StateFeedbackGains
AffineStateSpaceModel
StateSpaceModel
OutputResponse
StateResponse

Control Systems

Model and Regulate a Wilberforce Pendulum

Model a rotating mass suspended from a spring

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Set up the Lagrangian for the system, using the following symbols:

Symbol	Description
z	massverticalposition
θ	massangularposition
m	massweight
ℐ	momentofinertia
k	springconstant
δ	springtorsionconstant
ϵ	couplingconstant

In[1]:=

ℒ=

z'[t]

-k

z[t]

+ℐ

θ'[t]

-δ

θ[t]

-ϵz[t]θ[t]);

The equations of motion are derived using the Euler-Lagrange formula:

In[2]:=

eqns=Table[

∂

D[ℒ,q'[t]]-D[ℒ,q[t]]

[t],{q,{z,θ}}]/.{

f,

τ}//Simplify

Out[2]=

kz[t]+

ϵθ[t]+m

′′

[t]f[t],

ϵz[t]+δθ[t]+ℐ

′′

[t]τ[t]

Assign numerical values for the model parameters:

In[3]:=

pars={m0.05,ℐ6.281

,δ0.75,ϵ0.8,k1.2}

Out[3]=

{m0.05,ℐ6.281,δ0.75,ϵ0.8,k1.2}

Construct an affine state space model of the system:

In[4]:=

assm=AffineStateSpaceModel[eqns/.pars,{z[t],z'[t],θ[t],θ'[t]},{f[t],τ[t]},{z[t],θ[t]},t]//FullSimplify

Out[4]=

z[t]	0.+1. x. 1 [t]	0	0
x. 1 [t]	0.-24.z[t]-8.θ[t]	20.	0
θ[t]	0.+1. x. 2 [t]	0	0
x. 2 [t]	0.-0.0636841z[t]-0.119408θ[t]	0	0.15921
	z[t]	0	0
	θ[t]	0	0

Linearize the model:

In[5]:=

ssm=StateSpaceModel[assm/.pars,SystemsModelLabels{{f,τ},{z,θ},{z,z',θ,θ'}}]

Out[5]=

					f	τ
z	0	1.	0	0	0	0
′ z	-24.	0	-8.	0	20.	0
θ	0	0	0	1.	0	0
′ θ	-0.0636841	0	-0.119408	0	0	0.15921
z	1	0	0	0	0	0
θ	0	0	1	0	0	0

Simulate the output response of the affine model to a disturbance in the angle θ:

In[6]:=

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

Out[6]=

Find the eigenvalues of the linear system:

In[7]:=

Eigenvalues[First@Normal@ssm]//Chop

Out[7]=

{0.+4.90115,0.-4.90115,0.+0.313197,0.-0.313197}

The system is controllable with f and θ:

In[8]:=

Table[ControllableModelQ[SystemsModelExtract[ssm,i]],{i,{All,1,2}}]

Out[8]=

{True,True,True}

Set f as the sole feedback input and θ as a disturbance input:

In[9]:=

sspec="InputModel"ssm,"FeedbackInputs"1,"ExogenousInputs"2;

Compute a pole placement controller that places the poles at the specified locations:

In[10]:=

=StateFeedbackGains[sspec,{-2,-3,-3.5,-4},"Data"]

Out[10]=

SystemsModelControllerData

Design: pole placement

Feedback inputs count: 1



Connect the controller to the affine model to obtain the closed-loop system:

In[11]:=

csys=[{"ClosedLoopSystem","InputModel"assm}]

Out[11]=

z[t]	0.+1. x. 1 [t]	0	0
x. 1 [t]	0.-24.z[t]-8.θ[t]+20.(0.-1.66903z[t]+60.9711θ[t]-0.625 x. 1 [t]+89.1175 x. 2 [t])	20.	0
θ[t]	0.+1. x. 2 [t]	0	0
x. 2 [t]	0.-0.0636841z[t]-0.119408θ[t]	0	0.15921
	0.+1.z[t]	0	0
	0.+1.θ[t]	0	0

The mass's position and angle are regulated with respect to non-zero initial conditions:

In[12]:=

sr1=StateResponse[{csys,{0.1,0,0.2,0}},{0,0},{t,0,100}];Plot[Evaluate@%〚{1,3}〛,{t,0,10},PlotRangeAll,PlotLegends{z,θ}]

Out[12]=

It is also regulated to an external disturbance on its angular position:

In[13]:=

sr2=StateResponsecsys,0,UnitBox

-1,{t,0,100};Plot[Evaluate@%〚{1,3}〛,{t,0,10},PlotRangeAll,PlotLegends{z,θ}]

Out[13]=

The control effort:

In[14]:=

Table[Plot[-["FeedbackGains"].sr,{t,0,10},PlotRangeAll],{sr,{sr1,sr2}}]

Out[14]=

External Links

Prof. Vladimir Dobrushkin, MATHEMATICA TUTORIAL for the Second Course. Part III: Spring Pendulum

Related Symbols

StateFeedbackGains
AffineStateSpaceModel
StateSpaceModel
OutputResponse
StateResponse

Publisher Information

Contributed by: Wolfram Controls Team