Lecture 13 - Handling Nonlinearity

Transcription

1 Lecture 3 - Handling Nonlinearit Nonlinearit issues in control practice Setpoint scheduling/feedforward path planning repla - linear interpolation Nonlinear maps B-splines Multivariable interpolation: polnomials/splines/rbf Neural Networks Fuzz logic Gain scheduling Local modeling EE39m - Winter 003 Control Engineering 3-

2 Nonlinearit in control practice Here are the nonlinearities we alread looked into Constraints - saturation in control anti-windup in PID control MPC handles the constraints Control program, path planning Static optimization Nonlinear dnamics dnamic inversion nonlinear IMC nonlinear MPC One additional nonlinearit in this lecture Controller gain scheduling EE39m - Winter 003 Control Engineering 3-

3 Dealing with nonlinear functions Analtical epressions models are given b analtical formulas, computable as required rarel sufficient in practice Models are computable off line pre-compute simple approimation on-line approimation Models contain data identified in the eperiments nonlinear maps interpolation or look-up tables Advanced approimation methods neural networks EE39m - Winter 003 Control Engineering 3-3

4 EE39m - Winter 003 Control Engineering 3-4 Path planning Real-time repla of a pre-computed reference traector d t or feedforward vt Reproduce a nonlinear function d t in a control sstem Path planner, data arras Y,Θ d t t d t Y t Y t Θ n n d n d d Y Y Y Y M M, Code:. Find, such that. Compute t

5 Linear interpolation vs. table look-up linear interpolation is more accurate requires less data storage simple computation EE39m - Winter 003 Control Engineering 3-5

6 Empirical models Aerospace - most developed nonlinear approaches automotive and process control have second place Aerodnamic tables Engine maps et turbines automotive Process maps, e.g., in semiconductor manufacturing Empirical map for a attenuation vs. temperature in an optical fiber Eample TEFTrailing Edge Flap EE39m - Winter 003 Control Engineering 3-6

7 Approimation Interpolation: compute function that will provide given values in the nodes not concerned with accurac in-between the nodes Approimation compute function that closel corresponds to given data, possibl with some error might provide better accurac throughout Y EE39m - Winter 003 Control Engineering 3-7

8 B-spline interpolation st-order look-up table, nearest neighbor nd-order linear interpolation d t Y B t n-th order: Piece-wise n-th order polnomials, matched n- derivatives zero outside a local support interval support interval etends to n nearest neighbors EE39m - Winter 003 Control Engineering 3-8

9 B-splines Accurate interpolation of smooth functions with relative few nodes For -D function the gain from using high-order B-splines is not worth an added compleit Introduced and developed in CAD for -D and 3-D curve and surface data Are used for defining multidimensional nonlinear maps EE39m - Winter 003 Control Engineering 3-9

10 Multivariable B-splines Regular grid in multiple variables Tensor product B-splines Used as a basis of finite-element models u, v w, k, kb u Bk v EE39m - Winter 003 Control Engineering 3-0

11 EE39m - Winter 003 Control Engineering 3- Linear regression for nonlinear map Linear regression Multidimensional B-splines Multivariate polnomials RBF - Radial Basis Functions T φ ϕ k n n k n,, K K ϕ K c a e c R ϕ n M

12 Linear regression approimation Nonlinear map data available at scattered nodal points Y [ N K ] K N Linear regression map Linear regression approimation Y [ N ] T φ K φ Φ T regularized least square estimate of the weight vector ˆ T T ΦΦ ri ΦY Works ust the same for vector-valued data! EE39m - Winter 003 Control Engineering 3-

13 Nonlinear map eample - Epi Epitaial growth semiconductor process process map for run-to-run control EE39m - Winter 003 Control Engineering 3-3

14 EE39m - Winter 003 Control Engineering 3-4 Linear regression for Epi map Linear regression model for epitaial grouth 0 p c p c w w w w p { { { { w c w c w c w c p c { { { { w c c v c v c v p c,,, T φ ϕ

15 Neural Networks An nonlinear approimator might be called a Neural Network RBF Neural Network Polnomial Neural Network Linear in parameters B-spline Neural Network Wavelet Neural Network MPL - Multilaered Perceptron Nonlinear in parameters Works for man inputs w,0 f w,, w,0 f w, f e f EE39m - Winter 003 Control Engineering 3-5

16 Multi-Laered Perceptrons Network parameter computation training data set parameter identification F ; Noninear LS problem V F Iterative NLS optimization Levenberg-Marquardt Backpropagation ; variation of a gradient descent min Y [ N K ] K N EE39m - Winter 003 Control Engineering 3-6

17 Fuzz Logic Function defined at nodes. Interpolation scheme Fuzzfication/de-fuzzfication interpolation Linear interpolation in -D µ µ µ Marketing communication and social value Computer science: emphasis on interaction with a user EE - emphasis on mathematical computations EE39m - Winter 003 Control Engineering 3-7

18 Neural Net application Internal Combustion Engine maps Eperimental map: data collected in a stead state regime for various combinations of parameters -D table NN map approimation of the eperimental map MLP was used in this eample works better for a smooth surface spark advance RPM EE39m - Winter 003 Control Engineering 3-8

19 Linear feedback in a nonlinear plant Simple eample f u k g u d u ff Linearizing propert of feedback for k >> d k g f g u ff d Eample: Control design requires varing k, u ff, d process These variables are gain scheduled on d u Controller Plant EE39m - Winter 003 Control Engineering 3-9

20 Gain scheduling Single out several regimes - model linearization or eperiments Design linear controllers in these regimes: setpoint, feedback, feedforward Approimate controller dependence on the regime parameters Nonlinear sstem Linear interpolation: Y Θ ϕ Θ EE39m - Winter 003 Control Engineering 3-0 Y Y vec A vec B vec C vec D

21 Gain scheduling - eample Flight control Flight envelope parameters are used for scheduling Shown Approimation nodes Evaluation points Ke assumption Attitude and Mach are changing much slower than time constant of the flight control loop EE39m - Winter 003 Control Engineering 3-

22 Local Modeling Based on Data Data mining in the loop Honewell product Multidimensional Data Cube Heat Loads Relational Database Heat demand Quer point What if? Outdoor temperature Time of da Forecasted variable Eplanator variables EE39m - Winter 003 Control Engineering 3-