Multivariable Feedback with Geometric Tools

I remember Toni for his personal warmth and innate respect; for a friendship you could feel. He would give important and less important bits of advice, throwing in the odd comment, or dropping a hint, never insisting, to show he meant no offence. I remember how attached he was to us in Bologna. During the first research doctorate the university seats of Bologna, Florence and Padua had formed a consortium, but he would still take the train to Bologna to hold his lectures because he liked to breathe the atmosphere of the city and walk across its main square, Piazza Maggiore. I keep in my library a very simple, clear and attractive small book with a nice friendly hand-written dedication to me by Toni. The book presents feedback in simple terms but reflects Toni’s in-depth knowledge, highlighting the presence of feedback in many branches of the scientific world. Inspired by this book and recalling that many years ago Toni probably read our first paper on geometric tools in Italian, I decided to dedicate this contribution to feedback in geometric terms. In this short monograph I will try to explain how these tools not only enable a neat extension to the multivariable case of the most basic features of feedback control for single variable systems, including the internal model of the exosystem, hence steady-state robustness, but they are just within arm’s reach using a certain number of algorithms available in a specific Geometric Approach toolbox for Matlab. Although this approach sounds didactic and somewhat out of standard, it nevertheless contains an original idea: A proposal to extend model-following control to non minimal phase systems. Being a non optimal, but only pseudo-optimal solution, the proposal explains the detailed illustration of this toolbox and its use in the synthesis of multivariable control systems. The geometric approach to system analysis and control suffered from a very slow and inconsistent growth for about four decades, with papers by numerous authors in many different styles, often aiming to impress readers (or reviewers) with intricate mathematics rather than giving them the most direct insight and feeling.