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Simulate lane-keeping assistance using adaptive model predictive controller

**Library:**Model Predictive Control Toolbox / Automated Driving

The Lane Keeping Assist System block simulates a lane keeping assist (LKA) system that keeps an ego vehicle traveling along the center of a straight or curved road by adjusting the front steering angle. The controller reduces the lateral deviation and relative yaw angle of the ego vehicle with respect to the lane centerline. The block computes optimal control actions while satisfying steering angle constraints using adaptive model predictive control (MPC).

To customize your controller, for example to use advanced MPC features or modify
controller initial conditions, click **Create LKA subsystem**.

`Curvature`

— Road curvaturescalar

Road curvature, specified as 1/*R*, where *R* is the radius of the curve in meters.

The road curvature is:

Positive when the road curves toward the positive Y axis of the global coordinate system.

Negative when the road curves toward the negative Y axis of the global coordinate system.

Zero for a straight road.

The controller models the road curvature as a measured disturbance with previewing. You can specify the curvature as a:

Scalar signal — Specify the curvature for the current control interval. The controller uses this curvature value across the prediction horizon.

Vector signal with length less than or equal to the

**Prediction Horizon**— Specify the current and predicted curvature values across the prediction horizon. If the length of the vector is less than the prediction horizon, then the controller uses the final curvature value in the vector for the remainder of the prediction horizon.

`Longitudinal velocity`

— Ego vehicle velocitynonnegative scalar

Ego vehicle velocity in m/s.

`Lateral deviation`

— Ego vehicle lateral deviationscalar

Ego vehicle lateral deviation in meters from the centerline of the lane. The lateral deviation
*e*_{1} is positive when the ego vehicle is to
the right of the centerline and negative when the ego vehicle is to the left of the
centerline.

`Relative yaw angle`

— Angle from lane centerlinescalar

Ego vehicle longitudinal axis angle in radians from the centerline of the lane, defined as:

$${e}_{2}={\theta}_{e}-{\theta}_{c}$$

Here, *θ _{e}* is the ego vehicle angle and

`Minimum steering angle`

— Minimum front steering anglescalar

Minimum front steering angle constraint in radians. Use this input port when the minimum steering angle varies at run time.

To enable this port, select **Use external source** for the **Minimum steering angle** parameter.

`Maximum steering angle`

— Maximum front steering anglescalar

Maximum front steering angle constraint in radians. Use this input port when the maximum steering angle varies at run time.

To enable this port, select **Use external source** for the **Maximum steering angle** parameter.

`Enable optimization`

— Controller optimization enable signalscalar

Controller optimization enable signal. When this signal is:

Nonzero, the controller performs optimization calculations and generates a

**Steering angle**control signal.Zero, the controller does not perform optimization calculations. In this case, the

**Steering angle**output signal remains at the value it had when the optimization was disabled. The controller continues to update its internal state estimates.

To enable this port, select the **Use external signal to
enable or disable optimization** parameter.

`External control signal`

— Steering angle applied to ego vehiclescalar

Actual steering angle in radians applied to the ego vehicle. The controller uses this signal to estimate the ego vehicle model states. Use this input port when the control signal applied to the ego vehicle does not match the optimal control signal computed by the model predictive controller. This mismatch can occur when, for example:

The Lane Keeping Assist System is not the active controller. Maintaining an accurate state estimate when the controller is not active prevents bumps in the control signal when the controller becomes active.

The steering actuator fails and does not provide the correct control signal to the ego vehicle.

To enable this port, select the **Use external control
signal for bumpless transfer between PFC and other
controllers** parameter.

`Vehicle dynamics matrix A`

— State matrix of ego vehicle predictive modelsquare matrix

State matrix of ego vehicle predictive model. The number of rows in the state matrix corresponds to the number of states in the predictive model. This matrix must be square.

The ego vehicle predictive model defined by **Vehicle dynamics matrix
A**, **Vehicle dynamics matrix B**, and **Vehicle
dynamics matrix C** must be minimal.

To enable this port, select the **Use vehicle model**
parameter.

`Vehicle dynamics matrix B`

— Input-to-state matrix of ego vehicle predictive modelcolumn vector

Input-to-state matrix of ego vehicle predictive model. The number of rows in this signal must match the number of rows in **Vehicle dynamics matrix A**.

The ego vehicle predictive model defined by **Vehicle dynamics matrix A**, **Vehicle dynamics matrix B**, and **Vehicle dynamics matrix C** must be minimal.

To enable this port, select the **Use vehicle model** parameter.

