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FindSystemModelEquilibrium
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FindSystemModelEquilibrium

[Experimental]

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searches for an equilibrium to the model model.

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FindSystemModelEquilibrium[model,{{{x1,x10},…},{{u1,u10},…},{{y1,y10},…}}]

searches for an equilibrium, starting from the points xi=xi0, ui=ui0 and yi=yi0.

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FindSystemModelEquilibrium[model,{x1v1,…},…]

searches for an equilibrium, with variable xi constrained to have the value vi etc.

Details and Options

  • FindSystemModelEquilibrium is typically used to find values of states, inputs and outputs for which the dynamics of a system model stop.
  • The model can be a SystemModel object, a full model name string or a shortened model name accepted by SystemModel.
  • FindSystemModelEquilibrium returns a list {{{x1,},…},{{u1,},…},{{y1,},…}}, where , and are the computed equilibrium values for states, inputs and outputs.
  • With no explicit starting point given, SystemModel[model]["GroupedInitialValues"] is used.
  • An equilibrium for a differential algebraic system is a value and such that .
  • FindSystemModelEquilibrium will attempt to find a local equilibrium point. In general, many equilibrium points may exist for a system.
  • The following option can be given:
  • ProgressReporting $ProgressReportingcontrol display of progress

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Find an equilibrium, starting the search at initial values:

In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-um2ih
Out[2]=2

Use given start values for states:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-h3ippb
Out[1]=1

Find an equilibrium for one of the included introductory hierarchical examples:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-ksqhmc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-g2fl34
Out[2]=2

Scope  (3)Survey of the scope of standard use cases

Give start values for states, inputs and outputs:

In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-igq4gn
Out[2]=2

Use constraints on inputs and outputs, and start values for states:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-ce95lg
Out[1]=1

Find an equilibrium point with given constraints:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-buoot4
Out[1]=1

Options  (1)Common values & functionality for each option

ProgressReporting  (1)

Control progress reporting with ProgressReporting:

In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-e6sn70
Out[2]=2

Applications  (5)Sample problems that can be solved with this function

Find an equilibrium point for a single water tank with inflow and outflow:

In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-0n2hm
Out[2]=2

Linearize a model around an equilibrium point:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-cstvh6
In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-fk9xhw
In[3]:=3
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https://wolfram.com/xid/0b7eymuf8mnpzeky-b5resv
Out[3]=3

Linearize around an equilibrium point and analyze the stability:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-qtpokw
In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-kuf28t
In[3]:=3
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https://wolfram.com/xid/0b7eymuf8mnpzeky-jx158e
In[4]:=4
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https://wolfram.com/xid/0b7eymuf8mnpzeky-u0cq0
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0b7eymuf8mnpzeky-fu2ml
Out[5]=5

Design a PI controller for keeping the level in a tank with inflow and outflow constant:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-b9ecat

Find the equilibrium where the level "h" is constrained to be 2:

In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-f8ir1v
In[3]:=3
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https://wolfram.com/xid/0b7eymuf8mnpzeky-5h5bm
Out[3]=3

Linearize and close the loop around a PI controller:

In[4]:=4
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https://wolfram.com/xid/0b7eymuf8mnpzeky-i0ayve
In[5]:=5
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https://wolfram.com/xid/0b7eymuf8mnpzeky-vms3h
In[6]:=6
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https://wolfram.com/xid/0b7eymuf8mnpzeky-l6jhv5

Show the closed-loop step response for a family of PI controllers:

In[7]:=7
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https://wolfram.com/xid/0b7eymuf8mnpzeky-jy6cjv
Out[7]=7

Simple pendulum swinging through any angle:

Equilibrium with the pendulum hanging straight down:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-273n1
In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-i1ts2b
Out[2]=2

Pendulum standing straight up above its axis:

In[3]:=3
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https://wolfram.com/xid/0b7eymuf8mnpzeky-c27433
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0b7eymuf8mnpzeky-jdtndz
Out[4]=4

Level curves of the first integral give the potential energy of the system:

In[5]:=5
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https://wolfram.com/xid/0b7eymuf8mnpzeky-jhvhcn
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0b7eymuf8mnpzeky-bf0fd4

The pendulum has one stable and two unstable equilibrium points:

In[7]:=7
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https://wolfram.com/xid/0b7eymuf8mnpzeky-jql54c
Out[7]=7

Properties & Relations  (2)Properties of the function, and connections to other functions

Equilibrium points , for an ODE satisfy :

In[2]:=2
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https://wolfram.com/xid/0b7eymuf8mnpzeky-doisqu
In[3]:=3
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https://wolfram.com/xid/0b7eymuf8mnpzeky-i42qvh
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0b7eymuf8mnpzeky-qn10wt

Find an equilibrium point and :

In[5]:=5
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https://wolfram.com/xid/0b7eymuf8mnpzeky-jf3yp5
Out[5]=5

Verify :

In[6]:=6
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https://wolfram.com/xid/0b7eymuf8mnpzeky-xbcdb
Out[6]=6

Many equilibrium points may exist:

In[1]:=1
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https://wolfram.com/xid/0b7eymuf8mnpzeky-pgp99
Out[1]=1
Wolfram Research (2018), FindSystemModelEquilibrium, Wolfram Language function, https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html.
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Wolfram Research (2018), FindSystemModelEquilibrium, Wolfram Language function, https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html.

Text

Wolfram Research (2018), FindSystemModelEquilibrium, Wolfram Language function, https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html.

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Wolfram Research (2018), FindSystemModelEquilibrium, Wolfram Language function, https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html.

CMS

Wolfram Language. 2018. "FindSystemModelEquilibrium." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html.

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Wolfram Language. 2018. "FindSystemModelEquilibrium." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html.

APA

Wolfram Language. (2018). FindSystemModelEquilibrium. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html

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Wolfram Language. (2018). FindSystemModelEquilibrium. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html

BibTeX

@misc{reference.wolfram_2025_findsystemmodelequilibrium, author="Wolfram Research", title="{FindSystemModelEquilibrium}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html}", note=[Accessed: 16-April-2025 ]}

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@misc{reference.wolfram_2025_findsystemmodelequilibrium, author="Wolfram Research", title="{FindSystemModelEquilibrium}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html}", note=[Accessed: 16-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_findsystemmodelequilibrium, organization={Wolfram Research}, title={FindSystemModelEquilibrium}, year={2018}, url={https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html}, note=[Accessed: 16-April-2025 ]}

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@online{reference.wolfram_2025_findsystemmodelequilibrium, organization={Wolfram Research}, title={FindSystemModelEquilibrium}, year={2018}, url={https://reference.wolfram.com/language/ref/FindSystemModelEquilibrium.html}, note=[Accessed: 16-April-2025 ]}