# Path Following Control System

Simulate path-following control using adaptive model predictive controller

• Library:
• Model Predictive Control Toolbox / Automated Driving

• ## Description

The Path Following Control System block simulates a path-following control (PFC) system that keeps an ego vehicle traveling along the center of a straight or curved road while tracking a set velocity and maintaining a safe distance from a lead vehicle. To do so, the controller adjusts both the longitudinal acceleration and front steering angle of the ego vehicle. The block computes optimal control actions while satisfying safe distance, velocity, acceleration, and steering angle constraints using adaptive model predictive control (MPC).

This block combines the capabilities of the Lane Keeping Assist System and Adaptive Cruise Control System blocks into a single controller. To customize your controller, for example to use advanced MPC features or modify controller initial conditions, click .

## Ports

### Input

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Ego vehicle velocity setpoint in m/s. When there is no lead vehicle, the controller tracks this velocity.

Safe time gap in seconds between the lead vehicle and the ego vehicle. This time gap is used to calculate the minimum safe following distance constraint. For more information, see Safe Following Distance.

Distance in meters between lead vehicle and ego vehicle. To calculate this signal, subtract the ego vehicle position from the lead vehicle position.

Velocity difference in meters per second between lead vehicle and ego vehicle. To calculate this signal, subtract the ego vehicle velocity from the lead vehicle velocity.

Ego vehicle velocity in m/s.

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

• 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.

Ego vehicle lateral deviation in meters from the centerline of the lane.

Ego vehicle longitudinal axis angle in radians from the centerline of the lane.

Minimum ego vehicle longitudinal acceleration constraint in m/s2. Use this input port when the minimum acceleration varies at run time.

#### Dependencies

To enable this port, select Use external source for the Minimum longitudinal acceleration parameter.

Maximum ego vehicle longitudinal acceleration constraint in m/s2. Use this input port when the maximum acceleration varies at run time.

#### Dependencies

To enable this port, select Use external source for the Maximum longitudinal acceleration parameter.

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

#### Dependencies

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

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

#### Dependencies

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

Controller optimization enable signal. When this signal is:

• Nonzero, the controller performs optimization calculations and generates the Longitudinal acceleration and Steering angle control signals.

• Zero, the controller does not perform optimization calculations. In this case, the Longitudinal acceleration and Steering angle output signals remain at the values they had when the optimization was disabled. The controller continues to update its internal state estimates.

#### Dependencies

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

Actual control signals applied to the ego vehicle. The first element of this signal is the longitudinal acceleration in m/s2, and the second element is the steering angle in radians. The controller uses these signals to estimate the ego vehicle model states. Use this input port when the control signals applied to the ego vehicle do not match the optimal control signals computed by the model predictive controller. This mismatch can occur when, for example:

• The Path Following Control System is not the active controller. Maintaining an accurate state estimate when the controller is not active prevents bumps in the control signals when the controller becomes active.

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

#### Dependencies

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

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.

#### Dependencies

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

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.

#### Dependencies

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

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.

#### Dependencies

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

### Output

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Acceleration control signal in m/s2 generated by the controller.

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.

## Parameters

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### Parameters Tab

#### Ego Vehicle

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 longitudinal acceleration and front steering angle to the longitudinal velocity, lateral velocity, and yaw angle rate.

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

• Longitudinal acceleration tracking time constant

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

Selecting this parameter clears the Use vehicle model parameter.

Select this parameter to define the state-space matrices of the ego vehicle model used by the MPC controller. The ego vehicle model is the linear model from the longitudinal acceleration and front steering angle to the longitudinal velocity, lateral velocity, and yaw angle rate.

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 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.

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

Selecting this parameter clears the Use vehicle parameters parameter.

Ego vehicle mass in kg.

#### Dependencies

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

Moment of inertia about the ego vehicle vertical axis in mNs2.

#### Dependencies

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

Distance from the ego vehicle center of mass to its front tires in meters, measured along the longitudinal axis of the vehicle.

#### Dependencies

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

Distance from the ego vehicle center of mass to its rear tires in meters, measured along the longitudinal axis of the vehicle.

#### Dependencies

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

Front 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.

#### Dependencies

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

Rear 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.

#### Dependencies

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

Time constant for tracking longitudinal acceleration, specified in seconds.

#### Dependencies

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

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.

#### Dependencies

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

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.

#### Dependencies

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

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.

#### Dependencies

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

Initial velocity of the ego vehicle model in m/s, which can differ from the actual ego vehicle initial velocity.

This value is used to configure the initial conditions of the model predictive controller. For more information, see Initial Conditions.

### 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`.

Total 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 model is:

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

#### Spacing Control

To configure the safe following distance, set the Default spacing parameter. For more information on the safe following distance used by the controller, see Safe Following Distance.

Minimum spacing in meters between the lead vehicle and the ego vehicle. This value corresponds to the target relative distance between the ego and lead vehicles when the ego vehicle velocity is zero.

This value is used to calculate the:

#### Dependencies

To enable this parameter, select the Maintain safe distance between lead vehicle and ego vehicle parameter.

### Controller Tab

#### Path Following Controller Constraints

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.

#### Dependencies

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

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.

#### Dependencies

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

Minimum ego vehicle longitudinal acceleration constraint in m/s2.

If the minimum acceleration varies over time, add the Minimum longitudinal acceleration input port to the block by selecting Use external source.

Maximum ego vehicle longitudinal acceleration constraint in m/s2.

If the maximum acceleration varies over time, add the Maximum longitudinal acceleration input port to the block by selecting Use external source.

#### Model Predictive Controller Settings

Controller sample time in seconds.

Controller prediction horizon steps. The controller prediction time is the product of the sample time and the prediction horizon.

Controller control horizon, specified as one of the following:

• Positive integer less than or equal to the Prediction horizon parameter. In this case, the controller computes m free control moves occurring at times k through k+m-1, and holds the controller output constant for the remaining prediction horizon steps from k+m through k+p-1. Here, k is the current control interval.

• Vector of positive integers, [m1, m2, …], where the sum of the integers equals the Prediction horizon parameter. In this case, the controller computes M blocks of free moves, where M is the length of the control horizon vector. The first free move applies to times k through k+m1-1, the second free move applies from time k+m1 through k+m1+m2-1, and so on. Using block moves can improve the robustness of your controller.

#### Controller Behavior

Tuning weight for longitudinal velocity tracking. To produce smaller velocity-tracking errors, increase this weight.

Tuning weight for lateral error. To produce smaller lateral errors, increase this weight.

Tuning weight for changes in longitudinal acceleration. To produce less-aggressive vehicle acceleration, increase this weight.

Tuning weight for changes in steering angle. To produce less-aggressive steering angle changes, increase this weight.

### Block Tab

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.

#### Dependencies

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

Maximum number of controller optimization iterations.

#### Dependencies

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

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

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

Generate a custom PFC subsystem, which you can modify for your application. The 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.

• Use different application settings, such as a custom safe following distance definition.

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