Fractional-order systems and controls. Fundamentals and applications.

*(English)*Zbl 1211.93002
Advances in Industrial Control. London: Springer (ISBN 978-1-84996-334-3/hbk; 978-1-84996-335-0/ebook). xxvi, 414 p. (2010).

The main goal of the monograph is to present a concise and insightful view of the current knowledge on fractional-order control by emphasizing fundamental concepts, giving the basic tools to understand why fractional calculus is useful in control, to understand its terminology, and to illuminate the key points of its applicability.

The monograph is divided into six parts. The introductory Part I is devoted to mathematical concepts of fractional calculus and the system and control implications of those concepts. In Chapter 1 after a short historical review of the fractional calculus development the authors show how the necessity of defining and using the fractional-order differential and integral operators arise from very common and practical problems and applications and give a first approach to the definitions of these operators by using only well-known mathematical tools. In Chapter 2 after establishing the fundamental definitions of fractional calculus and determining the kind of solutions of Differential Equations (DEs) of fractional order the authors deal with the analysis of the systems described by such equations, similarly to the classical control theory framework. They start with the input-output models or representations of these systems in different domains, study their performance and steady state, and discuss the conditions and criteria for stability. The principles presented in Chapter 2 for the study of fractional-order dynamical systems expressed in the input-output representation are extended in Chapter 3 to the state space representation. Linear time-invariant commensurate-order systems are considered. The first part of Chapter 3 is devoted to the study of the representation and analysis of continuous systems, and the second one to discrete systems. For both kinds of systems results about stability analysis, controllability and observability are obtained. In Chapter 4 the fundamentals of fractional-order control are introduced. After a review of the evolution of fractional-order control strategies in control systems the effects of the fractional-order in the basic control actions, derivative and integral, are investigated. The generalized Proportional Integral Derivative (PID) controller is discussed in the frequency domain.

Part II: ‘Fractional-order PID-type controllers’ consists of Chapters 5–7. Chapter 5 begins with Fractional-Order control of First-Order Plus Dead-Time (FOPDT) plants which are widely seen in some indusrial process environments with focus on fractional-order \(PI^{\lambda}\) controllers. It is shown how to develop a set of practical tuning rules for \(PI^{\lambda}\) control of FOPDT plants. The tuning is optimal in the sense that the load disturbance rejection is optimized yet with a constraint of the maximum sensitivity. The so-called MIGO (\(M_{s}\) constrained Integral Gain Optimization) based controller tuning method is generalized to handle the \(PI^{\lambda}\) case, called F-MIGO, given the fractional order \(\alpha\). The F-MIGO method is then used to develop tuning rules for the FOPDT class of dynamical systems. The final tuning rules developed only apply the relative dead time \(\tau\) of the FOPDT model to determine the best fractional order \(\alpha\) and, at the same time, the best \(PI^{\lambda}\) gains. Simulation results show the practical nature of these tuning rules which are applicable for other more general classes than FOPDT. In Chapter 6 the new tuning method is presented for fractional-order proportional and derivative controllers \(PD^{\mu}\) for a class of second-order plants. The \(PD^{\mu}\) controller is tuned to ensure that the given gain crossover frequency and the phase margin are achieved, and also to guarantee the robustness of the systems to gain variations. The given simulation and experimental results show that the closed-loop system can achieve favorable dynamic performance and robustness. Chapter 7 deals with the design of fractional-order \(PI^{\lambda}D^{\mu}\) controllers, where the orders of the integral \(\lambda\) and derivative parts \(\mu\) are fractional with the aim to fulfill five different additional specifications of design, ensuring a robust performance of the controlled system with suspect to gain variations and noise. A method for tuning the \(PI^{\lambda}D^{\mu}\) controller is proposed to fulfill these specifications for closed-loop systems, which is based on a nonlinear function minimization subject to some given nonlinear constraints.

Part III: ‘Fractional-order lead-lag compensators’ consists of two chapters, concentrated on the tuning of fractional-order lead-lag compensators, that is the generalization of the classic lead-lag compensator. The similarities between this structure and the fractional-order \(PI^{\lambda}D^{\mu}\) controller in the frequency domain are discussed. The proposed design method avoids the nonlinear minimization and initial conditions of the Part II. In Chapter 8 a design method for Fractional-Order Lead-Lag Compensators (FOLLC) is presented. Simple relations between the parameters of the fractional-order controller are obtained and specifications of the steady-state error constant \(K_{ss}\), phase margin \(\varphi_{m}\) and gain crossover frequency \(\omega_{cg}\) are fulfilled following a robustness criterion based on the flatness of the phase curve of the compensator. The proposed tuning method is the first step for the generalization of these lead-lag compensators to the fractional-order \(PI^{\lambda}D^{\mu}\) controllers considered in Chapter 9.

Part IV: ‘Other fractional-order control strategies’ consists of Chapters 10, 11. It provides an overview of other fractional-order control strategies, showing their achievements and analyzing the challenges for further work. Chapter 10 reviews some important fractional-order robust control techniques, such as “Commande Robuste d’Ordre Non Entier” (CRONE) and Quantitative Feedback Theory (QFT). Chapter 11 presents some nonlinear fractional-order control strategies.

Part V: ‘Implementations of fractional-order controllers’ provides methods and tools for the implementation of fractional-order controllers. Chapter 12 deals with continuous and discrete-time implementations of these types of controllers and Chapter 13 with numerical issues and MATLAB implementations.

Part VI: ‘Real applications’ (Chapters 14–18) is devoted to some applications of fractional-order systems and controls. In Chapter 14 the identification problem of an electrochemical process and a flexible structure is presented; in Chapter 15 – the position control of a single-link flexible robot; in Chapter 16 – the automatic control of a hydraulic canal; Chapter 17 considers mechatronic applications; Chapter 18 presents fractional-order control strategies for power electronic buck converters.

