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Compute MPC control law

MPC Simulink Library

The MPC Controller block receives the current measured output signal
(`mo`

), reference signal (`ref`

), and optional
measured disturbance signal (`md`

). The block computes the optimal
manipulated variables (`mv`

) by solving a quadratic programming problem
using either the default KWIK solver or a custom QP solver. For more information, see
QP Solver.

To use the block in simulation and code generation, you must specify an
`mpc`

object, which defines a model predictive controller. This
controller must have already been designed for the plant that it controls.

Because the MPC Controller block uses MATLAB Function
blocks, it requires compilation each time you change the MPC object and block. Also,
because MATLAB^{®} does not allow compiled code to reside in any MATLAB product folder, you must use a non-MATLAB folder to work on your Simulink^{®} model when you use MPC blocks.

The MPC Controller block has the following parameter groupings:

You must provide an `mpc`

object that defines an implicit
MPC controller. To do so:

Enter the name of an

`mpc`

object in the**MPC Controller**edit box. This object must be present in the MATLAB workspace.If you want to modify the controller settings in a graphical environment, open the

**MPC Designer**app by clicking**Design**. For example, you can:Import a new prediction model.

Change horizons, constraints, and weights.

Evaluate MPC performance with a linear plant.

Export the updated controller to the MATLAB workspace.

To see how well the controller works for the nonlinear plant, run a closed-loop Simulink simulation.

If you do not have an existing

`mpc`

object in the MATLAB workspace, leave the**MPC controller**field empty. With the MPC Controller block connected to the plant, click**Design**to open**MPC Designer**. Using the app, linearize the Simulink model at a specified operating point, and design your controller. For more information, see Design MPC Controller in Simulink and Linearize Simulink Models Using MPC Designer.To use this design approach, you must have Simulink Control Design™ software.

Once you specify a controller in the **MPC Controller**
field, you can review your design for run-time stability and robustness issues
by clicking **Review**. For more information, see Review Model Predictive Controller for Stability and Robustness Issues.

Specifies the initial controller state. If this parameter is
left blank, the block uses the nominal values that are defined in
the `Model.Nominal`

property of the `mpc`

object.
To override the default, create an `mpcstate`

object
in your workspace, and enter its name in the field.

**Measured output**or**State estimate**If your controller uses default state estimation, this inport is labeled

`mo`

. Connect this inport to the measured plant output signals. The MPC controller uses measured plant outputs to improve its state estimates.To enable custom state estimation, in the

**General**section, check**Use custom estimated states instead of measured outputs**. Checking this option changes the label on this inport to`x[k|k]`

. Connect a signal that provides estimates of the controller state (plant, disturbance, and noise model states). Use custom state estimates when an alternative estimation technique is considered superior to the built-in estimator or when the states are fully measurable.**Reference**The

`ref`

dimension must not change from one control instant to the next. Each element must be a real number.When

`ref`

is a 1-by-*n*signal, where_{y}*n*is the number of outputs, there is no reference signal previewing. The controller applies the current reference values across the prediction horizon._{y}To use signal previewing, specify

`ref`

as an*N*-by-*n*signal, where_{y}*N*is the number of time steps for which you are specifying reference values. Here, $$1<N\le p$$, and*p*is the prediction horizon. Previewing usually improves performance, since the controller can anticipate future reference signal changes. The first row of`ref`

specifies the*n*references for the first step in the prediction horizon (at the next control interval_{y}*k*= 1), and so on for*N*steps. If*N*<*p*, the last row designates constant reference values for the remaining*p*-*N*steps.For example, suppose

*n*= 2 and_{y}*p*= 6. At a given control instant, the signal connected to the`ref`

inport is:[2 5 ← k=1 2 6 ← k=2 2 7 ← k=3 2 8] ← k=4

The signal informs the controller that:

Reference values for the first prediction horizon step

*k*= 1 are`2`

and`5`

.The first reference value remains at

`2`

, but the second increases gradually.The second reference value becomes

`8`

at the beginning of the fourth step*k*= 4 in the prediction horizon.Both values remain constant at

`2`

and`8`

respectively for steps 5–6 of the prediction horizon.

`mpcpreview`

shows how to use reference previewing in a specific case. For calculation details on the use of the reference signal, see Optimization Problem.