`Vehicle dynamics matrix C`

— State-to-output matrix of ego vehicle predictive modelmatrix with two rows

State-to-output matrix of ego vehicle predictive model. The number of columns in this signal must match the number of rows in **Vehicle dynamics matrix A**.

The ego vehicle predictive model defined by **Vehicle dynamics matrix A**, **Vehicle dynamics matrix B**, and **Vehicle dynamics matrix C** must be minimal.

To enable this port, select the **Use vehicle model** parameter.

`Steering angle`

— Front steering angle control signalscalar

Front steering angle control signal in radians generated by the controller. The front steering angle is the angle of the front tires from the longitudinal axis of the vehicle. The steering angle is positive towards the positive lateral axis of the ego vehicle.

`Use vehicle parameters`

— Define ego vehicle model using vehicle properties`on`

(default) | `off`

Select this parameter to define the ego vehicle model used by the MPC controller by specifying properties of the ego vehicle. The ego vehicle model is the linear model from the front steering angle to the lateral velocity and yaw angle rate. For more information, see Ego Vehicle Predictive Model.

To define the vehicle model, specify the following block parameters:

**Total mass****Yaw moment of inertia****Longitudinal distance from center of gravity to front tires****Longitudinal distance from center of gravity to rear tires****Cornering stiffness of front tires****Cornering stiffness of rear tires**

For more information on the ego vehicle model, see Ego Vehicle Predictive Model.

Selecting this parameter clears the **Use vehicle
model** parameter.

Block Parameter:
`ModelType` |

Type: string, character
vector |

Default:
`"Use vehicle parameters"` |

`Use vehicle model`

— Define ego vehicle model using state-space matrices`off`

(default) | `on`

Select this parameter to define the state-space matrices of the ego vehicle model used by the MPC controller. This model is the linear model from the front steering angle in radians to the lateral velocity in meters per second and yaw angle rate in radians per second. For more information on the ego vehicle model, see Ego Vehicle Predictive Model.

To define the initial internal model, specify the
**A**, **B**, and
**C** state-space matrices. The internal model must
be a minimal realization with no direct feedthrough, and the dimensions
of **A**, **B**, and
**C** must be consistent.

Typically, the ego vehicle steering model is velocity-dependent, and
therefore, it varies over time. To update the internal model at run
time, use the **Vehicle dynamics A**, **Vehicle
dynamics B**, and **Vehicle dynamics C**
input ports.

Selecting this parameter clears the **Use vehicle
parameters** parameter.

Block Parameter:
`ModelType` |

Type: string, character
vector |

Default:
`"Use vehicle parameters"` |

`Total mass`

— Ego vehicle mass`1575`

(default) | positive scalarEgo vehicle mass in kg.

To enable this parameter, select the **Use vehicle parameters** parameter.

Block Parameter:
`VehicleMass` |

Type: string, character vector |

Default:
`"1575"` |

`Yaw moment of inertia`

— Moment of inertia about the ego vehicle vertical axis`2875`

(default) | positive scalarMoment of inertia about the ego vehicle vertical axis in
Kg·m^{2}.

To enable this parameter, select the **Use vehicle parameters** parameter.

Block
Parameter:
`VehicleYawInertia` |

Type:
string, character vector |

Default:
`"2875"` |

`Longitudinal distance from center of gravity to front tires`

— Distance from the ego vehicle center of mass to its front tires`1.2`

(default) | positive scalarDistance from the ego vehicle center of mass to its front tires in meters, measured along the longitudinal axis of the vehicle.

To enable this parameter, select the **Use vehicle parameters** parameter.

Block Parameter:
`LengthToFront` |

Type: string, character vector |

Default:
`"1.2"` |

`Longitudinal distance from center of gravity to rear tires`

— Distance from the ego vehicle center of mass to its rear tires`1.6`

(default) | positive scalarDistance from the ego vehicle center of mass to its rear tires in meters, measured along the longitudinal axis of the vehicle.

To enable this parameter, select the **Use vehicle parameters** parameter.

Block Parameter:
`LengthToRear` |

Type: string, character vector |

Default:
`"1.6"` |

`Cornering stiffness of front tires`

— Front tire stiffness`19000`

(default) | positive scalarFront tire stiffness in N/rad, defined as the relationship between the side force on the front tires and the angle of the tires to the longitudinal axis of the vehicle.

To enable this parameter, select the **Use vehicle parameters** parameter.

Block Parameter:
`FrontTireStiffness` |

Type: string, character vector |

Default:
`"19000"` |

`Cornering stiffness of rear tires`

— Rear tire stiffness`33000`

(default) | positive scalarRear tire stiffness in N/rad, defined as the relationship between the side force on the rear tires and the angle of the tires to the longitudinal axis of the vehicle.

