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_Principle.jpg


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.


Process_Response.jpg

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 .





No comments
The Top Plant program honors outstanding manufacturing facilities in North America. View the 2013 Top Plant.
The Product of the Year program recognizes products newly released in the manufacturing industries.
The Leaders Under 40 program features outstanding young people who are making a difference in manufacturing. View the 2013 Leaders here.
The new control room: It's got all the bells and whistles - and alarms, too; Remote maintenance; Specifying VFDs
2014 forecast issue: To serve and to manufacture - Veterans will bring skill and discipline to the plant floor if we can find a way to get them there.
2013 Top Plant: Lincoln Electric Company, Cleveland, Ohio
Case Study Database

Case Study Database

Get more exposure for your case study by uploading it to the Plant Engineering case study database, where end-users can identify relevant solutions and explore what the experts are doing to effectively implement a variety of technology and productivity related projects.

These case studies provide examples of how knowledgeable solution providers have used technology, processes and people to create effective and successful implementations in real-world situations. Case studies can be completed by filling out a simple online form where you can outline the project title, abstract, and full story in 1500 words or less; upload photos, videos and a logo.

Click here to visit the Case Study Database and upload your case study.

Bring focus to PLC programming: 5 things to avoid in putting your system together; Managing the DCS upgrade; PLM upgrade: a step-by-step approach
Balancing the bagging triangle; PID tuning improves process efficiency; Standardizing control room HMIs
Commissioning electrical systems in mission critical facilities; Anticipating the Smart Grid; Mitigating arc flash hazards in medium-voltage switchgear; Comparing generator sizing software

Annual Salary Survey

Participate in the 2013 Salary Survey

In a year when manufacturing continued to lead the economic rebound, it makes sense that plant manager bonuses rebounded. Plant Engineering’s annual Salary Survey shows both wages and bonuses rose in 2012 after a retreat the year before.

Average salary across all job titles for plant floor management rose 3.5% to $95,446, and bonus compensation jumped to $15,162, a 4.2% increase from the 2010 level and double the 2011 total, which showed a sharp drop in bonus.

2012 Salary Survey Analysis

2012 Salary Survey Results

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.