## Design MPC Controller for Position Servomechanism

This example shows how to design a model predictive controller for a
position servomechanism using **MPC Designer**.

### System Model

A position servomechanism consists of a DC motor, gearbox, elastic shaft, and load.

The differential equations representing this system are

$$\begin{array}{l}{\dot{\omega}}_{L}=-\frac{{k}_{T}}{{J}_{L}}\left({\theta}_{L}-\frac{{\theta}_{M}}{\rho}\right)-\frac{{\beta}_{L}}{{J}_{L}}{\omega}_{L}\\ {\dot{\omega}}_{M}=\frac{{k}_{M}}{{J}_{M}}\left(\frac{V-{k}_{M}{\omega}_{M}}{R}\right)-\frac{{\beta}_{M}{\omega}_{M}}{{J}_{M}}+\frac{{k}_{T}}{\rho {J}_{M}}\left({\theta}_{L}-\frac{{\theta}_{M}}{\rho}\right)\end{array}$$

where,

*V*is the applied voltage.*T*is the torque acting on the load.$${\omega}_{L}={\dot{\theta}}_{L}$$ is the load angular velocity.

$${\omega}_{M}={\dot{\theta}}_{M}$$ is the motor shaft angular velocity.

The remaining terms are constant parameters.

**Constant Parameters for Servomechanism Model**

Symbol | Value (SI Units) | Definition |
---|---|---|

k_{T} | 1280.2 | Torsional rigidity |

k_{M} | 10 | Motor constant |

J_{M} | 0.5 | Motor inertia |

J_{L} | 50 | Load inertia |

ρ | 20 | Gear ratio |

β_{M} | 0.1 | Motor viscous friction coefficient |

β_{L} | 25 | Load viscous friction coefficient |

R | 20 | Armature resistance |

If you define the state variables as

$${x}_{p}={\left[\begin{array}{cccc}{\theta}_{L}& {\omega}_{L}& {\theta}_{M}& {\omega}_{M}\end{array}\right]}^{T},$$

then you can model the servomechanism as an LTI state-space system.

$$\begin{array}{c}{\dot{x}}_{p}=\left[\begin{array}{cccc}0& 1& 0& 0\\ -\frac{{k}_{T}}{{J}_{L}}& -\frac{{\beta}_{L}}{{J}_{L}}& \frac{{k}_{T}}{\rho {J}_{L}}& 0\\ 0& 0& 0& 1\\ \frac{{k}_{T}}{\rho {J}_{M}}& 0& -\frac{{k}_{T}}{{\rho}^{2}{J}_{M}}& -\frac{{\beta}_{M}+\frac{{k}_{M}^{2}}{R}}{{J}_{M}}\end{array}\right]{x}_{p}+\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{{k}_{M}}{R{J}_{M}}\end{array}\right]V\\ {\theta}_{L}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]{x}_{p}\\ T=\left[\begin{array}{cccc}{k}_{T}& 0& -\frac{{k}_{T}}{\rho}& 0\end{array}\right]{x}_{p}\end{array}$$

The controller must set the angular position of the load,
*θ _{L}*, at a desired value by adjusting the
applied voltage,

*V*.

However, since the elastic shaft has a finite shear strength, the torque,
*T*, must stay within the range |*T*| ≤ 78.5 Nm. Also, the voltage source physically limits the applied voltage to the
range |*V*| ≤ 220 V.

### Construct Plant Model

Specify the model constants (units are in MKS).

Kt = 1280.2; % Torsional rigidity Km = 10; % Motor constant Jm = 0.5; % Motor inertia Jl = 50*Jm; % Load inertia N = 20; % Gear ratio Bm = 0.1; % Rotor viscous friction Bl = 25; % Load viscous friction R = 20; % Armature resistance

Define the state-space matrices derived from the model equations.

