tilstandsligninger

Tilstandsligninger are a mathematical form used to describe the evolution of a dynamical system in state-space. They express how the internal state variables change over time in response to external inputs, and how these states relate to observable outputs. The approach is widely used in engineering, physics, and economics to model complex, time-dependent behavior.

In continuous time, the state equation is typically written as dx/dt = f(x, u, t), where x is the state vector, u is the input vector, and y = h(x, u, t) gives the outputs. In discrete time, common in digital control, the form is x[k+1] = f(x[k], u[k], k), with y[k] = h(x[k], u[k], k). For linear time-invariant systems, these become simpler linear forms: dx/dt = A x + B u, y = C x + D u in continuous time, and x[k+1] = A x[k] + B u[k], y[k] = C x[k] + D u[k] in discrete time.

State-space models are often preferred because they handle multivariable systems, time-varying dynamics, and nonuniform inputs naturally. They are also central to estimation and control, enabling concepts like controllability and observability. Under suitable conditions, linear state-space models relate to transfer functions via the formula G(s) = C(sI − A)^{-1}B + D.

Nonlinear and time-varying extensions use general f and h, with linearization around operating points providing local LTI approximations. Numerical methods (Euler, Runge-Kutta) are used to simulate state equations when analytical solutions are unavailable.

Common applications include mechanical and electrical systems, aerospace, robotics, and process control, where state-space representations support modeling, simulation, controller design, and state estimation such as Kalman filtering.