To enable this parameter, select the **Use vehicle parameters** parameter.

Block Parameter:
`RearTireStiffness` |

Type: string, character vector |

Default:
`"33000"` |

`A`

— Initial state matrix of ego vehicle predictive modelsquare matrix

Initial state matrix of ego vehicle predictive model. The number of rows in the state matrix corresponds to the number of states in the predictive model. This matrix must be square.

The initial ego vehicle predictive model defined by **A**, **B**, and **C** must be minimal.

Typically, the ego vehicle model varies over time. To update the state matrix at run time, use the **Vehicle dynamics A** input port.

To enable this parameter, select the **Use vehicle model** parameter.

Block Parameter:
`EgoModelMatrixA` |

Type: string, character
vector |

Default:
```
"[-4.4021
,-12.4603;1.3913,-5.1868]"
``` |

`B`

— Initial input-to-state matrix of ego vehicle predictive modelcolumn vector

Initial input-to-state matrix of ego vehicle predictive model. The number of rows in this parameter must match the number of rows in **A**.

The initial ego vehicle predictive model defined by **A**, **B**, and **C** must be minimal.

Typically, the ego vehicle model varies over time. To update the input-to-state matrix at run time, use the **Vehicle dynamics B** input port.

To enable this parameter, select the **Use vehicle model**
parameter.

Block Parameter:
`EgoModelMatrixB` |

Type: string, character vector |

Default:
`"[24.1270;15.8609]"` |

`C`

— Initial state-to-output matrix of ego vehicle predictive modelmatrix with two rows

Initial state-to-output matrix of ego vehicle predictive model. The number of columns in this parameter must match the number of rows in **A**.

The initial ego vehicle predictive model defined by **A**, **B**, and **C** must be minimal.

Typically, the ego vehicle model varies over time. To update the state-to-output matrix at run time, use the **Vehicle dynamics C** input port.

To enable this parameter, select the **Use vehicle model**
parameter.

Block Parameter:
`EgoModelMatrixC` |

Type: string, character vector |

Default:
`"[1,0;0,1]"` |

`Initial longitudinal velocity`

— Initial velocity of the ego vehicle`15`

(default) | positive scalarInitial velocity of the ego vehicle model when the lane-keeping assist is enabled in m/s. This velocity can differ from the actual ego vehicle initial velocity.

**Note**

A very small initial velocity, for example `eps`

,
can produce a nonminimal realization for the controller plant model,
causing an error. To prevent this error, set the initial velocity to
a larger value, for example `1e-3`

.

Block Parameter:
`InitialLongVel` |

Type: string, character
vector |

Default:
`"15"` |

`Transport lag between model inputs and outputs`

— Total transport lag in ego vehicle model`0`

(default) | nonnegative scalarTotal transport lag, *τ*, in the ego vehicle model in
seconds. This lag includes actuator, sensor, and communication lags. For
each input-output channel, the transport lag is approximated by:

$$\frac{1}{\tau s+1}$$

Block Parameter:
`TransportLag` |

Type: string, character
vector |

Default:
`"0"` |

`Minimum steering angle`

— Minimum front steering angle`-0.26`

(default) | scalar between `-pi/2`

and `pi/2`

Minimum front steering angle constraint in radians.

If the minimum steering angle varies over time, add the **Minimum steering
angle** input port to the block by selecting **Use external
source**.

This parameter must be less than the **Maximum steering angle** parameter.

Block
Parameter:
`MinSteering` |

Type:
string, character vector |

Default:
`"-0.26"` |

`Maximum steering angle`

— Maximum front steering angle`0.26`

(default) | scalar between `-pi/2`

and `pi/2`

Maximum front steering angle constraint in radians.

If the maximum steering angle varies over time, add the **Maximum steering
angle** input port to the block by selecting **Use external
source**.

This parameter must be greater than the **Minimum steering angle** parameter.

Block
Parameter:
`MaxSteering` |

Type:
string, character vector |

Default:
`"0.26"` |

`Sample time`

— Controller sample time`0.1`

(default) | positive scalarController sample time in seconds.

Block Parameter:
`Ts` |

Type: string, character vector |

Default:
`"0.1"` |

`Prediction horizon`

— Controller prediction horizon`10`

(default) | positive integerController prediction horizon steps. The controller prediction time is the product of the sample time and the prediction horizon.

