sensitivity
Calculate the value of a performance metric and its sensitivity to the diagonal weights of an MPC controller
Syntax
Description
[
calculates the value J
,sens
] = sensitivity(mpcobj
,PerfFcn
,PerfWeights
,Ns
,r
,v
,SimOptions
,utarget
)J
and sensitivity sens
of a
predefined closed-loop, cumulative performance metric with respect to the diagonal weights
defined in the MPC controller object mpcobj
. You chose the shape of the
performance metric, among the available options, using PerfFcn
. The
optional arguments PerfWeights
, Ns
,
r
, v
, SimOptions
, and
utarget
specify the performance metric weights, number of simulation
points, reference and disturbance signals, simulation options, and manipulated variables
targets, respectively. If you omit any of these arguments, then default values are
used.
[
calculates the value J
,sens
] = sensitivity(mpcobj
,customPerFcn
,Par1,...,ParN
)J
and sensitivity sens
of the
performance metric defined in the custom function customPerFcn
, with
respect to the diagonal weights defined in the MPC controller object
mpcobj
. The remaining input arguments
Par1,Par2,...,ParN
specify the value of the parameters needed by
customPerFnc
.
Examples
Calculate Value of Predefined Performance Metric and Its Sensitivity to Controller Weights
Fix the random number generator seed for reproducibility.
rng(0)
Define a third-order plant model with three manipulated variables and two controlled outputs. Then create an MPC controller for the plant, with sample time of 1
.
plant = rss(3,2,3); plant.D = 0; mpcobj = mpc(plant,1);
-->"PredictionHorizon" is empty. Assuming default 10. -->"ControlHorizon" is empty. Assuming default 2. -->"Weights.ManipulatedVariables" is empty. Assuming default 0.00000. -->"Weights.ManipulatedVariablesRate" is empty. Assuming default 0.10000. -->"Weights.OutputVariables" is empty. Assuming default 1.00000.
Specify an integral absolute error performance function and set the performance weights. The performance weights emphasize tracking the first output variable.
PerfFunc = 'IAE';
PerfWts.OutputVariables = [2 0.5];
PerfWts.ManipulatedVariables = zeros(1,3);
PerfWts.ManipulatedVariablesRate = zeros(1,3);
Define a 20
second simulation scenario with a unit step as setpoint for the first output and zero as a setpoint for the second output.
Tstop = 20; r = [1 0];
Calculate the closed-loop performance metric, J
, and its sensitivities, sens
, to the weights defined in mpcobj
, for the specified simulation scenario. For this example, do not specify the last three input argument of sensitivity
. This means that no disturbance signal or simulation option is used and the nominal value of the manipulated variables is kept to its default value of zero.
[J,sens] = sensitivity(mpcobj,PerfFunc,PerfWts,Tstop,r)
-->Converting model to discrete time. -->Assuming output disturbance added to measured output #1 is integrated white noise. -->Assuming output disturbance added to measured output #2 is integrated white noise. -->"Model.Noise" is empty. Assuming white noise on each measured output.
J = 2.2977
sens = struct with fields:
OutputVariables: [-0.1897 -0.0894]
ManipulatedVariables: [0.0311 0.0826 0.0014]
ManipulatedVariablesRate: [0.4268 2.3482 0.0149]
The positive, and relatively higher, values of the sensitivities to the first and last manipulated variable rate suggest that decreasing the corresponding weights defined in mpcobj
would contribute the most to decrease the IAE
performance metric defined by PerfWts
. At the same time, since the sensitivity to the weight of the second manipulated variable is negative, increasing the corresponding weight would also contribute to decrease the performance metric.
Modify the manipulated variable rate weights in mpcobj
and recalculate the value of the performance metric.
mpcobj.Weights.ManipulatedVariablesRate = [1e-2 1 1e-2]; sensitivity(mpcobj,PerfFunc,PerfWts,Tstop,r)
-->Converting model to discrete time. -->Assuming output disturbance added to measured output #1 is integrated white noise. -->Assuming output disturbance added to measured output #2 is integrated white noise. -->"Model.Noise" is empty. Assuming white noise on each measured output.
ans = 2.0116
As expected the value of the performance metric decreased, indicating an improved tracking performance.
