Descriptor systems state-space conditions to guarantee negative imaginary properties without minimality restrictions

Abstract

This paper is concerned with negative imaginary lemmas for descriptor systems. Without the minimal state-space realization assumption, sufficient conditions are developed for a negative imaginary lemma and a strictly negative imaginary lemma, respectively. As corollaries, sufficient conditions are derived to ensure the systems are both admissible and negative imaginary. Also, new sufficient negative imaginary lemmas are found for standard linear systems as special cases. The developed negative imaginary lemmas are applicable to descriptor systems with impulse modes. Two examples are used to illustrate the theory.

Negative imaginary synthesis via dynamic output feedback and static state feedback: A Riccati approach

“…Therefore, we start by developing an SNI lemma which only requires the assumption of no modes at the origin. This builds on and extends previous results [30,11,20,29]. Moreover, the SNI lemma proposed here is based on an algebraic Riccati equation (ARE) with a unique positive semi-definite solution, even though the associated Hamiltonian is singular.…”

“…Yet, an SNI lemma without the assumption of minimal state-space realization is essential for controller synthesis. While [11,29] have relaxed the minimality assumption, [11] imposes conditions at infinity and at zero frequency which restrict the SNI class to the subclass known as SSNI. Since we are interested in providing a solution to the complete SNI synthesis problem, the SSNI Lemma [11] cannot be used here.…”

“…Since we are interested in providing a solution to the complete SNI synthesis problem, the SSNI Lemma [11] cannot be used here. On the other hand, in [29,Corollary 4] the system matrix is required to be Hurwitz as an assumption instead of being implied by the conditions, therefore it is not useful, on its own, to guarantee the internal stability of the resulting closedloop. Also, [26] provides an NI Lemma without minimal assumptions based on LMIs, but the system is allowed to be marginally stable which is not desirable.…”

“…the sizes of the Jordan blocks to which an eigenvalue belongs) of the zero eigenvalues of the Hamiltonian (1) are all even [2,8]. Furthermore, assumption R > 0 allows us to establish the necessity of (iv), but this assumption can be removed when only sufficiency of (iv) is of interest as shown in [29].…”

On negative imaginary synthesis via solutions to Riccati equations

“…SSNI systems are SNI systems with particular properties at zero and infinity frequencies which restrict the SNI system class. Moreover, [11] offered sufficient conditions for a system to be SNI, but the matrix A is required to be Hurwitz as an assumption instead of being Box 09-01-5863, Guayaquil, Ecuador. gapasalc@espol.edu.ec implied by the conditions.…”