The Problem 2(47) Nonlinear System Identification: A Palette from Tunn: measurement; Tjock: simulation 5 y 1 0 −5 0 1 2 3 4 5 6 7 8 9 10 0.5 Off-white to Pit-black y 2 0 −0.5 0.5 0 1 2 3 4 5 6 7 8 9 10 y 3 0 −0.5 0 1 2 3 4 5 6 7 8 9 10 100 y 4 0 −100 0 1 2 3 4 5 6 7 8 9 10 200 Missile Dynamics: y 5 0 −200 0 1 2 3 4 5 6 7 8 9 10 [s] OUTPUT #1 25 20 15 10 5 0 100 200 300 400 500 600 INPUT #1 20 15 Pulp Buffer Vessel: 10 0 100 200 300 400 500 600 Lennart Ljung OUTPUT #1 4 2 0 -2 Automatic Control, ISY, Linköpings -4 0 100 200 300 400 500 600 INPUT #1 1.5 1 Universitet 0.5 0 Forest Crane: -0.5 -1 0 100 200 300 400 500 600 AUTOMATIC CONTROL AUTOMATIC CONTROL Lennart Ljung Brussels Workshop, April 25, 2017 Lennart Ljung Brussels Workshop, April 25, 2017 REGLERTEKNIK REGLERTEKNIK Nonlinear System Identification LINKÖPINGS UNIVERSITET Nonlinear System Identification LINKÖPINGS UNIVERSITET This Presentation ... A Common Frame 3(47) 4(47) The world of nonlinear models is very diverse. A common framework: Discrete time observations of inputs and outputs: Z t = [ u ( 1 ) , u ( 2 ) , ..., u ( t ) , y ( 1 ) , y ( 2 ) , ..., y ( t )] ... aims at A model is a parameterized predictor of the next output y ( t ) made at a display of the essence of the problem of non-linear time t − 1 : identification y ( t | θ ) = h ( Z t − 1 , θ ) a color-coded overview of typical parametric approaches y ( t | t − 1, θ ) = ˆ ˆ The parameters can be estimated using the prediction error method: ˆ � y ( t ) − h ( Z t − 1 , θ ) � 2 θ ∑ θ = arg min t (could be Maximum Likelihood) AUTOMATIC CONTROL AUTOMATIC CONTROL Lennart Ljung Brussels Workshop, April 25, 2017 Lennart Ljung Brussels Workshop, April 25, 2017 REGLERTEKNIK REGLERTEKNIK LINKÖPINGS UNIVERSITET LINKÖPINGS UNIVERSITET Nonlinear System Identification Nonlinear System Identification
What’s Special with Nonlinear Models? The Model Surface 5(47) 6(47) Let us take Z t − 1 = [ u ( t − 1 ) , u ( t − 2 )] and a scalar output y ( t ) . A model is then a surface in the space spanned by [ y ( t ) , u ( t − 1 ) , u ( t − 2 )] and the estimation task is to estimate this y ( t | θ ) = h ( Z t − 1 , θ ) is a nonlinear function of Z . What makes the ˆ surface. nonlinear problem much more difficult and rich than the linear 5 x 10 3 4 problem? 2.5 3 2 * * 2 * * * * * * * * * * * * 1.5 * * * * * * * * 1 * * Two major reasons: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * 1 * * * * * * * 0 * * * * * * * y * * * * * * y * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0.5 * * * * −1 * * * * * * * * * * * * * * * * * * * 0 * * −2 * * * * * * * * The richness of the model surface * * * * −0.5 −3 * −1 −4 6 6 5 5 5 5 4 4 Propagation of noise signals to the output not immediate 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 u(t−2) u(t−2) u(t−1) u(t−1) Linear: Nonlinear: y ( t ) = a 1 u ( t − 1 ) + a 2 u ( t − 2 ) ˆ y ( t ) = h ( u ( t − 1 ) , u ( t − 2 )) ˆ The observations Z t are points in this space. AUTOMATIC CONTROL AUTOMATIC CONTROL Lennart Ljung Brussels Workshop, April 25, 2017 Lennart Ljung Brussels Workshop, April 25, 2017 REGLERTEKNIK REGLERTEKNIK