Flexible structures universal approximators Pit-black: Non-Parametric Smoothing 6 u(t ) Form the model surface h(ϕ(t)) by smoothing over the observation points in the space! Even Blacker! Huge literature Mostly in the statistical community and now also in machine learning Important to find lower dimensional manifolds ( counterpart of PCA in linear modelling). Concepts like Manifold Learning and Local Linear Embedding become central. u(t ) Lennart Ljung Brussels Workshop, April, 7 Lennart Ljung Brussels Workshop, April, 7 Conclusions 7(7) Confusingly many approaches! A user-oriented roadmap would be excellent! Lennart Ljung Brussels Workshop, April, 7