Control of dynamic systems by digital computer. Characterization of discrete-time systems, discrete state space, Z transforms, time domain analysis of discrete-time control systems. Effects of sampling time. Discrete root locus. Frequency domain methods for compensator design. Laboratory experiences in the computer control of electromechanical systems with C/C++ programming, LabView and programmable logic controls (PLCs).
Ogata Digital Control Systems
This text is designed for senior undergraduate and first-year graduate level engineering courses on discrete-time control systems or digital control systems. The text provides a comprehensive treatment of the analysis and design of discrete-time control systems. MATLAB's ease-of-use for studying discrete-time control systems is demonstrated through problems involving vector-matrix operations, plots response curves, and system design based on quadratic optimal control.
Digital control is a branch of control theory that uses digital computers to act as system controllers.Depending on the requirements, a digital control system can take the form of a microcontroller to an ASIC to a standard desktop computer.Since a digital computer is a discrete system, the Laplace transform is replaced with the Z-transform. Since a digital computer has finite precision (See quantization), extra care is needed to ensure the error in coefficients, analog-to-digital conversion, digital-to-analog conversion, etc. are not producing undesired or unplanned effects.
Since the creation of the first digital computer in the early 1940s the price of digital computers has dropped considerably, which has made them key pieces to control systems because they are easy to configure and reconfigure through software, can scale to the limits of the memory or storage space without extra cost, parameters of the program can change with time (See adaptive control) and digital computers are much less prone to environmental conditions than capacitors, inductors, etc.
Although a controller may be stable when implemented as an analog controller, it could be unstable when implemented as a digital controller due to a large sampling interval. During sampling the aliasing modifies the cutoff parameters. Thus the sample rate characterizes the transient response and stability of the compensated system, and must update the values at the controller input often enough so as to not cause instability.
When substituting the frequency into the z operator, regular stability criteria still apply to discrete control systems. Nyquist criteria apply to z-domain transfer functions as well as being general for complex valued functions. Bode stability criteria apply similarly.Jury criterion determines the discrete system stability about its characteristic polynomial.
The digital controller can also be designed in the s-domain (continuous). The Tustin transformation can transform the continuous compensator to the respective digital compensator. The digital compensator will achieve an output that approaches the output of its respective analog controller as the sampling interval is decreased.
Digital control theory is the technique to design strategies in discrete time, (and/or) quantized amplitude (and/or) in (binary) coded form to be implemented in computer systems (microcontrollers, microprocessors) that will control the analog (continuous in time and amplitude) dynamics of analog systems. From this consideration many errors from classical digital control were identified and solved and new methods were proposed:
The digital controller can also be designed in the z-domain (discrete). The Pulse Transfer Function (PTF) G ( z ) \displaystyle G(z) represents the digital viewpoint of the continuous process G ( s ) \displaystyle G(s) when interfaced with appropriate ADC and DAC, and for a specified sample time T \displaystyle T is obtained as:[7]
Where Z ( ) \displaystyle Z() denotes z-Transform for the chosen sample time T \displaystyle T . There are many ways to directly design a digital controller D ( z ) \displaystyle D(z) to achieve a given specification.[7] For a type-0 system under unity negative feedback control, Michael Short and colleagues have shown that a relatively simple but effective method to synthesize a controller for a given (monic) closed-loop denominator polynomial P ( z ) \displaystyle P(z) and preserve the (scaled) zeros of the PTF numerator B ( z ) \displaystyle B(z) is to use the design equation:[8]
R1: Knowing the pros and cons of digital control with respect to analog control.
R2: Knowing the fundamentals of digital control.
R3: Analyzing sampled linear systems, studying their stability and their transient and steady state response.
R4: Using basic techniques of digital controller design, from both the time-domain and the frequency domain perspective.
