Key Takeaway: This chapter introduces the concept of controllability and observability in state linear systems, and demonstrates their generalizations using invariant subspaces and Lyapunov equations.

Abstract

In this chapter the system differential equation \(\dot{z}(t) = A z(t) + B u(t)\), \(y(t) = C z(t) + D u(t)\) is introduced. For this state linear system the concepts of controllability and observability are defined, and it shown that there are different generalisations of their finite-dimensional counterparts. Using the characterisation of invariant subspaces of Chaps. 2 and 3, tests for controllability and observability are derived for the different classes of systems. Since the controllability and observability gramian satisfy a Lyapunov equation, a section on Lyapunov equations is also part of this chapter. The chapter ends with a set of 28 exercises and a notes and references section.