Deep Koopman MPC Control

• The paper introduces DK-MPC, which unifies the Koopman operator approach with MPC to control nonlinear systems efficiently.
• It leverages data-driven lifting of state-space dynamics into a higher-dimensional space to construct finite-dimensional linear predictors using convex optimization.
• Numerical results on systems like the Van der Pol oscillator and KdV PDE demonstrate improved prediction accuracy and computational efficiency compared to traditional methods.

A Deep Koopman Operator with Model Predictive Control (DK-MPC) framework unifies operator-theoretic concepts with convex optimization for the purpose of controlling nonlinear dynamical systems. By leveraging a data-driven lifting of the state-space dynamics into a higher-dimensional function space—where the nonlinear evolution is approximated as linear—DK-MPC enables the application of efficient linear MPC techniques to highly nonlinear systems. This approach generalizes to controlled systems via systematic lifting of both states and inputs, constructing finite-dimensional linear predictors through regression in the lifted space, and embedding these models directly into a real-time MPC optimization. DK-MPC offers precise and computationally efficient control, even in settings traditionally handled by computationally intensive nonlinear control schemes.

1. Extension of the Koopman Operator to Controlled Systems

The Koopman operator, denoted as $\mathcal{K}$, provides a linear infinite-dimensional representation of a nonlinear, autonomous dynamical system as $(\mathcal{K}\psi)(x) = \psi(f(x))$ for any chosen observable $\psi$. For controlled systems, the extension requires augmenting the state with the entire future control sequence $u = [u(0), u(1), \ldots]$ and defining an extended state $\chi = [x; u]\in\mathbb{R}^n \times \ell(U)$. The evolution is then specified as

$\chi^+ = F(\chi) = [f(x, u(0)); S u],$

where $S$ is the left-shift operator on the control sequence. The generalized operator is

$(\mathcal{K}\varphi)(\chi) = \varphi(F(\chi))$

for $$\varphi : \mathbb{R}^n \times \ell(U) \to \mathbb{R}$$. This allows the controlled dynamics to be encoded linearly with respect to suitable lifted observables. The structure enables direct relation between predicted lifted observables and original system state via an output map, preserving the ability to impose constraints and recover original state trajectories in the control synthesis.

The significance of this construction is that it introduces full control authority into the operator-theoretic formalism, permitting rigorous and practical use of Koopman-based models in the context of model predictive control. The design choice of observables, including those derived from nonlinear functions of the state and inputs, critically affects the accuracy of subsequent predictions and the tractability of the controller.

2. Data-Driven Construction of Finite-Dimensional Linear Predictors

To discretize the infinite-dimensional Koopman operator for practical application, DK-MPC employs a data-driven methodology. A set of lifting functions $\{\psi_i\}_{i=1}^N$ is chosen, which could include nonlinear transformations such as radial basis functions, polynomials, or neural network-based feature maps. The state is lifted via $z = \psi(x) = [\psi_1(x), ..., \psi_N(x)]^\top$. The linear predictor in the lifted space takes the form: $z^+ = A z + B u, \qquad \hat{x} = C z,$ with $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times m}$, and $C \in \mathbb{R}^{n \times N}$. The matrices are identified from data snapshots $\{x_j, u_j, y_j = f(x_j, u_j)\}$ by solving a regularized least-squares regression: $$\min_{A,B} \sum_{j} \|\psi(y_j) - A\psi(x_j) - B u_j\|^2.$$ If the state is a subset of the lifted observables, $C$ can be set accordingly; otherwise, a projection is learned such that $x \approx C\psi(x)$.

The operational simplicity, computational scalability, and task-adaptive flexibility of this identification procedure are noteworthy. It enables the construction of finite-dimensional Koopman models for high-dimensional or partially observed systems directly from trajectory data without explicit knowledge of the underlying system equations.