Learning Outcome Contents Formative activity Evaluation method Knowing the pros and cons of digital control with respect to analog control. Part I A-1,A-7 Quiz Knowing the fundamentals of digital control. Parts I,II, and III A-1,A-2,A-5,A-6,A-7,A-8 Quiz, Exam Analyzing sampled linear systems, studying their stability and theirtransient and steady state response. Part II A-1,A-2,A-5,A-6,A-7,A-8 Quiz, Exam Using basic techniques of digital controller design, from both the time-domain and the frequency domain perspective. Part III A-1,A-2,A-5,A-6,A-7,A-8 Quiz, Exam
Classroom lectures will consist of theoretical explanations of the fundamental aspects of each lesson, as well as solutions to questions raised by the students based on their own self-study. There will be two kinds of laboratory sessions: simulation on a software package specialized on dynamical systems and control and real-time control of industrial devices, where the students will test the validity of the theoretical apparatus of the course.
A final exam is held at the end of the semester. Its purpose is to assess the student's ability to undertake the design of a digital control system. Theoretical contents are not evaluated. Rather, the idea is to apply the aquired knowledge in the analysis of a simplified control problem, and in the design of digital controllers for it. Simultaneously, the student will have the chance to pass the quizzes that he or she has not passed yet.
Module I: Discrete, sampled and hybrid systems In this module, the simple digital control loop is presented and the effect of the introduction of the computer in such loop is studied. A/D and D/A converters are introduced and the main advantages and disadvantages of digital control systems are analyzed in comparison to the analog control ones. The different types of systems appearing in the digital control loop are presented and the necessary tools are introduced for their study (Z transform, Modified Z Transform, ...) Module II: Analysis The techniques needed to study the stability and behavior of discrete systems in the temporal and frequency domains are introduced. The behavior of discrete and sampled systems is characterized, and it is shown how to obtain a discrete model that verifies a set of specifications. The effect of sampling from the frequency point of view is analyzed, as well as the effect of the sampling and retention process on a signal by means of a Zero Order Hold. Module III. Design and synthesis of controllers Digital controller design techniques are introduced, both by emulation of analog controllers and by direct digital design. Various techniques for controller synthesis are presented, such as the Truxal method and pole placement. They are generalized for the case of two-degree-of-freedom control systems, so that systems can be obtained that meet both reference tracking and disturbance rejection specifications.
Introduction: realizability of a digital controller. Performance specifications and choice of the samplig time. Design of digital controllers via continuous design: presentation of the method and feasibility conditions. Discretization of a continuous controller.
0. Introduction to Matlab
1. Sampling and reconstruction. Aliasing.
2. Simulation of discrete systems.
3. Other control structures.
4. Simulation of disturbance inputs.
5. Frequency response.
6. Discretization of analog controllers.
In this chapter, potential instability and performance limitation due to the use of an attractive and non-invariant switching sector in a number of discrete variable structure controllers (DVSCs) for linear systems with parametric uncertainties are discussed. Analytical explanation as well as counter examples are given to support the arguments. A state feedback DVSC based on the concept of switching sector and capable of avoiding the potential pitfalls is proposed. It is shown that global uniform asymptotic stability can be achieved despite the non-attractiveness and non-invariance of the switching sector. Conservativeness of the proposed controller is investigated and a numerical example is presented.
Abstract— A simple control strategy was studied for harmonic disturbance rejection in magnetic field compensation systems for lowfield magnetic resonance techniques. The strategy is based on the simultaneous action of a conventional PID and a selective harmoniccompensation controller. The system consists of a set of compensating coils fed by independent current sources driven by a digital controller. A series of hall magnetic sensors close the control loop. Despite its simplicity, it is shown that the performance of the dual controller improves within the frequency range where the waterbed effect becomes dominant, by selectively enhancing the rejection of the harmonic component. The proposed solution is particularly useful for selective harmonic rejection of slowly varying frequency and amplitude dependent harmonic perturbations. An extension to multiple-harmonic components perturbations is possible. 2ff7e9595c
Comments