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Variants of Model Descriptions
The model given above is called an ARX model. There are a handful of variants of this model known as Output-Error (OE) models, ARMAX models, FIR models, and Box-Jenkins (BJ) models. These are described later on in the manual. At a basic level it is sufficient to think of them as variants of the ARX model allowing also a characterization of the properties of the disturbances e.
Linear state-space models are also easy to work with. The essential structure variable is just a scalar: the model order. This gives just one knob to turn when searching for a suitable model description. See below.
General linear models can be described symbolically by
which says that the measured output y(t) is a sum of one contribution that comes from the measured input u(t) and one contribution that comes from the noise He. The symbol G then denotes the dynamic properties of the system, that is, how the output is formed from the input. For linear systems it is called the transfer function from input to output. The symbol H refers to the noise properties, and is called the disturbance model. It describes how the disturbances at the output are formed from some standardized noise source e(t).
State-space models are common representations of dynamical models. They describe the same type of linear difference relationship between the inputs and the outputs as in the ARX model, but they are rearranged so that only one delay is used in the expressions. To achieve this, some extra variables, the state variables, are introduced. They are not measured, but can be reconstructed from the measured input-output data. This is especially useful when there are several output signals, i.e., when y(t) is a vector. Tutorial, gives more details about this. For basic use of the toolbox it is sufficient to know that the order of the state-space model relates to the number of delayed inputs and outputs used in the corresponding linear difference equation. The state-space representation looks like
Here x(t) is the vector of state variables. The model order is the dimension of this vector. The matrix K determines the disturbance properties. Notice that if K = 0, then the noise source e(t) affects only the output, and no specific model of the noise properties is built. This corresponds to H = 1 in the general description above, and is usually referred to as an Output-Error model. Notice also that D = 0 means that there is no direct influence from u(t) to y(t). Thus the effect of the input on the output all passes via x(t) and will thus be delayed at least one sample. The first value of the state variable vector x(0) reflects the initial conditions for the system at the beginning of the data record. When dealing with models in state-space form, a typical option is whether to estimate D, K, and x(0) or to let them be zero.
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