# Process Controllers Predict the Future

## A feedback controller can steer a process variable toward the desired setpoint only if it can somehow predict the future effects of its current control efforts. A model-based controller does so with the help of a mathematical representation of the process’s behavior. A well-tuned PID loop uses an implicit model characterized by the values of the controller’s P (proportional), I (int...

03/01/2008

 Sidebars: Convolution

A feedback controller can steer a process variable toward the desired setpoint only if it can somehow predict the future effects of its current control efforts. A model-based controller does so with the help of a mathematical representation of the process’s behavior. A well-tuned PID loop uses an implicit model characterized by the values of the controller’s P (proportional), I (integral), and D (derivative) parameters.

These two techniques compute their control efforts differently, but they both rely on the linearity of the process to anticipate how it is going to respond. A process is said to be linear if the process variable increases by a factor of u when the control effort is increased by the same amount. And if two separate sequences of control efforts are added together and applied to a linear process, the resulting values of the process variable will always equal the sum of the values that would have resulted had the two control efforts been applied separately.

## Superposition

This predictability gives rise to the Superposition Principle which governs the behavior of all linear processes. The “Superposition Principle” graphic shows how it works in four situations where a computer-based controller with a cycle time ofΔ t seconds has applied a different sequence of control efforts to the same linear process.

In case A, the controller has applied a single impulse with a magnitude of 1 unit (percent, degree, PSI, whatever) and a width of one cycle time (Δt seconds). The resulting fluctuations in the process variable are known as the process’s impulse response or, in this particular case, its unity impulse response .

The process in this example happens to be somewhat sluggish, so its unity impulse response rises and falls relatively slowly as the effects of the impulse wear off. This could represent any number of industrial processes, such as the temperature in a vat after a heating element has been turned on then off again, or the flow rate in a pipe after a valve has been opened then closed.

Case B shows how increasing the magnitude of the impulse increases the magnitude of the impulse response but not its general shape. The second impulse is three times as large as the first, so the magnitude of the impulse response has been tripled.

In case C, both impulses have been applied to the process, but at different times. The process’s net response after the second impulse equals the sum of the two impulse responses added together point by point. The second impulse response has been effectively “superimposed” on the first, hence the name of the principle that describes this phenomenon.

Case D shows that a contiguous sequence of impulses with magnitudes of u (0), u (1), u (2), ... applied to the process at times 0,Δ t , 2Δ t , ... has the same additive effect. Each new impulse response adds to the impulse responses already in progress, and the magnitude of each is determined by the magnitude of the impulse that caused it. The process’s net response at any time is the sum of all the impulse responses that have been initiated up to that point.

## Equivalent calculation

Thanks to the Superposition Principle, a controller can predict how a linear process will respond to any sequence of control efforts, not just impulses. It also gives an algorithm for computing the resulting values of the process variable, as shown in “Calculating the Process Response.” This graphic depicts the same four situations, except that the control efforts and the corresponding process responses are represented by their numerical values rather than trend charts. Each data stream has been sampled and recorded once everyΔ t seconds, hence the expression sampling interval often used to describe the controller’s cycle timeΔ t .

Case D shows the calculations required to compute the values of the process variable y (0), y (1), y (2), ... that would result from an arbitrary sequence of control efforts u (0), u (1), u (2), ... Specifically,

y (0)= u (0) h (0)

y (1)= u (0) u (1) h (0)

y (2)= u (0) h (2)+ u (1) h (1)+ u (2) h (0)

etc. Each calculation gets successively longer as more and more impulses figure into the result. Fortunately, there’s a convenient way to organize all these multiplication and addition operations, as shown in the “Convolution” table, where two infinitely long “numbers”

H = h (0), h (1), h (2), ...

and

U = u (0), u (1), u (2), ...

are “multiplied” together to compute

Y = y (0), y (1), y (2), ...

using the familiar long multiplication algorithm, but with data points h (0), h (1), h (2), ... and u (0), u (1), u (2), ... instead of individual digits. This calculation, known as convolution , is actually the mirror image of long multiplication. The multiplication and addition steps are the same, but it does not involve any carry-over from one column to the next. It is typically written as Y=H*U where “*” is the convolution operator .