The
`mv`

outport provides a signal defining the $${n}_{u}\ge 1$$ manipulated variables for controlling the plant. At each control
instant, the controller updates its `mv`

outport by solving a
quadratic programming problem using either the default KWIK solver or a custom QP
solver. For more information, see QP Solver.

If the controller detects an infeasible
optimization problem or encounters numerical difficulties in solving an
ill-conditioned optimization problem, `mv`

remains at its most
recent successful solution; that is, the controller output freezes.

Otherwise, if the optimization problem is feasible
and the solver reaches the specified maximum number of iterations without finding an optimal
solution, `mv`

:

Remains at its most recent successful solution if the

`Optimizer.UseSuboptimalSolution`

property of the controller is`false`

.Is the suboptimal solution reached after the final iteration if the

`Optimizer.UseSuboptimalSolution`

property of the controller is`true`

. For more information, see Suboptimal QP Solution.

Add an inport (`md`

) to which you connect a
measured disturbance signal. The number of measured disturbances defined
for your controller, $${n}_{md}\ge 1$$, must match the
dimensions of the connected disturbance signal.

The number of measured disturbances must not change from one control instant to the next, and each disturbance value must be a real number.

When `md`

is a 1-by-*n _{md}* signal,
there is no measured disturbance previewing. The controller applies the current disturbance values
across the prediction horizon.

To use disturbance previewing, specify `md`

as an
*N*-by-*n _{md}* signal, where

`md`

specifies the For example, suppose *n _{md}* =
2 and

`md`

inport is:[2 5 ← k=0 2 6 ← k=1 2 7 ← k=2 2 8] ← k=3

This signal informs the controller that:

The current

`MD`

values are`2`

and`5`

at*k*= 0.The first

`MD`

remains at`2`

, but the second increases gradually.The second

`MD`

becomes`8`

at the beginning of the third step*k*= 3 in the prediction horizon.Both values remain constant at

`2`

and`8`

respectively for steps 4–6 of the prediction horizon.

`mpcpreview`

shows how to use `MD`

previewing
in a specific case.

For calculation details, see MPC Modeling and QP Matrices.

Add an inport (`ext.mv`

) to which you connect a vector signal
that contains the actual *n _{u}*
manipulated variables (MV) used in the plant. Use this option when the MV
applied to the plant between time

Controller state estimation assumes that the MV is piecewise constant. At time
*t _{k}*, the

`ext.mv`

value must be the effective MV between times
The following example, from the model `mpc_bumpless`

,
includes a switch that can override the controller output with a signal supplied
by the operator. Also, the controller output may saturate. Feeding back the
actual MV used in the plant (labeled `u(t)`

in the example)
improves the accuracy of controller state estimates.

If the external MV option is inactive or the `ext.mv`

inport
in unconnected, the controller assumes that its MV output is used in the plant
without modification.

Using this option can cause an algebraic loop in the Simulink model, since there is direct feedthrough from the
`ext.mv`

inport to the `mv`

outport.
To prevent such algebraic loops, insert a Memory block or
Unit Delay block.

If you want one or more manipulated variables (MV) to track
target values that change with time, use this option to add an `mv.target`

inport.
Connect this port to a target signal with dimension *n _{u}*,
where

For this to be effective, the corresponding MV(s) must have nonzero penalty weights (these weights are zero by default).

Add an outport (`cost`

) that provides the optimal
quadratic programming objective function value at the current time
(a nonnegative scalar). If the controller is performing well and no
constraints have been violated, the value should be small. If the
optimization problem is infeasible, however, the value is meaningless.
(See `qp.status`

.)

Add an outport (`qp.status`

) that allows you to monitor the
status of the QP solver.

If a QP problem is solved successfully at a given control interval, the
`qp.status`

output returns the number of QP solver iterations used in
computation. This value is a finite, positive integer and is proportional to the time
required for the calculations. Thus, a large value means a relatively slow block execution
for this time interval.

The QP solver can fail to find an optimal solution for the following reasons:

`qp.status = 0`

— The QP solver cannot find a solution within the maximum number of iterations specified in the`mpc`

object. In this case, if the`Optimizer.UseSuboptimalSolution`

property of the MPC controller is`false`

, the block holds its`mv`

output at the most recent successful solution. Otherwise, it uses the suboptimal solution found during the last solver iteration.`qp.status = -1`

— The QP solver detects an infeasible QP problem. See Monitoring Optimization Status to Detect Controller Failures for an example where a large, sustained disturbance drives the OV outside its specified bounds. In this case, the block holds its`mv`

output at the most recent successful solution.`qp.status = -2`

— The QP solver has encountered numerical difficulties in solving a severely ill-conditioned QP problem. In this case, the block holds its`mv`

output at the most recent successful solution.

