This chapter collects mathematical topics needed later which is useful to cover beforehand for reference purposes: function Γ and combinations of a things, b at a time in section 1.1, the basics of calculus in section 1.2, the Laplace transformation in section 1.3, and continued fractions in section 1.4.

This chapter concerns how operator D can be generalised to real orders. After a couple of simple examples in section 2.1, the relevant definitions and some of its properties are given in section 2.2. Laplace transforms are studied in section 2.3. The way of calculating fractional derivatives is addressed in sections 2.4 (analytical results) and 2.5 (numerical results).

As in the integer case, it is expedient to start the study of dynamical systems by their input-output representation, that is to say, by using transfer functions. Section 3.1 presents the basic definitions; sections 3.2 and 3.4 address time and frequency responses, respectively. Results on stability are split between sections 3.3 and 3.5 because some are needed before frequency responses are studied but other can only be established thereafter.

It is possible to find integer transfer functions that approximate the dynamic behaviour of a given fractional transfer function. Such approximations are useful for two reasons. Firstly, while there are numerical methods to solve fractional differential equations, those for differential equations with integer orders only are better known and it may be desirable to use software where these methods are the sole implemented or where they are implemented with a better support. Secondly, hardware implementations of fractional controllers are possible, but it is often easier and cheaper to implement in hardware integer transfer functions only.

The chapter addresses methods to identify a model for a plant from its time response, from its frequency response, or from the phase of its frequency response. This latter option is particularly important for the design of Crone controllers (which are the subject of Chapters 6 and 15), and is developed, for this reason, so as to find models given by integer, discrete-time or fractional transfer functions, while the two former possibilities are developed for models given by fractional transfer functions only: the way of finding integer and discrete-time transfer functions from such data is well known.

First and second-generation Crone controllers are feedback loop controllers for SISO plants. Their purpose is to have a frequency behaviour of the open loop F(s), as seen in Figure 6.1, formed by the Crone controller C(s) and the plant G(s), with a constant phase, within a desired frequency range. According to (3.43), a constant phase means F(s) = sa (or at least F(s) behaves so within the frequency range of interest); a will be integer only if the constant phase is an integer multiple of 90°. This chapter addresses the purpose of these controllers in section 6.1, the first and second generations in sections 6.2 and 6.3, and their eventual combination with pre-compensating filters in section 6.4.

Fractional PIDs are generalisations of widely used PID controllers. After the necessary definitions in section 7.1, this chapter addresses methods for tuning fractional PIDs: analytical ones in sections 7.2 and 7.3, numerical ones in section 7.4 and tuning rules in section 7.5.

Fractional reset control is a non-linear control technique that consists in resetting to zero the value of the fractional derivatives in a fractional controller, whenever their input becomes zero. The effects of reset control in the control action are shown in Figure 8.1 for a derivative of order -1 (the integer case, also known as Clegg integrator, for which this type of control was originally developed).

H2 and H∞ controllers minimise a performance function, defined using either the H2 norm (addressed in section 9.1) or the H∞ norm (addressed in section 9.2), and reflecting how much the transfer function satisfies some frequency response requirements (as seen in section 9.3). This is true for both the integer and the fractional case, though in the latter the calculations involved are far more difficult.

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.

Fractional sliding mode control is a control technique for MIMO plants that can be linear or non-linear. It is expedient to consider first the case in which the plant is SISO and commensurable (in section 11.1), and then generalise the results for noncommensurable plants (in section 11.2), commensurable MIMO plants (in section 11.3) and non-commensurable MIMO plants (in section 11.4).

This chapter concerns a control subject that does not consist in obtaining a controller, but rather in finding a trajectory in space for an actuator (such as a robotic arm) to follow, avoiding obstacles in the way. For simplicity, the subject will be presented assuming a two-dimensional space. Its extension to the three-dimensional space is straightforward.

In this chapter, the results of the previous chapters are extended to complex orders. In what follows, the differentiation order will no longer be a ∊ R, but rather ζ ∊ C, with R(ζ) = a and F(ζ) = b; that is to say, ζ = a + jb.

The definitions of fractional and commensurable transfer functions from section 3.1 can easily be adapted to the complex case.

Third generation Crone control is a generalisation of the principles of first and second generation Crone control to the case of plants known with uncertainties (of all kinds, not just gain variations as was the case for first and second generation Crone controllers). It can be applied both to single-input, single-output (SISO) and square multiple-input, multiple-output (MIMO) plants, as seen in sections 15.1 and 15.2, respectively.

In this chapter, the results of Chapter 13 are extended to time-varying orders. In what follows, the differentiation order will no longer be z E C, but rather z(t) E C, with R(z(t)) = a(t) and F(z(t)) = b(t); that is to say, z(t) = a(t) + jb(t). There are three obvious ways of extending the definitions of section 13.2.

This chapter concerns time-varying transfer functions (in section 17.1), their approximations (in section 17.2), and their use in adaptive control (in section 17.3).

Fractional control is a blanket term for control techniques that in one way or another make use of fractional derivatives. These can be employed in different ways. Among the several control techniques addressed in this book, the following lead to controllers that are fractional in the sense that they have a dynamic behaviour that involves fractional derivatives.

From the several existing software packages dedicated to fractional control, the two most widespread are the CRONE toolbox and the NINTEGER toolbox, both freely available for MATLAB.