TutorialPCET – Polynomial Chaos Expansion Toolbox

Stefan Streif Felix Petzke Ali Mesbah

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PCET Overview

What’s PCET?• MATLAB™ based toolbox for uncertainty quantification using the

polynomial chaos expansion (PCE)• simple syntax for definition of models, uncertainties, …• general purpose code and mid-level functions• free for academic purposes:

www.TU-Chemnitz.de/etit/control/research/PCET

Possible applications and problem setup• systems: continuous- or discrete-time systems with uncertain initial

conditions, parameters and inputs and outputs

• problems and applications: stochastic MPC, stochastic active FDI, experiment-design, parameter and state estimation, uncertainty quantification, …

Aims of this tutorial?explain main concepts and typical work flow at simple example

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Outline

1. Quick guide

2. Typical workflow illustrated at a simple example

3. Examples overview

4. References

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Quick guide – installation, files, folders, documentation

InstallationUnpack ZIP-file into current MATLAB™ folder and run

installPCET

generates documentation, sets paths, checks dependencies, …

Help and documentationFor short introduction to syntax etc. type

help PCET

For further and more detailed information on different functions, extended functionality, please see examples, help browser or type e.g.:

help PCETcompose

doc PCETcompose

Please contact Stefan.Streif@etit.TU-Chemnitz.de if you need further help.

Files and folders• main files (starting with PCET…) in main folder• auxiliary files in subfolders (shouldn’t be used directly)• examples in folders ./PCET/examples/ex…

mailto:Stefan.Streif@etit.TU-Chemnitz.de

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Typical workflow

7. Modify inputs, uncertaintiesof parameters and initial cond.

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6. Visualize or analyze results

5. Simulate PCE-system or perform MC simulations

4. Set options for simulation, integration, …

3. Write files for simulation

2. Compose the PCE-system

1. Define dynamical system and uncertainties, inputs, …

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Define system, inputs, uncertainties, …

1. Define dynamical system and uncertainties, inputs, …

states(1).name = 'x';

states(1).dist = 'uniform';

states(1).data = [0.9 1.3];

states(1).rhs = 'a*x + u';

parameters(1).name = 'a';

parameters(1).dist = 'normal';

parameters(1).data = [-5 0.1];

outputs(1).name = 'y';

outputs(1).rhs = '3*x^2';

inputs(1).name = 'u';

inputs(1).rhs = 'piecewise(ut,uv,t)';

inputs(1).ut = 1; % time of step

inputs(1).uv = 1; % height of step

Example ODE:

Implementation:

state

parameter

output

input(here piecewise constant)

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Compose and export problem

2. Compose the PCE-systemPCE is performed for the specified system, uncertainties, …

may take a long time depending on the number of uncertainties and PCE order• returns: structure sys that will be used by subsequent functions

3. Write files for simulationr.h.s. of continuous or discrete-time dynamics (PCE-system or Monte-Carlo) andof output equations have to written to files (used later by simulation functions etc.)

sys = PCETcompose(states,parameters, ...

inputs,outputs,pce_order)

PCETwriteFiles(sys,'ex1_ODE.m','ex1_OUT.m', ...

'ex1_MCODE.m','ex1_MCOUT.m')

r.h.s. of PCE system(for simulation)

r.h.s of PCE output equations

r.h.s. of MC system(for simulation)

r.h.s of MC output equations

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Set options and update of uncertainties

4. Set options for simulation, integration, …

simoptions.tspan = [0 3];

simoptions.dt = 0.05;

Continuous-time:simoptions.setup = odeset;

simoptions.solver = ‘ode15s';

Discrete-time:simoptions.solver = ‘discrete-time';

7. Update/modify uncertainties of initial conditions or parameter

set simulation interval

set time step-size for (simulation and/orevaluation

set integrator options

choose integrator

choose discrete-timesimulation

sys = PCETupdate(sys,‘x',[1.1 1.4],‘a',[-6 0.1]);

update ‘data’ (see step 1) of variables ‘x’ and ‘a’

Optional and possibly at a later time-point:

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Simulation

5a. Simulate PCE system (and output equations) using Galerkin projectionPCEresults = ...

