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In this note, we present a simple and readily computable formula for the calculation of the real controllability/stabilizability and observability/detectability radii for linear time-invariant systems. Index Terms--Linear time-invariant (LTI) systems, real contronability/stabilizability radius, real observability/detectability radius.

In the design of conventional control systems for a multivariable system, using robust/adaptive control techniques, the motivation is to design a controller which "works satisfactorily" in the presence of plant uncertainty. Unfortunately, however, if large unanticipated structural changes subsequently occur in the system, severe limitations in practical performance may occur, since such conventional control schemes usually do not have the ability to control systems which are subject to unplanned extreme changes. Moreover, for the realistic situation when control input constraints exist, few results for continuous time multivariable systems are available. In this paper, a new class of self-tuning proportional-integral-derivative switching controllers, which is an extension of the self-tuning integral controller of Miller and Davison, is described, and has the property that it is robust to unplanned extreme changes in the plant and satisfies any feasible control signal input constraints. Results of this self-tuning controller when applied to an experimental multivariable system also are described. Index Terms--Adaptive switching control, extreme perturbations, proportional-integral-derivative (PID), self-tuning controllers, unanticipated perturbations.

Fundamental limitations for error tracking/regulation are obtained for the robust servomechanism problem (RSP) for a sampled system. In studying this problem, the cheap control problem for a multi-input/multi-output discrete time system is considered, and explicit expressions are obtained for the limiting steady state solution of its associated algebraic Riccati equation (ARE), as the weight on the control energy tends to zero. Application of these results is then made to obtain explicit expressions for the limiting performance costs associated with error tracking/regulation in the RSP. These limitations can be characterized by the system order, the dimension of the outputs, the number of the system's transmission zeros and the location of the system's nonminimum phase transmission zeros. Index Terms--Cheap control problem, nonminimum phase, optimal control, performance limitations, sampled systems, servomechanism problem.

A new family of metrics, called pointwise gap metrics, in the space of real rational matrices of fixed size is developed in this paper. These metrics are then used to study open-loop and closed-loop stability robustness of linear time-invariant finite-dimensional continuous-time systems. It is shown that pointwise gap metrics have the desired qualitative properties for the study of stability robustness. Necessary and sufficient conditions on the open- and closed-loop stability robustness are obtained in terms of the radii of the pointwise gap metric balls centered at the nominal plant and/or the nominal controller. Comparison of the new metric with the available metrics, e.g., the gap metric and the graph metric, is made. All of these metrics induce the same topology. Surprisingly, it is shown that many of the quantitative properties of pointwise gap metrics are the same as those of the gap metric, although they differ in value. A notable distinct property of pointwise gap metrics is that in the scalar case, they have a very simple expression which is potentially useful to access the relationship between the uncertainty of physical parameters and uncertainty measured by pointwise gap metrics.