Lecture 3 slides

The third lecture treats stability, robustness and the sensitivity function.

Reading tip: KJA_pp206_235.pdf

The covered material is treated in the hand-in exercise 1.4

Control System Design

Feedback, if properly designed, can be used to make complex systems behave predictably, even when operating in noisy and uncertain environments. We start by introducing and discussing the common objectives of feedback design.

Process Disturbances

Systems in the real world operate in unpredictable environments. Designing to mitigate the effect of such disturbances is an important phase of control system design.

Measurement Noise

Feedback is used to reduce uncertainty, but the process of feedback also introduces additional uncertainty through measurement noise. We discuss this, and some common approaches to reducing its effect.

Process Uncertainty

System models, when they are available at all, are rarely perfect, often only approximately describing the true input-output behaviour of a system. Yet despite this, simple control systems (typically PID) are able to obtain robust and predictable performance. We address this apparent contradiction by studying the robustness properties of feedback loops.

Bode's Sensitivity Integral

Feedback is all about trade-offs. This is illustrated clearly by Bode's sensitivity integral: the conservation law of feedback.

Bode's Sensitivity Integral can be derived with some machinery from complex analysis. We give a quick sketch of the case with no open loop unstable poles to avoid getting into too many messy details about different branches of the complex logarithm. Hopefully the video will give you a flavour of the proof!

Some Extra Stuff

The existing lecture notes treated linearisation around trajectories. This is of course nice stuff, but a bit tangential to the rest of the course. If you're interested, here is a link to part of the nonlinear control course where you can see videos on this topic. You can also see this theory in action through this video on the stabilising of an inverted pendulum by oscillation.

Inverted pendulum with a vertically oscillated pivot.