Dissipativity Analysis of Nonlinear Systems: A Linear--Radial Kernel-based Approach

The Big Picture: Predicting the "Energy" of a Black Box

Imagine you have a mysterious machine (a nonlinear system). It takes inputs (like pushing a lever) and produces outputs (like a spinning wheel). You don't know the blueprints, the physics equations, or how the gears work inside. You only have a notebook of observations: "When I pushed here, it went there."

The paper asks a crucial question: Can we figure out if this machine is "safe" and "stable" just by looking at our notebook of observations?

In engineering, "safety" and "stability" are often described by a concept called Dissipativity. Think of it like a bank account for energy:

Storage Function: How much energy is currently stored in the machine (the balance).
Supply Rate: How much energy is being added or taken out by your actions (deposits and withdrawals).
The Rule: A safe machine is one where the energy stored never grows faster than the energy you put in. If you stop pushing, the machine should eventually calm down, not explode.

The Problem: The "Shape" of the Solution

For simple machines (linear systems), figuring out this energy balance is like drawing a perfect circle or a straight line. It's easy math.

But for complex, wobbly machines (nonlinear systems), the energy balance looks like a twisted, knotted piece of spaghetti. Traditional math struggles to find a formula for that spaghetti. If you try to force it into a simple shape (like a flat sheet), you might miss the danger spots, or you might be so cautious that you think the machine is broken when it's actually fine.

The Solution: The "Magic Lens" (Linear-Radial Kernel)

The authors propose a new way to look at the data. They use a mathematical tool called a Reproducing Kernel Hilbert Space (RKHS).

The Analogy: The Magic Lens
Imagine looking at a messy, tangled ball of yarn through a special Magic Lens.

Without the lens: The yarn looks like a chaotic mess.
With the lens: The yarn suddenly looks like a neat, organized grid of straight lines.

The "Magic Lens" in this paper is a specific type of math function called a Linear-Radial Kernel.

Why "Linear-Radial"? The authors realized that near the "center" of the machine (where it's supposed to be at rest), the energy usually looks like a bowl (quadratic). Far away, it might look different.
The lens is designed so that near the center, it forces the math to look like a smooth bowl (safe and stable). Far away, it allows the math to stretch and twist to fit the complex reality.

This ensures that the solution isn't just a random guess; it respects the fundamental physics that "near the center, things should behave nicely."

The Method: Turning Chaos into a Puzzle

Here is how they actually do it:

Lift the Data: They take their messy data points and "lift" them into a higher-dimensional world (like taking a 2D shadow and turning it into a 3D object). In this new world, the complex, twisted rules of the machine look like simple, straight lines.
The "Bank Account" Check: They set up a giant puzzle. They are looking for a "Storage Function" (the energy balance) that satisfies a simple rule: Energy Out ≤ Energy In.
The Convex Optimization: Because they used their "Magic Lens," this puzzle becomes a Convex Optimization problem.
- Analogy: Imagine trying to find the lowest point in a landscape. If the landscape is full of hills and valleys (non-convex), you might get stuck in a small dip thinking it's the bottom. But if the landscape is a perfect, smooth bowl (convex), you can roll a ball down, and it will guarantee to find the absolute lowest point.
- The authors turned the messy problem into a perfect bowl, so the computer can find the best answer quickly and reliably.

The Guarantee: "Probably Correct"

Since they are using data (which is never perfect), they can't promise the machine is 100% safe everywhere. But they provide a Statistical Guarantee.

The Metaphor: Imagine you are testing a new bridge by driving cars over it. You can't drive every possible car at every possible speed.
The authors say: "We tested 1,000 cars. Based on our math, we are 99% sure that if you drive a car we haven't seen yet, the bridge will hold, unless you drive it extremely far from the center of the road."
They proved that any "mistakes" (violations of the safety rule) get smaller the further you get from the center, and they vanish as you collect more data.

Real-World Tests (The Case Studies)

The authors tested this on three different "machines":

A Synthetic Polynomial System: A made-up math problem where they knew the answer. Their method found the exact answer, proving it works.
An Inverted Pendulum: A stick balancing on a cart. This is notoriously unstable. Their method found a "storage function" that was much less conservative (less fearful) than traditional physics-based guesses, meaning it could handle more aggressive control without breaking.
A Bioreactor: A tank growing bacteria. This is a messy, chemical system with no simple formula. Their method successfully found a safety rule where human intuition failed.

The Bottom Line

This paper gives engineers a data-driven flashlight to find safety rules for complex, unknown machines.

It doesn't need the blueprints.
It uses a special "lens" to make the math solvable.
It guarantees that the machine is safe, with a mathematically proven margin of error that shrinks as you collect more data.

It's like teaching a computer to understand the "laws of physics" for a black box just by watching it play, ensuring it won't crash the plane or explode the reactor.

1. Problem Statement

The paper addresses the challenge of estimating the dissipativity of nonlinear dynamical systems using empirical data, particularly when an accurate analytical model is unavailable.

Context: Dissipativity is a fundamental property linking system stability, energy conservation, and input-output behavior. While well-understood for linear systems (via Linear Matrix Inequalities, LMIs) and polynomial systems (via Sum-of-Squares), extending these methods to general nonlinear systems is computationally intractable due to the need to search for complex, non-parametric storage and supply rate functions.
Gap: Existing data-driven approaches often struggle with the "curse of dimensionality," lack of behavioral theory for nonlinear systems, or rely on conservative polynomial approximations that may not capture the system's true dynamics.
Goal: To develop a data-driven framework that formulates dissipativity estimation as a convex optimization problem, ensuring the learned storage function is "locally at least quadratic" near the equilibrium point (a requirement for stability analysis) while generalizing conventional quadratic forms.

