Questions: State-Space Representation

Question 1

For the state-space system ẋ = Ax + Bu, y = Cx + Du, what do the eigenvalues of the A matrix represent?

• A. The steady-state output values for a given constant input
• B. The system's natural modes and poles — they determine stability and the character of the transient response
• C. The number of independent inputs the system can accept simultaneously
• D. The DC gain from each input to each output

Answer: (B)

The eigenvalues of A are the poles of the system — they are the values of s where det(sI - A) = 0, which is the characteristic equation. Eigenvalues with negative real parts correspond to stable decaying modes; positive real parts indicate instability; imaginary parts produce oscillation. This is exactly the same information as the poles of the transfer function H(s) = C(sI-A)⁻¹B + D.

Question 2

For a given physical system, there is exactly one valid set of state variables, and those variables should correspond to measurable physical quantities like position and velocity.

• T. True
• F. False

Answer: False

State variables are not unique. Any invertible linear transformation T applied to a valid state vector x yields new state variables x̃ = Tx, with transformed matrices à = TAT⁻¹, B̃ = TB, C̃ = CT⁻¹ — a different but equally valid representation of the same system. The input-output behavior (transfer function) is identical for all equivalent representations. State variables are mathematical constructs chosen for computational convenience, not necessarily tied to physical quantities.

Question 3

What can a state-space model reveal about a system that a transfer function might conceal?

Model answer: A transfer function can have pole-zero cancellations that hide internal modes that are either uncontrollable (inputs cannot excite them) or unobservable (they don't affect outputs). These hidden modes still exist in the physical system and can cause problems — an unobservable unstable mode means the system will blow up even though the transfer function appears stable. State-space models expose all internal dynamics, including these hidden modes.

The transfer function H(s) = Y(s)/U(s) is formed by a ratio where cancellations can occur. If a mode is both uncontrollable and unobservable, it appears as a pole-zero pair that cancels, vanishing from H(s). The state-space A matrix retains all eigenvalues, making every mode visible. This is why controllability and observability analysis is done in state-space, not from the transfer function.