Robust controllability and observability degrees of polynomially uncertain systems

This paper deals with the class of polynomially uncertain continuous-time linear time-invariant (LTI) systems whose uncertainties belong to a semi-algebraic set. The objective is to determine the minimum of the smallest singular value of the controllability or observability Gramian over the uncertainty region. This provides a quantitative measure for the robust controllability or observability degree of the system. To this end, it is shown that the problem can be recast as a sum-of-squares (SOS) problem. In the special case when the uncertainty region is polytopic, the corresponding SOS formulation can be simplified significantly. One can apply the proposed method to any large-scale interconnected system in order to identify those inputs and outputs that are more effective in controlling the system, in a robust manner. This enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak. A numerical example is presented to demonstrate the efficacy of the results

Determination of Design of Optimal Actuator Location Based on Control Energy

The thesis deals with the selection of the sets of inputs and outputs using the energy properties of the controllability and observability of a system and aims to define input and output structures which require minimization of the energy for control and state reconstruction. Such a study explores the energy dimension of the properties of controllability and observability, develops computations for the controllability and observability Gramians for stable and unstable systems and examines measures of the degree of controllability and observability properties using SVD (Singular Value Decomposition) of Gramians to compute the maximal and minimal energy requirements. These characterize the relative degree of controllability and observability under conditions where the available energy is constrained. The notion of energy surfaces in the state space is introduced and this enables the characterization of restricted notions of controllability and observability when the available energy is bounded. The maximal and minimal energy requirements for different input vectors is demonstrated and this provides the basis for the development of strategies and methodologies for selection of systems of inputs and outputs to minimize the energy required for control, respectively state reconstruction. These results enable the development of input, output structure selection methodology using a novel optimization method. This thesis contributes in the further development of the area of systems, or global instrumentation, developed so far based on the assignment of structural characteristics by incorporating the role of energy requirements. The research provides energy based tools for the selection of input and outputs schemes with a main criterion the minimization of the energy required for control and observation and thus provide an alternative approach based on quantitative system properties in characterizing control and state observation as functions of given sets of inputs and output sets. The methodologies developed may be used as design tools where apart from energy requirements other design criteria may be also incorporated for the selection of inputs and outputs. The methodology that is used is based on linear systems theory and tools from numerical linear algebra. The solution to the problems considered here is an integral part of the effort to develop an integrated approach to control and global process instrumentation