# Feedback Stabilization Using Koopman Operator

@article{Huang2018FeedbackSU, title={Feedback Stabilization Using Koopman Operator}, author={Bowen Huang and Xu Ma and Umesh Vaidya}, journal={2018 IEEE Conference on Decision and Control (CDC)}, year={2018}, pages={6434-6439} }

In this paper, we provide a systematic approach for the design of stabilizing feedback controllers for nonlinear control systems using the Koopman operator framework. [...] Key Method The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Expand

#### 26 Citations

Data-Driven Nonlinear Stabilization Using Koopman Operator

- Mathematics, Computer Science
- 2020

The proposed approach is data-driven and relies on the use of time-series data generated from the control dynamical system for the lifting of a nonlinear system in the Koopman eigenfunction coordinates to construct a finite-dimensional bilinear representation of a control-affine nonlinear Dynamical system. Expand

Closed-loop stabilization of nonlinear systems using Koopman Lyapunov-based model predictive control

- Computer Science
- 2020 59th IEEE Conference on Decision and Control (CDC)
- 2020

This work considers the problem of stabilizing feedback control design for nonlinear systems and shows via an inverse mapping that the designed controller translates the stability of the Koopman bilinear system to the original closed-loop system. Expand

Data-driven feedback stabilization of nonlinear systems: Koopman-based model predictive control

- Computer Science, Engineering
- ArXiv
- 2020

The proposed feedback control design remains completely data-driven and does not require any explicit knowledge of the original system, and due to the bilinear structure of the Koopman model, seeking a CLF is no longer a bottleneck for LMPC. Expand

A convex data-driven approach for nonlinear control synthesis

- Engineering, Computer Science
- Mathematics
- 2021

This work proposes a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible, built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Expand

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

- Mathematics
- 2020

This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A… Expand

Koopman Lyapunov‐based model predictive control of nonlinear chemical process systems

- Mathematics
- AIChE Journal
- 2019

Funding information National Science Foundation, Grant/Award Number: CBET-1804407 Abstract In this work, we propose the integration of Koopman operator methodology with Lyapunov-based model… Expand

Optimal Quadratic Regulation of Nonlinear System Using Koopman Operator

- Mathematics, Computer Science
- 2019 American Control Conference (ACC)
- 2019

The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite dimensional bilinear system to solve the optimal quadratic regulation problem for nonlinear systems. Expand

Discrete System Linearization using Koopman Operators for Predictive Control and Its Application in Nano-positioning

- Computer Science
- 2020 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)
- 2020

This work proposes to linearize the nonlinear system model using Koopman operators and then use the obtained linear parameter-varying model for predictive control, which can significantly decrease the overall approximation error within the MPC prediction horizon, and thus, lead to improved tracking performance. Expand

Koopman NMPC: Koopman-based Learning and Nonlinear Model Predictive Control of Control-affine Systems

- Computer Science, Engineering
- 2021 IEEE International Conference on Robotics and Automation (ICRA)
- 2021

The benefits for control-affine dynamics compared to existing Koopman-based methods are highlighted through an example of a simulated planar quadrotor, and Prediction error is greatly reduced and closed loop performance similar to NMPC with full model knowledge is achieved. Expand

Koopman Operator Based Modeling and Control of Rigid Body Motion Represented by Dual Quaternions

- Computer Science, Engineering
- ArXiv
- 2021

This paper systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation and shows that an LQR type (linear) controller can steer the rigidBody to a desired state while its performance is commensurate with that of a nonlinear controller. Expand

#### References

SHOWING 1-10 OF 26 REFERENCES

Koopman operator-based model reduction for switched-system control of PDEs

- Computer Science, Mathematics
- Autom.
- 2019

A new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs) is presented and it is shown that thevalue of the K-ROM based objective function converges in measure to the value of the full objective function. Expand

Data-Driven Optimal Control Using Perron-Frobenius Operator

- Computer Science
- ArXiv
- 2018

A data-driven approach for control of nonlinear dynamical systems relies on transfer Koopman and Perron-Frobenius (P-F) operators for linear representation and control, and exploits the positivity and Markov property of these operators and their finite-dimensional approximation to provide an optimally stabilizing feedback controller. Expand

Nonlinear stabilization via control-Lyapunov measure

- Mathematics, Computer Science
- 2007 46th IEEE Conference on Decision and Control
- 2007

This paper poses and solves the co-design problem of jointly obtaining the control Lyapunov measure and a controller and provides a proof of existence for a stochastic version of such a controller while the deterministic restriction is posed as the solution of a related integer programming problem. Expand

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control

- Mathematics, Computer Science
- Autom.
- 2018

This work extends the Koopman operator to controlled dynamical systems and applies the Extended Dynamic Mode Decomposition (EDMD) to compute a finite-dimensional approximation of the operator in such a way that this approximation has the form of a linearcontrolled dynamical system. Expand

Optimal Stabilization Using Lyapunov Measures

- Computer Science, Mathematics
- IEEE Transactions on Automatic Control
- 2014

Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this technical note and it is shown the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program. Expand

Model-Based Control Using Koopman Operators

- Computer Science, Mathematics
- Robotics: Science and Systems
- 2017

It is illustrated how the Koopman operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Expand

Linear observer synthesis for nonlinear systems using Koopman Operator framework

- Mathematics
- 2016

Abstract: In this paper we develop a new approach for observer synthesis for discrete time autonomous nonlinear systems based on Koopman operator theoretic framework. Koopman operator is a linear but… Expand

Stabilization with relaxed controls

- Mathematics
- 1983

cannot in general be stabilized using a continuous closed loop control U(X), even if each state separately can be driven asymptotically to the origin. (An example is analyzed in Section 2.) In this… Expand

Control Lyapunov functions: new ideas from an old source

- Mathematics
- Proceedings of 35th IEEE Conference on Decision and Control
- 1996

A control design method for nonlinear systems based on control Lyapunov functions and inverse optimality is analyzed. This method is shown to recover the LQ optimal control when applied to linear… Expand

Data-driven discovery of Koopman eigenfunctions for control

- Physics, Computer Science
- 2017

This work illustrates a fundamental closure issue of this approach and argues that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigen Functions, termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). Expand