IntermediateTopic 60 of 101Signals, Control & Comms
Controllability and observability
Whether states can be driven and inferred from outputs.
In depth
Controllability: can inputs drive the state anywhere? Rank of [B AB A²B…] must be n. Observability: can states be inferred from outputs? Rank of [C; CA; CA²…] must be n. Duality between them. Needed for pole placement and observers; uncontrollable modes can't be changed.
Key points
- Kalman rank tests
- Hautus test
- Minimal realisation
ExampleAn unmeasured, uncontrolled unstable mode can make a system impossible to stabilise.
What is Controllability and observability?
Whether states can be driven and inferred from outputs.
Why is it important to understand controllability and observability?
Controllability and observability say whether you can steer a system and infer its state, so they decide whether a design is possible.
What does controllable mean?
All states can be driven to any desired value in finite time using the inputs.
What does observable mean?
All states can be determined from the outputs over time, which allows estimation of hidden internal states.
How is Controllability and observability used in practice?
Typical applications include checking whether a multi-tank system can be driven to any level with its pumps, deciding whether an estimator can work with the available sensors and finding which state cannot be seen from a temperature sensor alone.
What should I learn before Controllability and observability?
Study "State-space models" first (topic 59 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Controllability and observability?
Kalman rank tests. Hautus test. Minimal realisation.
Can you give an example of Controllability and observability?
An unmeasured, uncontrolled unstable mode can make a system impossible to stabilise.
What comes after Controllability and observability?
The next topic is "State feedback and pole placement", which builds directly on this one.
How difficult is Controllability and observability?
It is classed as Intermediate in Signals, Control & Comms (topic 60 of 101). It is an intermediate topic. Work through the beginner topics in Signals, Control & Comms first.