Controldirection

Controldirection is a term used in control theory to refer to the directional influence that control inputs exert on a system’s state or output. In linear time-invariant models, the state equation x' = Ax + Bu and the output equation y = Cx + Du encode how inputs steer state trajectories and how those trajectories project to outputs. The columns of B define basic input directions in the state space, and the combination of these directions with the dynamics specified by A determines which state directions can be reached or steered over time.

Mathematically, the controllable subspace is described by the controllability matrix [B, AB, A^2B, ..., A^{n-1}B]. If this matrix has full rank, all state directions are controllable; otherwise, only directions within the column space of that matrix can be reached by suitable inputs. In multi-input systems, the relative orientation and magnitude of the input directions influence the ease of steering along particular state trajectories. For nonlinear systems, the concept extends to direction fields and, in some analyses, to Lie brackets of control vector fields to characterize attainable directions.

Applications of controldirection include actuator placement, control allocation, and decoupling design, where engineers seek to maximize influence along desired state directions while minimizing cross-coupling. Related concepts are controllability, observability, reachability, and the controllability Gramian. The term controldirection is not universally standardized; it is often discussed implicitly through controllability and input–output mappings in control analysis and design.