Preface

Proportional–integral–derivative (PID) control, because of its simplicity in structure and its robustness with the uncertainties and disturbances, is such a popular control technology even today and has been widely used in many industrial processes. Furthermore, from a philosophical viewpoint, PID control is the embodiment of such a control idea that the control input can be determined based on the combination of the information at the present (P), the past (I), and the future (D), which is exactly the representation of the human being's wisdom, idea, and method to problem-solving.

Due to the popularity of PID control, a ton of content addressing this topic has already been made available by various publishers. At the same time, countless optimization opportunities have emerged with the development of science and technology, such as big data, artificial intelligence, machine learning, etc. There are also plenty of new fields that can leverage the power of control technologies. Some of the great examples include biological engineering, medical technologies, and financial technologies. The list is even extended to applications for social management and/or regulations, etc.

Therefore, PID control will continue to embrace new opportunities and face new challenges at the same time. This book highlights some of the new aspects of this subject and its related applications. Generally speaking, the better the cognition, the easier the realization of control. Yet, the relationship between cognition and control of complex systems can go both ways. We also know that the proper control method may be helpful for the improvement of the cognition of systems. That is to say, there is a mutual promotion between control and identification of complex systems that happens to reflect the law of the spiral rise in independent cognitive processes.

The main topics of this book include the improvement of PID controller, intelligent PID controller, optimization of PID controller, new forms of PID controller, development of PID controller, new application fields, etc. On the other hand, this book will also reveal that the proper control with a PID framework can also improve the cognition or identification for complex systems.

We intend to not only summarize the advance of science and technology on PID control but also extend the connotation and its extensions. We would like to bring more inspiration and benefits to the new fellows.

Based on the previous considerations, we present the book with the following chapters.

In Chapter 1, for some complex systems, the novelty method of the variable, fractional order PID controller is proposed. The main idea is to apply the split orders to discrete differentiation and summation functions, such that in the final time interval, the variable, fractional order PID controller must transform itself to the classical one that preserves the well-known stability conditions and zero steady-state error signal.

Chapter 2 focuses on the design and simulation of a hybrid LQR-PID controller used to stabilize the elevation, pitch, and travel axes of a helicopter system. An improvement in the performance of the hybrid LQR-PID controller is achieved by using a genetic algorithm (GA), which is adopted to obtain the best values of gain parameters for the LQR-PID controller. The strategy of the hybrid controller is based on the idea that the parameters of the PID controller are calculated using gain elements of the LQR optimal controller.

Chapter 3 presents the alternative optimized tuning approach of the PID controller. First, the PID controller parameters are optimized for the tracking response. Then, a simple disturbance estimator is introduced to substantially increase the disturbance-rejection performance. The proposed approach can still be used on the process data given either in parametric or nonparametric form. It will be shown how to achieve the best trade-off between performance and noise attenuation.

To solve the tuning problem in highly complex industrial processes, Chapter 4 concerns a controller adjustment method based on the internal product of PID terms. A propagation matrix (PM) is generated by the numerator coefficients of the plant transfer function (TF). In the proposed method, each term of the PID controller is influenced by each of the numerator and the denominator coefficients. Mathematical models of practical plants were employed to evaluate the proposed method. The obtained results demonstrated an assertive improvement in the adjustment gains from PID actions, thereby validating it as a promising alternative to conventional methods.

In Chapter 5, an application to the magnetic levitation (maglev) system is considered. To account for the complexity in the design procedure, this chapter presents a practical controller for the high-positioning performance of a magnetic levitation system. Three strategies of the proposed controller are the PI-PD controller is to enhance transient response, the model-based feedforward (FF) control is incorporated with the PI-PD controller to enhance the overshoot reduction characteristic in attaining a better transient response, and lastly, the disturbance compensator (Kz) is integrated as an additional feedback element to reduce the sensitivity function magnitude for robustness enhancement.

In Chapter 6, the estimation of multiparameters of complex systems based on the extended PID controllers is considered. Based on the results, with the introduction of a binary control mechanism, the integral item of the nonlinear PID controller could deal with the uncertain part of the complex system, which can also be called the new stripping principle (NSP), the new multiparameter estimation methods are given. Such kind of effort will improve the identification or cognition for certain kinds of complex systems, and it will also provide a completely different and effective way for the estimation methods in mathematical statistics.

Ever since the idea of publishing this book has been issued, we have received more than ten proposals and suggestions on the relevant chapters, given the current pandemic situation. We have accepted six of them to be included in this book. The intention of this book is to provide an active spur that will hopefully induce someone to come forward with his/her valuable contributions and to offer more relevant achievements in the future.

As this book is about to be published, I would like to express my deepest gratitude to all the authors of this book for their tremendous efforts in submitting their own

**V**

chapters. I would like to give my sincere appreciation for the support from the IntechOpen publishing working team, and Miss Sara Debeuc in particular, for their

**Wei Wang**

Beijing, China

School of Mathematics, Renmin University of China,

kind support throughout the entire publishing process.

chapters. I would like to give my sincere appreciation for the support from the IntechOpen publishing working team, and Miss Sara Debeuc in particular, for their kind support throughout the entire publishing process.

