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:
| System type | State-space model |
| Continuous time-invariant | |
| Continuous time-variant | |
| Explicit discrete time-invariant | |
| Explicit discrete time-variant | |
| Laplace domain of continuous time-invariant |
|
| Z-domain of discrete time-invariant |
Read more about this topic: State Space Representation
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