# sensitivity

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

## Syntax

``[J,sens] = sensitivity(MPCobj,PerfFunc,PerfWeights,Ns,r,v,SimOptions,utarget)``

## Description

example

````[J,sens] = sensitivity(MPCobj,PerfFunc,PerfWeights,Ns,r,v,SimOptions,utarget)` calculates the user-defined, closed-loop, cumulative scalar performance metric `J`, and its sensitivity `sens` to the diagonal weights defined in the MPC controller object `MPCobj`. `PerfFunc` specifies the shape of the performance metric, while the optional arguments `PerfWeights`, `Ns`, `r`, `v`, `SimOptions`, and `utarget` specify the performance metric weights, simulation steps, reference and disturbance signals, simulation options, and manipulated variables targets, respectively.```

## Examples

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Define a third-order plant model with three manipulated variables and two controlled outputs.

```plant = rss(3,2,3); plant.D = 0;```

Create an MPC controller for the plant.

`mpcobj = mpc(plant,1);`
```-->The "PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon = 10. -->The "ControlHorizon" property of the "mpc" object is empty. Assuming 2. -->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming default 0.00000. -->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming default 0.10000. -->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.00000. ```

Specify an integral absolute error performance function and set the performance weights.

```PerfFunc = 'IAE'; PerfWts.OutputVariables = [1 0.5]; PerfWts.ManipulatedVariables = zeros(1,3); PerfWts.ManipulatedVariablesRate = zeros(1,3);```

Define a `20` second simulation scenario with a unit step in the output 1 setpoint and a setpoint of zero for output 2.

```Tstop = 20; r = [1 0];```

Define the nominal values of the manipulated variables to be zeros.

`utarget = zeros(1,3);`

Calculate the closed-loop performance metric, `J`, and its sensitivities, `sens`, to the weight defined in `mpcobj`, for the specified simulation scenario.

`[J,sens] = sensitivity(mpcobj,PerfFunc,PerfWts,Tstop,r,[],[],utarget)`
```-->Converting model to discrete time. -->Assuming output disturbance added to measured output channel #1 is integrated white noise. -->Assuming output disturbance added to measured output channel #2 is integrated white noise. -->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on each measured output channel. ```
```J = 1.1426 ```
```sens = struct with fields: OutputVariables: [-0.0041 -0.1285] ManipulatedVariables: [0.0378 -0.0465 0.0523] ManipulatedVariablesRate: [0.4007 0.2433 0.6805] ```

The positive, and relatively higher, values of the sensitivities to the manipulated variable rates suggest that decreasing the weights that are defined in `mpcobj` for the manipulated variable rates would contribute the most to decrease the `IAE` performance metric defined by `PerfWts`.

## Input Arguments

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Model predictive controller, specified as an MPC controller object. To create an MPC controller, use `mpc`.

Performance metric function shape, specified as one of the following:

• `'ISE'` (integral squared error), for which the performance metric is

`$J=\sum _{i=1}^{Ns}\left(\sum _{j=1}^{{n}_{y}}{\left({w}_{j}^{y}{e}_{yij}\right)}^{2}+\sum _{j=1}^{{n}_{u}}\left[{\left({w}_{j}^{u}{e}_{uij}\right)}^{2}+{\left({w}_{j}^{\Delta u}\Delta {u}_{ij}\right)}^{2}\right]\right)$`
• `'IAE'` (integral absolute error), for which the performance metric is

`$J=\sum _{i=1}^{Ns}\left(\sum _{j=1}^{{n}_{y}}|{w}_{j}^{y}{e}_{yij}|+\sum _{j=1}^{{n}_{u}}\left(|{w}_{j}^{u}{e}_{uij}|+|{w}_{j}^{\Delta u}\Delta {u}_{ij}|\right)\right)$`
• `'ITSE'` (integral of time-weighted squared error), for which the performance metric is

`$J=\sum _{i=1}^{Ns}i\Delta t\left(\sum _{j=1}^{{n}_{y}}{\left({w}_{j}^{y}{e}_{yij}\right)}^{2}+\sum _{j=1}^{{n}_{u}}\left[{\left({w}_{j}^{u}{e}_{uij}\right)}^{2}+{\left({w}_{j}^{\Delta u}\Delta {u}_{ij}\right)}^{2}\right]\right)$`
• `'ITAE'` (integral of time-weighted absolute error), for which the performance metric is

`$J=\sum _{i=1}^{Ns}i\Delta t\left(\sum _{j=1}^{{n}_{y}}|{w}_{j}^{y}{e}_{yij}|+\sum _{j=1}^{{n}_{u}}\left(|{w}_{j}^{u}{e}_{uij}|+|{w}_{j}^{\Delta u}\Delta {u}_{ij}|\right)\right)$`

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'`

Performance function weights w, specified as a structure with the following fields:

• `OutputVariables`ny-element row vector that contains the ${w}_{j}^{y}$ values

• `ManipulatedVariables`nu-element row vector that contains the ${w}_{j}^{u}$ values

• `ManipulatedVariablesRate`nu-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`. In general, however, 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 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.

Number of simulation steps, specified as a positive integer.

If you omit `Ns`, the default value is the row size of whichever of the following arrays has the largest row size:

• The input argument `r`

• The input argument `v`

• The `UnmeasuredDisturbance` property of `SimOptions`, if specified

• The `OutputNoise` property of `SimOptions`, if specified

Example: `100`

Reference signal, specified as an array. This array has `ny` columns, where `ny` is the number of 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 `r` is empty or unspecified, it defaults to the nominal value of the plant output, `MPCobj.Model.Nominal.Y`.

Example: `ones(100,1)`

Measured 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)]`

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`.

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]`

## Output Arguments

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Depending on the `PerfFunc` 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 PerfFunc for more detail.

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`.