Negativeimaginary systems are broadly speaking stable and square (equal number of inputs and outputs) systems whose Nyquist plot lies underneath (never touches for strictly negativeimaginary 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, multilink robots, DC machines, active filters, etc. In this thesis, robustness analysis and controller synthesis methods for uncertain negativeimaginary systems are explored.Two new reformulation techniques are proposed that facilitate both the robustness analysis and controller synthesis for uncertain negativeimaginary systems. These reformulations are based on the transformation from negativeimaginary systems to a boundedreal framework via the positivereal property. In the presence of strictly negativeimaginary uncertainty, the robust stabilization problem is posed in an equivalent $\mathcal{H}_{\infty}$ control framework; similarly, a negativeimaginary 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 boundedreal and negativeimaginary 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 negativeimaginary systems.In addition, a static statefeedback synthesis method is proposed to achieve robust stability of a system in the presence ofstrictly negativeimaginary 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 negativeimaginary stability theorem: A positive feedback interconnection of two negativeimaginary 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 negativeimaginary. Also, in order to handle nonstrict negativeimaginary uncertainties, a strongly strictly negativeimaginary lemma is proposed that helps to ensure the strictly negativeimaginary property of the nominal closedloop system for robustness. To this end, a statespace characterization for strictly negativeimaginary property is given for nonminimal 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 negativeimaginary systems that quite often arise in practical scenarios. In addition, they can be applied to quantify the worsecase performance for this class of systems. Therefore, the proposed results have important implications in controller synthesis for uncertain negativeimaginary systems that achieve not only robust stabilization but also robust performance.
Date of Award  1 Aug 2011 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Alexander Lanzon (Supervisor) 

 negativeimaginary systems, lightly damped systems, boundedreal, positivereal,
 $\mathcal{H}_{\infty}$ control, robust stability, robust performance, $\mu$ analysis, LMI
Robust analysis and synthesis for uncertain negativeimaginary systems
Song, Z. (Author). 1 Aug 2011
Student thesis: Phd