Longitudinal speedtracking controller
Powertrain Blockset / Vehicle Scenario Builder
Vehicle Dynamics Blockset / Vehicle Scenarios / Driver
The Longitudinal Driver block implements a longitudinal speedtracking controller. Based on reference and feedback velocities, the block generates normalized acceleration and braking commands that can vary from 0 through 1. You can use the block to model the dynamic response of a driver or to generate the commands necessary to track a longitudinal drive cycle.
Use the Control type, cntrlType parameter to specify one of these control options.
Setting  Block Implementation 

 Proportionalintegral (PI) control with tracking windup and feedforward gains. 
 PI control with tracking windup and feedforward gains that are a function of vehicle velocity. 
 Optimal singlepoint preview (look ahead) control model developed by C. C. MacAdam^{1, 2, 3}. The model represents driver steering control behavior during pathfollowing and obstacle avoidance maneuvers. Drivers preview (look ahead) to follow a predefined path. To implement the MacAdam model, the block:

Use the Shift type, shftType parameter to specify one of these shift options.
Setting  Block Implementation 

 No transmission. Block outputs a constant gear of 1. Use this setting to minimize the number of parameters you need to generate acceleration and braking commands to track forward vehicle motion. This setting does not allow reverse vehicle motion. 
 Block uses a Stateflow^{®} chart to model reverse, neutral, and drive gear shift scheduling. Use this setting to generate acceleration and braking commands to track forward and reverse vehicle motion using simple reverse, neutral, and drive gear shift scheduling. Depending on the vehicle state and vehicle velocity feedback, the block uses the initial gear and time required to shift to shift the vehicle up into drive or down into reverse or neutral. For neutral gears, the block uses braking commands to control the vehicle speed. For reverse gears, the block uses an acceleration command to generate torque and a brake command to reduce vehicle speed. 
 Block uses a Stateflow chart to model reverse, neutral, park, and Nspeed gear shift scheduling. Use this setting to generate acceleration and braking commands to track forward and reverse vehicle motion using reverse, neutral, park, and Nspeed gear shift scheduling. Depending on the vehicle state and vehicle velocity feedback, the block uses these parameters to determine the:
For neutral gears, the block uses braking commands to control the vehicle speed. For reverse gears, the block uses an acceleration command to generate torque and a brake command to reduce vehicle speed. 
 Block uses the input gear, vehicle state, and velocity feedback to generate acceleration and braking commands to track forward and reverse vehicle motion. For neutral gears, the block uses braking commands to control the vehicle speed. For reverse gears, the block uses an acceleration command to generate torque and a brake command to reduce vehicle speed. 
If you set the control type to PI
or Scheduled
PI
, the block implements proportionalintegral (PI) control with tracking
windup and feedforward gains. For the Scheduled PI
configuration, the block uses feed forward gains that are a function of vehicle
velocity.
To calculate the speed control output, the block uses these equations.
Setting  Equation 


$$y=\frac{{K}_{ff}}{{v}_{nom}}{v}_{ref}+\frac{{K}_{p}{e}_{ref}}{{v}_{nom}}+\left(\frac{{K}_{i}}{{v}_{nom}}+{K}_{aw}{e}_{out}\right){\displaystyle \int {e}_{ref}dt+{K}_{g}\theta}$$


$$y=\frac{{K}_{ff}(v)}{{v}_{nom}}{v}_{ref}+\frac{{K}_{p}(v){e}_{ref}}{{v}_{nom}}+\left(\frac{{K}_{i}}{{v}_{nom}}(v)+{K}_{aw}{e}_{out}\right){\displaystyle \int {e}_{ref}dt+{K}_{g}(v)\theta}$$

