Title:

Feedback stability and properties of negative imaginary systems

In recent years, negative imaginary systems have attracted great attention due to their unique properties. The notion of negative imaginary systems was inspired by inertial systems. If a negative imaginary system with a single input and single output is considered, the output of a system follows but lags behind any sinusoidal input to the system by not more than 180 degrees. Additionally, the relative degree of a negative imaginary system is between zero and two. Due to these characterisation of negative imaginary systems, some applications of negative imaginary systems are found in such areas, as large space structures, multiagent networked systems, and nanopositioning control, etc. In this thesis, some tests are derived to characterise negative imaginary properties. Additionally, internal stability and characterisation of a positive feedback interconnection of negative imaginary systems are also discussed. Firstly, continuoustime and discretetime negative imaginary lemmas, which are applicable on real, rational, proper negative imaginary systems available in the literature, are derived. These lemmas generalise previous results in order to accommodate negative imaginary systems with possible poles at the origin in continuoustime (respectively poles at z=+1 in discretetime). The results proposed in these lemmas easily reduce to earlier conditions when the same assumptions are imposed. The introduced necessary and sufficient conditions involving linear matrix inequality conditions give complete statespace characterisation for the full class of real, rational, proper, continuoustime, negative imaginary systems (respectively discretetime, negative imaginary systems). Moreover, the dimension of the linear matrix inequality Lyapunov variable in these negative imaginary lemmas is smaller than or equal to the McMillan degree of the negative imaginary system. Secondly, a general robust stability theory is formulated to unify all stability results in all theories related to continuoustime negative imaginary systems to date. In the literature, a dc loop gain condition was used to test the internal stability of a positive feedback interconnection of a continuoustime negative imaginary system without poles at the origin and a continuoustime strictly negative imaginary system under instantaneous gain assumptions. By removing these restrictive assumptions that were imposed in prior results, necessary and sufficient conditions are provided for continuoustime negative imaginary systems with poles on imaginary axis, excluding at the origin. The conditions for determining the internal stability of a feedback system directly depend on a mixture of steadystate and infinity gains of the system. On the basis of these stability theorems, general stability results are derived for the case where the most general class of continuoustime negative imaginary systems is interconnected in a feedback system. A key feature for deriving these theorems does not require any complicated matrix factorisations that make the results less intuitive. Additionally, the key stability results easily reduce to the dc loop gain condition under identical assumptions imposed in the literature of continuoustime negative imaginary. Under different but simple assumptions that are not provided in the existing literature, these general stability theorems also simplify to the previous simple condition, i.e., the dc loop gain condition. Specialisations of the general stability theorems under singleinput singleoutput settings are given as corollaries to provide simple but elegant tests for checking feedback stability. Despite the fact that the above theory has been developed in continuoustime, these general robust feedback stability theorems are developed for interconnected discretetime negative imaginary systems to fill the gap that exists in the literature and these conditions can be considered as discretetime counterparts of the results that have been obtained in continuoustime. Finally, when a feedback system is considered to be internally stable, continuoustime negative imaginary systems interconnected in a positive feedback have negative imaginary properties. Without imposing the internal stability assumption of the feedback system, necessary and sufficient conditions are derived for a feedback interconnection of continuoustime negative imaginary systems with possible poles at the origin to preserve the characterisation of negative imaginary systems. These results, together with general stability theorems, are applied to check the stability of a string of interconnected continuoustime negative imaginary subsystems. An example in the decentralised control of large vehicle platoons is given to illustrate the usefulness of some of the results in this thesis.
