BUILT-IN MATHEMATICA SYMBOL

# LQOutputRegulatorGains

LQOutputRegulatorGains[ssm, {q, r}]
gives the optimal state feedback gain matrix for the StateSpaceModel ssm and the quadratic cost function, with output and control weighting matrices q and r.

LQOutputRegulatorGains[ssm, {q, r, p}]
includes the output-control cross-coupling matrix p in the cost function.

LQOutputRegulatorGains[{ssm, sensors}, {...}]
specifies sensors as the measured outputs of ssm.

LQOutputRegulatorGains[{ssm, sensors, finputs}, {...}]
specifies finputs as the feedback inputs of ssm.

## Details and OptionsDetails and Options

• The standard state-space model ssm can be given as StateSpaceModel[{a, b, c, d}], where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
•  continuous-time system discrete-time system
• The descriptor state-space model ssm can be given as StateSpaceModel[{a, b, c, d, e}] in either continuous time or discrete time:
•  continuous-time system discrete-time system
• The argument sensors is a list of integers specifying the positions of the measured outputs in .
• Similarly, finputs is a list of integers specifying the positions of the feedback inputs in .
• LQOutputRegulatorGains[ssm, {...}] is equivalent to LQOutputRegulatorGains[{ssm, All, All}, {...}].
• The cost function is:
•  continuous-time system discrete-time system
• In LQOutputRegulatorGains[ssm, {q, r}], the cross-coupling matrix p is assumed to be zero.
• The optimal control is given by , where is the computed feedback gain matrix.
• For continuous-time systems, the optimal feedback gain is computed as , where and . The matrix is the solution of the continuous Riccati equation .
• For discrete-time systems, the optimal feedback gain is computed as , where and . The matrix is the solution of the discrete Riccati equation .
• The subscript denotes the submatrix associated with the feedback inputs , and the subscript denotes the submatrix associated with the sensors .

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

A set of optimal output-weighted state feedback gains for a continuous-time system:

 Out[1]//MatrixForm=

LQ output regulator gains for a discrete-time system:

 Out[1]=