Control the Pitch and Depth of a Submarine

Design a controller that tracks the submarine's depth

A state-space model of the submarine's longitudinal dynamics:

In[1]:=

submarine=StateSpaceModel

abcd

labels



Out[1]=

					δ 	δ s
	-0.038006	0.89604	0	0.0014673	-0.007542	-0.022859
	0.0017105	0.091976	0	-0.0056095	0.0017323	-0.0022217
	1	0	0	-3.0867	0	0
θ	0	1	0	0	0	0
	0	0	1	0	0	0
θ	0	0	0	1	0	0

Without a controller, the submarine's pitch and depth are unstable to a disturbance in the states:

In[2]:=

OutputResponse[{submarine,{1,0,0.5,0.2}},{0,0},{t,0,20}];Plot#1,{t,0,20},

plotOpts

&/@Thread[{%,{,θ}}]

Out[2]=

Set the depth



and pitch

, which are outputs 1 and 2, as tracked outputs:

In[3]:=

sspec="InputModel"submarine,"TrackedOutputs"{1,2};

Specify a set of control weights and a sampling period:

In[4]:=

wts=DiagonalMatrix[{5,5,10,10,

}],

IdentityMatrix[2];τ=0.2;

Compute the discrete-time LQR controller:

In[5]:=

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

Out[5]=

SystemsModelControllerData

Design: discrete linear quadratic regulator

Feedback inputs count: 2



Obtain the closed-loop system:

In[6]:=

dcsys=dlqr["DiscreteTimeClosedLoopSystem"]

Out[6]=

0.919669	0.228618	-0.0240311	0.20002	-0.00147154	-0.00459448	0.	0.
0.00396224	0.897949	0.000625924	-0.0231511	0.000349398	-0.000449225	0.	0.
0.191864	-0.0364497	0.997579	-0.596769	-0.000155544	-0.000449494	0.	0.
0.000395475	0.189828	0.0000625387	0.997691	0.0000348417	-0.0000447594	0.	0.
0.	0.	0.148418	-0.115275	1.	0.	-0.148418	0.115275
0.	0.	0.120215	0.150733	0.	1.	-0.120215	-0.150733
0.	0.	1.	0.	0.	0.	0.	0.
0.	0.	0.	1.	0.	0.	0.	0.

0.2

The submarine tracks the reference depth of –10 and the pitch angle

of 0:

In[7]:=

sr=StateResponse[dcsys,Table[{-10,0},750]];

In[8]:=

ListStepPlot#1,

plotOpts

&/@Thread[{sr〚3;;4〛,{,θ}}]

Out[8]=

The controller model:

In[9]:=

cm=dlqr["DiscreteTimeControllerModel"]

Out[9]=

1.	0.	-0.148418	0.115275	0.148418	-0.115275	0.	0.	0.	0.
0.	1.	-0.120215	-0.150733	0.120215	0.150733	0.	0.	0.	0.
1.	0.	0.	0.	0.	0.	21.7631	-253.95	6.03224	-84.2471
0.	1.	0.	0.	0.	0.	8.87235	70.7993	3.29841	-16.5103

0.2

The inputs to the controller model:

In[10]:=

cinps=Join[Table[{-10,0},750],sr〚{3,4,1,2,3,4}〛];

The control effort:

In[11]:=

ListStepPlotOutputResponse[cm,cinps],

plotOpts



Out[11]=

Source Metadata

Citation: Grimble, van der Molen and Liceaga-Castro 1993, Submarine Depth and Pitch Control

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Contributed by: Wolfram Controls Team

Control the Pitch and Depth of a Submarine

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