Question 1

What is the difference between controllability and observability?

Accepted Answer

Controllability asks: 'Can the input u drive every state x to any desired value?' The controllability matrix Cc = [B | AB | A²B | ...] must be full rank. Observability asks: 'Can all states x be estimated from the output y?' The observability matrix Co = [C; CA; CA²; ...]ᵀ must be full rank. Pole placement (state feedback) requires controllability; observer (Luenberger) design requires observability.

Question 2

What is Ackermann's formula for pole placement?

Accepted Answer

Ackermann's formula computes the state feedback gain K for a SISO controllable system so that the closed-loop poles match desired locations. The formula is K = eₙᵀ Cc⁻¹ φ(A), where eₙ is the last standard basis vector, Cc is the controllability matrix, and φ(A) is the desired characteristic polynomial evaluated at the system matrix A. It provides an exact closed-form solution for any order system, assuming the controllability matrix is invertible.

Question 3

Can an open-loop unstable system like an inverted pendulum be stabilized with state feedback?

Accepted Answer

Yes — provided the system is controllable. The inverted pendulum has open-loop eigenvalues with positive real parts (unstable poles), but with state feedback u = -Kx the closed-loop poles can be placed in the left-half plane, achieving stability. In practice, a full state observer (Luenberger observer) is needed to estimate unmeasured states. Select the Inverted Pendulum preset in this tool to explore the design.

Question 4

How does the transfer function H(s) relate to the state-space representation?

Accepted Answer

The transfer function is H(s) = C(sI - A)⁻¹B + D. The state-space realization for a given H(s) is not unique — controllable canonical form, observable canonical form, and diagonal form are all valid realizations. Importantly, pole-zero cancellations in H(s) can hide unstable or uncontrollable/unobservable modes that are visible in the full state-space representation but absent from the transfer function.

State Space Analysis — Controllability, Observability & Pole Placement

State Space Fundamentals