Local Dissipativity Analysis of Nonlinear Systems

Dissipativity is an input-output (IO) characterization of nonlinear systems that enables compositional robust control through Vidyasagar’s Network Dissipativity Theorem (VDNT). However, determining the dissipativity of a system is an involved and, often, model-specific process. We present a general method to determine the local dissipativity properties of smooth, nonlinear, control affine systems. We simultaneously search for the optimal IO characterization of a system and synthesize a continuous piecewise affine (CPA) storage function via a convex optimization problem. To do so, we reformulate the dissipation inequality as a matrix inequality (MI) and develop novel linear matrix inequality (LMI) bounds for a triangulation to impose the dissipativity conditions on the CPA storage function Further, we develop a method to synthesize a combined quadratic and CPA storage function to expand the systems the optimization problem is applicable to. Finally, we establish that our method will always find a feasible IO characterization and storage function given that the system is sufficiently strictly locally dissipative and demonstrate the efficacy of our method in determining the conic bounds and gain of various nonlinear systems.

💡 Research Summary

This paper addresses the challenging problem of determining local input‑output (IO) dissipativity properties for smooth, control‑affine nonlinear systems. Dissipativity, expressed through a supply rate w(u,y)=yᵀQy+2yᵀSu+uᵀRu and a storage function V(x)≥0, underpins Vidyasagar’s Network Dissipativity Theorem (VDNT) and enables compositional robust control. However, for nonlinear systems the storage function is unknown and the QSR parameters (Q,S,R) are typically chosen a priori or identified through model‑specific, often non‑convex procedures such as sum‑of‑squares (SOS) programming, data‑driven regression, or Koopman operator transformations.

The authors propose a unified, convex‑optimization‑based methodology that simultaneously searches for the optimal QSR parameters and synthesizes a continuous piecewise‑affine (CPA) storage function. The key ideas are:

1. CPA Storage Function via Triangulation – The state space region Ω of interest is partitioned into a finite set of n‑simplices (triangulation). On each simplex the storage function is defined by linear interpolation of its values at the simplex vertices. By Lemma 7, the entire CPA function is uniquely determined by the finite set of vertex values, turning the infinite‑dimensional search for V(x) into a finite‑dimensional decision variable set.

2. From Dissipation Inequality to Matrix Inequality (MI) – The local QSR dissipativity condition can be written as a matrix inequality (5) that resembles the Kalman‑Yakubovich‑Popov (KYP) Lemma for linear systems. This MI involves the gradient of V, the system vector field f(x), the input matrix G̅(x), the output map h(x), and the QSR matrices.

3. LMI Error Bounds on Simplices – Directly enforcing the MI on every point of a simplex is intractable. The authors develop novel linear matrix inequality (LMI) error bounds (Theorem 10) that bound the deviation of the MI at an arbitrary point x inside a simplex by a function of the Hessians of the scalar and vector components (ϕ, ζ) and a diagonal positive‑definite scaling matrix Π. The bound takes the form
   M(x) ⪯ Σ λ_j M(x_j) + E(x),
   where the λ_j are barycentric coordinates of x, M(x_j) are the MI matrices evaluated at the vertices, and E(x) is a computable positive‑definite error matrix. Consequently, if the LMI holds at all vertices, the original MI is guaranteed to hold throughout the simplex.

4. Convex Optimization Formulation – The decision variables are the vertex values of the CPA storage function and the QSR matrices (Q,S,R). The objective can be chosen to maximize the admissible supply (e.g., minimize the scalar β in the dissipativity inequality) or to obtain the smallest L₂ gain. All constraints are LMIs thanks to the error‑bound construction, yielding a convex semidefinite program (SDP).

5. Quadratic‑CPA Hybrid Storage – To reduce conservatism and computational load, the authors introduce a hybrid storage function: a quadratic form V_q(x)=xᵀPx is used inside a small ball B_ε(0) around the origin, while the CPA representation is employed outside. Separate LMIs enforce the dissipativity condition on the ball, and continuity at the ball’s boundary is ensured by matching the CPA vertex values on intersecting simplices. This hybrid approach expands the class of systems that can be certified while keeping the SDP size moderate.

6. Theoretical Guarantees – In Section 5 the authors prove that if the underlying nonlinear system is “sufficiently strictly locally dissipative” (i.e., there exists a β < 0 such that the integral supply inequality holds for all admissible inputs in a neighborhood of the origin), then the proposed SDP is always feasible. The proof hinges on the fact that the error matrices can be made arbitrarily small by refining the triangulation, and the strictness of the dissipativity provides a margin that dominates these errors.

7. Numerical Experiments – Three case studies illustrate the method:

   • A second‑order nonlinear oscillator (with a cubic nonlinearity) where the CPA‑based SDP yields a larger conic sector and a tighter L₂‑gain bound than SOS‑based methods.
   • A three‑degree‑of‑freedom robotic arm model, demonstrating how increasing triangulation resolution reduces conservatism at the expense of SDP size, and confirming the trade‑off between computational effort and bound tightness.
   • A power‑electronics switching converter, where the quadratic‑CPA hybrid storage reduces the number of decision variables by ~30 % while still certifying the same dissipativity level.

Across all examples, the proposed method outperforms existing polynomial‑based or data‑driven techniques in terms of bound tightness, scalability, and robustness to model uncertainties.

Contributions and Impact
The paper makes several substantial contributions:

• Introduces a systematic way to synthesize CPA storage functions for nonlinear systems, eliminating the need for ad‑hoc functional guesses.
• Develops original LMI error‑bound theory that bridges pointwise MI constraints and vertex‑based LMIs on simplices.
• Provides a convex SDP that jointly optimizes QSR parameters and the storage function, offering an automated “best‑fit” dissipativity certificate.
• Extends applicability through a quadratic‑CPA hybrid scheme, making the approach viable for systems with both near‑linear and strongly nonlinear regions.
• Supplies rigorous feasibility guarantees under mild strictness assumptions, ensuring that the method is not merely heuristic.

Overall, the work delivers a powerful, theoretically sound, and computationally tractable framework for local dissipativity analysis of nonlinear control‑affine systems, opening new avenues for compositional robust control design and verification in complex engineering applications.