Convolution is the basis for an entire mathematical discipline known as linear systems analysis . It gives control engineers a powerful tool for analyzing the behavior of linear processes and designing feedback controllers that can predict the future.

 Author Information Vance Van Doren, Ph.D., P.E., is senior editor for Control Engineering. He can be reached at vance@control.com .

The Top Plant program honors outstanding manufacturing facilities in North America. View the 2015 Top Plant.
The Product of the Year program recognizes products newly released in the manufacturing industries.
Each year, a panel of Control Engineering and Plant Engineering editors and industry expert judges select the System Integrator of the Year Award winners in three categories.
A new approach to the Skills Gap; Community colleges may hold the key for manufacturing; 2017 Engineering Leaders Under 40
Doubling down on digital manufacturing; Data driving predictive maintenance; Electric motors and generators; Rewarding operational improvement
2017 Lubrication Guide; Software tools; Microgrids and energy strategies; Use robots effectively
The cloud, mobility, and remote operations; SCADA and contextual mobility; Custom UPS empowering a secure pipeline
Infrastructure for natural gas expansion; Artificial lift methods; Disruptive technology and fugitive gas emissions
Mobility as the means to offshore innovation; Preventing another Deepwater Horizon; ROVs as subsea robots; SCADA and the radio spectrum
Power system design for high-performance buildings; mitigating arc flash hazards
Research team developing Tesla coil designs; Implementing wireless process sensing
Commissioning electrical systems; Designing emergency and standby generator systems; Paralleling switchgear generator systems

### Annual Salary Survey

Before the calendar turned, 2016 already had the makings of a pivotal year for manufacturing, and for the world.

There were the big events for the year, including the United States as Partner Country at Hannover Messe in April and the 2016 International Manufacturing Technology Show in Chicago in September. There's also the matter of the U.S. presidential elections in November, which promise to shape policy in manufacturing for years to come.

But the year started with global economic turmoil, as a slowdown in Chinese manufacturing triggered a worldwide stock hiccup that sent values plummeting. The continued plunge in world oil prices has resulted in a slowdown in exploration and, by extension, the manufacture of exploration equipment.

Read more: 2015 Salary Survey

Maintenance and reliability tips and best practices from the maintenance and reliability coaches at Allied Reliability Group.
The One Voice for Manufacturing blog reports on federal public policy issues impacting the manufacturing sector. One Voice is a joint effort by the National Tooling and Machining...
The Society for Maintenance and Reliability Professionals an organization devoted...
Join this ongoing discussion of machine guarding topics, including solutions assessments, regulatory compliance, gap analysis...
IMS Research, recently acquired by IHS Inc., is a leading independent supplier of market research and consultancy to the global electronics industry.
Maintenance is not optional in manufacturing. It’s a profit center, driving productivity and uptime while reducing overall repair costs.
The Lachance on CMMS blog is about current maintenance topics. Blogger Paul Lachance is president and chief technology officer for Smartware Group.
The maintenance journey has been a long, slow trek for most manufacturers and has gone from preventive maintenance to predictive maintenance.
Featured articles highlight technologies that enable the Industrial Internet of Things, IIoT-related products and strategies to get data more easily to the user.
This digital report will explore several aspects of how IIoT will transform manufacturing in the coming years.
Maintenance Manager; California Oils Corp.
Associate, Electrical Engineering; Wood Harbinger
Control Systems Engineer; Robert Bosch Corp.
This course focuses on climate analysis, appropriateness of cooling system selection, and combining cooling systems.
This course will help identify and reveal electrical hazards and identify the solutions to implementing and maintaining a safe work environment.
This course explains how maintaining power and communication systems through emergency power-generation systems is critical.