sensitivity

Calculate the value of a performance metric and its sensitivity to the diagonal weights of an MPC controller

Syntax

[J,sens] = sensitivity(mpcobj,PerfFcn,PerfWeights,Ns,r,v,SimOptions,utarget)

[J,sens] = sensitivity(mpcobj,customPerFcn,Par1,...,ParN)

Description

[J,sens] = sensitivity(mpcobj,PerfFcn,PerfWeights,Ns,r,v,SimOptions,utarget) calculates the value 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.

[J,sens] = sensitivity(mpcobj,customPerFcn,Par1,...,ParN) calculates the value 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 3.7530e-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

  $$J={\displaystyle \sum _{i=1}^{Ns}\left({\displaystyle \sum _{j=1}^{n_y}{(w_j^y{e}_{yij})}^2+{\displaystyle \sum _{j=1}^{n_u}[{(w_j^u{e}_{uij})}^2+{(w_j^{\Delta u}\Delta u_{ij})}^2]}}\right)}$$

• 'IAE' (integral absolute error), for which the performance metric is

  $$J={\displaystyle \sum _{i=1}^{Ns}\left({\displaystyle \sum _{j=1}^{n_y}\left|w_j^y{e}_{yij}\right|+{\displaystyle \sum _{j=1}^{n_u}(|w_j^u{e}_{uij}|+|w_j^{\Delta u}\Delta u_{ij}|)}}\right)}$$

• 'ITSE' (integral of time-weighted squared error), for which the performance metric is

  $$J={\displaystyle \sum _{i=1}^{Ns}i\Delta t\left({\displaystyle \sum _{j=1}^{n_y}{(w_j^y{e}_{yij})}^2+{\displaystyle \sum _{j=1}^{n_u}[{(w_j^u{e}_{uij})}^2+{(w_j^{\Delta u}\Delta u_{ij})}^2]}}\right)}$$

• 'ITAE' (integral of time-weighted absolute error), for which the performance metric is

  $$J={\displaystyle \sum _{i=1}^{Ns}i\Delta t}\left({\displaystyle \sum _{j=1}^{n_y}\left|w_j^y{e}_{yij}\right|+{\displaystyle \sum _{j=1}^{n_u}(|w_j^u{e}_{uij}|+|w_j^{\Delta u}\Delta u_{ij}|)}}\right)$$

In these expressions, n_y is the number of controlled outputs and n_u is the number of manipulated variables, e_yij is the difference between output j and its setpoint (or reference) value at time interval i, e_uij 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 — n_y-element row vector that contains the $$w_j^y$$ values

• ManipulatedVariables — n_u-element row vector that contains the $$w_j^u$$ values

• ManipulatedVariablesRate — n_u-element row vector that contains the $$w_j^{\Delta u}$$ 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:

• The input argument r

• The input argument v

• The UnmeasuredDisturbance property of SimOptions, if specified

• The OutputNoise property of SimOptions, if specified

• The greatest value between the prediction horizon mpcobj.P and 10.

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 n_u 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

J — Performance metric for the given controller

'double'

Depending on the PerfFcn argument, this performance measure can be a function of the integral (time-weighted or not) of either the square or the absolute value or the (output and input) error. See PerfFcn for more detail.

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