Question

The state model is _____, the transfer function of the system is ____

• A. Nonunique, unique
• B. Nonunique, nonunique
• C. Unique, nonunique
• D. Unique, unique

Hint:

A system's transfer function is a unique input-output description.
A system's state-space representation is not unique; many different sets of state variables can describe the same system.
The transfer function can be derived from any valid state-space model: \(H(s) = C(sI-A)^{-1}B + D\).

Answer 1

The Correct Option is A

State Model (State-Space Representation): For a given LTI system, its state-space representation is nonunique. Different choices of state variables can lead to different state matrices (A, B, C, D) that describe the same system input-output behavior. These different representations are related by similarity transformations.
Transfer Function: For a given LTI system, its transfer function \(H(s)\) (or \(H(z)\)) which describes the input-output relationship (ratio of Laplace/Z-transform of output to input, assuming zero initial conditions) is unique. While it can be written in different factored forms, the overall rational function is unique. Therefore, the state model is nonunique, and the transfer function of the system is unique. This corresponds to option (a). \[ \boxed{\text{Nonunique, unique}} \]