Nonlinear System Identification LINKÖPINGS UNIVERSITET Nonlinear System Identification LINKÖPINGS UNIVERSITET Propagation of Noise Signals The Palette of Nonlinear Models 7(47) 8(47) 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 White: Known model system. Off-white: Careful Physical Modeling w or w/o noise models For nonlinear systems, this in generally not possible. Smoke-grey: Semi-physical modeling (Could be used more!) Example: A linear system + noise, z = Gu + w is followed by a static nonlinearity f ( z ) . At the output we have Steel-grey: Local Linear Models Slate-grey: Block-oriented Models. y ( t ) = f ( Gu + w ) = f ( Gu ) + ˜ w Black: Flexible structures – universal approximators w = f ( Gu + w ) − f ( Gu ) ˜ Pit-black: Non-Parametric Smoothing 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. AUTOMATIC CONTROL AUTOMATIC CONTROL Lennart Ljung Brussels Workshop, April 25, 2017 Lennart Ljung Brussels Workshop, April 25, 2017 REGLERTEKNIK REGLERTEKNIK LINKÖPINGS UNIVERSITET LINKÖPINGS UNIVERSITET Nonlinear System Identification Nonlinear System Identification
Off-white Models: Physical Modeling Example: Missile 9(47) 10(47) Perform physical modeling (e.g. in M ODELICA ) and denote unknown physical parameters by θ . Collect the model equations as x ( 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 ˆ y ( t | θ ) which is the parameterized function from sampled data ( Z t − 1 = u t − 1 ) y ( t | θ ) = h ( Z t − 1 , θ ) ˆ in somewhat implicit form. To be a correct predictor this really assumes white measurement noise. Then the estimate is the θ that y ( t | θ ) � 2 minimizes the output error fit ∑ t � y ( t ) − ˆ AUTOMATIC CONTROL AUTOMATIC CONTROL Lennart Ljung Brussels Workshop, April 25, 2017 Lennart Ljung Brussels Workshop, April 25, 2017 REGLERTEKNIK REGLERTEKNIK 10 inputs, 5 outputs, 16 unknown parameters. Nonlinear System Identification LINKÖPINGS UNIVERSITET Nonlinear System Identification LINKÖPINGS UNIVERSITET The Equations Initial Fit between Model and Data 11(47) 12(47) Tunn: measurement; Tjock: simulation 20 function [dx, y] = missile(t, x, p, u); 0 y 1 MISSILE A non-linear missile system. −20 0 1 2 3 4 5 6 7 8 9 10 Output equation. y = [x(1); ... x(2); ... x(3); ... 1 -p(18)*u(4)*(p(1)*x(5)+p(2)*u(3))/p(22); ... y 2 0 -p(18)*u(4)*(p(3)*x(4)+p(4)*u(2))/p(22) ... ]; −1 0 1 2 3 4 5 6 7 8 9 10 5 State equations. dx = y 3 0 [1/p(19)*(p(17)*p(18)*(p(5)*x(5)+0.5*p(6)*p(17)*x(1)/u(5)+ ... % −5 Angular velocity around x-axis. 0 1 2 3 4 5 6 7 8 9 10 100 p(7)*u(1))*u(4)-(p(21)-p(20))*x(2)*x(3))+ ... p(23)*(u(6)-x(1)); ... 0 y 4 1/p(20)*(p(17)*p(18)*(p(8)*x(4)+0.5*p(9)*p(17)*x(2)/u(5)+ ... ... −100 0 1 2 3 4 5 6 7 8 9 10 p(1)-p(25) unknown parameters u, y : measured inputs and outputs 500 y 5 0 −500 0 1 2 3 4 5 6 7 8 9 10 [s] AUTOMATIC CONTROL AUTOMATIC CONTROL Lennart Ljung Brussels Workshop, April 25, 2017 Lennart Ljung Brussels Workshop, April 25, 2017 REGLERTEKNIK REGLERTEKNIK LINKÖPINGS UNIVERSITET LINKÖPINGS UNIVERSITET Nonlinear System Identification Nonlinear System Identification
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