> **Wei Wang** School of Mathematics, Renmin University of China, Beijing, China

**IV**

achievements in the future.

Chapter 2 focuses on the design and simulation of a hybrid LQR-PID controller used to stabilize the elevation, pitch, and travel axes of a helicopter system. An improvement in the performance of the hybrid LQR-PID controller is achieved by using a genetic algorithm (GA), which is adopted to obtain the best values of gain parameters for the LQR-PID controller. The strategy of the hybrid controller is based on the idea that the parameters of the PID controller are calculated using gain

Chapter 3 presents the alternative optimized tuning approach of the PID controller. First, the PID controller parameters are optimized for the tracking response. Then, a simple disturbance estimator is introduced to substantially increase the disturbance-rejection performance. The proposed approach can still be used on the process data given either in parametric or nonparametric form. It will be shown how to achieve the best trade-off between performance and noise attenuation.

To solve the tuning problem in highly complex industrial processes, Chapter 4 concerns a controller adjustment method based on the internal product of PID terms. A propagation matrix (PM) is generated by the numerator coefficients of the plant transfer function (TF). In the proposed method, each term of the PID controller is influenced by each of the numerator and the denominator coefficients. Mathematical models of practical plants were employed to evaluate the proposed method. The obtained results demonstrated an assertive improvement in the adjustment gains from PID actions, thereby validating it as a promising alternative

In Chapter 5, an application to the magnetic levitation (maglev) system is considered. To account for the complexity in the design procedure, this chapter presents a practical controller for the high-positioning performance of a magnetic levitation system. Three strategies of the proposed controller are the PI-PD controller is to enhance transient response, the model-based feedforward (FF) control is incorporated with the PI-PD controller to enhance the overshoot reduction characteristic in attaining a better transient response, and lastly, the disturbance compensator (Kz) is integrated as an additional feedback element to reduce the sensitivity function

In Chapter 6, the estimation of multiparameters of complex systems based on the extended PID controllers is considered. Based on the results, with the introduction of a binary control mechanism, the integral item of the nonlinear PID controller could deal with the uncertain part of the complex system, which can also be called the new stripping principle (NSP), the new multiparameter estimation methods are given. Such kind of effort will improve the identification or cognition for certain kinds of complex systems, and it will also provide a completely different

Ever since the idea of publishing this book has been issued, we have received more than ten proposals and suggestions on the relevant chapters, given the current pandemic situation. We have accepted six of them to be included in this book. The intention of this book is to provide an active spur that will hopefully induce someone to come forward with his/her valuable contributions and to offer more relevant

As this book is about to be published, I would like to express my deepest gratitude to all the authors of this book for their tremendous efforts in submitting their own

and effective way for the estimation methods in mathematical statistics.

elements of the LQR optimal controller.

to conventional methods.

magnitude for robustness enhancement.

**Chapter 1**

Method

**Abstract**

**1. Introduction**

*Piotr Ostalczyk and Piotr Duch*

Variable, Fractional-Order PID

The novelty method of the discrete variable, fractional order PID controller is proposed. The PID controllers are known for years. Many tuning continuous time PID controller methods are invented. Due to different performance criteria there are optimized three parameters: proportional, integral and differentiation gains. In the fractional order PID controllers there are two additional parameters: fractional order integration and differentiation. In the variable, fractional order PID controller fractional orders are generalized to functions. Nowadays all PID controllers are realized by microcontrollers in a discrete time version. Hence, the order functions are discrete variable bounded ones. Such controllers offer better transient characteristics of the closed loop systems. The choice of the order functions is still the open problem. In this Section a novelty intuitive idea is proposed. As the order functions one applies two spline functions with bounded functions defined for every time subinterval. The main idea is that in the final time interval the variable, fractional order PID controller transforms itself to the classical one preserving the stability conditions and zero steady-state error signal. This means that in the last time

A continuous-time proportional–integral–derivative controller (PID controller) [1] invented almost 100 years ago is one of the most widely applied controllers in the closed-loop systems [2] with many industrial applications [3–5]. Currently the continuous-time control is successively replaced by discrete-time one in which the integration is replaced by a summation and differentiation by a difference evaluation. So, in the discrete PID controller the classical integral is replaced by a sum and the derivative by a backward difference, [6]. The discrete controller's PID algorithm

At 70s of the 20-th Century the Fractional Calculus [8] with a great success started a considerable attention in mathematics and engineering [9–12]. Now, the fractional-order backward-difference (FOBD) and the fractional-order backward sum (FOBS) [6, 13] are applied in the dynamical system modeling [14] and discrete control algorithms. The continuous-time FOPID controllers are more difficult in a

Controller Synthesis Novelty

interval the discrete integration order is 1 and differentiation is 1.

**Keywords:** fractional-Calculus, PID controller, discrete system

is mainly realized by micro-controllers [7].

practical realization [15–18].

**1**
