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What Are State-Space Models?

Definition of State-Space Models



Continuous-Time Representation




Discrete-Time Representation


where T is the sampling interval, u(kT) is the input at time instant kT, and y(kT) is the output at time instant kT.


Discrete-time state-space models provide the same type of linear difference relationship between the inputs and outputs as the linear ARX model, but are rearranged such that there is only one delay in the expressions.


Relationship Between Continuous-Time and Discrete-Time State Matrices

The relationships between the discrete state-space matrices A, B, C, D, and K and the continuous-time state-space matrices F, G, H, D, and are given for piece-wise-constant input, as follows:

These relationships assume that the input is piece-wise-constant over time intervals .

The exact relationship between K and is complicated. However, for short sampling intervals T, the following approximation works well:

State-Space Representation of Transfer Functions

For linear models, the general model description is given by:



where Inx is the nx-by-nx identity matrix, Iny is the nx-by-nx identity matrix, and ny is the dimension of y and e. The state-space representation in the continuous-time case is similar.

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