Digital control
Encyclopedia
Digital control is a branch of control theory
that uses digital
computer
s 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
.
Also since a digital computer has finite precision (See quantization
), extra care is needed to ensure the error in coefficients, A/D conversion, D/A conversion, etc. are not producing undesired or unplanned effects.
The application of digital control can readily be understood in the use of feedback
.
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 for several reasons:
Typically, a digital controller requires:
The programs can take numerous forms and perform many functions
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.
transformation can transform the continuous compensator to the respective digital compensator. The digital compensator will achieve an output which approaches the output of its respective analog controller as the sampling interval is decreased.
approximation of the exponential function
:
And its inverse
We must never forget that the 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:
http://mtc-m18.sid.inpe.br/col/sid.inpe.br/mtc-m18@80/2008/03.17.15.17.24/doc/mirrorget.cgi?languagebutton=pt-BR&metadatarepository=sid.inpe.br/mtc-m18@80/2009/02.09.14.45.33&index=0&choice=full
http://mtc-m05.sid.inpe.br/col/sid.inpe.br/deise/1999/09.14.15.39/doc/homepage.pdf
http://www.sae.org/technical/papers/2002-01-3468
and
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
that uses digital
Digital
A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
s to act as system controllers.
Depending on the requirements, a digital control system can take the form of a microcontroller
Microcontroller
A microcontroller is a small computer on a single integrated circuit containing a processor core, memory, and programmable input/output peripherals. Program memory in the form of NOR flash or OTP ROM is also often included on chip, as well as a typically small amount of RAM...
to an ASIC
Application-specific integrated circuit
An application-specific integrated circuit is an integrated circuit customized for a particular use, rather than intended for general-purpose use. For example, a chip designed solely to run a cell phone is an ASIC...
to a standard desktop computer.
Since a digital computer is a discrete
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
system, the Laplace transform is replaced with the Z-transform
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....
.
Also since a digital computer has finite precision (See quantization
Quantization (signal processing)
Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...
), extra care is needed to ensure the error in coefficients, A/D conversion, D/A conversion, etc. are not producing undesired or unplanned effects.
The application of digital control can readily be understood in the use of feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
.
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 for several reasons:
- Inexpensive: under $5 for many microcontrollers
- Flexibile: easy to configure and reconfigure through software
- Scalable: programs can scale to the limits of the memory or storage space without extra cost
- Adaptable: parameters of the program can change with time (See adaptive controlAdaptive controlAdaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself...
) - Static operation: digital computers are much less prone to environmental conditions than capacitorCapacitorA capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s, inductorInductorAn inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s, etc.
Digital Controller Implementation
A digital controller is usually cascaded with the plant in a feedback system. The rest of the system can either be digital or analog.Typically, a digital controller requires:
- A/D conversion to convert analog inputs to machine readable (digital) format
- D/A conversion to convert digital outputs to a form that can be input to a plant (analog)
- A program that relates the outputs to the inputs
Output Program
- Outputs from the digital controller are functions of current and past input samples, as well as past output samples - this can be implemented by storing relevant values of input and output in registers. The output can then be formed by a weighted sum of these stored values.
The programs can take numerous forms and perform many functions
- A digital filterDigital filterIn electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
for low-pass filterLow-pass filterA low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
ing - A state spaceState space (controls)In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
model of a system to act as a state observerState observerIn control theory, a state observer is a system that models a real system in order to provide an estimate of its internal state, given measurements of the input and output of the real system. It is typically a computer-implemented mathematical model.... - A telemetryTelemetryTelemetry is a technology that allows measurements to be made at a distance, usually via radio wave transmission and reception of the information. The word is derived from Greek roots: tele = remote, and metron = measure...
system
Stability
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
Nyquist stability criterion
When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...
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.
Design of digital controller in s-domain
The digital controller can also be designed in the s-domain (continuous). The TustinArnold Tustin
Arnold Tustin, , was a British engineer, and Professor of Engineering at the University of Birmingham and at Imperial College London, who made important contributions to the development of control engineering and its application to electrical machines.- Biography :Arnold Tustin was born in 1899...
transformation can transform the continuous compensator to the respective digital compensator. The digital compensator will achieve an output which approaches the output of its respective analog controller as the sampling interval is decreased.
Tustin transformation deduction
Tustin is the Padé(1,1)Padé table
In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximantsto a given complex formal power series...
approximation of the exponential function
:And its inverse
We must never forget that the 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:
- Marcelo Tredinnick and Marcelo Souza and their new type of analog-digital mapping
http://mtc-m18.sid.inpe.br/col/sid.inpe.br/mtc-m18@80/2008/03.17.15.17.24/doc/mirrorget.cgi?languagebutton=pt-BR&metadatarepository=sid.inpe.br/mtc-m18@80/2009/02.09.14.45.33&index=0&choice=full
http://mtc-m05.sid.inpe.br/col/sid.inpe.br/deise/1999/09.14.15.39/doc/homepage.pdf
http://www.sae.org/technical/papers/2002-01-3468
- Yutaka Yamamoto and his "lifting function space model" http://wiener.kuamp.kyoto-u.ac.jp/~yy/Papers/yamamoto_cwi96.pdf
- Alexander Sesekin and his studies about impulsive systems. http://www.amazon.com/dp/0792343948
and
- M.U. Akhmetov and his studies about impulsive and pulse control. http://portal.acm.org/author_page.cfm?id=81100182444&coll=GUIDE&dl=GUIDE&trk=0&CFID=27536832&CFTOKEN=71744014
See also
- Sampled data systemsSampled data systemsA sampled-data system is a control system where continuous-time plant is controlled with a digital device. Under periodic sampling, the sampled-data system is time-varying but also periodic, and thus it may be modeled by a simplified discrete-time system obtained by discretizing the plant...
- Adaptive controlAdaptive controlAdaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself...
- Analog control
- Control theoryControl theoryControl theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
- DigitalDigitalA digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
- FeedbackFeedbackFeedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
, Negative feedbackNegative feedbackNegative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...
, Positive feedbackPositive feedbackPositive feedback is a process in which the effects of a small disturbance on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn produces more of A. In contrast, a system that responds to a perturbation in a way that reduces its effect is... - Laplace transform
- Real-time controlReal-time controlReal-time control is a popular term for a certain class of digital controllers. For effective digital control, it is critical that sample time be constant. Real-time control achieves nearly constant sample time....
- Z-transformZ-transformIn mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....