Digital control is a branch of
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
that uses
digital
Digital usually refers to something using discrete digits, often binary digits.
Technology and computing Hardware
*Digital electronics, electronic circuits which operate using digital signals
**Digital camera, which captures and stores digital i ...
computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
s to act as system controllers.
Depending on the requirements, a digital control system can take the form of a
microcontroller
A microcontroller (MCU for ''microcontroller unit'', often also MC, UC, or μC) is a small computer on a single VLSI integrated circuit (IC) chip. A microcontroller contains one or more CPUs (processor cores) along with memory and programmable i ...
to an
ASIC
An application-specific integrated circuit (ASIC ) is an integrated circuit (IC) chip customized for a particular use, rather than intended for general-purpose use, such as a chip designed to run in a digital voice recorder or a high-efficien ...
to a standard desktop computer.
Since a digital computer is a
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
system, the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
is replaced with the
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
. Since a digital computer has finite precision (''See
quantization''), extra care is needed to ensure the error in coefficients,
analog-to-digital conversion
In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provi ...
,
digital-to-analog conversion
In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function.
There are several DAC archit ...
, 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 Adaptive 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 consumpt ...
'') and digital computers are much less prone to environmental conditions than
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s,
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
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:
*Analog-to-digital conversion to convert analog inputs to machine-readable (digital) format
*Digital-to-analog 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 filter
In signal processing, 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, t ...
for
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
ing
*A
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the toy ...
model of a system to act as a
state observer In control theory, a state observer or state estimator is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and pr ...
*A
telemetry
Telemetry is the in situ data collection, collection of measurements or other data at remote points and their automatic data transmission, transmission to receiving equipment (telecommunication) for monitoring. The word is derived from the Gr ...
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 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
Tustin
Tustin is a city located in Orange County, California, in the Los Angeles metropolitan area. In 2020, Tustin had a population of 80,276. The city is located next to the county seat, Santa Ana, California, Santa Ana, and does not include the un ...
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.
Tustin transformation deduction
Tustin is the
Padé(1,1) approximation of the exponential function
:
:
And its inverse
:
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
*Yutaka Yamamoto and his "lifting function space model"
*Alexander Sesekin and his studies about impulsive systems.
*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 ]
Design of digital controller in z-domain
The digital controller can also be designed in the z-domain (discrete). The
Pulse Transfer Function (PTF)
represents the digital viewpoint of the continuous process
when interfaced with appropriate ADC and DAC, and for a specified sample time
is obtained as:
Where
denotes z-Transform for the chosen sample time
. There are many ways to directly design a digital controller
to achieve a given specification.
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
and preserve the (scaled) zeros of the PTF numerator
is to use the design equation:
Where the scalar term
ensures the controller
exhibits integral action, and a steady-state gain of unity is achieved in the closed-loop. The resulting closed-loop discrete transfer function from the z-Transform of reference input
to the z-Transform of process output
is then given by:
Since process time delay manifests as leading co-efficient(s) of zero in the process PTF numerator
, the synthesis method above inherently yields a predictive controller if any such delay is present in the continuous plant.
See also
*
Sampled data systems
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of so ...
*
Adaptive control Adaptive 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 consumpt ...
*
Analog control
An analog stick (or analogue stick in British English), sometimes called a control stick or thumbstick, is an input device for a controller (often a game controller) that is used for two-dimensional input. An analog stick is a variation of a joys ...
*
Control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
*
Digital
Digital usually refers to something using discrete digits, often binary digits.
Technology and computing Hardware
*Digital electronics, electronic circuits which operate using digital signals
**Digital camera, which captures and stores digital i ...
*
Feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
,
Negative feedback
Negative feedback (or balancing feedback) occurs when some function (Mathematics), function of the output of a system, process, or mechanism is feedback, fed back in a manner that tends to reduce the fluctuations in the output, whether caused by ...
,
Positive feedback
Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in the ...
*
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
*
Real-time control
Real-time computing (RTC) is the computer science term for hardware and software systems subject to a "real-time constraint", for example from event to system response. Real-time programs must guarantee response within specified time constrai ...
*
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
References
*FRANKLIN, G.F.; POWELL, J.D., Emami-Naeini, A., Digital Control of Dynamical Systems, 3rd Ed (1998). Ellis-Kagle Press, Half Moon Bay, CA
*KATZ, P. Digital control using microprocessors. Englewood Cliffs: Prentice-Hall, 293p. 1981.
*OGATA, K. Discrete-time control systems. Englewood Cliffs: Prentice-Hall,984p. 1987.
*PHILLIPS, C.L.; NAGLE, H. T. Digital control system analysis and design. Englewood Cliffs, New Jersey: Prentice Hall International. 1995.
*M. Sami Fadali, Antonio Visioli, (2009) "Digital Control Engineering", Academic Press, .
*JURY, E.I. Sampled-data control systems. New-York: John Wiley. 1958.
{{DEFAULTSORT:Digital Control
Control theory
de:Digitaler Regler