Bode Integrals
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## 1 Introduction to Bode Integrals[1]

I felt that the Dr. Stein article on Bode Integrals was excellent. Many of his explanations are also excellent so I borrow heavily from the article. I doubt I could write these sections better than Dr. Stein already has.

As the microprocessors become smaller, cheaper, and with lower power requirements they move into more and more systems. With microprocessors come the capacity for controlling otherwise unstable or marginally stable systems in real-time. Microprocessor control of unstable systems allows us to utilize (often to great effect) systems and processes that were previously unavailable to us because they nearly impossible to manually control or the operating environment was not suited for people. Microprocessor control of marginally stable systems allows broader use and, frequently, more efficient use of these marginally stable systems.

However, the drawbacks are simple and I quote Dr. Stein:

• Unstable systems are fundamentally, and quantifiably, more difficult to control than stable ones.
• Controllers for unstable systems are operationally critical.
• Closed-loop systems with unstable components are only locally stable.

Simply put the failure of a control system in an unstable system are usually catastrophic. People die, the machines are wrecked, and millions of dollars are lost.

### 1.1 Two Trends of Concern

Dr. Stein states 2 trends that he finds concerning, indeed a threat to the current respect controls engineers enjoy. These trends are

1. There are an increasing number of dangerous systems being employed with the use of controls. When the system is also naturally unstable then control system failure can be catastrophic.
2. The preference for abstract theoretical results without a thorough understanding of the real-world consequences.

I certainly felt that a majority of my controls education focused on the mathematics and process of calculating a controller from a suite of tools. Only after I began to analysis, model, and design control systems did I begin to fully assimilate the design trade-offs of different controller tools.

## 2 The Bode Integrals[2]

Here are the Bode integrals as presented by Stein.

 $LaTeX: \int_{0}^{\infty} \ln \left | s \left( j\omega \right) \right | d\omega = 0$ Bode Integral #1

 $LaTeX: \int_{0}^{\infty} \ln \left | s \left( j\omega \right) \right | d\omega = \pi \sum_{p \in P} \mbox{Re} \left( p \right)$ Bode Integral #2

Dr. Stein's interpretation of the Bode Integrals

The first integral applies to stable plants and the second to unstable plants. They are valid for every stabilizing controller, assuming only that both plant and controller have finite bandwidths. In words, the integrals state that the log of the magnitude of sensitivity function of a SISO feedback system, integrated over frequency, is constant. The constant is zero for stable plants, and it is positive for unstable ones. It becomes larger as the number of unstable poles increases and/or as the poles move farther into the right-half plane. (Technically, we must count all unstable poles here, including those in the compensator, if any.)

The integral over frequency is a constant dependent on the plant. As a result, the more disturbance rejection your design achieves the more peaking (greater than 0 dB magnitude) the design will incorporate. Choose carefully where you put that peak.

A counter argument is made by some that any given design achieves high disturbance rejection in low frequencies but that the system has infinite bandwidth to spread out the peak. Since the bandwidth is inifinite the peaking can be very thin, inconsequential. Dr. Stein counters that the real systems have finite available bandwidths where he defines available bandwidth as the range of frequencies where the system's behavior is linear and what I call "well behaved".

## 3 The Take Away

Unstable systems fail catastrophically when the control system fails. The controller cannot be designed arbitrarily using automated tools without regard for the implementation.

## 4 References

• Stein, Gunter "Respect the Unstable," IEEE Control Systems Magazine, Vol. 23, Num. 4, pp. 12-25, Aug. 2003.

### 4.1 Notes

1. Stein, pp. 12-15
2. Stein, pg. 14