Control a 2-DOF Helicopter Model

Set numerical values for the axes' moments of inertia, damping coefficients and stiffness as well as the pitch and yaw rotor thrust gains and axes' coupling coefficients:

In[1]:=

parameters=



p

0.0219,



y

0.022,



p

0.00711,



y

0.022,



sp

0.0375,



pp

0.0011,



py

0.0021,



sy

0.00005,



yy

0.0022,



yp

0.0027;

Define the differential forced equations that govern the pitch (θ) and yaw (ψ) angles in terms of the pitch torque

(

τ

p

)

and yaw torque

(

τ

y

)

:

In[2]:=

diffEqns=



p

θ''[t]+



p

θ'[t]+



sp

θ[t]



p

[t],



y

ψ''[t]+



y

ψ'[t]+



sy

ψ[t]



y

[t];

Replace the pitch and yaw torques with their corresponding coupled components; the thrust gains from the input voltages

(

V

p,y

)

and the disturbance inputs



δ

p,y

):

In[3]:=

motors=



p

[t]



pp

V

p

[t]+



py

V

y

[t]-



p

δ

p

[t]+



py



y

δ

y

[t],



y

[t]



yp

V

p

[t]+



yy

V

y

[t]-



yp



p

δ

p

[t]+



y

δ

y

[t];

Replace the symbolic parameters in the differential equation with their corresponding numerical values:

In[4]:=

numEqns=diffEqns/.motors/.parameters

Out[4]=

{0.0375θ[t]+0.00711

′

θ

[t]+0.0219

′′

θ

[t]0.0011

V

p

[t]+0.0021

V

y

[t]-0.0219

δ

p

[t]+0.0000462

δ

y

[t],0.00005ψ[t]+0.022

′

ψ

[t]+0.022

′′

ψ

[t]0.0027

V

p

[t]+0.0022

V

y

[t]-0.00005913

δ

p

[t]+0.022

δ

y

[t]}

Obtain a state space representation of the model:

In[5]:=

ssm=StateSpaceModelnumEqns,{θ[t],ψ[t],θ'[t],ψ'[t]},{

V

p

[t],

V

y

[t],

δ

p

[t],

δ

y

[t]},{θ[t],ψ[t]},t,

labels



Out[5]=

					V p	V y	δ p	δ y
θ	0.	0.	1.	0.	0.	0.	0.	0.
ψ	0.	0.	0.	1.	0.	0.	0.	0.
′ θ	-1.71233	0.	-0.324658	0.	0.0502283	0.0958904	-1.	0.00210959
′ ψ	0.	-0.00227273	0.	-1.	0.122727	0.1	-0.00268773	1.
θ	1	0	0	0	0	0	0	0
ψ	0	1	0	0	0	0	0	0

Discretize the state space model with a sampling period of

0.1

:

In[6]:=

dssm=ToDiscreteTimeModel[ssm,τ=0.1]

Out[6]=

					V p	V y	δ p	δ y
θ	0.99161	0.	0.0979899	0.	0.00246093	0.00469814	-0.0489949	0.000103359
ψ	0.	0.999989	0.	0.0952376	0.00584412	0.00476188	-0.000127986	0.0476188
′ θ	-0.167791	0.	0.959797	0.	0.0492187	0.0939629	-0.979899	0.00206718
′ ψ	0.	-0.000216449	0.	0.904752	0.116882	0.0952376	-0.00255973	0.952376
θ	0.0995805	0.	0.00489949	0.	0.000123047	0.000234907	-0.00244975	5.16796× -6 10
ψ	0.	0.0999995	0.	0.00476188	0.000292206	0.000238094	-6.39932× -6 10	0.00238094

0.1

Assess the stability of the system by computing it's eigenvalues:

In[7]:=

eigs=Eigenvalues[First@Normal@dssm]

Out[7]=

{0.999772+0.,0.975704+0.127235,0.975704-0.127235,0.904969+0.}

Simulate the open-loop response of the system to a disturbance-step input on the pitch axis:

In[8]:=

OutputResponse[dssm,{0,0,UnitStep[t],0},{t,100}];ListStepPlot#1,

plotOpts

&/@Thread[{%,{θ,ψ}}]

Out[8]=



,



Repeat the simulation for the input to the yaw axis:

In[9]:=

OutputResponse[dssm,{0,UnitStep[t]},{t,10}];ListStepPlot#1,

plotOpts

&/@Thread[{%,{θ,ψ}}]

Out[9]=



,



Set the system specifications by designating inputs 1 and 2 as feedback inputs:

In[10]:=

sspec="InputModel"dssm,"FeedbackInputs"{1,2};

Specify a set of process and noise covariance matrices

cvs

:

In[11]:=

cvs=

10	0
0	10

,

0.025	0
0	0.025

;

Using Bryson's method, specify a set of state weight and control cost matrices

wts

for the weight specification

wtspec

:

In[12]:=

wtspec=

θ

max



π

18

,

ϕ

max



7π

36

,

θ

vmax



π

9

,

ϕ

vmax



7π

180

,

v

max

24;

In[13]:=

wts=

1 2 θ max	0	0	0
0	1 0.05 2 ϕ max	0	0
0	0	1 2 θ vmax	0
0	0	0	1 2 ϕ vmax

,

1 2 v max	0
0	1 2 v max

/.wtspec;

Compute the LQG controller:

In[14]:=

lqg=LQGRegulator[sspec,cvs,wts,"Data"]

Out[14]=

SystemsModelControllerData

Design: linear quadratic gaussian regulator

»

Measured outputs count: 2



Obtain the closed-loop system:

In[15]:=

csys=lqg["ClosedLoopSystem"]//Chop

Out[15]=

δ

p

δ

y

◼

θ

◼

ψ

◼

0.1

Generate a pair of noisy signals

u

w

to simulate the process noises for the pitch and yaw:

In[16]:=

n=100;

u

w

=RandomVariateNormalDistribution0,

2

-5

10

,{2,n+1};ListStepPlot[%,DataRange{0,n-1},PlotRangeAll,PlotLegends{"

u

w

θ","

u

w

ψ"}]

Out[16]=

	u w θ
	u w ψ

And another pair for the pitch and yaw position sensors' noise

y

v

:

In[17]:=

y

v

=RandomVariateNormalDistribution0,

5

-5

10

,{2,n+1};ListStepPlot[%,DataRange{0,n-1},PlotRangeAll,PlotLegends{"

y

v

θ","

y

v

ψ"}]

Out[17]=

	y v θ
	y v ψ

Simulate the response of the closed-loop system, first for the pitch angle:

In[18]:=

orclθ=OutputResponse[csys,Join[

u

w

,ConstantArray[#,n+1]&/@{0,1},

y

v

]];ListStepPlot[%,DataRange{0,99τ},PlotRangeAll,PlotLegends{θ,ψ}]

Out[18]=

	θ
	ψ

Simulate it for the yaw angle:

Obtain the controller model to assess the control effort:

Related Symbols

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