Calculate Value of Custom Performance Metric and Its Sensitivity to Controller Weights
Define a third-order plant model with three manipulated variables and two controlled outputs. Then create an MPC controller for the plant, with sample time of 1
.
plant = rss(3,2,3); plant.D = 0; mpcobj = mpc(plant,1);
-->"PredictionHorizon" is empty. Assuming default 10. -->"ControlHorizon" is empty. Assuming default 2. -->"Weights.ManipulatedVariables" is empty. Assuming default 0.00000. -->"Weights.ManipulatedVariablesRate" is empty. Assuming default 0.10000. -->"Weights.OutputVariables" is empty. Assuming default 1.00000.
Define a custom performance function and write it to a file. The function must take an MPC object as a first input argument. The simulation time and the output set point are the second and third input arguments, respectively. Internally, the function performs a closed loop simulation using the given MPC object, simulation time and set point. The norm of the difference between the set point and the output signal is then returned as the value of the performance metric (note that this norm depends on the number of simulation steps).
% write a function to the char vector "str" str = ['function J = mypfun(mpcobj,T,ySetPnt)', ... newline, ... 'y = sim(mpcobj,T,ySetPnt); J = norm(ySetPnt-y);', ... newline, ... 'end']; % create the function file fid=fopen('mypfun.m','w'); % open a file for writing fwrite(fid,str,'char'); % write "str" to the file fclose(fid); % close the file
Calculate the custom performance metric, J
, and its sensitivities, sens
, to the weights defined in mpcobj
, using a simulation time of 10
seconds and an output setpoint of [1
1
].
[J,sens] = sensitivity(mpcobj,'mypfun',10,[1 1])
-->Converting model to discrete time. -->Assuming output disturbance added to measured output #1 is integrated white noise. -->Assuming output disturbance added to measured output #2 is integrated white noise. -->"Model.Noise" is empty. Assuming white noise on each measured output.
J = 1.5145
sens = struct with fields:
OutputVariables: [-0.0507 -0.0888]
ManipulatedVariables: [0.0013 0.0055 3.7533e-05]
ManipulatedVariablesRate: [0.1918 1.2004 0.0034]
The comparatively higher values of the sensitivities to the manipulated variable rates suggest that decreasing the corresponding weights defined in mpcobj
would contribute the most to decrease the custom performance metric calculated in the function mypfun
.
Input Arguments
mpcobj
— Model predictive controller
mpc
object
Model predictive controller, specified as an MPC controller
object. To create an MPC controller, use mpc
.
PerfFcn
— Performance metric function shape
'ISE'
| 'IAE'
| 'ITSE'
| 'ITAE'
Performance metric function shape, specified as one of the following:
'ISE'
(integral squared error), for which the performance metric is'IAE'
(integral absolute error), for which the performance metric is'ITSE'
(integral of time-weighted squared error), for which the performance metric is'ITAE'
(integral of time-weighted absolute error), for which the performance metric is
In these expressions, ny is the number of controlled outputs and nu is the number of manipulated variables, eyij is the difference between output j and its setpoint (or reference) value at time interval i, euij is the difference between the manipulated variable j and its target at time interval i.
The w parameters are nonnegative performance weights defined by
the structure PerfWeights
.
Example: 'ITAE'
PerfWeights
— Performance function weights
mpcobj.Weights
(default) | structure
Performance function weights w, specified as a structure with the following fields:
OutputVariables
— ny-element row vector that contains the valuesManipulatedVariables
— nu-element row vector that contains the valuesManipulatedVariablesRate
— nu-element row vector that contains the values
If PerfWeights
is empty or unspecified, it defaults to the
corresponding weights in mpcobj
.
