MinimalStateSpaceModel

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`MinimalStateSpaceModel`

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MinimalStateSpaceModel[sys]

gives the minimal state-space model of the state-space model sys.

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MinimalStateSpaceModel[sys,{z₁,…}]

specifies the new coordinates z_i.

Details and Options

The minimal state-space model is both controllable and observable, and has the minimal number of states.
The system sys can be a standard or descriptor StateSpaceModel or AffineStateSpaceModel.

Examples

open allclose all

Basic Examples (1)Summary of the most common use cases

Compute a minimal realization:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-7v0u9p

Out[1]=1

Scope (6)Survey of the scope of standard use cases

Obtain a model with the minimal number of states:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-e7vz05

Out[1]=1

A minimal realization of a symbolic model:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-f7giam

Out[1]=1

A minimal realization does not contain uncontrollable modes:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-qjc4db

Out[1]=1

A minimal realization does not contain unobservable modes:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-4h9ewv

Out[1]=1

The minimal realization of a descriptor system:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-bier89

Out[1]=1

The minimal realization of an affine system:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-nxrtha

In[2]:=2

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-vu3et9

Out[2]=2

Specify the new variable:

In[3]:=3

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-wrgavx

Out[3]=3

Applications (2)Sample problems that can be solved with this function

Remove any uncontrollable or unobservable modes:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-4imwz

Out[1]=1

A parallel mass-spring-damper system is not completely controllable and observable. Design a controller based on the minimal system:

A model of the system:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-ce0ddv

In[2]:=2

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-6upitu

Compute the minimal model:

In[3]:=3

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-c6sknc

Out[3]=3

Design a controller based on the minimal system:

In[4]:=4

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-klrkba

In[5]:=5

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-hvgz65

Out[5]=5

The closed-loop system:

In[6]:=6

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-nsl8qe

Out[6]=6

Simulate the response of the closed-loop system and compare it with the open-loop system:

In[7]:=7

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-3ln6tm

In[8]:=8

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-m6m88r

Out[8]=8

Properties & Relations (4)Properties of the function, and connections to other functions

MinimalStateSpaceModel computes the subspace that is both controllable and observable:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-ud4cp3

In[2]:=2

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-9d43g

Out[2]=2

A minimal realization is both controllable and observable:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-oockzq

In[2]:=2

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-ltgf1c

Out[2]=2

Minimal realizations are not unique:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-dbj5se

In[2]:=2

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-oi88m2

Out[2]=2

Both are minimal realizations of the same transfer function:

In[3]:=3

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-ys3sh3

Out[3]=3

MinimalStateSpaceModel is equivalent to TransferFunctionCancel for scalar systems:

In[1]:=1

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-yxq3y5

In[2]:=2

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https://wolfram.com/xid/0me6tph2oqj4m4mcbh-hjp0qg

Out[2]=2

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