Application Areas

Build Regulators and Observers for Systems

The trajectories of the states and a Luenberger observer's state estimates.

In[1]:=

{a, b, c} = {( {

{0.81, -0.23, -0.045},

{0.09, 0.98, -0.0023},

{0.005, 0.1, 1}

} ), ( {

{0.09},

{0.0047},

{0.00016}

} ), ( {

{1, 3.5, 3}

} )}; ss = StateSpaceModel[{a, b, c}, SamplingPeriod -> 0.1];

l = EstimatorGains[ss, {0.5, 0.6, 0.75}];

ssobsv = StateSpaceModel[{a - l.c, l, IdentityMatrix[3]},

SamplingPeriod -> 0.1];

stateResp = StateResponse[{ss, {1, -0.75, 0.5}}, {0}, {t, 6}];

y = OutputResponse[{ss, {1, -0.75, 0.5}}, {0}, {t, 6}];

obsvResp = OutputResponse[ssobsv, y];

tspan = Range[0, 6, 0.1]; ListPlot[

Replace[Join[stateResp, obsvResp],

pts : {__} :> Thread[{tspan, pts}], {1}], PlotRange -> All,

PlotMarkers -> {"*", "o", "@", "+", "#", "x"},

Epilog ->

Inset[Framed[

Style[Column[{"*----\!\(\*SubscriptBox[\(x\), \(1\)]\)",

"+---\!\(\*SubscriptBox[\(x\), \(1  obs\)]\)",

"o----\!\(\*SubscriptBox[\(x\), \(2\)]\)",

"#----\!\(\*SubscriptBox[\(x\), \(2  obs\)]\)",

"@----\!\(\*SubscriptBox[\(x\), \(3\)]\)",

"x---\!\(\*SubscriptBox[\(x\), \(3  obs\)]\)"}], 10]], {5,

0.6}], AxesLabel -> {"t"}, ImageSize -> 350, Joined -> True]

Out[1]=