MATLAB: How to remove unwanted states from a linearized Simulink dynamic model

Simulink Control Design

I have a non-linear Simulink model of a dynamic system. When I linearize the model using the function "linearize" at an operating point (trim), I get a state space system with 30 states. That is expected because my model indeed has 30 states. But not all these states have significant dynamic effects on the linearized model, my use case, and the input/output (I/O) of the system. What is the best way to remove the unwanted states from my state space model and reduce it to the necessary 12 states that are significant for my system dynamics?

Best Answer

For the following use case, the "linearize" function generates a 30 dimensional state space model stored in an object, for example, called 'linsys'. 'linsys' stores a 30x1 vector of states. If you do not know the exact rows that correspond to the unwanted states in the state vector, you can execute the function "balreal" to perform a Gramian-based input/output balancing of state-space realizations in order to identify which states have negligible effect on the input and output. It returns the corresponding vector row index for the negligible states. Then, you can use the function "modred" to eliminate the states that do not affect the system I/O.

An example of how to achieve the above workflow can be found in the link below:

https://www.mathworks.com/help/control/ref/modred.html#f3-209417

The documentation for the functions "linearize", "balreal", and "modred" can be found below respectively:

https://www.mathworks.com/help/slcontrol/ug/linearize.html

https://www.mathworks.com/help/control/ref/balreal.html

https://www.mathworks.com/help/control/ref/modred.html

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MATLAB: How to estimate the inital state with an ssest output model

In short, you cannot. SSEST is a black-box identification function (unless you pass in a full initialized @idss model as input), which means that the state vector used by the identified model are arbitrary.

Note:

If you need to check if you really need 8 states, you can reduce an identified model using BALRED and then refine

sys = ssest(data, 8);
sys2 = balred(sys, 5);
sys3 = ssest(data, sys2)

You can linearize your original model at the given operating point. Let's say that that linearized model is called LinSys. LinSys should be state-consistent with the original model (states of LinSys are deviations from the equilibrium values of those of the nonlinear system), as long as block reduction did not remove any states. Then, you can pass LinSys as initial model to SSEST, along with the deviation experiment data. This was you can update the matrices of LinSys to better fit the data.

sys = ssest(data, LinSys)

MATLAB: State space system gives different bode plot then transfer function matrix

The sys2 created is not the minimal realization of the system, and if you perform minimal realization you will get back the same system:

>> minreal(sys2)
ans =
z^3 - 0.1 z^2 + 0.48 z - 0.34
-----------------------------
z^3 - 1.8 z^2 + 1.21 z - 0.41
Sample time: 0.01 seconds
Discrete-time transfer function.

which is the same as the sys3

>> sys3
sys3 =
z^3 - 0.1 z^2 + 0.48 z - 0.34
-----------------------------
z^3 - 1.8 z^2 + 1.21 z - 0.41
Sample time: 0.01 seconds
Discrete-time transfer function.

There are recommended working with model conversions: http://www.mathworks.com/help/control/ug/conversion-between-model-types.html#f3-1039600

And I understand this is what you are witnessing:

In addition, converting back and forth between model types can introduce additional states or orders, or introduce numeric inaccuracies. For example, conversions to state space are not uniquely defined, and are not guaranteed to produce a minimal realization for MIMO models.

Thanks, Sandip

Best Answer

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MATLAB: How to estimate the inital state with an ssest output model

MATLAB: State space system gives different bode plot then transfer function matrix

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