Yksikkötilamuotoja

Yksikkötilamuotoja is a Finnish term used in dynamical systems and control theory to refer to standard, canonical state-space representations of linear systems. A state-space model describes system dynamics with x' = Ax + Bu and y = Cx + Du, where x is the state vector, u the input, and y the output. Yksikkötilamuotoja emphasize particular arrangements of the matrices A, B, C and D that highlight properties such as controllability and observability and that ease analysis, realization, and design tasks.

Common forms

- Controllable canonical form: a realization constructed to make the system explicitly controllable and easy to perform pole placement and observer design.

- Observable canonical form: a realization arranged to emphasize observability of the state vector.

- Companion forms (including controllable and observable variants): special structures derived from the denominator (and sometimes numerator) of a transfer function.

- Diagonal or Jordan forms: realizations where A is diagonal or put into Jordan form when possible, often via similarity transforms.

Key ideas

- Existence and non-uniqueness: many systems admit multiple equivalent tilamuotoja related by similarity transforms; matrices change accordingly, but the input-output behavior remains the same.

- Realization from a transfer function: given a single-input single-output transfer function, canonical forms provide systematic methods to construct a state-space model.

- Applications: design of observers, state feedback, model reduction, and control synthesis rely on choosing convenient tilamuotoja.

Context and limits

While especially common for linear time-invariant systems, the concept of tilamuotoja extends to broader modeling contexts, with nonlinear systems requiring different realizations and approximations.