# FindSystemModelEquilibrium

FindSystemModelEquilibrium[model]

searches for an equilibrium to the model model.

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

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

FindSystemModelEquilibrium[model,{x1v1,},]

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

# Details and Options • 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 \$ProgressReporting control display of progress

# Examples

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## Basic Examples(3)

Find an equilibrium, starting the search at initial values:

Use given start values for states:

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

## Scope(3)

Give start values for states, inputs and outputs:

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

Find an equilibrium point with given constraints:

## Options(1)

### ProgressReporting(1)

Control progress reporting with ProgressReporting:

## Applications(5)

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

Linearize a model around an equilibrium point:

Linearize around an equilibrium point and analyze the stability:

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

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

Linearize and close the loop around a PI controller:

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

Simple pendulum swinging through any angle: Equilibrium with the pendulum hanging straight down:

Pendulum standing straight up above its axis:

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

The pendulum has one stable and two unstable equilibrium points:

## Properties & Relations(2)

Equilibrium points , for an ODE satisfy :

Find an equilibrium point and :

Verify :

Many equilibrium points may exist: