# How Does the Derivative Term Affect PID Controller Performance?

## The PID’s Derivative Term Can Improve Control Loop Performance, But Often at a Cost

Derivative is the third term within the PID. In mathematical terms the word derivative is defined as the slope of a curve. Seen in the context of strip chart data derivative represents the rate of change in error – the difference between the Process Variable (PV) and Set Point (SP). Like the proportional and integral terms within a PID controller, the derivative term seeks to correct for error. Valuable as the third term can be in maintaining effective control, experience suggests that appropriate uses of derivative are not entirely clear.

Each term of the PID seeks to complement the others and adds incremental value toward controlling process dynamics. Whereas the proportional term measures “how far” the PV is away from SP and the integral term sums error to determine “how long” PV has been away from SP, the derivative term assesses “how fast” error in the process is changing. As the rate of error either increases or decreases so too does the size of the derivative response. This aspect of derivative makes it ideal for some uses, but the same feature makes it entirely impractical for the majority of industrial applications.

When considering the use of derivative it is helpful to keep the following in mind:

**The Math of Measurement**

Although “derivative on error” is technically correct “derivative on measurement” is the more suitable form of the PID equation for industrial applications. From a practical standpoint, the math associated with “derivative on error” can result in excessive volatility – spikes in the Controller Output’s behavior often referred to as derivative kick. In contrast “derivative on measurement” applies a level of sensitivity to changes in SP that is more appropriate for practical applications.

**Turn Down the Noise**

Noise is a random source of error within the PV signal. Noise presents a significant challenge for derivative as the additional, excited variability in the PV signal results in equally agitated, derivative-driven responses to the CO. Typically the end result is excessive wear and tear on the associated control loop’s final control element (FCE). For most practitioners, the cost of accelerating wear and tear outpaces any improvements in control loop performance achieved through the use of derivative.

**A Small World**

Since PV volatility presents practical challenges for derivative the range of industrial applications becomes quite narrow. Suitable loops include those used in temperature control, some used in pH control, as well as others that can be characterized as having a high degree of inertia. The dynamics of such loops are slow and they allow derivative to correct appropriately for error. Most other loops – flow, pressure, level, etc. – can be too dynamic such that derivative negatively affects the FCE and other process instrumentation.

**Too Much Complexity**

Whereas tuning a controller using only the proportional and integral terms is relatively simple and straightforward, the addition of derivative makes the process difficult. The addition of a third variable expands the array of possibilities exponentially. As a result, additional testing is usually required which can waste limited resources and result in lost productivity. More often than not the costs outweigh the benefit.[vc_row css=”.vc_custom_1518713121206{margin-top: 1% !important;}”][vc_column][vc_column_text]In spite of these challenges derivative can play a meaningful role in improved control loop performance. To assist with evaluating the pros and cons of derivative, various PID tuning software packages simulate the responsiveness of the different forms of the controller (i.e. P-Only, PI, PID, and PID with Filter) and assess the impact on the associated FCE. It’s important to note, however, that most control loop tuning software products struggle to accurately model noisy process data. That is particularly true of products that apply frequency-based modeling.

In terms of the added complexity, training workshops on controller tuning best-practices can be helpful. Most detail the challenges of derivative while offering solutions that are both proven and practical. And again commercial tuning software can be useful and mitigate the added difficulty. One product in particular has been proven to handle noisy, highly oscillatory process dynamics and can provide improved controller tuning parameters using either open-loop or closed-loop process data.

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