# PID math demystified, part 1

## You’ve seen the equations, but have you thought about how those elements work together? Part 1: The basic concepts and proportional control.

05/06/2013

Most process control engineers have been exposed to the basic equation in a form that looks something like this:

More than you want to swallow in one bite? Let’s break this down into the major components:

Output: u(t)  is the output of the controller at the end of the scan. If the output of the controller is a valve, then the output is the valve position that the controller is requesting after it has seen the inputs. In most controllers, this is actually the change in output from 50%. So if u(t) = 0 then the valve output is 50%; if u(t) = 1 then the valve output is 51%; and if u(t) = -2 then the valve output is 48%. You get the idea. But what’s important is that it’s not a change in output from the previous scan, but a new output.

Proportional

Kpe(t) is the proportional component, the P in PID. If you have a controller configured as proportional only, this is it. So let’s look at how this works.

Let’s start with my own misconception of how I thought it worked. When I imagine a controller, I picture myself turning a valve while watching a gage. I look at the gage, decide if I need more or less, turn that valve a little more or a little less, and then repeat the process until the gage shows the value I want. That sounds fundamentally logical, but it is not how a proportional only controller works. It’s more like if I were to look at the gage, subtract what it reads from what I want it to read, and then take that error over to a chart to look up a new value for the valve.

e(t) usually called error, is simply the difference between the setpoint and the process variable. It is the difference between where you are and where you want to be, right now, at this instant.

Kp gain, is a factor that is multiplied by the error to give you the new output, the new valve position. It’s that simple. The error at that instant of the scan is calculated and the new output is calculated.

Let’s look at an example of pseudo code to explore how this works:

Error = Setpoint - ProcessValue;

Output = K * Error;

This control algorithm is deceptively simple, yet it gives an immediate response to a setpoint change or a disturbance in the process. And if K is set correctly, will quickly move the process toward the setpoint. But, it won’t get the process to the setpoint because there has to be some error if the output is anything other than 50%.

Note: On some systems (though not often in modern systems), gain is expressed as proportional band. Proportional band is defined as the amount of change in the controlled variable required to drive the loop output from 0 to 100%. To convert between the two, gain = 100/PB.

Next week: Understanding I and D.

This post was written by Scott Hayes. Scott is a senior engineer at MAVERICK Technologies, a leading system integrator providing industrial automation, operational support, and control systems engineering services in the manufacturing and process industries. MAVERICK delivers expertise and consulting in a wide variety of areas including industrial automation controls, distributed control systems, manufacturing execution systems, operational strategy, and business process optimization. The company provides a full range of automation and controls services – ranging from PID controller tuning and HMI programming to serving as a main automation contractor. Additionally MAVERICK offers industrial and technical staffing services, placing on-site automation, instrumentation and controls engineers.

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.