getCompSensitivity
Complementary sensitivity function at specified point
using slLinearizer
or slTuner
interface
Syntax
Description
returns the complementary sensitivity
function at the specified analysis point for the model associated
with the linsys
= getCompSensitivity(s
,pt
)slLinearizer
or slTuner
interface, s
.
The software enforces all the permanent openings
specified for s
when it calculates
linsys
. If you configured either
s.Parameters
, or s.OperatingPoints
, or
both, getCompSensitivity
performs multiple linearizations
and returns an array of complementary sensitivity functions.
considers additional, temporary, openings at the point specified by
linsys
= getCompSensitivity(s
,pt
,temp_opening
)temp_opening
. Use an opening, for example, to calculate
the complementary sensitivity function of an inner loop with the outer loop
open.
returns a subset of the batch linearization results.
linsys
= getCompSensitivity(___,mdl_index
)mdl_index
specifies the index of the linearizations of
interest, in addition to any of the input arguments in previous syntaxes.
Use this syntax for efficient linearization, when you want to obtain the complementary sensitivity function for only a subset of the batch linearization results.
Examples
Obtain Complementary Sensitivity Function at Analysis Point
Obtain the complementary sensitivity function, calculated at the plant output, for the ex_scd_simple_fdbk
model.
Open the ex_scd_simple_fdbk
model.
mdl = 'ex_scd_simple_fdbk';
open_system(mdl);
In this model:
Create an slLinearizer
interface for the model.
sllin = slLinearizer(mdl);
To calculate the complementary sensitivity function at the plant output, use the y
signal as the analysis point. Add this point to sllin
.
addPoint(sllin,'y');
Obtain the complementary sensitivity function at y
.
sys = getCompSensitivity(sllin,'y');
tf(sys)
ans = From input "y" to output "y": 3  s + 8 Continuoustime transfer function.
The software adds a linearization output at y
, followed by a linearization input, dy
.
sys
is the transfer function from dy
to y
, which is equal to .
Specify Temporary Loop Opening for Complementary Sensitivity Function Calculation
For the scdcascade
model, obtain the complementary sensitivity function for the innerloop at y2
.
Open the scdcascade
model.
mdl = 'scdcascade';
open_system(mdl)
Create an slLinearizer
interface for the model.
sllin = slLinearizer(mdl);
To calculate the complementary sensitivity transfer function for the inner loop at y2
, use the y2
signal as the analysis point. To eliminate the effects of the outer loop, break the outer loop at y1m
. Add both these points to sllin
.
addPoint(sllin,{'y2','y1m'});
Obtain the complementary sensitivity function for the inner loop at y2
.
sys = getCompSensitivity(sllin,'y2','y1m');
Here, 'y1m'
, the third input argument, specifies a temporary opening for the outer loop.
Obtain Complementary Sensitivity Function for Specific Parameter Combination
Suppose you batch linearize the scdcascade
model for multiple transfer functions. For most linearizations, you vary the proportional (Kp2
) and integral gain (Ki2
) of the C2
controller in the 10% range. For this example, calculate the complementary sensitivity function for the inner loop for the maximum value of Kp2
and Ki2
.
Open the scdcascade
model.
mdl = 'scdcascade';
open_system(mdl);
Create an slLinearizer
interface for the model.
sllin = slLinearizer(mdl);
Vary the proportional (Kp2
) and integral gain (Ki2
) of the C2
controller in the 10% range.
Kp2_range = linspace(0.9*Kp2,1.1*Kp2,3); Ki2_range = linspace(0.9*Ki2,1.1*Ki2,5); [Kp2_grid,Ki2_grid]=ndgrid(Kp2_range,Ki2_range); params(1).Name = 'Kp2'; params(1).Value = Kp2_grid; params(2).Name = 'Ki2'; params(2).Value = Ki2_grid; sllin.Parameters = params;
To calculate the complementary sensitivity of the inner loop, use the y2
signal as the analysis point. To eliminate the effects of the outer loop, break the outer loop at y1m
. Add both these points to sllin
.
addPoint(sllin,{'y2','y1m'})
Determine the index for the maximum values of Ki2
and Kp2
.
mdl_index = params(1).Value == max(Kp2_range) & params(2).Value == max(Ki2_range);
Obtain the complementary sensitivity transfer function at y2
.
sys = getCompSensitivity(sllin,'y2','y1m',mdl_index);
Obtain Offsets from Complementary Sensitivity Function
Open Simulink model.
mdl = 'watertank';
open_system(mdl)
Create a linearization option set, and set the StoreOffsets
option.
opt = linearizeOptions('StoreOffsets',true);
Create slLinearizer
interface.
sllin = slLinearizer(mdl,opt);
Add an analysis point at the tank output port.
addPoint(sllin,'watertank/WaterTank System');
Calculate the complementary sensitivity function at y
, and obtain the corresponding linearization offsets.