Note
The performance index is not related to the quadratic cost function that the MPC controller tries to minimize by choosing the values of the manipulated variables.
One clear difference is that the performance index is based on a closed loop simulation running until a time that is generally different than the prediction horizon, while the MPC controller calculates the moves which minimize its internal cost function up to the prediction horizon and in open loop fashion. Furthermore, even when the performance index is chosen to be of ISE type, its weights should be squared to match the weights defined in the MPC cost function.
Therefore, the performance weights and those used in the controller have different purposes; define these weights accordingly.
Ns
— Number of simulation points, including zero
positive integer
Number of simulation points, including zero, specified as a positive integer. The
simulation runs for Ns-1
steps, between 0
and
Ts*(Ns-1)
.
If you omit Ns
, the default value is the number of rows of
whichever of the following arrays has the largest number of rows:
Example: 20
r
— Reference signal
mpcobj.Model.Nominal.Y
(default) | double
array
Reference signal, specified as an array. This array has ny
columns, where ny
is the number of total (measured and unmeasured)
plant outputs. r
can have anywhere from 1 to
Ns
rows. If the number of rows is less than
Ns
, the missing rows are set equal to the last row.
If Ns is empty or unspecified, it defaults to the nominal output vector
mpcobj.Model.Nominal.Y
.
Example: ones(20,3)
v
— Measured input disturbance signal
mpcobj.Model.Nominal.U(md)
(default) | double
array
Measured input disturbance signal, specified as an array. This array has
nv
columns, where nv
is the number of measured
input disturbances. v
can have anywhere from 1 to
Ns
rows. If the number of rows is less than
Ns
, the missing rows are set equal to the last row.
If v
is empty or unspecified, it defaults to the nominal value of
the measured input disturbance, mpcobj.Model.Nominal.U(md)
, where
md
is the vector containing the indices of the measured disturbance
signals, as defined by setmpcsignals
.
Example: [zeros(50,1);ones(50,1)]
SimOptions
— Simulation options object
[]
(default) | mpcsimopt
object
Use a simulation options objects to specify options such as noise and disturbance
signals that feed into the plant but are unknown to the controller. You can also use
this object to specify an open loop scenario, or a plant model in the loop that is
different from the one in mpcobj.Model.Plant
.
For more information, see mpcsimopt
.
utarget
— Target for manipulated variables
mpcobj.Model.Nominal.U
(default) | vector
The optional input utarget
is a vector of
nu manipulated variable targets. Their
defaults are the nominal values of the manipulated variables.
Example: [0.1;0;-0.2]
customPerFcn
— Name of the custom performance function
character vector
Name of the custom performance function, specified as a character vector. The
character vector must be different than 'ISE'
,
'IAE'
, 'ITSE'
, or 'ITAE'
, and
specify the name of a file in the MATLAB® path containing a custom function.
The custom function must have the following signature:
J
=
customPerFcn(mpcobj
,Par1,...,ParN
)
where J
is a scalar indicating the value of the performance
index mpcobj
is an mpc
object. The remaining arguments Par1,...,ParN
are
parameters that, if needed by customPerFcn
, you must pass to
sensitivity
after the customPerFcn
argument.
For example, inside customPerFcn
, you can use
mpcobj
and, if needed, Par1,...,ParN
, to
perform a simulation and calculate J
based on the simulation
results.
Example: 'myPerfFcn(mpcobj,Ts,Setpoint)'
Par1,...,ParN
— Values of the parameters used by the custom performance function
values of any needed parameter
Values of the parameters used by the custom performance function
customPerFcn
, specified as needed.
Example: 10,[1 1]
Output Arguments
sens
— Sensitivity of the performance metric
structure
This structure contains and the numerical partial derivatives of the performance
measure J
with respect to its diagonal weights. These partial
derivatives, also called sensitivities, suggest weight adjustments
that should improve performance; that is, reduce J
.
Version History
Introduced in R2009a
See Also
Functions
Objects
Blocks
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