In a real-time application, you can use `qp.status`

to set an alarm or
take other special action.

The following diagram shows how to use the status indicator to monitor the MPC Controller block in real time. For more information, see Monitoring Optimization Status to Detect Controller Failures.

Add an outport (`est.state`

) to receive the
controller state estimates, `x[k|k]`

, at each control
instant. These include the plant, disturbance, and noise model states.

Add an outport (`mv.seq`

) that provides the predicted optimal MV
adjustments (moves) over the entire prediction horizon from `k`

to
`k+p`

, where `k`

is the current time and
*p* is the prediction horizon. This signal is a
(*p*+1)-by-*n _{u}* matrix,
where and

`mv.seq`

contains the calculated optimal MV moves at time
`k+i-1`

, for `i = 1,...,p`

. The first row of
`mv.seq`

is identical to the `mv`

outport signal,
which is the current MV adjustment applied at time `k`

. Since the
controller does not calculate optimal control moves at time `k+p`

, the last
row of `mv.seq`

duplicates the previous row.

Add an outport (`x.seq`

) that provides the predicted optimal state
variable sequence over the entire prediction horizon from `k`

to
`k+p`

, where `k`

is the current time and
*p* is the prediction horizon. This signal is a
(*p*+1)-by-*n _{x}* matrix,
where

`x.seq`

contains the calculated optimal state values at time
`k+i`

, for `i = 1,...,p`

. The first row of
`x.seq`

contains the current states at time `k`

as
determined by state estimation.

Add an outport (`y.seq`

) that provides the predicted optimal output
variable sequence over the entire prediction horizon from `k`

to
`k+p`

, where `k`

is the current time and
*p* is the prediction horizon. This signal is a
(*p*+1)-by-*n _{y}* matrix,
where and

`y.seq`

contains the calculated optimal output values at time
`k+i-1`

, for `i = 1,...,p+1`

. The first row of
`y.seq`

contains the current outputs at time `k`

based
on the estimated states and measured disturbances; it is not the measured output at time
`k`

.

Replace `mo`

with the `x[k|k]`

inport for
custom state estimation as described in Required Inports.

Add inport `umin`

that you can connect to a run-time constraint signal for
manipulated variable lower bounds. This signal is a vector with
*n _{u}* finite values. The

`i`

th element of `umin`

replaces the
`ManipulatedVariables(i).Min`

property of the controller at run
time.If a manipulated variable does not have a lower bound specified in the controller object, then the corresponding connected signal value is ignored.

If this parameter is not selected, the block uses the constant constraint values stored
within its `mpc`

object.

You cannot specify time-varying constraints at run time using a matrix signal.

If the `ManipulatedVariables(i).Min`

property of the controller is
specified as a vector (that is, the constraint varies over the prediction horizon), the
`i`

th element of `umin`

replaces the first finite
entry in this vector, and the remaining values shift to retain the same constraint
profile.

Add inport `umax`

that you can connect to a run-time constraint signal for
manipulated variable upper bounds. This signal is a vector with
*n _{u}* finite values. The

`i`

th element of `umax`

replaces the
`ManipulatedVariables(i).Max`

property of the controller at run
time.If a manipulated variable does not have an upper bound specified in the controller object, then the corresponding connected signal value is ignored.

If this parameter is not selected, the block uses the constant constraint values stored
within its `mpc`

object.

You cannot specify time-varying constraints at run time using a matrix signal.

If the `ManipulatedVariables(i).Max`

property of the controller is
specified as a vector (that is, the constraint varies over the prediction horizon), the
`i`

th element of `umax`

replaces the first finite
entry in this vector, and the remaining values shift to retain the same constraint
profile.

Add inport `ymin`

that you can connect to a run-time constraint signal for
output variable lower bounds. This signal is a vector with
*n _{y}* finite values. The

`i`

th element of `ymin`

replaces the
`OutputVariables(i).Min`

property of the controller at run time.If an output variable does not have a lower bound specified in the controller object, then the corresponding connected signal value is ignored.

If this parameter is not selected, the block uses the constant constraint values stored
within its `mpc`

object.

You cannot specify time-varying constraints at run time using a matrix signal.

