State Space Representation - Linear Systems

Linear Systems

The most general state-space representation of a linear system with inputs, outputs and state variables is written in the following form:

where:

is called the "state vector", ;

is called the "output vector", ;

is called the "input (or control) vector", ;

is the "state matrix", ,

is the "input matrix", ,

is the "output matrix", ,

is the "feedthrough (or feedforward) matrix" (in cases where the system model does not have a direct feedthrough, is the zero matrix), ,

.

In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in the common LTI case, matrices will be time invariant. The time variable can be continuous (e.g. ) or discrete (e.g. ). In the latter case, the time variable is usually used instead of . Hybrid systems allow for time domains that have both continuous and discrete parts. Depending on the assumptions taken, the state-space model representation can assume the following forms: