Bode's Integral

There are fundamental limitations on the achievable sensitivity function that is expressed by Bode’s Integral.[^1] The Bode Integrals have been characterized as a form of Jensen’s Theorem and can be thought of as a conservation law for control systems.[^1] Bode’s Integral has two forms, one for stable plants and one for unstable plants.[^1] These integrals are valid for every controller as long as both the plant and controller have a finite bandwidth. [^1]

\int_0^\infty ln |s(j\omega)|d\omega = 0 \\ \int_0^\infty ln|s(j\omega)|d\omega=\pi\sum_{p\in P}Re(p) \end{matrix}

These integrals state that for a stable plant, the magnitude of the log of the sensitivity function integrated over frequency is constant.[^1] For unstable plants, the magnitude of the log of the sensitivity function is positive.[^1] And it becomes more positive the further into the right-half plane the poles are, and the more unstable poles there are.[^1] Sensitivities less than 1 are better than the open-loop response and sensitivities greater than 1 are worse than the open-loop performance.[^1] The implications of these integrals are that the sensitivity decreases across different frequencies must be compensated by increased sensitivity in other frequencies.[^1] Bode’s Integral was first used for feedback amplifier design in 1945.[^2] It clarifies the connection between open-loop stability margins and closed-loop bandwidth.[^2] This shows that there is a strong tradeoff between disturbance rejection below the design bandwidth and amplification just above the design bandwidth.[^2] The bode integrals are the integrals of the sensitivity and complementary sensitivity functions.[^2] These integrals can provide insights for control system engineers who are designing controllers for difficult unstable plants.[^2]

Sensitivity Function
Jensen’s Inequality
Available Bandwidth - can cause problems by increasing sensitivity in undesirable frequencies
X-29 Sensitivity Function Prototype - derived using the bode integral
Disadvantages of the State-Space Model - state-space models obscure effects of sensor noise and loop bandwidths
Nominal Sensitivity Peak -
Interactive Loop Shaping - the act of adjusting the sensitivity functions at different frequencies.
X-29 Prototype Bode Plot - uses a bandwidth restriction from Bode’s Integral
Minimum Sensitivity for an Inverted Pendulum - shows a graph of the bode integral
Hendrick Wade Bode

steinPracticalPhysicalSometimes2003 [^1]
#ruthWhatNewWhat2010[^2]