If the `OutputVariables(i).Min`

property of the controller is
specified as a vector (that is, the constraint varies over the prediction horizon), the
`i`

th element of `ymin`

replaces the first finite
entry in this vector, and the remaining values shift to retain the same constraint
profile.

Add inport `ymax`

that you can connect to a run-time constraint signal for
output variable upper bounds. This signal is a vector with
*n _{y}* finite values. The

`i`

th element of `ymax`

replaces the
`OutputVariables(i).Max`

property of the controller at run time.If an output variable does not have an upper bound specified in the controller object, then the corresponding connected signal value is ignored.

`mpc`

object.

You cannot specify time-varying constraints at run time using a matrix signal.

If the `OutputVariables(i).Max`

property of the controller is
specified as a vector (that is, the constraint varies over the prediction horizon), the
`i`

th element of `ymax`

replaces the first finite
entry in this vector, and the remaining values shift to retain the same constraint
profile.

Add inports `E`

, `F`

, `G`

, and `S`

to the block that you can connect to the following run-time custom constraint matrix signals:

`E`

— Manipulated variable constraint matrix with size*n*-by-_{c}*n*, where_{u}*n*is the number of custom constraints_{c}`F`

— Controlled output constraint matrix with size*n*-by-_{c}*n*_{y}`G`

— Custom constraint matrix with size 1-by-*n*_{c}`S`

— Measured disturbance constraint matrix, with size*n*-by-_{c}*n*, where_{v}*n*is the number of measured disturbances._{v}`S`

is added only if the`mpc`

object has measured disturbances.

These constraints replace the custom constraints previously set using `setconstraint`

.

If you define `E`

, `F`

, `G`

, or `S`

in the `mpc`

object, you must connect a signal to the corresponding inport, and that signal must have the same dimensions as the array specified in the controller. If an array is not defined in the controller object, use a zero matrix with the correct size.

The custom constraints are of the form `E`

*u* + `F`

*y* + `S`

*v* <= `G`

, where:

*u*is a vector of manipulated variable values.*y*is a vector of predicted plant output values.*v*is a vector of measured plant disturbance input values.

For more information, see Constraints on Linear Combinations of Inputs and Outputs.

A controller intended for real-time applications should have "knobs" you can use to tune its performance when it operates with the real plant. This group of optional inports serves that purpose.

The diagram shown below shows three of the MPC Controller tuning
inports. In this simulation context, the inports are tuned using pre-stored signals
(the `ywt`

, `duwt`

, and `ECRwt`

variables in the From Workspace blocks). In practice, you would
connect a knob or similar manual adjustment.

You cannot specify time-varying weights at run time using a matrix signal.

Add an inport (`y.wt`

) that you can connect to a run-time output variable
(OV) weight signal. This signal overrides the `Weights.OV`

property of the
`mpc`

object, which establishes the relative importance of OV
reference tracking.

To use the same tuning weights over the prediction horizon, connect
`y.wt`

to a vector signal with
*n _{y}* elements, where

To vary the tuning weights over the prediction horizon, connect `y.wt`

to
a matrix signal with *n _{y}* columns and up to

If you do not connect a signal to the `y.wt`

inport, the block uses the OV
weights specified in your `mpc`

object.

Add an inport (`u.wt`

) that you can connect to a run-time manipulated
variable (MV) weight signal. This signal overrides the `Weights.MV`

property of the `mpc`

object, which establishes the relative importance
of MV target tracking.

To use the same tuning weights over the prediction horizon, connect
`u.wt`

to a vector signal with
*n _{u}* elements, where

To vary the tuning weights over the prediction horizon, connect `u.wt`

to
a matrix signal with *n _{u}* columns and up to

If you do not connect a signal to the `u.wt`

inport, the block uses the
MV weights specified in your `mpc`

object.

Add an inport (`du.wt`

) that you can connect to a run-time manipulated
variable (MV) rate weight signal. This signal overrides the
`Weights.MVrate`

property of the `mpc`

object, which
establishes the relative importance of MV changes.

To use the same tuning weights over the prediction horizon, connect
`du.wt`

to a vector signal with
*n _{u}* elements, where

To vary the tuning weights over the prediction horizon, connect `du.wt`

to a matrix signal with *n _{u}* columns and up to

If you do not connect a signal to the `du.wt`

inport, the block uses the
MV rate weights specified in your `mpc`

object.

