sensitivity

Compute effect of controller tuning weights on performance

Syntax

[J,sens] = sensitivity(MPCobj,PerfFunc,PerfWeights,Tstop,r,v,simopt,utarget) [J,sens] = sensitivity(MPCobj,'perf_fun',param1,param2,...)

Description

The sensitivity function is a controller tuning aid. J specifies a scalar performance metric. sensitivity computes J and its partial derivatives with respect to the controller tuning weights. These sensitivities suggest tuning weight adjustments that should improve performance; that is, reduce J.

[J,sens] = sensitivity(MPCobj,PerfFunc,PerfWeights,Tstop,r,v,simopt,utarget) calculates the scalar performance metric, J, and sensitivities, sens, for the controller defined by the MPC controller object MPCobj.

PerfFunc must be one of the following:

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

$J = \sum_{i = 1}^{T s t o p} (\sum_{j = 1}^{n_{y}} {(w_{j}^{y} e_{y i j})}^{2} + \sum_{j = 1}^{n_{u}} [{(w_{j}^{u} e_{u i j})}^{2} + {(w_{j}^{Δ u} Δ u_{i j})}^{2}])$

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

$J = \sum_{i = 1}^{T s t o p} (\sum_{j = 1}^{n_{y}} | w_{j}^{y} e_{y i j} | + \sum_{j = 1}^{n_{u}} (| w_{j}^{u} e_{u i j} | + | w_{j}^{Δ u} Δ u_{i j} |))$

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

$J = \sum_{i = 1}^{T s t o p} i Δ t (\sum_{j = 1}^{n_{y}} {(w_{j}^{y} e_{y i j})}^{2} + \sum_{j = 1}^{n_{u}} [{(w_{j}^{u} e_{u i j})}^{2} + {(w_{j}^{Δ u} Δ u_{i j})}^{2}])$

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

$J = \sum_{i = 1}^{T s t o p} i Δ t (\sum_{j = 1}^{n_{y}} | w_{j}^{y} e_{y i j} | + \sum_{j = 1}^{n_{u}} (| w_{j}^{u} e_{u i j} | + | w_{j}^{Δ u} Δ u_{i j} |))$

In the above 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 manipulated variable j and its target at time interval i.

The w parameters are nonnegative performance weights defined by the structure PerfWeights, which contains 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 $w_{j}^{Δ u}$ values

If PerfWeights is unspecified, it defaults to the corresponding weights in MPCobj. In general, however, the performance weights and those used in the controller have different purposes and should be defined accordingly.

Inputs Tstop, r, v, and simopt define the simulation scenario used to evaluate performance. See sim for details.

Tstop is the integer number of controller sampling intervals to be simulated. The final time for the simulations will be Tstop × Δt, where Δt is the controller sampling interval specified in MPCobj.

The optional input utarget is a vector of n_u manipulated variable targets. Their defaults are the nominal values of the manipulated variables. Δu_ij is the change in manipulated variable j and its target at time interval i.

The structure variable sens contains the computed sensitivities (partial derivatives of J with respect to the MPCobj tuning weights.) Its fields are:

OutputVariables — n_y element row vector of sensitivities with respect to MPCobj.Weights.OutputVariables
ManipulatedVariables — n_u element row vector of sensitivities with respect to MPCobj.Weights.ManipulatedVariables
ManipulatedVariablesRate — n_u element row vector of sensitivities with respect to MPCobj.Weights.ManipulatedVariablesRate

For more information on the tuning weights controller property, see mpc.

[J,sens] = sensitivity(MPCobj,'perf_fun',param1,param2,...) employs a performance function 'perf_fun' to define J. Its function definition must be in the form

function J = perf_fun(MPCobj, param1, param2, ...)

That is, it must compute J for the given controller and optional parameters param1, param2, ... and it must be on the MATLAB^® path.

Note

While performing the sensitivity analysis, the software ignores time-varying, nondiagonal, and ECR slack variable weights.

Examples

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Compute Controller Performance and Sensitivity

Open Live Script

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 performance metric, J, and sensitivities, sens, for the specified controller and 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.

Documentation

sensitivity

Syntax

Description

Examples

Compute Controller Performance and Sensitivity

See Also

Model Predictive Control Toolbox Documentation

Support