$$\begin{array}{l}\text{where:}\\ \\ {e}_{ref}={v}_{ref}v\\ {e}_{out}={y}_{sat}y\\ \\ {y}_{sat}=\{\begin{array}{cc}1& y<1\\ y& 1\le y\le 1\\ 1& 1<y\end{array}\end{array}$$
The velocity error lowpass filter uses this transfer function.
$$H(s)=\frac{1}{{\tau}_{err}s+1}\text{for}{\tau}_{err}0$$
To calculate the acceleration and braking commands, the block uses these equations.
$$\begin{array}{l}{y}_{acc}=\{\begin{array}{cc}0& {y}_{sat}<0\\ {y}_{sat}& 0\le {y}_{sat}\le 1\\ 1& 1<{y}_{sat}\end{array}\\ \\ {y}_{dec}=\{\begin{array}{cc}0& {y}_{sat}>0\\ {y}_{sat}& 1\le {y}_{sat}\le 0\\ 1& {y}_{sat}<1\end{array}\end{array}$$
The equations use these variables.
v_{nom}  Nominal vehicle speed 
K_{p}  Proportional gain 
K_{i}  Integral gain 
K_{aw}  Antiwindup gain 
K_{ff}  Velocity feedforward gain 
K_{g}  Grade feedforward gain 
θ  Grade angle 
τ_{err}  Error filter time constant 
y  Nominal control output magnitude 
y_{sat}  Saturated control output magnitude 
e_{ref}  Velocity error 
e_{out}  Difference between saturated and nominal control outputs 
y_{acc}  Acceleration signal 
y_{dec}  Braking signal 
v  Velocity feedback signal 
v_{ref}  Reference velocity signal 
If you set the Control type, cntrlType parameter to
Predictive
, the block implements an optimal
singlepoint preview (look ahead) control model developed by C. C.
MacAdam^{1, 2, 3}. The model represents driver steering
control behavior during pathfollowing and obstacle avoidance maneuvers. Drivers
preview (look ahead) to follow a predefined path. To implement the MacAdam model,
the block:
Represents the dynamics as a linear single track (bicycle) vehicle
Minimizes the previewed error signal at a single point T* seconds ahead in time
Accounts for the driver lag deriving from perceptual and neuromuscular mechanisms
For longitudinal motion, the block implements these linear dynamics.
$$\begin{array}{l}{x}_{1}=v\\ \dot{x}={x}_{2}=\frac{{K}_{pt}}{m}g\mathrm{sin}(\theta )+{F}_{r}{x}_{1}\end{array}$$
In matrix notation:
$$\begin{array}{l}\dot{x}=Fx+g\overline{u}\\ \\ \text{where:}\\ \\ x=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]\\ F=\left[\begin{array}{cc}0& 1\\ \frac{{F}_{r}}{m}& 0\end{array}\right]\\ g=\left[\begin{array}{c}\begin{array}{c}0\\ \frac{{K}_{pt}}{m}\end{array}\end{array}\right]\\ \overline{u}=u\frac{{m}^{2}}{{K}_{pt}}g\text{sin}(\gamma )\end{array}$$
The block uses this equation for the rolling resistance.
${F}_{r}=\left[\mathrm{tanh}({x}_{1})(\frac{{a}_{r}}{{x}_{1}}+{c}_{r}{x}_{1})+{b}_{r}\right]$
The singlepoint model assumes a minimum previewed error signal at a single point T* seconds ahead in time. a* is the driver ability to predict the future vehicle response based on the current steering control input. b* is the driver ability to predict the future vehicle response based on the current vehicle state. The block uses these equations.
$$\begin{array}{l}a*=(T*){m}^{T}\left[I+{\displaystyle \sum _{n=1}^{\infty}\frac{{F}^{n}{(T*)}^{n}}{(n+1)!}}\right]ge\\ \\ b*={m}^{T}\left[I+{\displaystyle \sum _{n=1}^{\infty}\frac{{F}^{n}{(T*)}^{n}}{n!}}\right]\\ \\ \text{where:}\\ \\ {m}^{T}=\left[\begin{array}{cc}1& 1\end{array}\right]\end{array}$$
The equations use these variables.
a, b  Forward and rearward tire location, respectively 
m  Vehicle mass 
I  Vehicle rotational inertia 
a*, b*  Driver prediction scalar and vector gain, respectively 
x  Predicted vehicle state vector 
v  Longitudinal velocity 
F  System matrix 
K_{pt}  Tractive force and brake limit 
g  Control coefficient vector 
g  Gravitational constant 
T*  Preview time window 
ƒ(t+T*)  Previewed path input T* seconds ahead 
U  Forward vehicle velocity 
m^{T}  Constant observer vector; provides vehicle lateral position 
F_{r}  Rolling resistance 
a_{r}  Static rolling and driveline resistance 
b_{r}  Linear rolling and driveline resistance 
c_{r}  Aerodynamic rolling and driveline resistance 
The singlepoint model implemented by the block finds the steering command that minimizes a local performance index, J, over the current preview interval, (t, t+T).
$$J=\frac{1}{T}{\displaystyle {\int}_{t}^{t+T}{[f(\eta )y(\eta )]}^{2}}d\eta $$
To minimize J with respect to the steering command, this condition must be met.
$$\frac{dJ}{du}=0$$
You can express the optimal control solution in terms of a current nonoptimal and corresponding nonzero preview output error T* seconds ahead^{1, 2, 3}.
$${u}^{o}(t)=u(t)+\frac{e(t+T*)}{a*}$$
The equations use these variables.
ƒ(t+T*)  Previewed path input T* sec ahead 
y(t+T*)  Previewed plant output T* sec ahead 
e(t+T*)  Previewed error signal T* sec ahead 
u(t), u^{o}(t)  Steer angle and optimal steer angle, respectively 
J  Performance index 
The singlepoint model implemented by the block introduces a driver lag. The driver lag accounts for the delay when the driver is tracking tasks. Specifically, it is the transport delay deriving from perceptual and neuromuscular mechanisms. To calculate the driver transport delay, the block implements this equation.
$$H(s)={e}^{s\tau}$$
The equations use these variables.
τ  Driver transport delay 
y(t+T*)  Previewed plant output T* sec ahead 
e(t+T*)  Previewed error signal T* sec ahead 
u(t), u^{o}(t)  Steer angle and optimal steer angle, respectively 
J  Performance index 
[1] MacAdam, C. C. "An Optimal Preview Control for Linear Systems". Journal of Dynamic Systems, Measurement, and Control. Vol. 102, Number 3, Sept. 1980.
[2] MacAdam, C. C. "Application of an Optimal Preview Control for Simulation of ClosedLoop Automobile Driving ". IEEE Transactions on Systems, Man, and Cybernetics. Vol. 11, Issue 6, June 1981.
[3] MacAdam, C. C. Development of Driver/Vehicle Steering Interaction Models for Dynamic Analysis. Final Technical Report UMTRI8853. Ann Arbor, Michigan: The University of Michigan Transportation Research Institute, Dec. 1988.