Fractional-order Systems and Controls

Students of mathematics, sciences, and engineering encounter the differential operators d/dx, d²/dx², etc., but probably few of them ponder over whether it is necessary for the order of differentiation to be an integer. Why not be a rational, fractional, irrational, or even a complex number? At the very beginning of integral and differential calculus, in a letter to L’Hôpital in 1695, Leibniz himself raised the question: “Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?” L’Hôpital was somewhat curious about that question and replied by another question to Leibniz: “What if the order will be 1/2?” Leibniz in a letter dated September 30, 1695 replied: “It will lead to a paradox, from which one day useful consequences will be drawn.” The question raised by Leibniz for a non-integer-order derivative was an ongoing topic for more than 300 years, and now it is known as fractional calculus, a generalization of ordinary differentiation and integration to arbitrary (non-integer) order.

Before introducing fractional calculus and its applications to control in this book, it is important to remark that “fractional,” or “fractional-order,” are improperly used words. A more accurate term should be “non-integer-order,” since the order itself can be irrational as well. However, a tremendous amount of work in the literature use “fractional” more generally to refer to the same concept. For this reason, we are using the term “fractional” in this book.

Essentially, the mathematical problem for defining fractional-order derivatives and integrals consists of the following [2,7]: to establish, for each function f(z), z = x + jy of a general enough class, and for each number α (rational, irrational or complex), a correspondence with a function \( g(z) = \mathcal{D}^{\alpha}_{c} f(z) \) fulfilling the following conditions.

The principles presented in Chapter 2 for studying fractional-order dynamic systems expressed in the input-output representation are extended in this chapter to the state-space representation. This chapter considers only systems that exhibit the following features, which are similar to those considered in the previous chapter: (1) systems are linear, (2) systems are time invariant, and (3) systems are of commensurate-order. We will show that this last property enables models and properties that are a quite straightforward generalization of well known results for integer-order LTI (linear time invariant) systems to be obtained. This chapter is divided into two parts: the first is devoted to the study of the representation and analysis of continuous systems, and the second studies discrete systems. Similar results are obtained for both kinds of systems in modeling, solution of the differential (or difference) equations, stability analysis, controllability, and observability.

This chapter gives a historical review of fractional-order control. The main basis of the application of fractional calculus to control is given. In order for the reader to understand the effects of the generalized control actions (derivative and integral ones), a section is devoted to this topic.

PID (proportional integral derivative) controllers are the most popular controllers used in industry because of their simplicity, performance robustness, and the availability of many effective and simple tuning methods based on minimum plant model knowledge [82]. A survey has shown that 90% of control loops are of PI or PID structures [83, 84]. In control engineering, a dynamic field of research and practice, better performance is constantly demanded; therefore, developing better and simpler control algorithms is a continuing objective.

In Chapter 5, we focused on fractional-order proportional integral controller tuning. The plants to be controlled are assumed to be FOPDT plants. In this chapter, we focus on fractional-order proportional derivative controllers, PD ^μ , for another class of plants that are very common in motion control applications. A new tuning method for PD ^μ controllers is proposed to ensure that specifications of gain crossover frequency and phase margin are fulfilled. Furthermore, the derivative of the phase in Bode plot of the open-loop system with respect to the frequency is forced to be zero at the given gain crossover frequency so that the closed-loop system is robust to gain variations. The design method proposed is practical and simple to apply. Simulation and experimental results show that the closed-loop system can achieve favorable dynamic performance and robustness.

This chapter deals with the design of fractional-order PI ^λ D ^μ controllers, in which the orders of the integral and derivative parts, λ and μ respectively, are fractional. The purpose is to take advantage of the introduction of these two parameters and fulfil additional specifications of design, ensuring a robust performance of the controlled system with respect to gain variations and noise. A method for tuning the PI ^λ D ^μ controller is proposed to fulfil five different design specifications. Experimental results show that the requirements are fully met for the platform to be controlled.

This part of the book will concentrate on the tuning and auto-tuning of fractional-order lead-lag compensators, that is, a generalization of the classical lead-lag compensator. The similarities between this structure and the fractional-order PI ^λ D ^μ controller in the frequency domain will be discussed. The design method proposed here will avoid the nonlinear minimization and initial conditions problems presented in Part II of the book.

In this chapter, a design method for fractional-order lead-lag compensators (FOLLC) is presented. Simple relations among the parameters of the fractional-order controller are obtained and specifications of steady-state error constant K_ss, phase margin ϕ_m, and gain crossover frequency ω_cg are fulfilled, following a robustness criterion based on the flatness of the phase curve of the compensator. The tuning method proposed will be taken as a first step for a later generalization of these lead-lag compensators to the fractional-order PI ^λ D ^μ controllers, as will be explained in Chapter 9.

In this chapter, an auto-tuning method for the FOLLC described in the previous chapter will be discussed. We will study to what extent the auto-tuning method proposed here can be used for the tuning of the PI ^λ D ^μ controller, avoiding in this way the nonlinear minimization and initial conditions problems presented in Part II of the book.

