On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems | Sami Leon Noel Aziz Hanna et al. | ResearchPod

Transcript: On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems

Alex: Welcome to another episode of ResearchPod. Sam, today we're looking at a paper on using data to control tricky systems like robotic limbs.

Sam: The paper tackles a key puzzle in robot control. It shows that a simple stability tool—a quadratic control Lyapunov function—only works for data-learned models if the system stays balanced with a fixed control signal that never changes. This matters for real systems like neuroprosthetic arms, which need changing inputs to stay steady. These models often work in very high dimensions from data.

Alex: So why do these straightforward stability checks fail in the complicated spaces from learning robot dynamics from data?

Sam: Researchers turn data into a higher-dimensional view of nonlinear robot movements—like stretching a wiggly path on a map into straight lines. They call this the Koopman operator approach. With controls added, it becomes a bilinear system, linear separately in position and control. But quadratic control Lyapunov functions—think of them as energy bowls the robot rolls down to rest—are too basic for these high spaces. The paper proves they exist only if a constant control stabilizes everything.

Alex: And that's a problem for a prosthetic limb, where you can't just apply fixed stimulation—you need it to adapt as the person moves.

Sam: Precisely. In neuroprosthetics, data-driven bilinear models create high-dimensional approximations. Yet quadratic versions demand constant control works, which it rarely does for nonlinear behaviors. The authors show this limit using a quadratically constrained quadratic program, relaxed to a solvable semidefinite program.

Alex: So the core issue is that these simple quadratic tools break down when data makes dimensions explode?

Sam: That's it. In high dimensions, these energy bowls can't capture stability unless a fixed control—like a steady electrical pulse—holds the system on its own. They frame the search for such a bowl as finding a positive matrix shape where, from any state, you can choose a control to drop the energy toward zero.

Alex: They're testing if this energy bowl exists by solving a math puzzle tailored to the bilinear setup?

Sam: Yes. Imagine checking the highest hill around a bowl to ensure it's always downhill—maximizing energy rise under the worst control over states on a unit sphere. If that maximum is negative, a bowl exists. This creates a non-convex quadratically constrained quadratic program, hard to solve.

Alex: And they make it solvable without losing the exact answer?

Sam: They relax it to a convex semidefinite program by allowing more flexible shapes but bounding the same maximum from above. If the SDP says no good bowl, no quadratic one exists. The paper shows this is exact for single controls, by a theorem ensuring the solution collapses to rank one. It holds empirically when dimensions far exceed inputs, like in data-driven models.

On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems

Sami Leon Noel Aziz Hanna, Nicolas Hoischen, Sandra Hirche, Armin Lederer

Apr 10, 2026·4:44·arxiv·Original Paper

Koopman OperatorBilinear SystemsControl Lyapunov Functions (CLFs)Quadratic CLF ExistenceQCQP and SDP RelaxationExtended Dynamic Mode Decomposition (EDMD)

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5 min

Briefing

This paper tackles a core challenge in data-driven control of nonlinear systems: how to design stabilizing controllers for high-dimensional approximations learned via the Koopman operator. Traditional first-principles modeling fails for complex systems like neuroprosthetics or biological dynamics, so researchers turn to Koopman methods to 'lift' nonlinear dynamics into a higher-dimensional linear or bilinear form using data alone. But these lifted models are often huge—hundreds or thousands of dimensions—making control design tricky. The paper focuses on quadratic control Lyapunov functions (CLFs), the simplest and most popular tool for proving stabilizability and synthesizing controllers.

Koopman Operator and Bilinear Lifting

The Koopman operator is a clever mathematical trick: instead of evolving states nonlinearly, it evolves observables (functions of the state) linearly in an infinite-dimensional space. For control-affine nonlinear systems (\dot{x} = f(x) + g(x)u), mild conditions allow lifting to bilinear form (\dot{z} = A z + \sum B_i z u_i) in lifted coordinates (z = \mathcal{G}(x)), where (z) lives in (\mathbb{R}^n) with (n \gg \dim(x)). Data-driven methods like Extended Dynamic Mode Decomposition (EDMD) approximate (A, B_i) by regressing on trajectories: pick a dictionary of observables (e.g., monomials, radial basis functions), then solve a least-squares problem on data snapshots. This bilinear structure simplifies control design compared to general nonlinear MPC, but high (n) demands simple CLFs.