2. Methodology

The authors propose a framework based on Koopman operator theory within a specific Reproducing Kernel Hilbert Space (RKHS).

A. Theoretical Foundation: Koopman Operators on RKHS

The nonlinear system $x_{t+1} = f(x_t, u_t)$ is "lifted" into an infinite-dimensional linear space using the Koopman operator $K$ .
Instead of standard radial kernels, the authors utilize a Linear–Radial Kernel. This kernel is defined as $\check{\kappa}(z, z') = (z^\top z')\rho(|z-z'|)$ , where $\rho$ is a radial function.
Key Property: This specific kernel construction ensures that the RKHS contains functions that are "locally at least linear" near the origin. Consequently, the quadratic forms defined on this space are "locally at least quadratic," satisfying the necessary conditions for Lyapunov stability analysis without forcing a global polynomial structure.

B. Formulation as Linear Operator Inequality

Storage and Supply Functions: The storage function $v(x)$ and supply rate $s(y, u)$ are expressed as Kernel Quadratic Forms:
$v(x) = \langle \check{\phi}_x, P \check{\phi}_x \rangle, \quad s(y, u) = \langle \check{\phi}_{(x,u)}, S \check{\phi}_{(x,u)} \rangle$
where $P$ and $S$ are self-adjoint Hilbert–Schmidt operators, and $\check{\phi}$ represents the canonical feature map in the RKHS.
Dissipative Inequality: The standard dissipative inequality $v(f(x,u)) - v(x) \leq s(h(x,u), u)$ is transformed into a Linear Operator Inequality (LOI):
$K^* P K - E P E^* - S \preceq 0$
where $K$ is the Koopman operator and $E$ is an embedding operator. This converts the functional inequality into a constraint on linear operators.

C. Data-Driven Estimation

Finite-Rank Approximation: Using a dataset of $n$ snapshots $\{(x_i, u_i, x_i^+)\}$ , the infinite-dimensional operators $P$ and $S$ are approximated by finite-rank operators.
Convex Optimization: The problem is reduced to finding coefficient matrices $\Theta_P$ and $\Theta_S$ that satisfy a Linear Matrix Inequality (LMI) on the data points:
$\text{diag}(G_{xx^+}^\top \Theta_P G_{xx^+} - G_{xx}^\top \Theta_P G_{xx} - G_{xu}^\top \Theta_S G_{xu}) \leq 0$
where $G$ matrices are kernel matrices derived from the linear-radial kernel.
Supply Rate Estimation: The supply rate operator $S$ is estimated using Kernel Ridge Regression (KRR) to approximate the underlying functions from data.

D. Statistical Guarantees

The authors derive a statistical learning bound on the violation of the dissipative inequality.
They prove that the violation $w(x, u)$ scales with the distance from the equilibrium point (specifically $|(x, u)|^2$ or $|(x, u)|$ ).
As the sample size $n \to \infty$ , the probability of the dissipativity estimate being approximately correct converges to 1, with the error bound decaying at a rate of $O(n^{-1/3} \log n)$ .

3. Key Contributions

Linear–Radial Kernel RKHS: Introduction of a specific kernel structure that inherently encodes the equilibrium point, ensuring the learned storage functions possess the correct local quadratic behavior required for stability, while remaining expressive enough to capture complex nonlinearities.
Operator-Theoretic Formulation: Reformulating the dissipative inequality as a linear operator inequality, bridging the gap between infinite-dimensional operator theory and finite-dimensional convex optimization.
Data-Driven Convex Optimization: Providing a computationally tractable method to estimate dissipativity certificates directly from data without requiring explicit system identification or Koopman operator matrix inversion.
Generalization Bounds: Deriving rigorous statistical error bounds that guarantee the dissipativity property holds not just on training data, but with high probability over the entire state-input space, with violations vanishing near the equilibrium.

4. Experimental Results

The method was validated on three case studies:

Case 1 (Polynomial System): The method successfully learned the exact ground-truth quartic storage function from data, outperforming simple quadratic approximations and matching the performance of an "oracle" lifting.
Case 2 (Inverted Pendulum): Applied to a system with unknown friction. The learned storage function was significantly less conservative than the physically intuitive mechanical energy function, successfully verifying dissipativity with a lower bound on the supply rate parameter ( $\beta$ ).
Case 3 (Bioreactor): Applied to a complex chemical process where physical intuition for storage functions is difficult. The method identified a feasible storage function and supply rate, with dissipativity violations bounded by the squared distance from the origin, confirming the theoretical error bounds.

5. Significance and Implications

Model-Free Control: This approach enables the certification of stability and the design of controllers for nonlinear systems without requiring an explicit parametric model.
Robustness: The statistical bounds provide a rigorous way to quantify the confidence in the dissipativity certificate, which is crucial for safety-critical applications.
Scalability: By avoiding the explicit calculation of the Koopman operator matrix (which can be ill-conditioned) and relying on kernel matrices, the method is computationally efficient.
Future Directions: The authors note that the current method requires full state measurements. Future work aims to extend this to input-output only data using data-driven state observers.

In summary, this paper presents a mathematically rigorous and computationally efficient framework for data-driven dissipativity analysis, leveraging the unique properties of linear-radial kernels to ensure local stability properties while capturing global nonlinear dynamics.