[sys,info] = getCompSensitivity(sllin,'watertank/WaterTank System');
View offsets.
info.Offsets
ans = struct with fields: x: [2x1 double] dx: [2x1 double] u: 1 y: 1 StateName: {2x1 cell} InputName: {'watertank/WaterTank System'} OutputName: {'watertank/WaterTank System'} Ts: 0
Input Arguments
s
— Interface to Simulink^{®} model
slLinearizer
interface  slTuner
interface
Interface to a Simulink model, specified as either an slLinearizer
interface or an slTuner
interface.
pt
— Analysis point signal name
character vector  string  cell array of character vectors  string array
Analysis point signal name, specified as:
Character vector or string — Analysis point signal name.
To determine the signal name associated with an analysis point, type
s
. The software displays the contents ofs
in the MATLAB^{®} command window, including the analysis point signal names, block names, and port numbers. Suppose that an analysis point does not have a signal name, but only a block name and port number. You can specifypt
as the block name. To use a point not in the list of analysis points fors
, first add the point usingaddPoint
.You can specify
pt
as a uniquely matching portion of the full signal name or block name. Suppose that the full signal name of an analysis point is'LoadTorque'
. You can specifypt
as'Torque'
as long as'Torque'
is not a portion of the signal name for any other analysis point ofs
.For example,
pt = 'y1m'
.Cell array of character vectors or string array — Specifies multiple analysis point names. For example,
pt = {'y1m','y2m'}
.
To calculate linsys
, the software adds a linearization output, followed
by a linearization input at pt
.
Consider the following model:
Specify pt
as 'y'
:
The software computes linsys
as the transfer function from
dy
to y
.
If you specify pt
as multiple signals,
for example pt = {'u','y'}
, the software adds a
linearization output, followed by a linearization input at each point.
du
and dy
are linearization inputs, and
u
and y
are linearization outputs.
The software computes linsys
as a MIMO transfer
function with a transfer function from each linearization input to each
linearization output.
temp_opening
— Temporary opening signal name
character vector  string  cell array of character vectors  string array
Temporary opening signal name, specified as:
Character vector or string — Analysis point signal name.
temp_opening
must specify an analysis point that is in the list of analysis points fors
. To determine the signal name associated with an analysis point, types
. The software displays the contents ofs
in the MATLAB command window, including the analysis point signal names, block names, and port numbers. Suppose that an analysis point does not have a signal name, but only a block name and port number. You can specifytemp_opening
as the block name. To use a point not in the list of analysis points fors
, first add the point usingaddPoint
.You can specify
temp_opening
as a uniquely matching portion of the full signal name or block name. Suppose that the full signal name of an analysis point is'LoadTorque'
. You can specifytemp_opening
as'Torque'
as long as'Torque'
is not a portion of the signal name for any other analysis point ofs
.For example,
temp_opening = 'y1m'
.Cell array of character vectors or string array — Specifies multiple analysis point names. For example,
temp_opening = {'y1m','y2m'}
.
mdl_index
— Index for linearizations of interest
array of logical values  vector of positive integers
Index for linearizations of interest, specified as:
Array of logical values — Logical array index of linearizations of interest. Suppose that you vary two parameters,
par1
andpar2
, and want to extract the linearization for the combination ofpar1 > 0.5
andpar2 <= 5
. Use:params = s.Parameters; mdl_index = params(1).Value>0.5 & params(2).Value <= 5;
The expression
params(1).Value>0.5 & params(2).Value<5
uses logical indexing and returns a logical array. This logical array is the same size asparams(1).Value
andparams(2).Value
. Each entry contains the logical evaluation of the expression for corresponding entries inparams(1).Value
andparams(2).Value
.Vector of positive integers — Linear index of linearizations of interest. Suppose that you vary two parameters,
par1
andpar2
, and want to extract the linearization for the combination ofpar1 > 0.5
andpar2 <= 5
. Use:params = s.Parameters; mdl_index = find(params(1).Value>0.5 & params(2).Value <= 5);
The expression
params(1).Value>0.5 & params(2).Value<5
returns a logical array.find
returns the linear index of every true entry in the logical array
Output Arguments
linsys
— Complementary sensitivity function
statespace model
Complementary sensitivity function, returned as described in the following:
If you did not configure
s.Parameters
ands.OperatingPoints
, the software calculateslinsys
using the default model parameter values. The software uses the model initial conditions as the linearization operating point.linsys
is returned as a statespace model.If you configured
s.Parameters
only, the software computes a linearization for each parameter grid point.linsys
is returned as a statespace model array of the same size as the parameter grid.If you configured
s.OperatingPoints
only, the software computes a linearization for each specified operating point.linsys
is returned as a statespace model array of the same size ass.OperatingPoints
.If you configured
s.Parameters
and specifieds.OperatingPoints
as a single operating point, the software computes a linearization for each parameter grid point. The software uses the specified operating point as the linearization operating point.linsys
is returned as a statespace model array of the same size as the parameter grid.If you configured
s.Parameters
and specifieds.OperatingPoints
as multiple operating point objects, the software computes a linearization for each parameter grid point. The software requires thats.OperatingPoints
is the same size as the parameter grid specified bys.Parameters
. The software computes each linearization using corresponding operating points and parameter grid points.linsys
is returned as a statespace model array of the same size as the parameter grid.If you configured
s.Parameters
and specifieds.OperatingPoints
as multiple simulation snapshot times, the software simulates and linearizes the model for each snapshot time and parameter grid point combination. Suppose that you specify a parameter grid of sizep
andN
snapshot times.linsys
is returned as a statespace model array of sizeN
byp
.
