| # State-space models of linear dynamical systems Matrices > Applications > State-space models ## Definition Many discrete-time dynamical systems can be modeled via linear state-space equations, of the form where is the state, which encapsulates the state of the system at time , contains control variables, contains specific outputs of interest, and are matrices of appropriate size. In effect, a linear dynamical model postulates that the state at the next instant is a linear function of the state at past instants, and possibly other ‘‘exogeneous’’ inputs; and that the output is a linear function of the state and input vectors. A continuous-time model would take the form of a differential equation Finally, the so-called time-varying models involve time-varying matrices (see an example below). ## Motivation The main motivation for state-space models is to be able to model high-order derivatives in dynamical equations, using only first-order derivatives, but involving vectors instead of scalar quantities.
| Consider for example the second-order differential equation which describes the evolution of a damped mass-spring system, with the external force acting on the mass, and the vertical position. (Here and are the first and second derivatives of , respectively.)
The above involves second-order derivatives of a scalar function . We can express it in an equivalent form involving only first-order derivatives, by defining the state vector to be The price we pay is that now we deal with a vector equation instead of a scalar equation: The position is a linear function of the state: with . ## A nonlinear system In the case of non-linear systems, we can also use state-space representations. In the case of autonomous systems (no external input) for example, these come in the form where is now a non-linear map. Now assume we want to model the behavior of the system near an equilibrium point (such that ). Let us assume for simplicity that . Using the first-order approximation of the map , we can write a linear approximation to the above model: where
| The motion of a pendulum can be described by the dimensionless nonlinear equation The linearization around the equilibrium point yields . The linearization around the equilibrium point yields .