# DiscreteLQRegulatorGains

DiscreteLQRegulatorGains[ssm,{q,r},τ]

gives the optimal discrete-time state feedback gain matrix with sampling period τ for the continuous-time StateSpaceModel ssm and the quadratic cost function, with state and control weighting matrices q and r.

DiscreteLQRegulatorGains[ssm,{q,r,p},τ]

includes the state-control cross-coupling matrix p in the cost function.

DiscreteLQRegulatorGains[{ssm,finputs},{},τ]

specifies the feedback inputs of ssm.

# Details and Options • The standard state-space model ssm can be given as StateSpaceModel[{a,b,}], where a and b represent the state and input matrices in the continuous-time system .
• The descriptor continuous-time state-space model ssm defined by can be given as StateSpaceModel[{a,b,c,d,e}].
• The argument finputs is a list of integers specifying the positions of the feedback inputs in .
• DiscreteLQRegulatorGains[ssm,{},τ] is equivalent to DiscreteLQRegulatorGains[{ssm,All},{},τ].
• The cost function is given by .
• In DiscreteLQRegulatorGains[ssm,{q,r},τ], the cross-coupling matrix is assumed to be zero.
• DiscreteLQRegulatorGains computes the regulator gains based on the emulated system with cost function .
• The matrix is the submatrix of associated with the feedback inputs .
• The emulated closed-loop system with the computed feedback gain matrix k can be obtained from SystemsModelStateFeedbackConnect[ToDiscreteTimeModel[ssm,τ,Method->"ZeroOrderHold"],k]

# Examples

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## Basic Examples(1)

Compute a set of discrete-time regulator gains for a continuous-time system:

## Scope(3)

Compute a set of discrete LQ regulator gains for a continuous-time state-space model:

Only use the first input as the feedback input:

Compute the gains with state-control coupling in the cost function:

Use only inputs 1 and 3 for feedback:

Find the optimal gains for a descriptor state-space model:

## Applications(1)

Design an optimal controller by emulation:

The step response of the emulated system:

## Properties & Relations(1)

A continuous-time system:

Compute the weights for the emulated discrete-time system:

Compute the discrete LQ gains using the emulated system and corresponding weights:

DiscreteLQRegulatorGains directly gives the same result:

## Possible Issues(1)

It is not possible to compute an optimal regulator for a system that is not stabilizable: The second mode is not stabilizable:

Introduced in 2010
(8.0)
|
Updated in 2012
(9.0)