A = [ 0 1 0 0; -Kt/Jl -Bl/Jl Kt/(N*Jl) 0; 0 0 0 1; Kt/(Jm*N) 0 -Kt/(Jm*N^2) -(Bm+Km^2/R)/Jm]; B = [0; 0; 0; Km/(R*Jm)]; C = [ 1 0 0 0; Kt 0 -Kt/N 0]; D = [0; 0];

Create a state-space model.

plant = ss(A,B,C,D);

### Open MPC Designer App

mpcDesigner

### Import Plant and Define Signal Configuration

In **MPC Designer**, on the **MPC Designer** tab, select
**MPC Structure**.

In the Define MPC Structure By Importing dialog box, select the
`plant`

plant model, and assign the plant I/O channels to the following
signal types:

Manipulated variable — Voltage,

*V*Measured output — Load angular position,

*θ*_{L}Unmeasured output — Torque,

*T*

Click **Import**.

**MPC Designer** imports the specified plant and creates an MPC controller and
a simulation scenario:

`mpc1`

— Default MPC controller created using`plant`

as its internal model.`scenario1`

— Default simulation scenario. The results of this simulation are displayed in the**Input Response**and**Output Response**plots.

Plants, controllers and simulation scenarios are accessible via the data browser, on
the on the left hand side of **MPC Designer**.

### Define Input and Output Channel Attributes

On the **MPC Designer** tab, in the **Structure**
section, click **I/O Attributes**.

In the Input and Output Channel Specifications dialog box, for each input and output channel:

Specify a meaningful

**Name**and**Unit**.Keep the

**Nominal Value**at its default value of`0`

.Specify a

**Scale Factor**for normalizing the signal. Select a value that approximates the predicted operating range of the signal:Channel Name Minimum Value Maximum Value Scale Factor `Voltage`

–220 V 220 V `440`

`Theta`

–π radians π radians `6.28`

`Torque`

–78.5 Nm 78.5 Nm `157`

Click **OK** to update the channel attributes and close the dialog
box.

### Modify Scenario To Simulate Angular Position Step Response

In the **Scenario** section, **Edit Scenario**
drop-down list, select `scenario1`

to modify the default
simulation scenario.

In the Simulation Scenario dialog box, keep a **Simulation duration**
of `10`

seconds.

In the **Reference Signals** table, keep the default configuration
for the first channel. These settings create a `Step`

change of
`1`

radian in the angular position setpoint at a
**Time** of `1`

second.

For the second output, in the **Signal** drop-down list, select
`Constant`

to keep the torque setpoint at its nominal
value.

Click **OK**.

The app runs the simulation with the new scenario settings and updates the
**Input Response** and **Output Response**
plots.

### Specify Controller Sample Time and Horizons

On the **Tuning** tab, in the **Horizon** section,
specify a **Sample time** of `0.1`

seconds.

For the specified sample time, *T _{s}*, and a
desired response time of

*T*= 2 seconds, select a prediction horizon,

_{r}*p*, such that:

$${T}_{r}\approx p{T}_{s}.$$

Therefore, specify a **Prediction horizon** of
`20`

.

Specify a **Control horizon** of `5`

.

As you update the sample time and horizon values, the **Input
Response** and **Output Response** plots update automatically.
Both the input voltage and torque values exceed the constraints defined in the system
model specifications.

### Specify Constraints

In the **Design** section, select
**Constraints**.

In the Constraints dialog box, in the **Input Constraints** section,
specify the **Min** and **Max** voltage values for the
manipulated variable (MV).

In the **Output Constraints** section, specify
**Min** and **Max** torque values for the unmeasured
output (UO).

There are no additional constraints, that is the other constraints remain at their
default maximum and minimum values, `—Inf`

and
`Inf`

respectively

Click **OK**.

The response plots update to reflect the new constraints. In the **Input
Response** plot, there are undesirable large changes in the input
voltage.

### Specify Tuning Weights

In the **Design** section, select
**Weights**.

In the Weights dialog box, in the **Input Weights** table, increase
the manipulated variable **Rate Weight**.

The tuning **Weight** for the manipulated variable (MV) is
`0`

. This weight indicates that the controller can allow the input
voltage to vary within its constrained range. The increased **Rate
Weight** limits the size of manipulated variable changes.

