construct the static gain (Luenberger) observer for a given system and observer gain - MapleSim Help

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ControlDesign[StateObserver][Observer] - construct the static gain (Luenberger) observer for a given system and observer gain

Calling Sequence

Observer(sys, L, opts)

Parameters

sys

-

System; system object;

L

-

Matrix or Vector; observer gain

opts

-

(optional) equation(s) of the form option = value; specify options for the Observer command

Description

• 

The Observer command constructs the static gain (Luenberger) observer for the given system sys and observer gain L.

• 

The system sys is a continuous or discrete-time linear system object created using the DynamicSystems package. The system object must be in state-space (SS) form.

• 

All sys inputs are assumed known (deterministic), and all sys outputs are assumed to be measured.

• 

The Observer command returns the observer object in the state-space form. The observer system object gets the input(s) and output(s) of sys as inputs and generates the estimates of the state(s) and output(s) of sys as its outputs.

• 

Suppose the sys state-space realization is given as

x=Ax+Buy=Cx+Du

The observer equations are given by

xˆ=ALCx^+BLDLuy

x^y^=ICx^+00D0uy

• 

In the discrete-time domain, the Observer command uses the same state-space matrices as in the continuous-time case to generate estimates for state(s) xn and output(s) yn as x^n|n1 and y^n|n1, respectively, based on the past measurements up to yn-1. The sampling time of the generated observer object is the same as the sampling time of the system sys.

Examples

withControlDesign:

withDynamicSystems:

State-space system in continuous time (5 states, 3 inputs and 2 outputs)

AmatMatrix1,2,3,5,12,0,4,1,2,3,2,5,7,4,3,2,4,3,8,7,19,14,1,4,7:

BmatMatrix3,6,3,9,7,5,5,2,1,4,1,7,6,8,2:

CmatMatrix3,9,5,4,6,1,0,4,5,7:

DmatMatrix1,2,0,3,0,1:

sysStateSpaceAmat,Bmat,Cmat,Dmat:

PrintSystemsys

State Spacecontinuous2 output(s); 3 input(s); 5 state(s)inputvariable=u1t,u2t,u3toutputvariable=y1t,y2tstatevariable=x1t,x2t,x3t,x4t,x5ta=1−23−51204−12−3257−43−2438719141−47b=3639755214−176−82c=3954610−457d=1−2030−1

(1)

Desired observer poles

p3,4,5+3I,53I,7:

Get the observer gain Matrix L for the desired poles

LStateObserver:-PolePlacementsys,p

L:=5.613879995032444.252888651567703.338518762086812.638950666309372.084444931839580.8573782145235160.7438101676115210.2976171958150695.062902321183923.86362016334737

(2)

Construct the observer object for sys with gain L

ObsvStateObserver:-Observersys,L:

PrintSystemObsv

State Spacecontinuous7 output(s); 5 input(s); 5 state(s)inputvariable=u1t,u2t,u3t,y1t,y2toutputvariable=x1_obs_outt,x2_obs_outt,x3_obs_outt,x4_obs_outt,x5_obs_outt,y1t,y2tstatevariable=x1_obst,x2_obst,x3_obst,x4_obst,x5_obsta=20.0945286366650252.524919955291958.05784536889137548.7199632379682751.45350053116856512.6545069525697934.046668858781255.1367911451965428.54882837989408635.503767236686445.1107130100422513.76000438655620.00728819889617593516.62467079997589515.5083170927020820.529047698649632610.6942915085036895.52858205479732912.4633266495214313.5461813763746120.05232712689912588631.566120890655278.86003095253011243.5697101014725350.4227550705351b=15.37254594973555317.2277599900648757.2528886515677045.61387999503243854.25288865156770420.2553707610149230.32296247582638852.36104933369062753.33851876208680582.63895066630937250.343420424589873546.1688898636791561.85737821452351632.0844449318395780.85737821452351635.6366617550567282.4876203352230426.7023828041849310.74381016761152110.2976171958150692410.653762811226032.12580464236783765.863620163347375.0629023211839193.863620163347371c=10000010000010000010000013954610−457d=00000000000000000000000001−200030−100

(3)

See Also

ControlDesign, ControlDesign[StateFeedback][PolePlacement], ControlDesign[StateObserver][Ackermann], ControlDesign[StateObserver][PolePlacement]


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