ABSTRACT

In Chapter 3 we analyzed the Kalman decomposition, which allows us to determine the controllable and observable part of a system. In this chapter we deal with determining how controllable and observable a system is. In simplifying the poles and zeros of a system, the Kalman decomposition takes into account the scenario where there is an exact simplification between pole and zero, whereas the technique discussed in this chapter examines the case in which poles and zeros are very close to each other. The role of both the controllability and observability gramians is studied. The discussion leads to remark the concept of system invariants, in particular the singular values of a linear dynamical system are presented. The subject is considered one of the most important in the book and therefore both the theoretical aspects and the algorithms to derive the open-loop balanced representation are discussed. The case of discrete-time systems is also reported. In this part the principal component analysis procedure is also discussed. The chapter includes several worked examples.