Tilayhtälöt

Tilayhtälöt, or state equations, are a fundamental concept in control theory and system analysis. They provide a mathematical framework for describing the behavior of a dynamic system over time. Instead of focusing solely on the input-output relationship, tilayhtälöt represent the internal state of the system as a vector. This state vector encapsulates all the necessary information to predict the future behavior of the system given its current state and any future inputs.

A typical tilayhtälö representation consists of two main equations: the state equation and the output equation. The state equation describes how the state vector evolves over time, usually as a function of the current state vector, the input vector, and potentially a time variable. The output equation relates the system's output to its current state vector and input vector. This formulation is particularly useful for systems with memory or internal dynamics that cannot be fully captured by an input-output mapping alone.

Linear time-invariant (LTI) systems are commonly represented using tilayhtälöt in a compact matrix form. The state equation is expressed as x_dot = Ax + Bu, where x is the state vector, u is the input vector, A is the system matrix, and B is the input matrix. The output equation is y = Cx + Du, where y is the output vector, C is the output matrix, and D is the feedforward matrix. This matrix representation simplifies analysis and allows for the application of powerful linear algebra techniques. Tilayhtälöt are essential for understanding system stability, controllability, observability, and for designing controllers.