Robust analysis and synthesis for uncertain negative-imaginary systems

  • Zhuoyue Song

Student thesis: Phd


Negative-imaginary systems are broadly speaking stable and square (equal number of inputs and outputs) systems whose Nyquist plot lies underneath (never touches for strictly negative-imaginary systems) the real axis when the frequency varies in the open interval $0$ to $\infty$.This class of systems appear quite often in engineering applications,for example, in lightly damped flexible structures with collocatedposition sensors and force actuators, multi-link robots, DC machines, active filters, etc. In this thesis, robustness analysis and controller synthesis methods for uncertain negative-imaginary systems are explored.Two new reformulation techniques are proposed that facilitate both the robustness analysis and controller synthesis for uncertain negative-imaginary systems. These reformulations are based on the transformation from negative-imaginary systems to a bounded-real framework via the positive-real property. In the presence of strictly negative-imaginary uncertainty, the robust stabilization problem is posed in an equivalent $\mathcal{H}_{\infty}$ control framework; similarly, a negative-imaginary robust performance analysis problem is cast into an equivalent $\mu$-framework. The latter framework also allows robust stability analysis when the perturbations are a mixture of bounded-real and negative-imaginary uncertainties. The proposed two techniques pave the way for existing $\mathcal{H}_{\infty}$ control and $\mu$ theory to be applied to robustness analysis and controller synthesis for negative-imaginary systems.In addition, a static state-feedback synthesis method is proposed to achieve robust stability of a system in the presence ofstrictly negative-imaginary uncertainties. The method is developed in the LMI framework, which can be solved efficiently using convex optimization techniques. The controller synthesis method is based on the negative-imaginary stability theorem: A positive feedback interconnection of two negative-imaginary systems is internally stable if and only if the DC loop gain is contractive and at least one of the systems in the interconnected loop is strictly negative-imaginary. Also, in order to handle non-strict negative-imaginary uncertainties, a strongly strictly negative-imaginary lemma is proposed that helps to ensure the strictly negative-imaginary property of the nominal closed-loop system for robustness. To this end, a state-space characterization for strictly negative-imaginary property is given for non-minimal systems where the conditions are convex and hence numerically attractive.The results in this thesis hence facilitate both the robustness analysis and controller synthesis for negative-imaginary systems that quite often arise in practical scenarios. In addition, they can be applied to quantify the worse-case performance for this class of systems. Therefore, the proposed results have important implications in controller synthesis for uncertain negative-imaginary systems that achieve not only robust stabilization but also robust performance.
Date of Award1 Aug 2011
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorAlexander Lanzon (Supervisor)


  • negative-imaginary systems, lightly damped systems, bounded-real, positive-real,
  • $\mathcal{H}_{\infty}$ control, robust stability, robust performance, $\mu$ analysis, LMI

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