Questions: State-Space Representation and Realization

Question 1

Two state-space models (A₁, B₁, C₁, D) and (A₂, B₂, C₂, D) produce exactly the same transfer function, but A₁ ≠ A₂. Which statement is correct?

• A. Both models are incorrect, since the transfer function uniquely determines the A matrix
• B. The models represent physically different systems that happen to have the same input-output behavior
• C. Both are valid realizations related by a similarity transformation T: A₂ = T⁻¹A₁T, B₂ = T⁻¹B₁, C₂ = C₁T
• D. At least one must be uncontrollable or unobservable, because only minimal realizations match

Answer: (C)

State-space realization is non-unique: infinitely many (A, B, C, D) quadruples produce the same transfer function. Any invertible coordinate transformation T applied to the state vector yields a different A but identical input-output behavior. This is because the transfer function H(s) = C(sI−A)⁻¹B + D depends on the eigenvalues of A (the poles), not on the specific coordinate representation. The eigenvalues of A₁ and A₂ will be identical (they are the same poles), even though the matrices look different. This non-uniqueness is a feature, not a bug — it lets engineers choose canonical forms optimized for different purposes.

Question 2

A control engineer wants to analyze a system's modes individually — understanding how each pole contributes independently to the total response. The most useful canonical realization is:

• A. Controllable canonical form, because it directly encodes the denominator polynomial
• B. Observable canonical form, because output measurements are physically meaningful
• C. Diagonal (modal) canonical form, because A is diagonal and each state variable evolves as a decoupled mode corresponding to one pole
• D. Any minimal realization, since all share identical modal structure

Answer: (C)

In diagonal (modal) form, A is a diagonal matrix with the system's poles on the diagonal. Each state variable xᵢ satisfies ẋᵢ = λᵢxᵢ + (coupling to input), meaning it evolves independently of the other state variables. The total response is a superposition of these decoupled modal responses. This makes the modal form ideal for analysis: you can see exactly which poles are fast/slow, well-damped/lightly-damped, and how strongly each is excited by the input. Controllable canonical form is useful for controller design but couples the state variables together in the A matrix.

Question 3

The eigenvalues of the A matrix in a state-space representation are identical to the poles of the corresponding transfer function.

• T. True
• F. False

Answer: True

The poles of the transfer function H(s) = C(sI−A)⁻¹B + D are the values of s where (sI−A)⁻¹ blows up, which occurs when det(sI−A) = 0. But det(sI−A) = 0 is exactly the characteristic equation of matrix A — its roots are the eigenvalues of A. This equivalence is one of the key bridges between the frequency-domain and state-space perspectives: poles, eigenvalues, and stability are all the same concept viewed from different angles.

Question 4

A transfer function uniquely determines the A, B, C, D matrices of its state-space realization.

• T. True
• F. False

Answer: False

Realization is non-unique: there are infinitely many state-space models that produce the same transfer function. Any similarity transformation T maps one valid realization (A, B, C, D) to another valid realization (T⁻¹AT, T⁻¹B, CT, D). This is why engineers choose among canonical forms — controllable canonical form, observable canonical form, diagonal modal form — each optimized for a different purpose. The transfer function captures only the input-output (external) behavior; the state-space model additionally encodes an internal coordinate representation, which is not unique.

Question 5

What information does a state-space representation reveal that a transfer function hides, and why does this matter for control design?

Model answer: A transfer function describes only the input-output relationship of a system, implicitly assuming zero initial conditions and hiding any internal structure. A state-space model makes the internal state explicit: the state vector x captures everything about the system's 'memory' at each instant. This reveals whether all internal modes are reachable from the input (controllability) and whether all modes can be inferred from the output (observability). Transfer functions can mask uncontrollable or unobservable modes — pole-zero cancellations in H(s) hide poles that still affect internal dynamics. State-space also handles MIMO systems and non-zero initial conditions naturally, making it the preferred representation for modern control design.

The practical consequence is that a transfer function with a pole-zero cancellation looks simpler than it is: the cancelled pole still exists in the internal dynamics, can be excited by initial conditions, and can cause internal instability even when the input-output map looks stable. State-space reveals this hidden mode. For control design, you need to know about all modes — not just the ones visible at the output — to place poles robustly and design observers.