PCETsimGalerkin(sys,'ex1_ODE','ex1_OUT',simoptions)

PCEresults/MCresults contain all simulation data which can be processed and analyzed further

5b.Simulate PCE system (and output equations) using collocation

and/or:

samples_col = PCETsample(sys,'basis',col_samples)

PCEresults = ...

PCETsimCollocation(sys,'ex1_MCODE', ...

'ex1_MCOUT',samples_col,simoptions)

5c. Monte-Carlo simulation (states and output equations)

and/or:

samples = PCETsample(sys,'variables',mc_samples)

MCresults = ...

PCETsimMonteCarlo(sys,'ex1_MCODE','ex1_MCOUT', ...

samples,simoptions)

draws col_samples samples, evaluates basis polyn. and uses collocation

draws mc_samples samples, evaluates uncert. and simulates system

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Visualize and analyze results

6a. MomentsMomMats = PCETmomentMatrices(sys,4)

compute moment matrices (up to 4th order)need not to recomputed (can be stored for repeated use)

6b.Histogram

and/or:

samples_basis = PCETsample(sys,'basis',n_samples)

samples_y = samples_basis' * PCEresults.y.pcvals

draws n_samples samples and evaluates basis polynomials

use MATLAB™ functions for histogram plots (e.g. histc)

PCEresults.y.moments = ...

PCETcalcMoments(sys,MomMats,results.y.pcvals)

use MATLAB™ function plot to display moments stored in PCEresults.y.moments

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Quick guide – examples

ex1_continuous_simulation

• uncertainty quantification for a nonlinear system• plotting of moments, histograms, …

ex2_lti_systems

• uncertainty quantification for a linear system• specification of system dynamics in matrix form

ex3_discrete_simulation

• uncertainty quantification for a discrete-time nonlinear system

ex4_fault_detection

• input design for active fault detection for a two-tank-system• distribution dissimilarity measure• embedding uncertainty quantification into an nonlinear optimization scheme

ex5_model_discrimination

• input design for model discrimination for biochemical reaction networks

[Mesbah et al. (2014)]

[Streif et al. (2014)]

ex6_SNMPC

• stochastic nonlinear model predictive control for a crystallization process• chance constraints

[Mesbah et al. (2014)]

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Selected PCE applications/publications

• J. A. Paulson, S. Streif, and A. Mesbah. Stability for receding-horizon stochastic model predictive control. In Proc. American Control Conference (ACC), Chicago, IL, July 2015.

• J. A. Paulson, A. Mesbah, S. Streif, R. Findeisen, and R. D. Braatz. Fast stochastic model predictive control of high-dimensional systems. In Proc. 53rd IEEE Conference on Decision and Control (CDC), Los Angeles, CA, December 2014.

• A. Mesbah, S. Streif, R. Findeisen, and R. D. Braatz. Stochastic nonlinear model predictive control with probabilistic constraints. In Proc. American Control Conference (ACC), Portland, Oregon, June 2014.

• A. Mesbah, S. Streif, R. Findeisen, and R. D. Braatz. Active fault diagnosis for nonlinear systems with probabilistic uncertainties. In Proc. 19th IFAC World Congress, Cape Town, South Africa, 2014.

• S. Streif, F. Petzke, A. Mesbah, R. Findeisen, and R. D. Braatz. Optimal experimental design for probabilistic model discrimination using polynomial chaos. In Proc. 19th IFAC World Congress, Cape Town, South Africa, 2014.

• S. Streif, M. Karl, and A. Mesbah. Stochastic nonlinear model predictive control with efficient sample approximation of chance constraints. arXiv: 1410.4535.

• A. Mesbah and S. Streif. A probabilistic approach to robust optimal experiment design with chance constraints. In. Proc. International Symposium on Advanced Control of Chemical Processes (ADCHEM), Whistler, Canada, 2015.

source codes available, see examples!