Block Parameter:
`PredictionHorizon` |

Type: string, character vector |

Default:
`"30"` |

`Controller behavior`

— Closed-loop controller performance`0.5`

(default) | scalar between `0`

and `1`

Closed-loop controller performance. The default parameter value provides a balanced controller design. Specifying a:

Smaller value produces a more robust controller with smoother control actions.

Larger value produces a more aggressive controller with a faster response time.

When you modify this parameter, the change is applied to the controller immediately.

Block Parameter:
`ControllerBehavior` |

Type: string, character
vector |

Default:
`"0.5"` |

`Use suboptimal solution`

— Apply suboptimal solution after specified number of iterations`off`

(default) | `on`

Configure the controller to apply a suboptimal solution after a specified maximum number of iterations, which guarantees the worst-case execution time for your controller.

For more information, see Suboptimal QP Solution.

After selecting this parameter, specify the **Maximum iteration number** parameter.

Block Parameter:
`suboptimal` |

Type: string, character vector |

Default:
`"off"` |

`Maximum iteration number`

— Maximum optimization iterations`10`

(default) | positive integerMaximum number of controller optimization iterations.

To enable this parameter, select the **Use suboptimal solution**
parameter.

Block
Parameter:
`maxiter` |

Type:
string, character vector |

Default:
`"10"` |

`Use external signal to enable or disable optimization`

— Add port for enabling optimization`off`

(default) | `on`

To add the **Enable optimization** input port to the block, select this
parameter.

Block
Parameter:
`optmode` |

Type:
string, character vector |

Default:
`"off"` |

`Use external signal for bumpless transfer between LKA and other controllers`

— Add external control signal input port`off`

(default) | `on`

To add the **External control signal** input port to
the block, select this parameter.

Block Parameter:
`trackmode` |

Type: string, character
vector |

Default:
`"off"` |

`Create LKA subsystem`

— Create custom controllerbutton

Generate a custom LKA subsystem, which you can modify for your
application. The controller configuration data for the custom controller
is exported to the MATLAB^{®} workspace as a structure.

You can modify the custom controller subsystem to:

Modify default MPC settings or use advanced MPC features.

Modify the default controller initial conditions.

The default ego vehicle predictive model is the following state-space model:

$$\begin{array}{l}A=\left[\begin{array}{cc}-2\left({C}_{F}+{C}_{R}\right)/m/{V}_{X}& -{V}_{X}-2\left({C}_{F}{L}_{F}-{C}_{R}{L}_{R}\right)/m/{V}_{X}\\ -2\left({C}_{F}{L}_{F}-{C}_{R}{L}_{R}\right)/{I}_{Z}/{V}_{X}& -2\left({C}_{F}{L}_{F}^{2}+{C}_{R}{L}_{R}^{2}\right)/{I}_{Z}/{V}_{X}\end{array}\right]\\ B=2{C}_{F}\left[\begin{array}{c}1/m\\ {L}_{F}/{I}_{Z}\end{array}\right]\\ C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ D=\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$$

Here:

*V*is the longitudinal velocity of the car. At the start of the simulation, this velocity is equal to the_{X}**Initial condition for longitudinal velocity**parameter. At run time, this velocity is equal to the**Longitudinal velocity**input signal.*m*is the**Total mass**parameter.*I*is the_{Z}**Yaw moment of inertia**parameter.*L*is the_{F}**Longitudinal distance from center of gravity to front tires**parameter.*L*is the_{R}**Longitudinal distance from center of gravity to rear tires**parameter.*C*is the_{F}**Cornering stiffness of front tires**parameter.*C*is the_{R}**Cornering stiffness of rear tires**parameter.

The input to this model is the steering angle in radians, and the outputs are the lateral velocity in meters per second and yaw angle rate in radians per second.

To define a different ego vehicle predictive model, select the **Use
vehicle model** parameter, and specify the initial state-space model.
Then, specify the run-time values of the state-space matrices using the
**Vehicle dynamics A**, **Vehicle dynamics
B**, and **Vehicle dynamics C** input signals.

The controller creates its internal predictive model by augmenting the ego vehicle dynamic model. The augmented model includes the road curvature as a measured disturbance input signal.

By default, the model predictive controller assumes the following initial conditions for the ego vehicle:

Longitudinal velocity is equal to the

**Initial longitudinal velocity**parameter.Lateral velocity is zero.

Steering angle is zero.

Yaw angle rate is zero.

If the initial conditions in your model do not match these conditions, the
**Steering angle** output can exhibit an initial bump at the
start of the simulation.

To modify the controller initial conditions to match your simulation, create a
custom lane-keeping control system by, on the **Block** tab,
clicking **Create LKA subsystem**.

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