[Paper Review] Output Feedback Controller Synthesis for Negative Imaginary Systems

James Dannatt, Ian R. Petersen|arXiv (Cornell University)|Mar 27, 2019

Piezoelectric Actuators and Control18 references3 citations

TL;DR

This paper presents necessary and sufficient conditions for synthesizing a strictly proper dynamic output feedback controller for negative imaginary (NI) systems using solutions to two dual algebraic Riccati equations (AREs). The method avoids singular Hamiltonians via Schur decomposition and Lyapunov equations, ensuring the closed-loop system maintains the NI property and enables robust stability under positive feedback when plant uncertainty is strictly negative imaginary (SNI).

ABSTRACT

This paper presents necessary and sufficient conditions for deriving a strictly proper dynamic controller which satisfies the negative imaginary output feedback control problem. Our synthesis method divides the output feedback control problem into a state feedback problem and a dual output injection problem. Thus, a controller is formed using the solutions to a pair of dual algebraic Riccati equations. Finally, an illustrative example is offered to show how this method may be applied.

Motivation & Objective

• Address the lack of necessary and sufficient conditions for dynamic output feedback controller synthesis in negative imaginary systems.
• Overcome limitations in prior methods that only provided sufficient conditions or failed to guarantee asymptotic stability.
• Ensure the closed-loop system remains negative imaginary and robustly stable under positive feedback when plant uncertainty is strictly negative imaginary (SNI).
• Develop a computationally tractable method using Schur decomposition and Lyapunov equations to solve the dual AREs without singular Hamiltonian issues.
• Provide a controller design framework that separates the output feedback problem into a state feedback and dual output injection problem, enabling systematic synthesis.

Proposed method

• Decompose the output feedback control problem into a state feedback problem and a dual output injection problem using system duality.
• Formulate the controller using solutions to two dual algebraic Riccati equations (AREs), ensuring the closed-loop system inherits the negative imaginary property.
• Apply Schur decomposition to the Hamiltonian-like matrix structure to avoid singularities commonly encountered in NI AREs.
• Solve two Lyapunov equations derived from the Schur decomposition to compute the solutions to the dual AREs.
• Construct the controller via state estimation and feedback using the solutions to the dual AREs, ensuring strict properness and stability.
• Use transformation matrices to diagonalize the system and isolate the key matrix equations governing the controller design.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a strictly proper dynamic controller to solve the output feedback control problem for negative imaginary systems?
• RQ2 How can the dual algebraic Riccati equations be solved robustly without encountering singular Hamiltonian problems common in NI systems?
• RQ3 Can the controller synthesis method ensure both the negative imaginary property and asymptotic stability of the closed-loop system?
• RQ4 How can the duality between state feedback and output injection be exploited to derive a systematic controller design procedure?
• RQ5 What conditions guarantee that the closed-loop system remains robustly stable under positive feedback when the plant uncertainty is strictly negative imaginary?

Key findings

• The paper establishes necessary and sufficient conditions for dynamic output feedback controller synthesis in negative imaginary systems, resolving a gap in prior work that only offered sufficient conditions.
• The controller design avoids singular Hamiltonians by using Schur decomposition and solving two Lyapunov equations, enabling reliable numerical computation.
• The closed-loop system is guaranteed to be negative imaginary, ensuring robust stability under positive feedback when the plant uncertainty is strictly negative imaginary (SNI), provided the DC gain is less than unity.
• The method successfully separates the output feedback problem into a state feedback and dual output injection problem, with solutions derived from dual AREs.
• The controller is strictly proper and ensures asymptotic stability, unlike earlier methods that could not rule out poles at the origin.
• The transformation matrices used in the derivation ensure that the resulting controller maintains the required structural properties, with explicit conditions on the matrix inverses and positive definiteness of the Riccati solutions.