Sampled Data Systems:

INTRODUCTION TO DIGITAL CONTROL:

The use of computers in process plant operations dates back to the mid-1960’s. Nowadays, process computer s are now common place in the process industries, performing a variety of manufacturing tasks. Several examples are given in the following schematics.

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We are interested in the use of computers to control process plants, within either a DDC or Supervisory control framework. The principles of controller design are identical in both cases.

Typical scenarios where computer based process control may be beneficial are:

Plants with large throughput – because utilities consumption is approximately proportional to throughput, a small improvement can result in large savings

Plants subject to frequent upsets – some plants requires quick responses to process upsets that operators cannot provide

Complex plants -some processes are too complex for operators to deal with the relationships between process variables and hence are unable to determine the best operating conditions, and maintaining consistent operation is difficult

Batch processes – some batch processes require frequent cycling or changes in product specification, and computer control can increase production rate and decrease labour cost

New processes – in plants with 40-50 control loops, a DCS may prove to be cheaper than an equivalent analog system.

SAMPLED DATA SYSTEMS

Because of the nature of digital devices, signals from plant have to be converted into a suitable form before it can be transferred for processing by a computer. Similarly, signals generated by a computer must be presented in a form suitable for receipt by the plant. The important pieces of hardware that achieve these tasks are the:

• sampler

• analog-to-digital converter (ADC)

• digital-to-analog converter (DAC)

• signal hold devices

The Sampler

The sampler is essentially a switch, operating usually at fixed intervals of time. When the ‘switch’ closes, it grabs or samples the output of the transmitting device. It then transfers the sampled signal to a receiver. The sampler can operate on both continuous and discrete signals.

Thus if the source signal is continuous, the output of the sampler is a series of pulses, and the magnitude of each pulse is equal to the magnitude of the continuous signal at the instant of sampling as shown in the figure below.

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ADCs and DACs

ADCs converts’ sampled voltage or current signals to their binary equivalent while DACs converts binary signals to continuous signals such as voltages or currents. These converters provide the interface between a computer and the external environment.

Signal Hold Devices

The output of a sampler is a train of pulses, regardless of whether the source is continuous or discrete. Thus the output of a computer after digital-to-analog conversion is also a train of pulses. If this is a control signal, then unless the device receiving this signal, say a pump or valve, has integration capabilities, then the process will be driven by pulses. This is obviously not acceptable. So, in process control applications, the signal from the DAC is always ‘held’ using hardware known as signal hold devices. The most common is the Zero-Order-Hold, where each pulse is held until the next pulse comes along, that is:

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Digital Controllers:

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. Also since a digital computer has finite precision, 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:

Inexpensive: under for many microcontrollers

Flexible: 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

Static operation: digital computers are much less prone to environmental conditions than capacitor, inductors, etc.

Digital PID Controller:

A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism(controller) widely used in industrial control systems. A PID controller calculates an “error” value as the difference between a measured process variable and a desired set point. The controller attempts to minimize the error by adjusting the process control inputs.

The PID controller algorithm involves three separate constant parameters, and is accordingly sometimes called three-term control: the proportional, the integral and derivative values, denoted P, I, and D. Simply put, these values can be interpreted in terms of time: P depends on the present error, I on the accumulation of past errors, and D is a prediction of future errors, based on current rate of change. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve, a damper, or the power supplied to a heating element.

In the absence of knowledge of the underlying process, a PID controller has historically been considered to be the best controller. By tuning the three parameters in the PID controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the set point, and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability.

Some applications may require using only one or two actions to provide the appropriate system control. This is achieved by setting the other parameters to zero. A PID controller will be called a PI, PD, P or I controller in the absence of the respective control actions. PI controllers are fairly common, since derivative action is sensitive to measurement noise, whereas the absence of an integral term may prevent the system from reaching its target value due to the control action.

PI controller:

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Fig: Basic block of a PI controller

A PI Controller (proportional-integral controller) is a special case of the PID controller in which the derivative (D) of the error is not used.

The controller output is given by

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where clip_image011 is the error or deviation of actual measured value (PV) from the set point (SP).

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A PI controller can be modelled easily in software such as Simulink or Xcos using a “flow chart” box involving Laplace operators:

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Where,

clip_image014 = proportional gain

clip_image015 = integral gain

Setting a value for clip_image016 is often a trade-off between decreasing overshoot and increasing settling time.

The lack of derivative action may make the system steadier in the steady state in the case of noisy data. This is because derivative action is more sensitive to higher-frequency terms in the inputs.

Without derivative action, a PI-controlled system is less responsive to real (non-noise) and relatively fast alterations in state and so the system will be slower to reach set point and slower to respond to perturbations than a well-tuned PID system may be.

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