System Identification Toolbox

Multivariable Systems

Systems with many input signals and/or many output signals are called multivariable. Such systems are often more challenging to model. In particular systems with several outputs could be difficult. A basic reason for the difficulties is that the couplings between several inputs and outputs lead to more complex models. The structures involved are richer and more parameters will be required to obtain a good fit.

Available Models

The System Identification Toolbox as well as the GUI handle general, linear multivariable models. All earlier mentioned models are supported in the single output, multiple input case. For multiple outputs, ARX models and state-space models are covered. Multi-output ARMAX and OE models are covered via state-space representations: ARMAX corresponds to estimating the K-matrix, while OE corresponds to fixing K to zero. (These are pop-up options in the GUI model order editor.)

Generally speaking, it is preferable to work with state-space models in the multivariable case, since the model structure complexity is easier to deal with. It is essentially just a matter of choosing the model order.

Working with Subsets of the Input-Output Channels

In the process of identifying good models of a system, it is often useful to select subsets of the input and output channels. Partial models of the system's behavior will then be constructed. It might not, for example, be clear if all measured inputs have a significant influence on the outputs. That is most easily tested by removing an input channel from the data, building a model for how the output(s) depends on the remaining input channels, and checking if there is a significant deterioration in the model output's fit to the measured one. See also the discussion under Step 3 above.

Generally speaking, the fit gets better when more inputs are included and often gets worse when more outputs are included. To understand the latter fact, you should realize that a model that has to explain the behavior of several outputs has a tougher job than one that just must account for a single output. If you have difficulties obtaining good models for a multi-output system, it might be wise to model one output at a time, to find out which are the difficult ones to handle.

Models that are just to be used for simulations could very well be built up from single-output models, for one output at a time. However, models for prediction and control will be able to produce better results if constructed for all outputs simultaneously. This follows from the fact that knowing the set of all previous output channels gives a better basis for prediction, than just knowing the past outputs in one channel. Also, for systems, where the different outputs reflect similar dynamics, using several outputs simultaneously will help estimating the dynamics.

Some Practical Advice

Both the GUI and command line operation will do useful bookkeeping for you, handling different channels. You could follow the steps of this agenda:

• Import data and create a data set with all input and output channels of interest. Do the necessary preprocessing of this set in terms of detrending, etc., and then select a Validation Data set with all channels.
• Then select a Working Data set with all channels, and estimate state-space models of different orders using n4sid for these data. Examine the resulting model primarily using the Model Output view.
• If it is difficult to get a good fit in all output channels or you would like to investigate how important the different input channels are, construct new data sets using subsets of the original input/output channels. Use the pop-up menu Preprocess > Select Channels for this. Don't change the Validation Data. The GUI will keep track of the input and output channels. It will "do the right thing" when evaluating the channel-restricted models using the Validation Data. It might also be appropriate to see if improvements in the fit are obtained for various model types, built for one output at a time.
• If you decide for a multi-output model, it is often easiest to use state-space models. Use n4sid as a primary tool and try out pem when a good order has been found.

Step 4: Fine Tuning Orders and Disturbance Structures

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