In the Appendix additional useful information is given, such as Laplace transform tables involving fractional-order operators.

The monograph is divided into six parts. The introductory Part I is devoted to mathematical concepts of fractional calculus and the system and control implications of those concepts. In Chapter 1 after a short historical review of the fractional calculus development the authors show how the necessity of defining and using the fractional-order differential and integral operators arise from very common and practical problems and applications and give a first approach to the definitions of these operators by using only well-known mathematical tools. In Chapter 2 after establishing the fundamental definitions of fractional calculus and determining the kind of solutions of Differential Equations (DEs) of fractional order the authors deal with the analysis of the systems described by such equations, similarly to the classical control theory framework. They start with the input-output models or representations of these systems in different domains, study their performance and steady state, and discuss the conditions and criteria for stability. The principles presented in Chapter 2 for the study of fractional-order dynamical systems expressed in the input-output representation are extended in Chapter 3 to the state space representation. Linear time-invariant commensurate-order systems are considered. The first part of Chapter 3 is devoted to the study of the representation and analysis of continuous systems, and the second one to discrete systems. For both kinds of systems results about stability analysis, controllability and observability are obtained. In Chapter 4 the fundamentals of fractional-order control are introduced. After a review of the evolution of fractional-order control strategies in control systems the effects of the fractional-order in the basic control actions, derivative and integral, are investigated. The generalized Proportional Integral Derivative (PID) controller is discussed in the frequency domain.

Part II: ‘Fractional-order PID-type controllers’ consists of Chapters 5–7. Chapter 5 begins with Fractional-Order control of First-Order Plus Dead-Time (FOPDT) plants which are widely seen in some indusrial process environments with focus on fractional-order \(PI^{\lambda}\) controllers. It is shown how to develop a set of practical tuning rules for \(PI^{\lambda}\) control of FOPDT plants. The tuning is optimal in the sense that the load disturbance rejection is optimized yet with a constraint of the maximum sensitivity. The so-called MIGO (\(M_{s}\) constrained Integral Gain Optimization) based controller tuning method is generalized to handle the \(PI^{\lambda}\) case, called F-MIGO, given the fractional order \(\alpha\). The F-MIGO method is then used to develop tuning rules for the FOPDT class of dynamical systems. The final tuning rules developed only apply the relative dead time \(\tau\) of the FOPDT model to determine the best fractional order \(\alpha\) and, at the same time, the best \(PI^{\lambda}\) gains. Simulation results show the practical nature of these tuning rules which are applicable for other more general classes than FOPDT. In Chapter 6 the new tuning method is presented for fractional-order proportional and derivative controllers \(PD^{\mu}\) for a class of second-order plants. The \(PD^{\mu}\) controller is tuned to ensure that the given gain crossover frequency and the phase margin are achieved, and also to guarantee the robustness of the systems to gain variations. The given simulation and experimental results show that the closed-loop system can achieve favorable dynamic performance and robustness. Chapter 7 deals with the design of fractional-order \(PI^{\lambda}D^{\mu}\) controllers, where the orders of the integral \(\lambda\) and derivative parts \(\mu\) are fractional with the aim to fulfill five different additional specifications of design, ensuring a robust performance of the controlled system with suspect to gain variations and noise. A method for tuning the \(PI^{\lambda}D^{\mu}\) controller is proposed to fulfill these specifications for closed-loop systems, which is based on a nonlinear function minimization subject to some given nonlinear constraints.

Part III: ‘Fractional-order lead-lag compensators’ consists of two chapters, concentrated on the tuning of fractional-order lead-lag compensators, that is the generalization of the classic lead-lag compensator. The similarities between this structure and the fractional-order \(PI^{\lambda}D^{\mu}\) controller in the frequency domain are discussed. The proposed design method avoids the nonlinear minimization and initial conditions of the Part II. In Chapter 8 a design method for Fractional-Order Lead-Lag Compensators (FOLLC) is presented. Simple relations between the parameters of the fractional-order controller are obtained and specifications of the steady-state error constant \(K_{ss}\), phase margin \(\varphi_{m}\) and gain crossover frequency \(\omega_{cg}\) are fulfilled following a robustness criterion based on the flatness of the phase curve of the compensator. The proposed tuning method is the first step for the generalization of these lead-lag compensators to the fractional-order \(PI^{\lambda}D^{\mu}\) controllers considered in Chapter 9.

Part IV: ‘Other fractional-order control strategies’ consists of Chapters 10, 11. It provides an overview of other fractional-order control strategies, showing their achievements and analyzing the challenges for further work. Chapter 10 reviews some important fractional-order robust control techniques, such as “Commande Robuste d’Ordre Non Entier” (CRONE) and Quantitative Feedback Theory (QFT). Chapter 11 presents some nonlinear fractional-order control strategies.

Part V: ‘Implementations of fractional-order controllers’ provides methods and tools for the implementation of fractional-order controllers. Chapter 12 deals with continuous and discrete-time implementations of these types of controllers and Chapter 13 with numerical issues and MATLAB implementations.

Part VI: ‘Real applications’ (Chapters 14–18) is devoted to some applications of fractional-order systems and controls. In Chapter 14 the identification problem of an electrochemical process and a flexible structure is presented; in Chapter 15 – the position control of a single-link flexible robot; in Chapter 16 – the automatic control of a hydraulic canal; Chapter 17 considers mechatronic applications; Chapter 18 presents fractional-order control strategies for power electronic buck converters.

In the Appendix additional useful information is given, such as Laplace transform tables involving fractional-order operators.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

26A33 | Fractional derivatives and integrals |

93C10 | Nonlinear systems in control theory |

93C95 | Application models in control theory |