[Paper Review] On boundedness of solutions of three-state Moore-Greitzer compressor model with nonlinear proportional-integral controller for the surge subsystem

Anton Shiriaev, Leonid B. Freidovich+3 more|arXiv (Cornell University)|Mar 4, 2026

Turbomachinery Performance and Optimization0 citations

TL;DR

The paper proves boundedness of all solutions for the closed-loop 3-state Moore–Greitzer compressor model with a nonlinear PI controller targeting surge dynamics, using circle-criterion and IQC-style arguments to derive explicit controller parameter conditions.

ABSTRACT

The work focuses on Lagrange stability of the origin for the three-state Moore-Greitzer compressor model in closed loop with a nonlinear PI controller, tuned only to stabilize a lower-dimensional invariant surge-dynamics subsystem.The linearization of the system is not stabilizable but the static nonlinearity satisfies a sector condition, and together with a structural property of the stall-dynamics subsystem, this plays an essential role in the analysis. The main contribution provides explicit conditions on the controller parameters together with analytical arguments that guarantee boundedness of all solutions of the closed-loop system. The analysis employs a non-standard application of circle-criterion-based arguments. Together with the additional arguments developed in the work, this stability test also shows that the closed-loop system is robust to certain perturbations and model uncertainties.

Motivation & Objective

• Motivate and analyze boundedness of the origin for the 3-MG compressor with nonlinear PI control in surge dynamics.
• Obtain explicit controller parameter conditions that guarantee boundedness of all closed-loop solutions.
• Leverage circle-criterion-based arguments and sector/IQC concepts to handle the nonlinearities.
• Show robustness of the closed-loop system to certain perturbations and model uncertainties.

Proposed method

• Model the closed-loop surge dynamics with a nonlinear PI controller (6) that includes a copy of the nonlinear term.
• Apply the circle criterion to the surge subsystem with a sector-bound nonlinearity to derive LMI-based conditions.
• Construct a Lyapunov-like argument via a quadratic form with a positive definite matrix P to establish stability properties.
• Introduce an augmented representation including stall dynamics and derive a non-strict LMI ensuring boundedness.
• Prove finite-time escape does not occur and derive bounds for the stall variable R.
• Use IQC-style frequency-domain conditions to obtain a matching condition that guarantees boundedness of all states.

Experimental results

Research questions

• RQ1 Under what conditions on the nonlinear PI controller parameters do all solutions of the closed-loop 3-MG model remain bounded?
• RQ2 Can circle-criterion and IQC-based techniques certify boundedness of the surge-stall coupled dynamics without relying on a Lyapunov function for the full system?
• RQ3 How do the stall dynamics interact with surge dynamics under the proposed controller, and what are the robustness implications?
• RQ4 What explicit parameter relations ensure the nonlinearity satisfies a sector condition leading to boundedness?

Key findings

• Explicit conditions on controller parameters ensure boundedness of all closed-loop solutions.
• The surge subsystem can be stabilized in a way compatible with the stall dynamics via a non-strict LMI-based approach.
• The circle criterion can be applied in a non-standard way to derive frequency-domain matching conditions for the nonlinear PI controller.
• The analysis demonstrates robustness to certain perturbations and model uncertainties.
• Finite-time escape is ruled out for the closed-loop system under the specified conditions.
• The results align with numerical observations that quadratic surge stabilization alone does not guarantee stability when stall dynamics are present.