diffeomorphism; geometric structure; output feedback; immersion
In this paper, we investigate the geometric structures of the stable time-varying and the stable static output feedback systems. Firstly, we give a parametrization of stabilizing time-varying output feedback gains subject to certain constraints, that is, the subset of stabilizing time-varying output feedback gains is diffeomorphic to the Cartesian product of the set of time-varying positive definite matrices and the set of time-varying skew symmetric matrices satisfying certain algebraic conditions. Further, we show how the Cartesian product satisfying certain algebraic conditions is imbedded into the Cartesian product of the set of time-varying positive definite matrices and the set of time-varying skew symmetric matrices. Then, we give some eigenvalue properties of the stable time-varying output feedback systems. Notice that the stable static output feedback system, which does not depend on the temporal parameter $t$, is just a special case of the stable time-varying output feedback system. Moreover, we use the Riemannian metric, the connections and the curvatures to describe the subset of stabilizing static output feedback gains. At last, we use a static output feedback system to illustrate our conclusions.
 S. Amari: Differential geometry of a parametric family of invertible linear systems-Riemannian metric, dual affine connections, and divergence
. Math. Systems Theory 20 (1987), 53–83. MR 0901894
| Zbl 0632.93017
 A. Ben-Israel and T. N. E Greville: Generalized Inverses. Wiley, New York 1972.
 D. F. Delchamps: Global structure of families of multivariable linear systems with an application to identification
. Math. Systems Theory 18 (1985), 329–380. MR 0818420
 A. Hotz and R. E. Skelton: Covariance control theory
. Internat. J. Control 46 (1987), 13–32. MR 0895691
 P. S. Krishnaprasad: Symplectic mechanics and rational functions
. Ricerche Automat. 10 (1979), 107–135. MR 0614258
 A. Ohara and T. Kitamori: Geometric structures of stable state feedback systems
. IEEE Trans. Automat. Control 38 (1993), 1579–1583. MR 1242914
 A. Ohara and S. Amari: Differential geometric structures of stable state feedback systems with dual connections
. Kybernetika 30 (1994), 369–386. MR 1303289
 A. Ohara, S. Nakazumi, and N. Suda: Relations between a parametrization of Stabilizing state feedback gains and eigenvalue locations
. Systems Control Lett. 16 (1991), 261–266. MR 1102211
 A. Ohara, N. Suda, and S. Amari: Dualistic Differential geometry of positive definite matrices and its applications to related problems
. Linear Algebra Appl. 247 (1996), 31–53. MR 1412739
 A. Ohara and T. Kitamori: Robust stabilization for structurally perturbed plants by assigning a Lyapunov equation’s solution. (In Japanese.) Trans. SICE 25 (1989), 682–689.
 F. Zhong, H. Sun, and Z. Zhang: Geometric structures of stable time-variant state feedback systems
. J. Beijing Institute of Technology 16 (2007), 500–504. MR 2375866