Kontrollvektor

Kontrollvektor (u) is a term used in control theory to denote the vector of control inputs that can be manipulated to influence the behavior of a dynamical system. It collects all actuating signals into a single vector, allowing the designer to describe how external actions affect the system’s state.

In a linear time-invariant (LTI) system, the state dynamics are commonly written as x'(t) = Ax(t) + Bu(t), where x(t) ∈ R^n is the state vector, u(t) ∈ R^m is the control vector, A ∈ R^{n×n} is the system matrix, and B ∈ R^{n×m} is the input or control matrix. Each column of B describes how a corresponding input channel influences the state. In discrete time, the model is x_{k+1} = Ax_k + Bu_k.

The control vector is central to how the system is steered. The number of control inputs m is the dimension of u, and their combined effect via B determines how states can be driven, stabilized, or tracked. In multivariable systems, interactions among inputs can be exploited to achieve desired performance.

Controllability is a key concept related to the control vector. A system is controllable if any initial state can be driven to any final state in finite time using a suitable control sequence. For continuous-time LTI systems, controllability is tested with the matrix [B AB A^2B ... A^{n-1}B]; if its rank is n, the system is controllable. The controllability Gramian Wc = ∫_0^T e^{At} B B^T e^{A^T t} dt is another measure of how effectively inputs influence the state over a horizon T.

In practice, the control vector is designed to achieve objectives such as stabilization, tracking, or optimal performance, using laws like u = -Kx in linear-quadratic regulator problems or more advanced techniques including model predictive control with constraints.