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## Über dieses Buch

This book covers a wide spectrum of systems such as linear and nonlinear multivariable systems as well as control problems such as disturbance, uncertainty and time-delays. The purpose of this book is to provide researchers and practitioners a manual for the design and application of advanced discrete-time controllers. The book presents six different control approaches depending on the type of system and control problem. The first and second approaches are based on Sliding Mode control (SMC) theory and are intended for linear systems with exogenous disturbances. The third and fourth approaches are based on adaptive control theory and are aimed at linear/nonlinear systems with periodically varying parametric uncertainty or systems with input delay. The fifth approach is based on Iterative learning control (ILC) theory and is aimed at uncertain linear/nonlinear systems with repeatable tasks and the final approach is based on fuzzy logic control (FLC) and is intended for highly uncertain systems with heuristic control knowledge. Detailed numerical examples are provided in each chapter to illustrate the design procedure for each control method. A number of practical control applications are also presented to show the problem solving process and effectiveness with the advanced discrete-time control approaches introduced in this book.

## Inhaltsverzeichnis

### Chapter 1. Introduction

In recent years there has been a rapid increase in the use of digital controllers in control systems. Digital controls are used for achieving optimal performance, e.g., in the form of maximum productivity, maximum profit, minimum cost, or minimum energy use.
Khalid Abidi, Jian-Xin Xu

### Chapter 2. Discrete-Time Sliding Mode Control

In this study, two different approaches to the sliding surface design are presented. First, a discrete-time integral sliding mode (ISM) control scheme for sampled-data systems is discussed. The control scheme is characterized by a discrete-time integral switching surface which inherits the desired properties of the continuous-time integral switching surface, such as full order sliding manifold with eigenvalue assignment, and elimination of the reaching phase. In particular, comparing with existing discrete-time sliding mode control, the scheme is able to achieve more precise tracking performance. It will be shown that, the control scheme achieves $$O(T^2)$$ steady-state error for state regulation and reference tracking with the widely adopted delay-based disturbance estimation. Another desirable feature is that the Discrete-time ISM control prevents the generation of overlarge control actions which are usually inevitable due to the deadbeat poles of a reduced order sliding manifold designed for sampled-data systems. Second, a terminal sliding mode control scheme is discussed. Terminal Sliding Mode (TSM) control is known for its high gain property nearby the vicinity of the equilibrium while retaining reasonably low gain elsewhere. This is desirable in digital implementation where the limited sampling frequency may incur chattering if the controller gain is overly high. The overall sliding surface integrates a linear switching surface with a terminal switching surface. The switching surface can be designed according to the precision requirement. The design is implemented on a specific SISO system example, but, the approach can be used in exactly the same way for any other system as long as it is SISO. The analysis and experimental investigation show that the TSM controller design outperforms the linear SM control.
Khalid Abidi, Jian-Xin Xu

### Chapter 3. Discrete-Time Periodic Adaptive Control

In this study a periodic adaptive control approach is discussed for a class of nonlinear discrete-time systems with time-varying parametric uncertainties which are periodic, and the only prior knowledge is the periodicity. The adaptive controller updates the parameters and the control signal periodically in a pointwise manner over one entire period, in the sequel achieves the asymptotic tracking convergence. The result is further extended to a scenario with mixed time-varying and time-invariant parameters, and a hybrid classical and periodic adaptation law is proposed to handle the scenario more appropriately. Extension of the periodic adaptation to systems with unknown input gain, higher order dynamics, and tracking problems are also discussed.
Khalid Abidi, Jian-Xin Xu

