[Paper Review] Negative Imaginary Systems Theory in the Robust Control of Highly Resonant Flexible Structures

Ian R. Petersen|arXiv (Cornell University)|Jan 16, 2013

Piezoelectric Actuators and Control48 references17 citations

TL;DR

This paper introduces a robust control framework for highly resonant flexible structures using Negative Imaginary (NI) systems theory, which guarantees stability against unmodeled dynamics and parameter uncertainties. The key contribution is a stability theorem showing that positive-feedback interconnection of a NI system and a strictly NI system is internally stable if the DC loop gain is less than one, enabling robust controller synthesis for structures with collocated sensors and actuators.

ABSTRACT

This paper covers recent developments in the theory of negative imaginary systems and their application to the control of highly resonant flexible structures. The theory of negative imaginary systems arose out of a desire to unify a number of classical methods for the control of lightly damped structures with collocated force actuators and position sensors including positive position feedback and integral force feedback. The key result is a stability result which shows why these methods are guaranteed to yield robust closed loop stability in the face of unmodelled spillover dynamics. Related results to be presented connect the theory of negative imaginary systems to positive real systems theory and a negative imaginary lemma has been established which is analogous to the positive real lemma. The paper also presents recent controller synthesis results based on the theory of negative imaginary systems.

Motivation & Objective

• Address the challenge of robust vibration control in highly resonant flexible structures, such as aerospace systems and nano-positioning devices.
• Unify classical control methods like positive position feedback and integral force feedback under a single theoretical framework.
• Provide a systematic method to analyze and design controllers that are robust to spillover dynamics and parameter uncertainties.
• Establish a connection between negative imaginary systems and positive real systems theory to enable stability analysis and controller synthesis.
• Develop synthesis techniques for state-feedback and output-feedback controllers that preserve the NI property and ensure closed-loop stability.

Proposed method

• Define negative imaginary (NI) transfer function matrices via frequency-domain conditions: the Hermitian-imaginary part of the frequency response is negative semidefinite for all frequencies.
• Establish a negative imaginary lemma analogous to the positive real lemma, enabling LMI-based stability and controller synthesis.
• Use the equivalence: a transfer function R(s) is NI if and only if sR(s) is positive real, linking NI systems to established positive real theory.
• Propose controller classes such as positive-position feedback, resonant, and integral resonant controllers, all proven to be strictly negative imaginary (SNI).
• Develop LMI-based state-feedback controller synthesis via the negative imaginary lemma, ensuring robust stability under unmodeled dynamics.
• Apply the framework to MIMO systems with collocated force actuators and position sensors, ensuring stability even with uncertain or unmodeled spillover dynamics.

Experimental results

Research questions

• RQ1 How can classical control methods like positive position feedback and integral force feedback be unified under a single theoretical framework?
• RQ2 What conditions guarantee robust stability of flexible structures with unmodeled dynamics and parameter uncertainties?
• RQ3 How can the negative imaginary property be used to derive a stability condition for positive-feedback interconnections of NI systems?
• RQ4 What is the relationship between negative imaginary systems and positive real systems, and how can this be exploited for controller design?
• RQ5 How can LMI-based synthesis techniques be applied to design robust state-feedback controllers for NI systems?

Key findings

• The paper establishes that a positive-feedback interconnection of a negative imaginary system and a strictly negative imaginary system is internally stable if the DC loop gain is less than one.
• All flexible structures with collocated force actuators and position sensors have a negative imaginary transfer function matrix, justifying the use of NI theory for such systems.
• Positive-position feedback controllers are proven to be strictly negative imaginary (SNI), ensuring robust stability against unmodeled dynamics and frequency uncertainties.
• The negative imaginary lemma provides an LMI characterization for NI systems, enabling systematic controller synthesis via convex optimization.
• State-feedback control laws can be synthesized using LMI conditions (Theorem 10) to ensure robust stability for uncertain NI systems with bounded DC gain and positive semidefinite uncertainty at DC.
• Integral resonant controllers of the form $ C(s) = [sI + \Gamma\Phi]^{-1}\Gamma $ are shown to be SNI when $ \Gamma $ and $ \Phi $ are positive definite, enabling robust force control in flexible systems.