PID Controller

This page documents the Proportional–integral–derivative (PID) controller, which can be used for both longitudinal and lateral control. Based on the PID controller, a controller for decoupled longitudinal and lateral motion control (PIDLongLat) is presented afterwards.

Control Law

In continuous-time, the control input at time $t$ is obtained by evaluating

\[ u(t)= K_p e(t) + K_i \int_{0}^{t} e(\tau) d\tau + K_d \dot{e}(t), \]

with the error $e(t) = y(t) - y_{\mathrm{ref}}(t)$ between the measured output $y(t)$ and the reference output $y_{\mathrm{ref}}(t)$ and the gains $K_{\square}$. To obtain the discrete-time version of the control law, we employ the following simplifications

\[ \int_{0}^{t} e(\tau) d\tau \approx \Delta t \sum_{j=0}^{k} e(j) \qquad t=k{\Delta t}, $$ $$ \dot{e}(t) = \frac{e(k) - e(k-1)}{\Delta t}, \]

and, thus, obtain $$ u(k)= K_p e(k) + K_i \Delta t \sum_{j=0}^{k} e(j) d\tau + K_d \frac{e(k) - e(k-1)}{\Delta t}. $$

Interface Implementation

The PIDControl class implements the ControllerInterface.

Constructor

pid = PIDControl(
    kp, 
    ki, 
    kd, 
    delta_t
)

with the parameters:

Parameter	Variable type	Description
`kp`	`float`	Proportional gain
`ki`	`float`	Integral gain
`kd`	`float`	Derivative gain
`delta_t`	`float`	Sampling time in seconds

Computation of Control Input

Given the measured output measured_output and the reference output reference_output, the control input is obtained by running:

u = pid.compute_control_input(
        measured_output,
        reference_output,
)

Decoupled Longitudinal and Lateral PID-Control

The PIDLongLat controller combines two PID controllers for longitudinal and lateral control, i.e., the coupling between the longitudinal and lateral motion is neglected when computing the control inputs. The following outputs for tracking and control inputs are chosen:

	Controlled variable	Control input
Longitudinal control	Velocity	Longitudinal acceleration
Lateral control	Lateral offset from reference trajectory	Steering angle velocity

Controller Interface Implementation

The PIDLongLat class implements the ControllerInterface.

Constructor

pid_long_lat = PIDLongLat(        
    kp_long,
    ki_long,
    kd_long,
    kp_lat,
    ki_lat,
    kd_lat,
    delta_t
)

with the parameters:

Parameter	Variable type	Description
`kp_long`	`float`	Proportional gain
`ki`	`float`	Integral gain
`kd`	`float`	Derivative gain
`delta_t`	`float`	Sampling time in seconds

Computation of Control Input

Given the measured output measured_output and the reference output reference_output, the control input is obtained by running:

u = pid.compute_control_input(
    measured_v_long,
    reference_v_long,
    measured_lat_offset,
    reference_lat_offset
)