For most models, linsys
is returned as an ss
object or an array of
ss
objects. However, if your model contains one of
the following blocks in the linearization path defined by
pt
, then linsys
returns the
specified type of statespace model.
Block  linsys Type 

Block with a substitution specified as a
genss object or tunable model
object  genss 
Block with a substitution specified as an uncertain
model, such as uss  uss (Robust Control Toolbox) 
Sparse Second Order block  mechss 
Descriptor StateSpace block configured to linearize to a sparse model  sparss 
info
— Linearization information
structure
Linearization information, returned as a structure with the following fields:
Offsets
— Linearization offsets
[]
(default)  structure  structure array
Linearization offsets, returned as []
if
s.Options.StoreOffsets
is false
.
Otherwise, Offsets
is returned as one of the
following:
If
linsys
is a single statespace model, thenOffsets
is a structure.If
linsys
is an array of statespace models, thenOffsets
is a structure array with the same dimensions aslinsys
.
Each offset structure has the following fields:
Field  Description 

x  State offsets used for linearization, returned as a column vector of length
n_{x}, where
n_{x} is the number of states in
linsys . 
y  Output offsets used for linearization, returned as a column vector of length
n_{y}, where
n_{y} is the number of outputs in
linsys . 
u  Input offsets used for linearization, returned as a column vector of length
n_{u}, where
n_{u} is the number of inputs in
linsys . 
dx  Derivative offsets for continuous time systems or updated state values for discretetime systems, returned as a column vector of length n_{x}. 
StateName  State names, returned as a cell array that contains
n_{x} elements that match the names
in linsys.StateName . 
InputName  Input names, returned as a cell array that contains
n_{u} elements that match the names
in linsys.InputName . 
OutputName  Output names, returned as a cell array that contains
n_{y} elements that match the names
in linsys.OutputName . 
Ts  Sample time of the linearized system, returned as a scalar that matches the sample time in
linsys.Ts . For continuoustime systems,
Ts is 0 . 
If Offsets
is a structure array, you can
configure an LPV System block using
the offsets. To do so, first convert them to the required format using getOffsetsForLPV
. For an example, see Approximate Nonlinear Behavior Using Array of LTI Systems.
Advisor
— Linearization diagnostic information
[]
(default)  LinearizationAdvisor
object  array of LinearizationAdvisor
objects
Linearization diagnostic information, returned as []
if
s.Options.StoreAdvisor
is false
.
Otherwise, Advisor
is returned as one of the
following:
If
linsys
is a single statespace model,Advisor
is aLinearizationAdvisor
object.If
linsys
is an array of statespace models,Advisor
is an array ofLinearizationAdvisor
objects with the same dimensions aslinsys
.
LinearizationAdvisor
objects store linearization
diagnostic information for individual linearized blocks. For an example of
troubleshooting linearization results using a
LinearizationAdvisor
object, see Troubleshoot Linearization Results at Command Line.
More About
Complementary Sensitivity Function
The complementary sensitivity function at a point is the transfer function from an additive disturbance at the point to a measurement at the same point. In contrast to the sensitivity function, the disturbance is added after the measurement.
To compute the complementary sensitivity function at an analysis
point, x
, the software adds a linearization output
at x
, followed by a linearization input, dx
.
The complementary sensitivity function is the transfer function from dx
to x
.
