Definition 10.1 (State variables). Consider a system with a known dynamical behaviour, and suppose that its inputs are known from an arbitrary time instant t on. The state variables of the system are those in a set of variables, with as few elements as possible, such that, knowing them at instant t, it is possible to calculate the system's future behaviour. Remark 10.1. As is well known, a system's state variables are not unique. If x(t) is an n x 1 vector with the system's state variables, and P is an n x n invertible matrix, then the variables in vector w(t) = Px(t) also are state variables. Fractional order systems have no state variables, as we shall see below; but it is possible to obtain for them representations similar to those that use the state variables of integer systems. This chapter addresses first the general case of multipleinput, multiple-output (MIMO) systems, and then the particular case of single-input, single-output (SISO) systems. The discretisation of these representations is also addressed.

Chapter Contents:

• 10.1 Pseudo-state-space representations of MIMO systems
• 10.1.1 Linear system
• [10.1.2 Commensurable system, 10.1.2 Commensurable system]
• [10.1.3 Fractional MIMO transfer function matrix, 10.1.3 Fractional MIMO transfer function matrix]
• [10.1.4 Non-linear fractional systems, 10.1.4 Non-linear fractional systems]
• [10.2 Pseudo-state-space representations of SISO systems, 10.2 Pseudo-state-space representations of SISO systems]
• [10.3 Discretisation, 10.3 Discretisation]
• [Further reading, Further reading]
• Appendix to Chapter 10

Inspec keywords: state-space methods; matrix inversion; MIMO systems

Other keywords: pseudo-state-space representations; MIMO systems; invertible matrix; dynamical behaviour; multiple-input multiple-output systems; single-input single-output systems; state variables; SISO systems; vector variables; integer systems; fractional order systems

Subjects: Multivariable control systems; Control system analysis and synthesis methods; Algebra