Add an inport (`ecr.wt`

), for a scalar nonnegative signal that overrides the
`MPCobj.Weights.ECR`

property of the `mpc`

controller.
This inport has no effect unless your controller object defines soft constraints whose
associated ECR values are nonzero.

If there are soft constraints, increasing the `ecr.wt`

value makes these
constraints relatively harder. The controller then places a higher priority on minimizing
the magnitude of the predicted worst-case constraint violation.

You may not be able to avoid violations of an output variable constraint. Thus, increasing the
`ecr.wt`

value is often counterproductive. Such an increase causes the
controller to pay less attention to its other objectives and does not help reduce constraint
violations. You usually need to tune `ecr.wt`

to achieve the proper balance
in relation to the other control objectives.

If you want to vary your prediction and control horizons at run time, select this parameter. Doing so adds the following input ports to the block:

`p`

— Prediction horizon, specified as positive integer signal. The prediction horizon signal value must be less than or equal to the**Maximum prediction horizon**parameter.`m`

— Control horizon, specified as one of the following:Positive integer signal less than or equal to the prediction horizon.

Vector signal of positive integers specifying blocking interval lengths. For more information, see Manipulated Variable Blocking.

At run time, the values of the `p`

and `m`

signals override the default horizons specified in the controller object.

For more information, see Adjust Horizons at Run Time.

Specify the maximum prediction horizon value when varying the prediction
horizon at run time. This value, *P _{max}*,
is used to define:

The number of rows in the optimal sequence output signals

`mv.seq`

,`x.seq`

, and`y.seq`

. When varying your prediction horizon at run time, these signals have*P*+1 rows._{max}The maximum number of rows for the time-varying weights input signals

`y.wt`

,`u.wt`

, and`du.wt`

. When varying your prediction horizon at run time, these signals can have at most*P*rows._{max}

To enable this parameter, select the **Adjust prediction horizon and
control horizon at run time** parameter.

For more information, see Adjust Horizons at Run Time.

Specify the default block sample time and signal dimensions for performing
simulation, trimming, or linearization. In these cases, the
`mv`

output signal remains at zero. You must specify default
condition values that are compatible with your Simulink model design.

These default conditions apply only if the **MPC Controller**
field is empty. If you specify a controller from the MATLAB workspace, the sample time and signal sizes from the specified
controller are used.

Specify the default controller sample time.

Specify the default signal dimensions for the following input signal types:

Manipulated variables

Unmeasured disturbances

Measured disturbances

### Note

You can specify the measured disturbances signal dimension only if, on the

**General**section, in the**Additional Inports**section, the**Measured disturbance**option is selected.

Specify the default signal dimensions for the following output signal types:

Measured outputs

Unmeasured outputs

Specify the block data type of the manipulated variables as one of the following:

`double`

— Double-precision floating point (default)`single`

— Single-precision floating pointIf you are implementing the block on a single-precision target, specify the output data type as

`single`

.

For an example of double-precision and single-precision simulation and code generation for an MPC controller, see Simulation and Code Generation Using Simulink Coder.

To view the port data types in a model, in the Simulink Editor,
select **Display > ****Signals
& Ports****Port Data Types**.

Use the sample time of the parent subsystem as the block sample time. Doing so allows you to conditionally execute this block inside Function-Call Subsystem or Triggered Subsystem blocks. For an example, see Using MPC Controller Block Inside Function-Call and Triggered Subsystems.

You must execute Function-Call Subsystem or Triggered Subsystem blocks at the sample rate of the controller. Otherwise, you can see unexpected results.

To view the sample time of a block, in the Simulink Editor,
select **Display > ****Sample
Time**. Select **Colors**, **Annotations**,
or **All**. For more information, see View Sample Time Information (Simulink).

Add an inport (`switch`

) whose input specifies
whether the controller performs optimization calculations. If the
input signal is zero, the controller behaves normally. If the input
signal is nonzero, the MPC Controller block turns off
the controller optimization calculations. This action reduces computational
effort when the controller output is not needed, such as when the
system is operating manually or another controller has taken over.
However, the controller continues to update its internal state estimates
in the usual way. Thus, it is ready to resume optimization calculations
whenever the `switch`

signal returns to zero. While
controller optimization is off, the MPC Controller block
passes the current `ext.mv`

signal to the controller
output. If the `ext.mv`

inport is not enabled, the
controller output is held at the value it had when optimization was
disabled.