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)

Amat:=Matrix1,2,3,5,12,0,4,1,2,3,2,5,7,4,3,2,4,3,8,7,19,14,1,4,7:

Bmat:=Matrix3,6,3,9,7,5,5,2,1,4,1,7,6,8,2:

Cmat:=Matrix3,9,5,4,6,1,0,4,5,7:

Dmat:=Matrix1,2,0,3,0,1:

sys:=StateSpaceAmat,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

p:=3,4,5+3I,53I,7:

Get the observer gain Matrix L for the desired poles

L:=StateObserver:-PolePlacementsys,p

L:=5.613879995032174.252888651567493.338518762086622.638950666309232.084444931839500.8573782145234570.7438101676114630.2976171958150205.062902321183713.86362016334721

(2)

Construct the observer object for sys with gain L

Obsv:=StateObserver:-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=20.09452863666401652.5249199552895558.05784536889088448.7199632379661751.453500531165512.65450695256908634.046668858779575.13679114519618128.5488283798926235.503767236684325.11071301004196613.7600043865555260.00728819889631537916.624670799975315.5083170927012190.529047698649410610.694291508503175.52858205479723512.46332664952095513.5461813763739230.05232712689835139531.5661208906534168.86003095252971443.5697101014709250.42275507053276b=15.37254594973465817.2277599900643467.2528886515674955.6138799950321734.25288865156749520.2553707610143050.32296247582676152.36104933369077143.33851876208661922.63895066630922860.34342042459012496.1688898636790061.85737821452345742.0844449318395030.85737821452345745.6366617550565242.48762033522292656.702382804184980.74381016761146330.297617195815020210.6537628112253522.12580464236742555.8636201633472125.0629023211837133.8636201633472127c=10000010000010000010000013954610−457d=00000000000000000000000001−200030−100

(3)

See Also

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


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