Tilamääräyhtälö

Tilamääräyhtälö, often translated as state equation or state-space equation, is a fundamental concept in systems theory and control engineering. It describes the behavior of a dynamic system using a set of first-order differential or difference equations. The core idea is to represent the system's internal state by a vector, which evolves over time according to these equations.

The general form of a linear time-invariant (LTI) continuous-time state-space equation is given by ẋ(t) = Ax(t) + Bu(t) and y(t) = Cx(t) + Du(t). Here, x(t) is the state vector, u(t) is the input vector, and y(t) is the output vector. The matrices A, B, C, and D characterize the system's dynamics, input coupling, output coupling, and direct feedthrough, respectively. For discrete-time systems, the equations take the form x[k+1] = Ax[k] + Bu[k] and y[k] = Cx[k] + Du[k], where k represents discrete time steps.

The state-space representation offers several advantages. It provides a complete description of the system's internal dynamics, not just its input-output behavior. This allows for a more in-depth analysis of system properties like stability, controllability, and observability. Furthermore, state-space methods are particularly well-suited for analyzing and designing controllers for multi-input, multi-output (MIMO) systems, which can be cumbersome to handle with traditional transfer function methods. The concept is widely applied in fields such as robotics, aerospace engineering, chemical process control, and economics.