Statespaceform

Statespaceform is a formal representation of a linear dynamical system using a state vector and first-order dynamics. In continuous time, it is written as dx/dt = A x + B u and y = C x + D u, where x is the state vector, u is the input, and y is the output. In discrete time, the corresponding form is x_{k+1} = A x_k + B u_k and y_k = C x_k + D u_k. The matrices A, B, C, and D define the system dynamics, input coupling, and how the state maps to the output.

The state vector x collects the internal variables needed to describe the system’s evolution, while the matrices encode how the current state and inputs drive future states and outputs. This framework is well suited for multi-input/multi-output systems and for combining dynamics with control strategies. Realistic models may include process or measurement noise, leading to stochastic state-space models of the form dx/dt = A x + B u + w and y = C x + D u + v, with w and v representing disturbances.

Key concepts in statespaceform include controllability and observability. A system is controllable if inputs can drive the state between any two configurations; it is observable if outputs reveal the entire state over time. These properties guide model reduction, observer design, and controller synthesis. A state-space model can realize a given transfer function via the relation G(s) = C(sI − A)^{-1}B + D, and can be transformed into various canonical forms through similarity transformations.

Applications include control design (state feedback, observers, and optimal controllers), system identification, and numerical simulation. The framework supports time-varying and discrete models, and underpins modern techniques such as Kalman filtering and subspace identification.