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ControlDesign[StateFeedback]

  

PolePlacement

  

calculate the state feedback gain for single-input or multiple-input systems according to desired closed-loop poles

 

Calling Sequence

Parameters

Options

Description

Examples

References

Calling Sequence

PolePlacement(Amat, Bmat, p, opts)

PolePlacement(sys, p, opts)

Parameters

sys

-

System; system object

p

-

list; list of desired closed-loop poles (real or complex). Complex poles including those containing symbolic parameters must be given in complex conjugate pairs. All symbolic parameters in the list are assumed to be real.

Amat

-

Matrix; system matrix of a state-space system

Bmat

-

Matrix or Vector; input matrix of a state-space system

opts

-

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

Options

• 

return_Kr =  true or false

  

True means return the direct gain Kr. The returned value becomes a sequence (Kc, Kr). The system must be specified with a system object sys. The default value is false.

• 

parameters = {list, set}(name = complexcons)

  

Specifies numeric values for the parameters of sys. These values override any parameters previously specified for sys. The numeric value on the right-hand side of each equation is substituted for the name on the left-hand side in the sys equations. The default is the value of sys given by DynamicSystems:-SystemOptions(parameters).

Description

• 

The PolePlacement command calculates the state feedback gain Kc for single-input or  multiple-input systems according to the desired closed-loop pole locations given by p.

• 

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 and must be controllable. If Amat and Bmat matrices are specified, the (Amat, Bmat) pair must be controllable.

• 

The pole placement is performed using the Sylvester equation method if open-loop and closed-loop poles are disjoint and there are no symbolic parameters in the A and B matrices and the p list. The closed-loop system matrix is then Ac=AB.Kc where Kc is the calculated feedback gain. The algorithm detects if any of the open-loop poles and desired closed-loop poles are at the same position (or numerically too close) or if any symbolic parameters exists and switches to the cyclic method in such cases.

• 

When the syntax PolePlacement(sys, p, return_Kr) is used, the direct gain Kr is computed as follows:

CD.Kc.1A+B.Kc.B+D (continuous time)

CD.Kc.1A+I+B.Kc.B+D (discrete time)

• 

If sys contains structural uncontrollable states, use ReduceSystem to remove them before computing the state feedback gain Kc.

Examples

withControlDesign:

withDynamicSystems:

Numeric example

• 

State-space system matrix Amat and input matrix Bmat (5 states and 3 inputs)

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:

• 

Desired poles

p11+I,0.1,3.581,1I,5.7720:

• 

Obtain the state feedback gain Kc:

KcStateFeedback:-PolePlacementAmat,Bmat,p1

Kc:=4.900102280219711.357239790329590.2988451425672843.758867224134042.980402604200630.7696630416132820.8056384138234610.03439114719697571.040288740336081.157335000007624.129175828396730.3100958145743410.5152005450315564.007540850252462.27773489827040

(1)
• 

Verify the closed-loop system matrix has the desired poles

LinearAlgebra:-EigenvaluesAmatBmat.Kc

5.77200000000035+0.I3.58099999999948+0.I0.100000000000087+0.I0.999999999999930+1.00000000000014I0.9999999999999301.00000000000014I

(2)

Symbolic example

• 

System object sys with symbolic entries (3 states, 3 inputs, 3 outputs)

sysStateSpaceMatrix1,2,3,x,4,1,2,5,7,Matrix3,6,9,7,5,2,Matrix1,0,0,0,1,0,0,0,1,Matrix3,2:

PrintSystemsys

State Spacecontinuous3 output(s); 2 input(s); 3 state(s)inputvariable=u1t,u2toutputvariable=y1t,y2t,y3tstatevariable=x1t,x2t,x3ta=123x4−1257b=369752c=100010001d=000000

(3)
• 

Desired symbolic poles

p24,b+cI,bcI:

• 

Obtain the state feedback gain Kc and the feed-forward gain Kr:

Kc,KrStateFeedback:-PolePlacementsys,p2,'return_Kr'=true,'parameters'=x=0:

evalfKc

0.01105243808b20.01105243808c20.07399413568b0.31477903520.01492746525b2+0.01492746525c20.05451317909b0.87747541210.01297231488b20.01297231488c20.2777819673b1.0559715000.03282239188b2+0.03282239188c2+0.2197401605b+0.93479834680.04433004833b20.04433004833c2+0.1618876227b+2.6058360720.03852384419b2+0.03852384419c2+0.8249282666b+3.135915362

(4)

evalfKr

0.008550987593b20.008550987593c20.001731055306b+0.048859583250.01803167490b2+0.01803167490c2+0.03516269174b0.99247809050.003750099818b20.003750099818c20.01136627340b+0.32081665980.02539384194b2+0.02539384194c2+0.005140709696b0.074344640630.05354861031b20.05354861031c20.1044225391b+1.5101526060.01113666007b2+0.01113666007c2+0.03375438766b0.4881539649

(5)
• 

Verify the closed-loop system matrix has the desired poles

sys:-asys:-ax=0|sys:-ax=0:

eigenLinearAlgebra:-Eigenvaluessys:-asys:-b.Kc

eigen:=4b+IcbIc

(6)

Use ReduceSystem

sys1StateSpaceMatrix4,0,3,5,3,0,0,3,1,0,1,1,0,0,0,0,Matrix5,1,1,1,3,1,0,2,6,0,0,z,Matrix1,0,3,5,3,0,5,7,Matrix1,0,0,0,1,1:

PrintSystemsys1

State Spacecontinuous2 output(s); 3 input(s); 4 state(s)inputvariable=u1t,u2t,u3toutputvariable=y1t,y2tstatevariable=x1t,x2t,x3t,x4ta=40−35300310−110000b=51−113−102600zc=1035−3057d=100011

(7)
• 

Get a subsystem considering the first two inputs.

subsys1Subsystemsys1,1,2:

PrintSystemsubsys1

State Spacecontinuous2 output(s); 2 input(s); 4 state(s)inputvariable=u1t,u2toutputvariable=y1t,y2tstatevariable=x1t,x2t,x3t,x4ta=40−35300310−110000b=51130200c=1035−3057d=1001

(8)
• 

Desired poles

p31+I3,1I3,7+I9,7I9

p3:=1+3I,13I,7+9I,79I

(9)
• 

Get the state-feedback gain for subsys1

KcStateFeedback:-PolePlacementsubsys1,p3

Error, (in ControlDesign:-StateFeedback:-PolePlacement) the given state-space realization is not controllable

• 

Remove the uncontrollable states if possible. ReduceSystem will remove the structural uncontrollable states.

rsubsysReduceSystemsubsys1,'reducedtype'=controllable:

PrintSystemrsubsys

State Spacecontinuous2 output(s); 2 input(s); 3 state(s)inputvariable=u1t,u2toutputvariable=y1t,y2tstatevariable=x1t,x2t,x3ta=40−330010−1b=511302c=103−305d=1001

(10)
• 

Verify the resulting subsystem is controllable

Controllablersubsys

true

(11)
• 

Desired poles for the controllable subsystem

p41,7+I9,7I9:

KcStateFeedback:-PolePlacementrsubsys,p4

Kc:=28.968527091212226.702047070317239.757667399160922.684716946284020.610371646918730.4970951956363

(12)

References

  

[1] T. Kailath, Linear Systems, Prentice-Hall, 1980.

  

[2] C. T. Chen, Linear System Theory and Design, 3rd Ed., Oxford University Press, 1999.

See Also

ControlDesign

ControlDesign[StateFeedback][Ackermann]

LinearAlgebra[SylvesterSolve]

 


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