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 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 - LQOutputRegulatorGains also accepts nonlinear systems specified by AffineStateSpaceModel and NonlinearStateSpaceModel.
- For nonlinear systems the operating values of state and input variables are taken into consideration, and the gains are computed based on the approximate Taylor linearization and returned as a vector.
- 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
.
Examples
open allclose allBasic Examples (2)
Scope (7)
Compute a set of optimal output-weighted state feedback gains of a continuous-time system:
The gains for a cost function with input-output coupling:
The gains for a discrete-time system:
The gains computed using only the first output as the measurement:
Use all outputs as measurements:
The gains using only the first input as the feedback input:
Find the optimal gains for a descriptor state-space model:
The gains for an AffineStateSpaceModel:
The closed-loop system is stabilized at the specified operating value:
Applications (2)
Properties & Relations (3)
Equivalent output regulator gains can be computed using LQRegulatorGains:
LQOutputRegulatorGains gives the same result:
Compute the LQ output regulator gains by solving the underlying Riccati equation:
LQOutputRegulatorGains gives the same result:
Compute the gains for a discrete-time system using DiscreteRiccatiSolve:
LQOutputRegulatorGains gives the same result:
Text
Wolfram Research (2010), LQOutputRegulatorGains, Wolfram Language function, https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html (updated 2014).
BibTeX
BibLaTeX
CMS
Wolfram Language. 2010. "LQOutputRegulatorGains." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html.
APA
Wolfram Language. (2010). LQOutputRegulatorGains. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html