Package: TuningGoal
Frequencyweighted H_{2} norm constraint for control system tuning
Use TuningGoal.WeightedVariance
to limit the
weighted H_{2} norm of the
transfer function from specified inputs to outputs. The H_{2} norm
measures:
The total energy of the impulse response, for deterministic inputs to the transfer function.
The square root of the output variance for a unitvariance whitenoise input, for stochastic inputs to the transfer function. Equivalently, the H_{2} norm measures the rootmeansquare of the output for such input.
You can use TuningGoal.WeightedVariance
for
control system tuning with tuning commands, such as systune
or looptune
.
By specifying this tuning goal, you can tune the system response to
stochastic inputs with a nonuniform spectrum such as colored noise
or wind gusts. You can also use TuningGoal.WeightedVariance
to
specify LQGlike performance objectives.
After you create a tuning goal object, you can configure it further by setting Properties of the object.
creates
a tuning goal Req
=
TuningGoal.Variance(inputname
,outputname
,WL,WR
)Req
. This tuning goal specifies that
the closedloop transfer function H(s)
from the specified input to output meets the requirement:
W_{L}(s)H(s)W_{R}(s)_{2} < 1.
When you are tuning a discretetime system, Req
imposes
the following constraint:
$$\frac{1}{\sqrt{{T}_{s}}}{\Vert {W}_{L}\left(z\right)T\left(z,x\right){W}_{R}\left(z\right)\Vert}_{2}<1.$$
The H_{2} norm is scaled by the square root of the sample time T_{s} to ensure consistent results with tuning in continuous time. To constrain the true discretetime H_{2} norm, multiply either W_{L} or W_{R} by $$\sqrt{{T}_{s}}$$.

Input signals for the tuning goal, specified as a character vector or, for multipleinput tuning goals, a cell array of character vectors.
For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design. 

Output signals for the tuning goal, specified as a character vector or, for multipleoutput tuning goals, a cell array of character vectors.
For more information about analysis points in control system models, see Mark Signals of Interest for Control System Analysis and Design. 

Frequencyweighting functions, specified as scalars, matrices, or SISO or MIMO numeric LTI models. The functions W_{L}(s)H(s)W_{R}(s)_{2} < 1. WL provides the weighting for
the output channels of H(s),
and WR provides the weighting for the input channels.
You can specify scalar weights or frequencydependent weighting. To
specify a frequencydependent weighting, use a numeric LTI model.
For example:
WL = tf(1,[1 0.01]); WR = 10; If you specify MIMO weighting functions, then If you are tuning in discrete time (that is, using a A value of 

Frequencyweighting function for the output channels of the
transfer function to constrain, specified as a scalar, a matrix, or
a SISO or MIMO numeric LTI model. The initial value of this property
is set by the 

Frequencyweighting function for the input channels of the transfer
function to constrain, specified as a scalar, a matrix, or a SISO
or MIMO numeric LTI model. The initial value of this property is set
by the 

Input signal names, specified as a cell array of character
vectors that identify the inputs of the transfer function that the
tuning goal constrains. The initial value of the 

Output signal names, specified as a cell array of character
vectors that identify the outputs of the transfer function that the
tuning goal constrains. The initial value of the 

Models to which the tuning goal applies, specified as a vector of indices. Use the Req.Models = 2:4; When Default: 

Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loopopening locations. The tuning goal is evaluated against the openloop configuration created by opening feedback loops at the locations you identify. If you are using the tuning goal to tune a Simulink model
of a control system, then If you are using the tuning goal to tune a generalized statespace
( For example, if Default: 

Name of the tuning goal, specified as a character vector. For example, if Req.Name = 'LoopReq'; Default: 
When you use this tuning goal to tune a continuoustime
control system, systune
attempts to enforce zero
feedthrough (D = 0) on the transfer that the tuning
goal constrains. Zero feedthrough is imposed because the H_{2} norm,
and therefore the value of the tuning goal (see Algorithms), is infinite for continuoustime
systems with nonzero feedthrough.
systune
enforces zero feedthrough by fixing
to zero all tunable parameters that contribute to the feedthrough
term. systune
returns an error when fixing these
tunable parameters is insufficient to enforce zero feedthrough. In
such cases, you must modify the tuning goal or the control structure,
or manually fix some tunable parameters of your system to values that
eliminate the feedthrough term.
When the constrained transfer function has several tunable blocks in series, the software’s approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice.
To fix parameters of tunable blocks to specified values, use
the Value
and Free
properties
of the block parametrization. For example, consider a tuned statespace
block:
C = tunableSS('C',1,2,3);
To enforce zero feedthrough on this block, set its D matrix value to zero, and fix the parameter.
C.D.Value = 0; C.D.Free = false;
For more information on fixing parameter values, see the Control
Design Block reference pages, such as tunableSS
.
This tuning goal imposes an implicit stability constraint
on the weighted closedloop transfer function from Input
to Output
,
evaluated with loops opened at the points identified in Openings
.
The dynamics affected by this implicit constraint are the stabilized
dynamics for this tuning goal. The MinDecay
and MaxRadius
options
of systuneOptions
control the bounds on these
implicitly constrained dynamics. If the optimization fails to meet
the default bounds, or if the default bounds conflict with other requirements,
use systuneOptions
to change
these defaults.
When you tune a control system using a TuningGoal
,
the software converts the tuning goal into a normalized scalar value f(x). x is
the vector of free (tunable) parameters in the control system. The
software then adjusts the parameter values to minimize f(x)
or to drive f(x) below 1 if
the tuning goal is a hard constraint.
For TuningGoal.WeightedVariance
, f(x)
is given by:
$$f\left(x\right)={\Vert {W}_{L}T\left(s,x\right){W}_{R}\Vert}_{2}.$$
T(s,x)
is the closedloop transfer function from Input
to Output
. $${\Vert \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\Vert}_{2}$$ denotes
the H_{2} norm (see norm
).
For tuning discretetime control systems, f(x) is given by:
$$f\left(x\right)=\frac{1}{\sqrt{{T}_{s}}}{\Vert {W}_{L}\left(z\right)T\left(z,x\right){W}_{R}\left(z\right)\Vert}_{2}.$$
T_{s} is the sample time of the discretetime transfer function T(z,x).
systune
 looptune
 systune
(for slTuner)
(Simulink Control Design)  looptune (for slTuner)
(Simulink Control Design)  TuningGoal.Gain
 TuningGoal.LoopShape
 slTuner
(Simulink Control Design)  norm
 TuningGoal.Variance