ELEC4632 Chap.3 Controllability, Observation and Pole Placement
Controllability, Observation and Pole Placement
State-feedback design depends on two structural questions: whether the input can reach every modelled state direction and whether the measured output reveals enough information to estimate every relevant mode.
This chapter transforms the tutorial basis b, Ab and A squared b into controllability coordinates, connects a desired pole polynomial to feedback feasibility, and explains how a state observer supports real-time output feedback.
What this chapter covers
- 01
Controllability matrix and full-state reachability
- 02
Coordinate transformation using a controllability basis
- 03
Similarity transforms that preserve eigenvalues
- 04
Desired characteristic polynomials and pole assignment
- 05
Observability as state inference from output history
- 06
Observer correction and estimation-error dynamics
- 07
Separation of controller poles and observer poles
Worked example · free
Transform a three-state model into controllability coordinates
- +1Set T = [b, Ab, A squared b]; independence makes T invertible.
- +1The transformed input T inverse b is the first coordinate vector.
- +1T inverse Ab is the second coordinate vector and T inverse A squared b is the third.
- +1The final transformed state column expresses A cubed b in the same basis through characteristic coefficients.
Key terms
- Controllability Matrix
- The controllability matrix collects successive input directions propagated by the state matrix.
- Similarity Transform
- A similarity transform changes state coordinates while preserving the system eigenvalues.
- State Feedback
- State feedback computes the control input from measured or estimated state variables.
- Estimation Error
- Estimation error is the difference between the actual state and the observer estimate.
- Output Feedback
- Output feedback uses measured output, often through an observer, to generate the control input.
Controllability, Observation and Pole Placement FAQ
Why check controllability before choosing poles?
A feedback gain acts through the input channel. If a state direction is unreachable, its associated mode cannot be moved by that gain. Full controllability establishes feasibility; pole selection then addresses the desired response and control-effort trade-off.
What problem does a state observer solve?
A controller may require a full state vector when sensors provide only an output. The observer propagates the model and corrects its estimate using the measured output, supplying an estimated state for the feedback law.
Are controller and observer poles the same design choice?
No. Controller poles shape the closed-loop plant motion, while observer poles shape estimation-error decay. They interact in implementation through the estimated state, but their roles and justification should remain separately labelled.
What does a controllability basis accomplish?
It uses successive input directions to define coordinates that expose how the input propagates through the state update. When those directions are independent, the basis is invertible and makes the relationship between desired characteristic coefficients and feedback design easier to inspect.
What evidence supports a combined observer-controller design?
Show that the plant response follows the intended closed-loop dynamics while estimation error decays from a nonzero initial mismatch. Also inspect the actual control input, sampling behaviour and sensitivity to measurement noise rather than validating controller and observer equations only in isolation.
Why label matrix dimensions during pole placement?
Dimensions expose whether the input, output, gain and basis matrices can form the stated products. They do not prove correctness, but they catch transposed channels and malformed gains before those errors propagate into a characteristic polynomial or simulation.
Exam move
Draw the flow from input to plant, measurement, observer and estimated-state feedback. Recreate the controllability-basis transformation symbolically and identify what each transformed column means. For any pole-placement question, write the rank result before the desired polynomial.
When discussing output feedback, name the controller and observer pole sets separately and state what simulation evidence would confirm state-estimate convergence. Practise the design as a decision sequence. First label the dimensions of A, b and the measured-output matrix. Build the controllability or observability matrix and state what full rank authorises. Expand the desired polynomial separately so signs remain visible.
Only then connect coefficients to a gain calculation. On an observer problem, write the estimation-error dynamics before selecting its poles; this keeps the estimator objective separate from state-feedback dynamics.
Build two short explanations of the separation principle: one for the algebraic roles of controller and observer poles, and one for implementation, where noise, saturation and model mismatch still couple their behaviour. In simulation, start the observer from an incorrect initial state so convergence can actually be seen. Plot the estimate, measured output and control input together.
Under time pressure, use a fixed closing checklist: feasibility established, poles labelled by role, gain dimensions checked, and combined-loop evidence named. That checklist turns a long formula derivation into a complete control argument. Practise one counterexample in words: describe what fails when a mode is uncontrollable and what fails when a mode is unobservable. This makes the two rank tests harder to confuse.
Then trace a measurement correction through the observer and into the feedback law. On the final review pass, check that every chosen pole set belongs to the correct dynamics and that the response claim includes both state motion and estimation-error decay.
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