StateFeedbackGains
NonlinearStateSpaceModel
PIDTune

Engineering

Stabilize an Inverted Pendulum on a Wheel

Create a feedback controller to stabilize the mechanical system

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Define the equations of motion for a pendulum on a wheel (like a Segway):

In[1]:=

diffeqs=-lmSin[θ[t]]

′

[t]

-(m+M)

′′

[t]+lmCos[θ[t]]

′′

[t]F[t],-gSin[θ[t]]-Cos[θ[t]]

′′

[t]+l

′′

[t]0

Out[1]=

-lmSin[θ[t]]

′

[t]

-(m+M)

′′

[t]+lmCos[θ[t]]

′′

[t]F[t],-gSin[θ[t]]-Cos[θ[t]]

′′

[t]+l

′′

[t]0

Create a dynamic model by specifying states, inputs, outputs and parameter values:

In[2]:=

system=NonlinearStateSpaceModel[diffeqs,{θ[t],x[t]},F[t],{θ[t],x[t]},t]/.{M10,m80,l1,g9.8}

Out[2]=

θ[t]	x. 1 [t]
x. 1 [t]	- 882.Sin[θ[t]] -90+80 2 Cos[θ[t]] + Cos[θ[t]](F[t]+80Sin[θ[t]] 2 x. 1 [t] ) -90+80 2 Cos[θ[t]]
x[t]	x. 2 [t]
x. 2 [t]	- 784.Cos[θ[t]]Sin[θ[t]] -90+80 2 Cos[θ[t]] + F[t]+80Sin[θ[t]] 2 x. 1 [t] -90+80 2 Cos[θ[t]]
	θ[t]
	x[t]

Choose poles in the left-half plane for the closed-loop system:

In[3]:=

p={-1+2I,-1-2I,-0.35+0.35I,-0.35-0.35I}

Out[3]=

{-1+2,-1-2,-0.35+0.35,-0.35-0.35}

Create a state feedback controller and closed-loop system which achieves those poles:

In[4]:=

control=StateFeedbackGains[system,p,"Data"]

Out[4]=

SystemsModelControllerData

Design: pole placement

Feedback inputs count: 1



View the closed-loop system:

In[5]:=

control["Properties"]

Out[5]=

{ClosedLoopPoles,ClosedLoopSystem,ControllerModel,Data,Design,DesignModel,FeedbackGains,FeedbackGainsModel,FeedbackInputs,InputModel,InputsCount,OpenLoopPoles,OutputsCount,OutputsCount,SamplingPeriod,StatesCount,TrackedOutputs}

In[6]:=

controlcsys=control["ClosedLoopSystem"]

Out[6]=

θ[t]	◼
x. 1 [t]	◼
x[t]	◼
x. 2 [t]	◼
	◼
	◼

Simulate the response of the stabilized system to a sudden input:

In[7]:=

or=OutputResponse[controlcsys,UnitBox[t-1],{t,0,20}];{Plot[#〚1〛,{t,0,20},ImageSizeSmall,PlotRangeAll,PlotLabel#〚2〛]&/@Thread[{or,{θ,x}}]}//Grid

Out[7]=

Related Symbols

StateFeedbackGains
NonlinearStateSpaceModel
PIDTune

Publisher Information

Contributed by: Stephen Wolfram

Wolfram Language Example Repository

Stabilize an Inverted Pendulum on a Wheel

See Also

Publisher Information