3. Embedding of Koopman Predictors in Model Predictive Control

The key to DK-MPC is the integration of lifted linear predictors into a standard linear MPC framework. Given the linear predictor, the control input sequence $\{u_k\}$ is optimized by solving a quadratic program: $$\begin{align*} \min_{\{u_i, z_i\}} \quad & z_n^\top Q_n z_n + q_n^\top z_n + \sum_{i=0}^{n-1} (z_i^\top Q_i z_i + u_i^\top R_i u_i + q_i^\top z_i + r_i^\top u_i) \ \text{subject to} \quad & z_{i+1} = A z_i + B u_i, \quad \forall i=0,...,n-1, \ & E_i z_i + F_i u_i \leq b_i, \ & z_0 = \psi(x_0). \end{align*}$$ Constraints that are nonlinear in the state (e.g., $g(x)\leq 0$) are lifted to linear inequalities in $z$ by including $g(x)$ in the lifting vector. Because the predictor is linear, so are the constraints and the cost function when written in terms of $z$ and $u$. The computational complexity is dominated by the number of control inputs and prediction horizon length, not the dimension $N$ of the lifted space if the MPC is implemented in dense form (i.e., by eliminating state variables).

Significantly, this architecture preserves the convexity and efficiency of linear MPC, admitting the use of high-performance quadratic programming solvers (e.g., qpOASES), and allows the controller to scale to systems where $N \gg n$.

4. Constraint and Cost Handling in the Lifted Space

DK-MPC possesses an intrinsic mechanism to handle complex state and input constraints arising from nonlinear physics or operational safety. Any constraint or cost that is a function of a chosen lifting function $\psi(x)$ becomes linear in $z$. For instance, imposing

$\cos(\|x\|_\infty) \leq c$

is achieved by including $\cos(\|x\|_\infty)$ as a lifting function, so the constraint becomes

$z^{(\text{new})} \leq c,$

which is linear in $z$. Similarly, nonlinear state cost terms can be incorporated without loss of convexity. Input constraints remain linear, as the input is not subject to a nonlinear lifting.

This ability allows the representation of far richer sets of operational requirements and objectives than conventional linear models, while retaining tractable online optimization.

5. Numerical Demonstrations and Comparative Performance

DK-MPC has been evaluated on a suite of nonlinear systems:

• Forced Van der Pol oscillator: With a 100-dimensional thin-plate spline RBF lifting, the Koopman-based predictor achieved the lowest relative RMSE compared to predictors from local linearization (origin and initial condition) and Carleman linearization over multiple trajectories.
• Bilinear DC motor model: Predictors constructed using time-delayed measurements achieved superior long-term prediction compared to local linearization, thereby enhancing the robustness and feasibility of MPC in tracking and stabilization.
• Korteweg–de Vries (KdV) PDE control: The approach was demonstrated on a PDE discretized to 128 spatial points. With a lifted dimension $N=385$ (composed of the state vector, its square, and cross-terms), closed-loop control with linear constraints and online computation times of 0.28 ms were achieved, demonstrating applicability to high-dimensional, infinite-dimensional systems.

In all examples, DK-MPC outperformed both local and classical linearization-based methods in prediction and closed-loop tracking error, and it consistently maintained the computational efficiency characteristic of linear MPC.

6. Practical Benefits, Scalability, and Deployment Considerations

Key practical implications include:

• Data-Driven and Non-Intrusive: No explicit knowledge of the system's equations is required; the approach is fully compatible with black-box, experimentally derived, or simulation-generated data.
• Computational Efficiency and Scalability: The separation between offline model identification and efficient online MPC optimization allows for application to large-scale systems, including high-dimensional PDEs and networks. The dimension of the lifting (potentially into the hundreds or thousands) is decoupled from the online optimization footprint.
• Versatile Constraint and Cost Representation: Nonlinear constraints and objectives are handled readily by appropriately extending the lifting functions, enabling the formulation of highly expressive, yet convex, predictive control problems.
• Technology Transfer: The algorithmic simplicity facilitates deployment in embedded, industrial, or experimental hardware where resource constraints and real-time requirements preclude complex nonlinear optimization.

The DK-MPC framework thus turns nonlinear optimal control problems—often conventionally intractable—into convex, high-speed predictive optimizations, without the need for local model reduction, sacrificing generality for tractability, or imposing significant modeling overhead.

---

In summary, DK-MPC formalizes the synthesis of data-driven, globally linear predictors for nonlinear controlled systems via lifting in the Koopman operator framework, and embeds those predictors into scalable, convex quadratic MPC controllers. The result is a class of nonlinear feedback controllers that combine the expressiveness and accuracy of nonlinear system representation with the computational and practical advantages of linear MPC, enabling rapid, constraint-aware predictive control for complex, high-dimensional, and even experimentally opaque systems (Korda et al., 2016).