Since the control objective is for the angular position of the load to track its
setpoint, the tuning **Weight** on the measured output is
`1`

. There is no setpoint for the applied torque, so the controller can
allow the second output to vary within its constraints. Therefore, the
**Weight** on the unmeasured output (UO) is `0`

, which
enables the controller to ignore the torque setpoint.

Click **OK**.

The response plots update to reflect the increased rate weight. The **Input
Response** is smoother with smaller voltage changes.

### Examine Output Response

In the **Output Response** plot, right-click the
**Theta** plot area, and select **Characteristics** > **Peak Response**.

The peak output response occurs at time of 3 seconds with a maximum overshoot of 3%. Since the reference signal step change is at 1 second, the controller has a peak time of 2 seconds.

### Improve Controller Response Time

Click and drag the **Closed-Loop Performance** slider to the right to
produce a more **Aggressive** response. The further you drag the slider
to the right, the faster the controller responds. Select a slider position such that the
peak response occurs at 2.6 seconds.

The final controller peak time is 1.6 seconds. Reducing the response time further results in overly-aggressive input voltage changes.

### Generate and Run MATLAB Script

In the **Analysis** section, click the **Export
Controller** arrow .

Under **Export Controller**, click ```
Generate
Script
```

.

In the Generate MATLAB^{®} Script dialog box, check the box next to
`scenario1`

.

Click **Generate Script**.

The app exports a copy of the plant model, `plant_C`

, to the
MATLAB workspace, along with simulation input and reference signals.

Additionally, the app generates the following code in the MATLAB Editor.

%% create MPC controller object with sample time mpc1 = mpc(plant_C, 0.1); %% specify prediction horizon mpc1.PredictionHorizon = 20; %% specify control horizon mpc1.ControlHorizon = 5; %% specify nominal values for inputs and outputs mpc1.Model.Nominal.U = 0; mpc1.Model.Nominal.Y = [0;0]; %% specify scale factors for inputs and outputs mpc1.MV(1).ScaleFactor = 440; mpc1.OV(1).ScaleFactor = 6.28; mpc1.OV(2).ScaleFactor = 157; %% specify constraints for MV and MV Rate mpc1.MV(1).Min = -220; mpc1.MV(1).Max = 220; %% specify constraints for OV mpc1.OV(2).Min = -78.5; mpc1.OV(2).Max = 78.5; %% specify overall adjustment factor applied to weights beta = 1.2712; %% specify weights mpc1.Weights.MV = 0*beta; mpc1.Weights.MVRate = 0.4/beta; mpc1.Weights.OV = [1 0]*beta; mpc1.Weights.ECR = 100000; %% specify simulation options options = mpcsimopt(); options.RefLookAhead = 'off'; options.MDLookAhead = 'off'; options.Constraints = 'on'; options.OpenLoop = 'off'; %% run simulation sim(mpc1, 101, mpc1_RefSignal, mpc1_MDSignal, options);

In the MATLAB Window, in the **Editor** tab, select
**Save**.

Complete the Save dialog box and then click **Save**.

In the **Editor** tab, click **Run**.

The script creates the controller, `mpc1`

, and runs the simulation
scenario. The input and output responses match the simulation results from the app.

### Validate Controller Performance In Simulink

If you have a Simulink^{®} model of your system, you can simulate your controller and validate its
performance.

Open the model.

`open_system('mpc_motor')`

This model uses an MPC Controller block to control a servomechanism
plant. The Servomechanism Model block is already configured to use the
`plant`

model from the MATLAB workspace.

The Angle reference source block creates a sinusoidal reference signal
with a frequency of `0.4`

rad/sec and an amplitude of
*π*.

Double-click the MPC Controller block.

In the MPC Controller Block Parameters dialog box, specify an **MPC
Controller** from the MATLAB workspace. Use the `mpc1`

controller created using the
generated script.

Click **OK**.

At the MATLAB command line, specify a torque magnitude constraint variable.

tau = 78.5;

The model uses this value to plot the constraint limits on the torque output scope.

In the Simulink model window, click **Run** to simulate the model.

In the **Angle** scope, the output response, yellow, tracks the
angular position setpoint, blue, closely.

## See Also

MPC
Designer | `mpc`

| MPC
Controller