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digital control: Definition from Answers.com

The use of digital or discrete technology to maintain conditions in operating systems as close as possible to desired values despite changes in the operating environment. Traditionally, control systems have utilized analog components, that is, controllers which generate time-continuous outputs (volts, pressure, and so forth) to manipulate process inputs and which operate on continuous signals from instrumentation measuring process variables (position, temperature, and so forth). In the 1970s, the use of discrete or logical control elements, such as fluidic components, and the use of programmable logic controllers to automate machining, manufacturing, and production facilities became widespread. In parallel with these developments has been the accelerating use of digital computers in industrial and commercial applications areas, both for logic-level control and for replacing analog control systems. The development of inexpensive mini- and microcomputers with arithmetic and logical capability orders of magnitude beyond that obtainable with analog and discrete digital control elements has resulted in the rapid substitution of conventional control systems by digital computer-based ones. With the introduction of microcomputer-based control systems into major consumer products areas (such as automobiles and video and audio electronics), it is clear that the digital computer will be widely used to control objects ranging from small, personal appliances and games up to large, commercial manufacturing and production facilities. See also Microcomputer; Programmable controllers.

The object that is controlled is usually called a device or, more inclusively, process. A characteristic of any digital control system is the need for a process interface to mate the digital computer and process, to permit them to pass information back and forth.

Measurements of the state of the process often are obtained naturally as one of two switch states; for example, a part to be machined is in position (or not), or a temperature is above (or below) the desired temperature. Control signals sent to the process often are expressed as one of two states as well; for example, a motor is turned on (or off), or a valve is opened (or closed). Such binary information can be communicated naturally to and from the computer, where it is manipulated in binary form. For this reason the binary or digital computer/process interface usually is quite simple.

Process information also must be dealt with in analog form; for example, a variable such as temperature can take on any value within its measured range, or, looked at conceptually, it can be measured to any number of significant figures by a suitable instrument. Furthermore, analog variables generally change continuously in time. Digital computers are not suited to handle arbitrarily precise or continuously changing information; hence, analog process signals must be reduced to a digital representation (discretized), both in terms of magnitude and in time, to put them into a useful digital form.

The magnitude discretization problem most often is handled by transducing and scaling each measured variable to a common range, then using a single conversion device—the analog-to-digital converter (ADC)—to put the measured value into digital form. See also Analog-to-digital converter.

Discretization in time requires the computer to sample the signal periodically, storing the results in memory. This sequence of discrete values yields a “staircase” approximation to the original signal, on which control of the process must be based. Obviously, the accuracy of the representation can be improved by sampling more often, and many digital systems simply have incorporated traditional analog control algorithms along with rapid sampling. However, newer control techniques make fundamental use of the discrete nature of computer input and output signals. Analog outputs from a computer most often are obtained from a digital-to-analog converter (DAC), a device which accepts a digital output from the computer, converts it to a voltage in several microseconds, and latches (holds) the value until the next output is converted. Usually a single DAC is used for each output signal. See also Digital-to-analog converter.

In order to be used as the heart of a control system, a digital computer must be capable of operating in real time. Except for very simple microcomputer applications, this feature implies that the machine must be capable of handling interrupts, that is, inputs to the computer's internal control unit which, on change of state, cause the computer to stop executing some section of program code and begin executing some other section. The ability to initiate operations on schedule and to respond to process interrupts in a timely fashion is the very basis of real-time computing; this feature must be available in any digital control system.

Computer control systems for large or complex processes may involve complicated programs with many thousands of computer instructions. Several routes have been taken to mitigate the difficulty of programming control computers. One approach is to develop a single program which utilizes data supplied by the user to specify both the actions to be performed on the individual process elements and the schedule to be followed. Another approach is to develop a rather sophisticated operating system to supervise the execution of user programs, scheduling individual program elements for execution as specified by the user or needed by the process. See also Digital computer.

Many applications, particularly machining, manufacturing, and batch processing, involve large or complex operating schedules. Invariably, these can be broken down into simple logical sequences. Some applications—in the chemical process industries, in power generation, and in aerospace areas—require the use of traditional automatic control algorithms.

Attempts to expand the digital control medium through development of strictly digital control algorithms is an important and continuing trend. Such algorithms typically attempt to exploit the sampled nature of process inputs and outputs, significantly decreasing the sampling requirements of the algorithm. See also Control systems.

Wikipedia: digital control

Digital control is a branch of control theory that uses digital computers to act as a system. 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:

Cheap: under $5 for many microcontrollers
Flexibility: easy to configure and reconfigure through software
Static operation: digital computers are much less prone to environmental conditions than capacitors, inductors, etc.
Scaling: programs can scale to the limits of the memory or storage space without extra cost
Adaptive: parameters of the program can change with time (See adaptive control)

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. Some examples of analog systems with a digital feedback controller are:

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 filter for low-pass filtering (analog filters are preferred because digital filters introduce more delay)
A state space model of a system to act as a state observer
A telemetry system

Stability

Note that 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. Thus the sample rate characterises 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.

Stability of digital control systems can be checked using a specific bilinear transform to the Laplace domain, allowing the use of the Routh-Hurwitz stability criterion. This bilinear transform is application specific, and can not be used to compare system attributes such as transient responses in the s and z domains.

Design of digital controller in s-domain:

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 which approaches the output of its respective analog controller as the sampling interval is decreased.

$s = \frac{2(z-1)}{T(z+1)}$

Tustin transformation deduction

Failed to parse (unknown function\begin): \begin{align} z &= e^{sT} \\ &= \frac{e^{sT/2}}{e^{-sT/2}} \\ &\approx \frac{1 + s T / 2}{1 - s T / 2} \end{align}

And its inverse

Failed to parse (unknown function\begin): \begin{align} s &= \frac{1}{T} \ln(z) \\ &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} \right)^3 + \frac{1}{5} \left( \frac{z-1}{z+1} \right)^5 + \frac{1}{7} \left( \frac{z-1}{z+1} \right)^7 + \cdots \right] \\ &\approx \frac{2}{T} \frac{z - 1}{z + 1} \\ &\approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}} \end{align}