Structure-preserving Koopman predictive control for memristive neural dynamics: input-exact and commutator-defect...

Area of Science:

• Control Theory
• Nonlinear Dynamics
• Computational Neuroscience

Background:

• Finite-dimensional Koopman Model Predictive Control (MPC) for nonlinear systems faces challenges with learned linear time-invariant (LTI) lifts.
• Accurate surrogate modeling is crucial for effective finite-horizon control strategies.

Purpose of the Study:

• To develop an input-exact Lie-lifting certificate for nonlinear controlled systems.
• To improve the accuracy and efficiency of Koopman MPC by addressing truncation errors.

Main Methods:

• Recasting the memristive Hindmarsh-Rose benchmark using an input-exact Lie-lifting certificate.
• Utilizing an augmented polynomial dictionary closed under Lie derivative.
• Representing the pure stimulation flow exactly with a nilpotent matrix exponential.
• Developing a bilinear, stimulation-aligned surrogate model.

Main Results:

• The controlled Koopman error is precisely defined as a finite shifted-drift defect.
• The proposed bilinear Lie model outperforms affine Extended Dynamic Mode Decomposition with Control (EDMDc)-MPC.
• Lower controlled-prediction error, closed-loop Root Mean Square Error (RMSE), and control energy were observed with the bilinear Lie model.

Conclusions:

• The Lie-lifting framework provides a conservative finite-horizon deterministic setting for Koopman MPC.
• The bilinear Lie model offers a superior approach for controlling nonlinear systems compared to existing methods.
• This method enhances the performance of Koopman MPC for neuroscience and control applications.