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Generating uncertainty sets of transfer operators for robust prediction & control

This project contains the experimental framework for generating perturbations of nonlinear dynamical systems via a nominal Koopman operator, per "An MCMC Method for Uncertainty Set Generation via Operator-Theoretic Metrics," by A. Srinivasan and N. Takeishi, submitted to IEEE CDC.

Below is the library organization. To run any module, simply use python -m FOLDER.MODULE.

├── sampler
│   ├── ugen.py 		# High-level MCMC procedure for uncertainty set generation
│   ├── hmc.py 			# PyTorch autograd-based Hamiltonian Monte Carlo for tensor-valued arguments with support for constraint-based reflection
│   ├── hmc_nuts.py 		# No U-Turn Sampler integrator for HMC (not used in experiments)
│   ├── hmc_parallel.py 	# Parallel HMC sampler from a specified prior over initial conditions 
│   ├── kernel.py 		# Positive-definite kernel over dynamical systems (autograd-compliant implementation of Ishikawa et al., https://arxiv.org/abs/1805.12324)
│   ├── reflections.py 		# Various boundary conditions for HMC 
│   ├── features.py 		# Observables & kernels for Koopman operator
│   ├── operators.py 		# Dynamic Mode Decomposition & variants
│   └── utils.py 	
├── experiments			# Examples of uncertainty set generation for prediction & control (see sections below)
└── ...

Please see the below sections for usage of the above procedures and replication of numerical examples, from prediction to control.

Perturbations of LTI systems

  1. Configure candidate systems in systems/lti2x2.py
  2. Run python -m experiments.2x2_perturb with method = 'discounted_kernel' to generate an uncertainty set
  3. Run python -m experiments.2x2_plot with the correct filename to produce trace-determinant plots

Perturbations of nonlinear systems

Duffing Oscillator

  1. Configure Duffing equation parameters in experiments/duffing_perturb.py (unforced only for poly obs.)
  2. Run python -m experiments.duffing_perturb with method = kernel' to generate perturbations
  3. Run python -m experiments.duffing_plot to generate attractor basin visualizations

Van der Pol Oscillator

  1. Configure VDP parameters in experiments/vdp_perturb.py
  2. Run python -m experiments.vdp_perturb with method = 'kernel' or method = 'constrained_kernel' to generate uncertainty set
  3. Run python -m experiments.vdp_plot to show phase-space results.

Perturbing a custom system (instructions)

  1. Compute a nominal Koopman operator for the system. (See experiments/duffing_perturb.py for an example using the Observable class from sampler/features.py. Any algorithm can be used here, including kernel DMD.)
  2. Call sampler.ugen.perturb(...) with model = my_koopman_op, method = 'kernel', and any other arguments specified in the file.
  3. Adjust parameters of parallel HMC (hmc_step, hmc_leapfrog, n_ics, ic_step, ic_leapfrog) until the desired stationary distribution is reached (i.e. MCMC is adequately mixed; sampler.ugen.perturb() will return the posterior distribution as its second result, which can be used for visual/numerical verification.)
  4. Use the resulting uncertainty set for robust prediction & control.

(In progress) Robust control & scenario optimization examples

We're still working on tuning the method to achieve cool improvements for model-predictive controllers facing heavy observation noise, please check back in later! Some preliminary results...