Analysis Point in Simulink Model  How getCompSensitivity Interprets
Analysis Point  Complementary Sensitivity Function 


 Transfer function from 
For example, consider the following model where you compute
the complementary sensitivity function at y
:
Here, the software adds a linearization output at y
,
followed by a linearization input, dy
. The complementary
sensitivity function at y
, T,
is the transfer function from dy
to y
. T is
calculated as follows:
$$\begin{array}{l}y=GK(y+dy)\\ \to y=GKyGKdy\\ \to (I+GK)y=GKdy\\ \to y={(}^{I}GKdy\\ \therefore T={(}^{I}GK\end{array}$$
Here I is an identity matrix of the same
size as GK.
The complementary sensitivity transfer function at y
is
equal to 1
times the closedloop transfer function
from r
to y
.
Generally, the complementary sensitivity function, T, computed from
reference signals to plant outputs, is equal to I–S. Here S is the sensitivity function at the
point, and I is the identity matrix of commensurate size.
However, because getCompSensitivity
adds the linearization
output and input at the same point, T, as
returned by getCompSensitivity
, is equal to S–I.
The software does not modify the Simulink model when it computes the complementary sensitivity function.
Analysis Point
Analysis points, used
by the slLinearizer
and slTuner
interfaces,
identify locations within a model that are relevant for linear analysis
and control system tuning. You use analysis points as inputs to the
linearization commands, such as getIOTransfer
, getLoopTransfer
, getSensitivity
, and getCompSensitivity
. As inputs to the
linearization commands, analysis points can specify any openloop
or closedloop transfer function in a model. You can also use analysis
points to specify design requirements when tuning control systems
using commands such as systune
.
Location refers to a specific block output port within a model or to a bus element in such an output port. For convenience, you can use the name of the signal that originates from this port to refer to an analysis point.
You can add analysis points to an slLinearizer
or slTuner
interface, s
,
when you create the interface. For example:
s = slLinearizer('scdcascade',{'u1','y1'});
Alternatively, you can use the addPoint
command.
To view all the analysis points of s
, type s
at
the command prompt to display the interface contents. For each analysis
point of s
, the display includes the block name
and port number and the name of the signal that originates at this
point. You can also programmatically obtain a list of all the analysis
points using getPoints
.
For more information about how you can use analysis points, see Mark Signals of Interest for Control System Analysis and Design and Mark Signals of Interest for Batch Linearization.
Permanent Loop Openings
Permanent openings,
used by the slLinearizer
and slTuner
interfaces,
identify locations within a model where the software breaks the signal
flow. The software enforces these openings for linearization and tuning.
Use permanent openings to isolate a specific model component. Suppose
that you have a largescale model capturing aircraft dynamics and
you want to perform linear analysis on the airframe only. You can
use permanent openings to exclude all other components of the model.
Another example is when you have cascaded loops within your model
and you want to analyze a specific loop.
Location refers to a specific block output port within a model. For convenience, you can use the name of the signal that originates from this port to refer to an opening.
You can add permanent openings to an slLinearizer
or slTuner
interface, s
,
when you create the interface or by using the addOpening
command. To remove a location
from the list of permanent openings, use the removeOpening
command.
To view all the openings of s
, type s
at
the command prompt to display the interface contents. For each permanent
opening of s
, the display includes the block name
and port number and the name of the signal that originates at this
location. You can also programmatically obtain a list of all the permanent
loop openings using getOpenings
.
Version History
Introduced in R2013bR2020b: Linearize Simulink model to a sparse statespace model
You can linearize and obtain a sparse model from a Simulink model that contains a Sparse Second Order or Descriptor StateSpace block.
mechss
model when you use a Sparse Second Order in your Simulink model.sparss
model when you use a Descriptor StateSpace block and select the Linearize to sparse model block parameter.
For more information, see Sparse Model Basics. For an example, see Linearize Simulink Model to a Sparse SecondOrder Model Object.
R2016b: Compute operating point offsets for model inputs, outputs, states, and state derivatives during linearization
You can compute operating point offsets for model inputs, outputs, states, and state derivatives when linearizing Simulink models. Thee offsets streamline the creation of linear parametervarying (LPV) systems.
To obtain operating point offsets, first create a linearizeOptions
or slTunerOptions
object and set the
StoreOffsets
option to true
. Then,
create an slLinearizer
or slTuner
interface for the
model.
You can extract the offsets from the info
output argument of
getCompSensitivity
and convert them into the required
format for the LPV System block using the getOffsetsForLPV
function.
See Also
slLinearizer
 slTuner
 addPoint
 addOpening
 getIOTransfer
 getLoopTransfer
 getSensitivity
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