Tutorial PCET Polynomial Chaos Expansion Toolbox. Stefan Streif Felix Petzke Ali Mesbah

Transcription

1 Tutorial PCET Polynomial Chaos Expansion Toolbox Stefan Streif Felix Petzke Ali Mesbah

2 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: 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 2

3 Outline 1. Quick guide 2. Typical workflow illustrated at a simple example 3. Examples overview 4. References 3

4 Quick guide installation, files, folders, documentation Installation Unpack ZIP-file into current MATLAB folder and run installpcet generates documentation, sets paths, checks dependencies, 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 Help and documentation For 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. 4

5 problem specific Stefan Streif TU Chemnitz PCET Tutorial initialization Typical workflow 1. Define dynamical system and uncertainties, inputs, 2. Compose the PCE-system 3. Write files for simulation 4. Set options for simulation, integration, 5. Simulate PCE-system or perform MC simulations 6. Visualize or analyze results 7. Modify inputs, uncertainties of parameters and initial cond. 5

6 Define system, inputs, uncertainties, 1. Define dynamical system and uncertainties, inputs, Example ODE: Implementation: states(1).name = 'x'; states(1).dist = 'uniform'; states(1).data = [ ]; 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 state parameter output input (here piecewise constant) 6

7 Compose and export problem 2. Compose the PCE-system PCE is performed for the specified system, uncertainties, sys = PCETcompose(states,parameters,... inputs,outputs,pce_order) 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 simulation r.h.s. of continuous or discrete-time dynamics (PCE-system or Monte-Carlo) and of output equations have to written to files (used later by simulation functions etc.) PCETwriteFiles(sys,'ex1_ODE.m','ex1_OUT.m',... r.h.s. of PCE system (for simulation) r.h.s of PCE output equations 'ex1_mcode.m','ex1_mcout.m') r.h.s. of MC system (for simulation) r.h.s of MC output equations 7

8 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'; set simulation interval set time step-size for (simulation and/or evaluation set integrator options choose integrator choose discrete-time simulation Optional and possibly at a later time-point: 7. Update/modify uncertainties of initial conditions or parameter sys = PCETupdate(sys, x',[ ], a',[-6 0.1]); update data (see step 1) of variables x and a 8

9 Simulation 5a. Simulate PCE system (and output equations) using Galerkin projection PCEresults =... PCETsimGalerkin(sys,'ex1_ODE','ex1_OUT',simoptions) and/or: 5b.Simulate PCE system (and output equations) using collocation samples_col = PCETsample(sys,'basis',col_samples) PCEresults =... PCETsimCollocation(sys,'ex1_MCODE',... 'ex1_mcout',samples_col,simoptions) draws col_samples samples, evaluates basis polyn. and uses collocation and/or: 5c. Monte-Carlo simulation (states and output equations) samples = PCETsample(sys,'variables',mc_samples) MCresults =... PCETsimMonteCarlo(sys,'ex1_MCODE','ex1_MCOUT',... samples,simoptions) draws mc_samples samples, evaluates uncert. and simulates system PCEresults/MCresults contain all simulation data which can be processed and analyzed further 9

10 Visualize and analyze results 6a. Moments MomMats = PCETmomentMatrices(sys,4) compute moment matrices (up to 4 th order) need not to recomputed (can be stored for repeated use) PCEresults.y.moments =... PCETcalcMoments(sys,MomMats,results.y.pcvals) use MATLAB function plot to display moments stored in PCEresults.y.moments and/or: 6b.Histogram samples_basis = PCETsample(sys,'basis',n_samples) draws n_samples samples and evaluates basis polynomials samples_y = samples_basis' * PCEresults.y.pcvals use MATLAB functions for histogram plots (e.g. histc) 10

11 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 [Mesbah et al. (2014)] stochastic nonlinear model predictive control for a crystallization process chance constraints 11

12 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 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 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 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, 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, S. Streif, M. Karl, and A. Mesbah. Stochastic nonlinear model predictive control with efficient sample approximation of chance constraints. arxiv: 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, source codes available, see examples! 12