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ControlDesign

  

LQRContinuous

  

design continuous-time linear quadratic state feedback regulator (LQR) for a given pair

 

Calling Sequence

Parameters

Options

Solvability Conditions

Description

Examples

Calling Sequence

LQRContinuous(A, B, Q, R, opts)

Parameters

A

-

Matrix; system matrix of a state-space system

B

-

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

Q

-

Matrix; state weighting matrix

R

-

Matrix; input weighting matrix

opts

-

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

Options

• 

N = Matrix or Vector

Weighting on the state-input multiplication term in the cost function. If omitted, a zero matrix with appropriate dimensions will be considered.

• 

poles   = true or false True means the eigenvalues of A-BK are returned. See LQR for details. The default value is false.    

• 

riccati  =  true or false True means the solution of the associated Riccati equation is returned. The infinite horizon solution of the following continuous-time Riccati equation (CARE) is returned.

ATX+XAXB+NR1XB+NT+Q=0

The LQR feedback gain is calculated as

K=R1XB+NT

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).

Solvability Conditions

• 

The pair A,B must be stabilizable.

• 

The pair QNR1NT,ABR1NT must have no unobservable modes on the imaginary axis.

• 

R>0 (positive definite) and QNR1NT0 (positive semidefinite).

Description

• 

The LQRContinuous command calculates the LQR state feedback gain for a given pair.

• 

The optimal state feedback gain, K, is calculated such that the quadratic cost function

Ju=0xTQx+uTRu+2xTΝuⅆt

is minimized by the feedback law u = -Kx subject to the system dynamics

x·=Ax+Bu

• 

Q and R are expected to be symmetric. If the input Q and/or R are not symmetric, their symmetric part will be considered since their antisymmetric (skew-symmetric) part has no role in the quadratic cost function.

• 

In addition to the state feedback gain, depending on the corresponding option values, the command also returns the closed-loop eigenvalues and the solution of the associated Riccati equation.

• 

Note that the time domain is inherent to the command regardless of the any time domain implications the input pair might have. To be more specific, this command calculates a continuous-time LQR feedback gain for the given input pair even if that pair is initially extracted from a discrete-time state-space.  If the gain is then used for feedback stabilization in discrete time domain, obviously such a mistake in command usage leads to a result which is not theoretically valid.     

Examples

withControlDesign:

withDynamicSystems:

csysNewSystemMatrix2s2+3s+1,1s+2,s1s2+5,7s+1s+4

csys:=Transfer Functioncontinuous2 output(s); 2 input(s)inputvariable=u1s,u2soutputvariable=y1s,y2s

(1)

sysStateSpacecsys

sys:=State Spacecontinuous2 output(s); 2 input(s); 7 state(s)inputvariable=u1t,u2toutputvariable=y1t,y2tstatevariable=x1t,x2t,x3t,x4t,x5t,x6t,x7t

(2)

sys:-a;sys:-b;sys:-c;sys:-d

010000000100000001000515630000000010000000100008147

00000010000001

1002045112211470

0000

(3)

Q1LinearAlgebra:-IdentityMatrix73;R2LinearAlgebra:-IdentityMatrix2

Q:=13000000013000000013000000013000000013000000013000000013

R:=2002

(4)

KLQRContinuoussys:-a,sys:-b,Q,R

K:=0.01663898109747050.2994436759550440.8677764521914780.3018509310763290.0.0.0.0.0.0.0.01040989379861050.01954395072894290.0146813589873455

(5)

KprLQRContinuoussys:-a,sys:-b,Q,R,poles=true,riccati=true:Kpr1;Kpr2;Kpr3

0.01663898109747050.2994436759550440.8677764521914780.3018509310763290.0.0.0.0.0.0.0.01040989379861050.01954395072894290.0146813589873455

2.62756665299744+0.I0.145624709217511+2.22784297750641I0.1456247092175112.22784297750641I0.383034859643869+0.I1.00185981437603+0.I1.98515367555330+0.I4.02766786905801+0.I

3.50357106780249636427356983347380869701730928710788.90631012705411115311675810756718031049477924614583.12838818122098082686058820294514592760627824361050.0332779621949409374206603236526689213529915396260090.0.0.8.906310127054111153116758107567180310494779246145827.01792983748675985061180093742507824269810008618110.9996867306090944849060693759024372654302363650350.598887351910087243343592782953780547783773573792700.0.0.3.128388181220980826860588202945145927606278243610510.9996867306090944849060693759024372654302363650358.75386079427495244526889789195148694929582021351381.73555290438295551200857175359974385723535498487790.0.0.0.0332779621949409374206603236526689213529915396260090.598887351910087243343592782953780547783773573792701.73555290438295551200857175359974385723535498487790.603701862152658441644654693320879940510941682759390.0.0.0.0.0.0.0.604587138927168509676898420793792313225217218130870.380945919753830599663964618798960532181765706634190.0208197875972210396136451371378709701895903592627870.0.0.0.0.380945919753830599663964618798960532181765706634190.664447437767019541498152361507087810149285907951570.0390879014578858512583266659427122986389872752131640.0.0.0.0.0208197875972210396136451371378709701895903592627870.0390879014578858512583266659427122986389872752131640.029362717974691029572229097828269653238134822396934

(6)

See Also

ControlDesign

ControlDesign[LQR]

ControlDesign[LQRDiscrete]

ControlDesign[LQROutput]

ControlDesign[StateFeedback][Ackermann]

ControlDesign[StateFeedback][PolePlacement]

 


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