### Chapter 4. Discrete-Time Adaptive Posicast Control

In this study, we discuss the discrete version of the Adaptive Posicast Controller (APC) that deals with parametric uncertainties in systems with input time-delays. The continuous-time APC is based on the Smith Predictor and Finite Spectrum Assignment with time-varying parameters adjusted online. Although the continuous-time APC showed dramatic performance improvements in experimental studies with internal combustion engines, the full benefits could not be realized since the finite integral term in the control law had to be approximated in computer implementation. It is shown in the literature that integral approximation in time-delay compensating controllers degrades the performance if care is not taken. In this study, we discuss a development of the APC in the discrete-time domain, eliminating the need for approximation. Rigorous and complete derivation is provided with a Lyapunov stability proof. The discussed discrete-time APC is developed in State Space to easily accommodate multivariable systems and also allow for the extension to nonlinear systems. In essence, this study presents a unified development of the discrete-time APC for systems that are linear/nonlinear with known input time-delays or linear systems with unknown but upper-bounded time-delays. Performances of the continuous-time and discrete-time APC, as well as conventional Model Reference Adaptive Controller (MRAC) for linear systems with known time-delay are compared in simulation studies. It is shown that discrete-time APC outperforms it’s continuous-time counterpart and MRAC. Further simulations studies are also presented to show the performance of the design for nonlinear systems and also for systems with unknown time-delay.
Khalid Abidi, Jian-Xin Xu

### Chapter 5. Discrete-Time Iterative Learning Control

In this study the convergence properties of iterative learning control (ILC) algorithms are discussed. The analysis is carried out in a framework using linear iterative systems, which enables several results from the theory of linear systems to be applied. This makes it possible to analyse both first-order and high-order ILC algorithms in both the time and frequency domains. The time and frequency domain results can also be tied together in a clear way. Illustrative examples are presented to support the analytical results.
Khalid Abidi, Jian-Xin Xu

### Chapter 6. Discrete-Time Fuzzy PID Control

In this study, a parallel structure of fuzzy PID control systems is presented. It is associated with a new tuning method which, based on gain margin and phase margin specifications, determines the parameters of the fuzzy PID controller. In comparison with conventional PID controllers, the presented fuzzy PID controller shows higher control gains when system states are away from equilibrium and, at the same time, retains lower profile of control signals. Consequently better control performance is achieved. With the presented formula, the weighting factors of a fuzzy logic controller can be systematically selected according to the plant under control. By virtue of using the simplest structure of fuzzy logic control, the stability of the nonlinear control system is able to be analyzed and a sufficient BIBO stability condition is given. The superior performance of the controller is demonstrated through both numerical and experimental examples.
Khalid Abidi, Jian-Xin Xu

### Chapter 7. Benchmark Precision Control of a Piezo-Motor Driven Linear Stage

In this study, practical application of ISM control and ILC is investigated using a piezo-motor driven stage as the platform. Theoretical developments have shown ISM control and ILC to be highly suitable for high-precision motion control problems. The experimental results show that ISM control and ILC can indeed produce superior performance to conventional control methods.
Khalid Abidi, Jian-Xin Xu

### Chapter 8. Advanced Control for Practical Engineering Applications

Although the theoretical results for periodic adaptive control are based on purely time-based uncertainties, in practical scenarios, state based periodic uncertainties can also be considered time-based if the system is at steady state. In this study, this is investigated and verified using a PM synchronous motor as a platform. In some systems with low quality components, it is unavoidable to have varying sampling rates at the various A/D ports of the system. Multirate ILC was developed with the intention to, as effectively as possible, attenuate the effects of the lower sampling rate components on the overall system performance. In this study, a Ball-and-Beam apparatus is used to verify that indeed, in spite of having lower sampling rates of certain components, superior performance is achievable. Fuzzy PID has been shown to be effective on highly nonlinear and difficult to model systems. In this study, a coupled tank apparatus is used as a test bed to investigate and verify the effectiveness of fuzzy PID. Experimental results show that even with the uncertain nature of the model, it is possible to achieve high performance of the system. Finally, using the fact that traffic patterns are repeatable, ILC is implemented to improve the flow of traffic in a freeway. It is shown by the simulation results that superior performance can be achieved using a very simple controller structure.
Khalid Abidi, Jian-Xin Xu

### Backmatter

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