Quadratic CLFs: Definition and Role

A quadratic CLF is (V_Q(z) = \frac{1}{2} z^T Q z) with (Q \succ 0), satisfying (V_Q(z) \geq 0) (obvious) and (\inf_u \dot{V}_Q(z,u) \leq 0) everywhere except the origin. For bilinear systems, the Lie derivative is (\dot{V}_Q = z^T Q (A z + \sum B_i z u_i)), so the infimum condition becomes: there exists (u) such that (z^T Q A z + \sum (z^T Q B_i z) u_i \leq 0) for all (z \neq 0). This is equivalent to solving a pointwise QP at every (z), but for existence, we need a global (Q). CLFs enable robust QP-based controllers (u = \arg\min \dot{V}_Q + |u|^2), handling model uncertainty crucial for learned models.

Exact Characterization via QCQP

The key insight: quadratic CLF existence is exactly characterized by a nonconvex quadratically constrained quadratic program (QCQP). Normalize over the unit sphere (\Vert z \Vert =1): maximize (z^T Q A z) subject to (z^T Q B_i z = 0) for all (i=1,\dots,m) (the control authority constraints). A CLF exists iff the optimal value (J^* \leq 0). Why? If (J^* > 0), there's a (z) where no (u) can make (\dot{V}_Q < 0). This is NP-hard, but reveals structure: existence implies stabilizability by constant control (u^\star)—a fixed input that drives all trajectories to zero. Proof sketch: if a quadratic CLF exists, the optimal constant (u^\star = - (B^T Q B)^{-1} B^T Q A z / \Vert z \Vert) (approximately) stabilizes.

SDP Relaxation and Practical Checks

To make it tractable, relax to semidefinite programming (SDP): lift to (Z = z z^T), yielding (\max \tr(A Q Z)) s.t. (\tr(B_i Q Z)=0), (Z \succeq 0), (\tr(Z)=1). If SDP optimal (J^*_{SDP} \leq 0), a rank-1 (Z) gives a CLF (sufficient). For single-input (m=1), Pataki's lemma guarantees rank-1 exactness, making it necessary and sufficient. Empirically holds for many multi-input Koopman models. Constant control stabilizability is nearly necessary: just pick (Q = I), check if some fixed (u) works via spectral radius (\rho(A + \sum u_i B_i) <1).

Implications for High-Dimensional Control

Quadratic CLFs are highly restrictive in high dimensions—requiring 'easy' dynamics stabilizable by constants, like biased equilibria. This explains why prior SDP/LMI methods [10-12] succeed only sometimes. For Koopman control, check SDP first; if feasible, great—use the (Q) for robust QP control. If not, fall back to richer CLF classes or certify non-existence. Bridges theory (stabilizability) with practice (data-driven bilinear control), warning against blindly assuming quadratic CLFs in learned high-D models.

Abstract

Koopman operator-based methods enable data-driven bilinear representations of unknown nonlinear control systems. Accurate representations often demand significantly higher dimensions than the original system, making control design challenging. Control Lyapunov Functions (CLFs) are widely used for controller synthesis, with quadratic CLF candidates being the most common due to their simplicity. Yet, we show that this class is highly restrictive, especially when the state dimension is large: under mild conditions, their existence implies stabilizability of the bilinear system by a constant input -- that is, the control remains fixed over time. We establish this result by formulating a quadratically constrained quadratic program (QCQP) that exactly characterizes valid CLFs. Since QCQPs are NP-hard, we propose a convex semidefinite relaxation that offers a sufficient validity condition. For single-input systems, we prove that a quadratic CLF requires constant control stabilizability, and empirically demonstrate that this extends to high-dimensional multi-input systems in many cases.

On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems: Research Paper Podcast