Simulate Control of a Toy Train

Model the position of a toy train and use a tracking controller to stop at stations

The toy train has a locomotor and a cart and can be modeled as a two-mass-spring system, with the locomotor exerting the input force f to displace the train.

The Lagrangian of a two-mass-spring system:

In[1]:=

ℒ=

[t],t]

(

[t]-

[t])

;

Derive the equations of motion:

In[2]:=

eqns=Table[

∂

D[ℒ,q'[t]]-D[ℒ,q[t]]



,{q,{

}}]/.{



f[t]-μ

'[t],



-μ

'[t]}

Out[2]=

{k(

[t]-

[t])+

′′

[t]f[t]-gμ

′

[t],-k(

[t]-

[t])+

′′

[t]-gμ

′

[t]}

Define a set of numerical values for the model's parameters:

In[3]:=

pars={

0.5,

0.25,k1,μ0.02,g9.8};

A state-space model of the toy train:

In[4]:=

train=StateSpaceModeleqns/.pars,{

[t],

'[t],

'[t]},f[t],{

[t],

[t]},t,

labels



Out[4]=

					f
x 1	0.	0.	1.	0.	0.
x 2	0.	0.	0.	1.	0.
′ x 1	-2.	2.	-0.196	0.	2.
′ x 2	4.	-4.	0.	-0.196	0.
x 1	1	0	0	0	0
x 2	0	1	0	0	0

A displacement of the train's locomotive's position

will cause unregulated oscillations:

In[5]:=

OutputResponse[{train,{0.1,0,0,0}},0,{t,0,30}];Plot[%,{t,0,30},PlotRangeAll,PlotLegends{

}]

Out[5]=

This is because its open-loop poles are lightly damped:

In[6]:=

Eigenvalues[First@Normal@train]//Chop

Out[6]=

{-0.098+2.44753,-0.098-2.44753,-0.196,0}

The train's route modeled as a piecewise function:

In[7]:=

tSig=Table[Splice[ConstantArray[i,40]],{i,{2,5,9}}];tSigPlot=ListStepPlot%,

plotOpts



Out[7]=

Specify the position of the train's locomotive as the tracked output:

In[8]:=

sspec="InputModel"train,"TrackedOutputs"1;

Specify a set of control weights and a sampling period:

In[9]:=

wts={DiagonalMatrix[{1,1,1,1,1}],{{0.1}}};τ=0.2;

Compute the controller:

In[10]:=

dlqr=DiscreteLQRegulatorGains[sspec,wts,τ,"Data"]

Out[10]=

SystemsModelControllerData

Design: discrete linear quadratic regulator

Feedback inputs count: 1



Obtain the closed-loop system:

In[11]:=

dcsys=dlqr["DiscreteTimeClosedLoopSystem"]

Out[11]=

0.765685	0.083751	0.0928256	-0.0327403	0.039222	0.
0.0748044	0.923196	0.00383002	0.190494	0.000520874	0.
-2.30733	0.821369	-0.0706863	-0.31043	0.387094	0.
0.701972	-0.741646	0.0498487	0.875861	0.0103352	0.
-0.288213	0.	0.	0.	1.	0.288213
1.	0.	0.	0.	0.	0.
0.	1.	0.	0.	0.	0.

0.2

The train follows the route and makes the 3 stops:

In[12]:=

sr=StateResponse[dcsys,tSig];

In[13]:=

Show[tSigPlot,ListStepPlot[Take[sr,2],PlotLegends{

}]]

Out[13]=

The engine and cart velocities during the route:

In[14]:=

ListStepPlot[sr〚{3,4}〛,PlotLegends{

′

}]

Out[14]=

Obtain the discrete-time controller model:

In[15]:=

dcm=dlqr["DiscreteTimeControllerModel"]

Out[15]=



1.	0.288213	-0.288213	0.	0.	0.	0.
1.	0.	0.	-4.98736	1.14859	-2.56798	-0.900621



0.2

The controller inputs:

In[16]:=

cinps=Join[{tSig},sr〚1;;1〛,sr〚1;;4〛];

Assess the control effort:

In[17]:=

ListStepPlot[OutputResponse[dcm,cinps]]

Out[17]=

Publisher Information

Contributed by: Wolfram Controls Team

Simulate Control of a Toy Train

Related Symbols

Publisher Information