Output feedback stabilization for symmetric control systems☆
Section snippets
Introduction and notation
There is a wide range of topics from several disciplines involving symmetric systems. In particular, a state-space symmetric system has a physical interpretation as an electrical circuit containing amplifiers, interconnected through a network of linear gains. The study of systems with the symmetry property attracted the attention of many authors, for example, a sufficient condition for global convergence of a class of symmetric neural circuits is given in [1], systems with zeros interlacing the
Symmetric systems
In this section, we firstly present some results about symmetric systems represented in both state-space and input–output form. Definition 2.1 (i) A state-space realization is symmetric if it satisfies the following conditions: and . (ii) A system is externally symmetric if it satisfies . (iii) A system is internally symmetric if there exists a symmetric minimal realization.
It is clear that internal symmetry implies the external one. However, a externally symmetric
The stabilization problem
In this section, given a symmetric system , the symmetric stabilization problem is studied using a symmetric feedback gain such that the control law stabilizes the closed-loop system. To prove a necessary and sufficient condition for this stabilization problem we need the following results. Lemma 3.1 A real matrix A is a symmetric matrix with if and only if is a symmetric positive definite matrix with . Proof It follows immediately that A is a symmetric matrix if and
Conclusions
We have studied symmetric control systems. A relationship was shown between internal symmetry and the external one of a system having n simple poles . This leads to obtain a symmetric diagonal realization of . We have given a necessary and sufficient condition to guarantee that an output feedback stabilizes a closed-loop symmetric control system. This result was also extended to generalized symmetric control system where the matrix E is a positive definite matrix.
References (13)
- et al.
Model reduction of systems with zeros interlacing the poles
Systems Control Lett.
(1997) - et al.
Model reduction for state-space symmetric systems
Systems Control Lett.
(1998) - et al.
Stabilization and control of symmetric systems: and explicit solution
Systems Control Lett.
(2001) - et al.
Decentralized control of symmetric systems
Systems Control Lett.
(2001) - et al.
A condition for global convergence of a class of symmetric neural circuits
IEEE Trans. Circuits Syst. I
(1992) - et al.
Reliable linear-quadratic control for symmetric composite systems
Int. J. Syst. Sci.
(2001)
Cited by (9)
Nonnegativity, stability, and regularization of discrete-time descriptor systems
2010, Linear Algebra and Its ApplicationsAn algorithm to check the nonnegativity of singular systems
2007, Applied Mathematics and ComputationOn positive realness and negative imaginariness of uncertain discrete-time state-space symmetric systems
2020, International Journal of Systems ScienceAn algorithm to study the nonnegativity, regularity and stability via state-feedbacks of singular systems of arbitrary index
2015, Linear and Multilinear AlgebraOn the dissipative analysis and control of state-space symmetric systems
2011, Proceedings of the IEEE Conference on Decision and ControlH<inf>∞</inf> norm computation for descriptor symmetric systems
2009, Proceedings of 2009 7th Asian Control Conference, ASCC 2009
- ☆
Partially supported by Generalitat Valenciana under Grant Grupos03/062.