In the study of linear time-invariant systems, a number of definitions, such as controllability, observability, not having decentralized fixed modes, minimum phase, etc., have been made. These definitions are highly useful in obtaining existence results for solving various types of control problems, but a drawback to these definitions is that they are binary, which simply determines whether a system is, for instance, either controllable or uncontrollable. In practical situations, however, there are many uncertainties in a system’s parameters caused by linearization, modelling errors, discretizations, and other numerical approximations and/or errors. So knowing that a system is controllable can sometimes be misleading if the controllable system is actually "almost" uncontrollable as a result of such uncertainties. Since an "almost" uncontrollable system poses significant difficulty in designing a quality controller, a continuous measure of controllability, called a controllability radius, is more desirable to use and has been widely studied in the past. The main focus of this thesis is to extend the development behind the controllability radius, with an emphasis on real parametric perturbations, to other definitions, replacing the traditional binary 'yes/no' metrics with continuous measures. We study four topics related to this development. First, we generalize the concept of real perturbation values of a matrix to the cases of matrix pairs and matrix triplets. By doing so, we are able to deal with more general perturbation structures and subsequently study, in addition to standard LTI systems, other types of systems such as LTI descriptor and time-delay systems. Second, we introduce the real decentralized fixed mode (DFM) radius, the real transmission zero at s radius, and the real minimum phase radius, which respectively measure how "close" i) a decentralized LTI system is to having a DFM, ii) a centralized system is to having a transmission zero at a particular point s in the complex plane, and iii) a minimum phase system is to being a nonminimum phase system. These radii are defined in terms of real parametric perturbations, and computable formulas for these radii are derived using a characterization based on real perturbation values and the aforementioned generalizations. Third, we present two efficient algorithms to i) solve the general real perturbation value problem, and ii) evaluate the various real LTI robustness radii introduced in this thesis. Finally as the last topic, we study the ability of a LTI system to achieve high performance control, and characterize the difficulty of achieving high performance control using a new continuous measure called the Toughness Index. A number of examples involving the various measures are studied in this thesis.

In the study of linear time-invariant systems, a number of definitions, such as controllability, observability, not having decentralized fixed modes, minimum phase, etc., have been made. These definitions are highly useful in obtaining existence results for solving various types of control problems, but a drawback to these definitions is that they are binary, which simply determines whether a system is, for instance, either controllable or uncontrollable. In practical situations, however, there are many uncertainties in a system’s parameters caused by linearization, modelling errors, discretizations, and other numerical approximations and/or errors. So knowing that a system is controllable can sometimes be misleading if the controllable system is actually "almost" uncontrollable as a result of such uncertainties. Since an "almost" uncontrollable system poses significant difficulty in designing a quality controller, a continuous measure of controllability, called a controllability radius, is more desirable to use and has been widely studied in the past. The main focus of this thesis is to extend the development behind the controllability radius, with an emphasis on real parametric perturbations, to other definitions, replacing the traditional binary 'yes/no' metrics with continuous measures. We study four topics related to this development. First, we generalize the concept of real perturbation values of a matrix to the cases of matrix pairs and matrix triplets. By doing so, we are able to deal with more general perturbation structures and subsequently study, in addition to standard LTI systems, other types of systems such as LTI descriptor and time-delay systems. Second, we introduce the real decentralized fixed mode (DFM) radius, the real transmission zero at s radius, and the real minimum phase radius, which respectively measure how "close" i) a decentralized LTI system is to having a DFM, ii) a centralized system is to having a transmission zero at a particular point s in the complex plane, and iii) a minimum phase system is to being a nonminimum phase system. These radii are defined in terms of real parametric perturbations, and computable formulas for these radii are derived using a characterization based on real perturbation values and the aforementioned generalizations. Third, we present two efficient algorithms to i) solve the general real perturbation value problem, and ii) evaluate the various real LTI robustness radii introduced in this thesis. Finally as the last topic, we study the ability of a LTI system to achieve high performance control, and characterize the difficulty of achieving high performance control using a new continuous measure called the Toughness Index. A number of examples involving the various measures are studied in this thesis.