In this paper a Riemannian geometric framework is given for the minimum sensitivity design problem and its solution using a natural optimization criterion. The theory is then applied to the case of linear systems to generate a class of minimum sensitivity realizations related to the so-called balanced realizations. In particular, conditions are given which are applicable in the cases of fixed point and floating point implementations.

A Geometric Approach to the Minimum Sensitivity Design Problem

Discrete Time Nonlinear Balancing

A Riemannian geometric framework is given for the minimum sensitivity design problem and its solution using a natural optimization criterion. The theory is then applied to the case of linear systems to generate a class of minimum sensitivity realizations related to the so-called balanced realizations. In particular, conditions are given which are applicable in the cases of fixed point and floating point implementations. >

A geometric approach to the minimum sensitivity design problem

The geometric approach is a mathematical concept developed to improve the analysis and synthesis of linear multivariable systems. In this article we summarized the foundations for the geometric approach to both linear and singular systems. A mathematical introduction to the controlled and conditioned invariant subspaces has been presented. Additional characteristic subspaces were introduced, such as maximum-controlled and minimumconditioned ones, as well as output-null invariant subspaces. The solution of the disturbance localization problem with respect to the system stability was described. The mathematical and the geometric tools applied in the control system theory were presented. The geometric approach was systematically applied to the controllability analysis and the pole assignment procedure. The existence and uniqueness of the solutions for singular systems was separately analyzed due to specific system characteristics which are a consequence of the singular matrix in the state space model. The purpose of the article was to give a comprehensive overview of the geometric approach to the control theory.

Survey of the Geometric Approach to the Modern Control Theory

In a multirate sampled-data system encompassing a continuous-time process and multiple output samplers with periods $n_{1}T$ and $n_{2}T$, coupled with a zero-order hold with a period of $mT$ for the input, we introduce an innovative approach that leverages non-uniform data collated through a coprime collaborative sensing mechanism. The ultimate aim is to identify the intricate dynamics governing the system. The predominant challenge - relating to the accurate identification and representation of the multirate system dynamics - is addressed by pioneering a lifted state-space model for the system. This model is achieved by building upon and extending the subspace system identification. Moving forth, using this elevated model as a foundational basis, we seamlessly extract the single-rate system through an eigenvalue decomposition process. The proposed methodology's efficacy is empirically tested through demonstrative examples with multiple orders and varying coefficients.

State-Space System Identification Beyond the Nyquist Frequency with Collaborative Non-Uniform Sensing Data

A Hankel matrix approach to singular system realization theory is presented. The focus is exclusively on discrete-time or descriptor systems. Based on definitions of reachability and observability, the notion of a system Hankel matrix is first defined. The system Hankel matrix is then shown to have reachability-observability matrix factorizations which can be used to solve the realization synthesis problem. The notion of a balanced realization for singular systems is derived using the singular value decomposition of the system Hankel matrix. The theory is applied to synthesizing realizations which have minimum parametric sensitivity properties. It is demonstrated that such realizations are related to the notion of balancing. >

A Hankel matrix approach to singular system realization theory

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A Hankel matrix approach to singular system realization theory