In Chapter 4 we introduced the idea of robust control and basic controllers to fulfil this characteristic. In the frequency domain, we can say that Bode’s ideal loop transfer function in Section 2.3.4 is the reference model to achieve a robust performance. The purpose is to obtain an open-loop characteristic similar to that of this reference model, ensuring in this way a constant phase margin around a frequency of interest and, therefore, a constant overshoot of the time responses to plant gain variations.

This idea was developed extensively by Oustaloup [50], who studied the fractional-order algorithms for the control of dynamic systems and demonstrated the superior performance of the CRONE (Commande Robuste d’Ordre Non Entier, meaning Non-integer-order Robust Control) method over the PID controller. There are three generations of CRONE controllers [51], and we will briefly review all of them in this chapter.

The purpose of this section is to present the consequences of combining fractional-order control (FOC) and sliding mode control (SMC) in two ways, taking into account the two components of the sliding mode design approach. So, FOC is introduced in SMC by using, on the one hand, fractional-order sliding surfaces or switching functions, and, on the other hand, control laws with fractional-order derivatives and integrals. For a clear illustration of the proposed control strategies, the well known double integrator is considered as the system to be controlled.

Theoretically speaking, an integer-order transfer function representation to a fractional-order operator s ^α is infinite-dimensional. However it should be pointed out that a band-limit implementation of fractional-order controller (FOC) is important in practice, i.e., the finite-dimensional approximation of the FOC should be done in a proper range of frequencies of practical interest [17, 51]. Moreover, the fractional-order can be a complex number as discussed in [51]. In this book, we focus on the case where the fractional order is a real number.

There are several relevant MATLAB toolboxes which can be used to handle fractional-order systems. The N-integer toolbox [162] is the one used widely by the researchers. We also developed useful MATLAB code and an object-oriented toolbox in [181–183] which solves similar problems. This chapter is designed to be self-contained and is presented in the sequence of modeling, analysis, and design of fractional-order systems. Readers can run the examples on their own computer and obtain the same results. Many illustrative examples are given in the chapter to demonstrate the modeling, analysis, and design problems in fractional-order systems. Also the code can easily be applied to tackle other problems in the other chapters.

It has been experimentally observed or analytically found that both the time domain and frequency domain behaviors of some linear systems and processes do not fit the standard laws, i.e., exponential evolution in time domain or integer-order slopes in their frequency responses. In the time domain, it has been shown that these complicated dynamics can be described by, (i.e., the solutions of the constitutive equations are) generalized hyperbolic functions, \( \mathcal{F}^{k}_{\alpha , \beta} (z) \), defined as

. (14.1)

In particular, the Mittag–Leffler function in two parameters is defined as

, (14.2)

from which we can obtain the standard exponential, hyperbolic, or time-scaling functions as particular cases.

In this chapter fractional-order control is applied to accurate positioning of the tip of a single-link lightweight flexible manipulator. This kind of robot exhibits the advantage of being very lightweight. But they present a drawback in that vibrations appear in the structure when they move that prevent precise positioning of the end effector. Moreover, these vibrations may substantially change their amplitudes and frequencies when the tip payload changes, which is quite usual in robotics. The control of this kind of mechanical structure is nowadays a very challenging and attractive research area. These robots have found application in the aerospace and building construction industries, among others. This chapter develops a fractional-order controller that removes the structural vibrations and is robust to payload changes. The proposed control system is based on Bode’s ideal transfer function described in Section 2.3.4. Properties of this transfer function are used to design a controller with the interesting feature that the overshoot of the controlled robot is independent of the tip mass. This allows a constant safety zone to be delimited for any given placement task of the arm, independently of the load carried, thereby making it easier to plan collision avoidance. Other considerations about noise and motor saturation issues are also presented throughout the chapter.

Let us now turn our attention to the problem of the automation of water transportation processes, which are often implemented by means of open hydraulic canals. We will show how simple fractional-order PI controllers like those described in Chapter 5 can substantially improve the robustness of standard PI or PID controllers. Hydraulic canals are a typical example of dynamical systems with important delays and whose parameters may vary over a large range. Fractional-order controllers are designed that improve phase and/or gain margins — which are classical indices that measure closed-loop process robustness — while keeping the desired closed-loop behavior of the canal with the nominal dynamics. Moreover, it is shown that, for canals with significant and variable time delays, fractional-order controllers behave better than standard controllers when all of them are combined with the Smith predictor.

In this chapter, the tuning and auto-tuning methods described in Part III of the book will be applied to the control of a real mechatronic laboratory platform consisting of position and velocity servos. This type of devices are very commonly used in industrial environments and many other processes have the same type of transfer functions modeling their dynamics. For this reason, this application is rather practical and representative of a class of industrial processes.

The experimental platform and the implementation of the control strategy are described in the following sections.

This chapter presents several alternative methods for the control of power electronic buck converters applying fractional-order control (FOC). For achieving this goal, the controller design will be carried out by two strategies. On the one hand, the design of a linear controller for the DC/DC buck converter will be considered. In that sense, the Bode’s ideal loop transfer function presented in Chapter 2 will be used as reference system. On the other hand, the fractional calculus is proposed in order to determine the switching surface applying a fractional sliding mode control (F_RSMC) scheme to the control of such devices. In that sense, switching surfaces based on fractional-order PID and PI structures are defined. An experimental prototype has been developed and the experimental and simulation results confirm the validity of the proposed control strategies.