Welcome to Introduction to Self-Driving Cars, the first course in University of Toronto’s Self-Driving Cars Specialization. This course will introduce you to the terminology, design considerations and safety assessment of self-driving cars. By the end of this course, you will be able to: - Understand commonly used hardware used for self-driving cars - Identify the main components of the self-driving software stack - Program vehicle modelling and control - Analyze the safety frameworks and current industry practices for vehicle development For the final project in this course, you will develop control code to navigate a self-driving car around a racetrack in the CARLA simulation environment. You will construct longitudinal and lateral dynamic models for a vehicle and create controllers that regulate speed and path tracking performance using Python. You’ll test the limits of your control design and learn the challenges inherent in driving at the limit of vehicle performance. This is an advanced course, intended for learners with a background in mechanical engineering, computer and electrical engineering, or robotics. To succeed in this course, you should have programming experience in Python 3.0, familiarity with Linear Algebra (matrices, vectors, matrix multiplication, rank, Eigenvalues and vectors and inverses), Statistics (Gaussian probability distributions), Calculus and Physics (forces, moments, inertia, Newton's Laws). You will also need certain hardware and software specifications in order to effectively run the CARLA simulator: Windows 7 64-bit (or later) or Ubuntu 16.04 (or later), Quad-core Intel or AMD processor (2.5 GHz or faster), NVIDIA GeForce 470 GTX or AMD Radeon 6870 HD series card or higher, 8 GB RAM, and OpenGL 3 or greater (for Linux computers).
Lesson 1: Proportional-Integral-Derivative (PID) Control
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Introduction to Self-Driving Cars
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Module 5: Vehicle Longitudinal Control
Longitudinal control of an autonomous vehicle involves tracking a speed profile along a fixed path, and can be achieved with reasonable accuracy using classic control techniques. This week, you will learn how to develop a baseline controller that is applicable for a significant subset of driving conditions, which include most non-evasive or highly-dynamic motions.
與講師見面
Steven WaslanderAssociate Professor
Aerospace Studies
Jonathan KellyAssistant Professor
Aerospace Studies
Welcome to week five of this course.
In the previous module,
you learned how to develop vehicle models to capture longitudinal and lateral dynamics.
In this module, we will go through the concepts of
longitudinal vehicle control to regulate the speed of our self-driving car.
Specifically, you'll review some of
the essential concepts from classical linear time-invariant control,
develop a PID control law for the longitudinal vehicle model and
combine feedforward and feedback control to improve desired speed tracking.
Design of the longitudinal speed control underpins all vehicle performance
on the road and is one of the fundamental components needed for autonomous driving.
In this video, we will briefly review some of the basics of
linear time-invariant control and the PID controller.
By the end of this video,
you'll be able to design a PID control for a linear time-invariant system.
Note that we will have to assume you're familiar with
classical control design including the use of transfer functions and the Laplace domain.
So, if you haven't seen these concepts before,
please check out some of the great controls courses on
Coursera listed in the supplemental materials. Let's get started.
In module three of this course,
we learned how to develop
the dynamic and kinematic models for a vehicle based on the bicycle model.
These models aim to capture how the dynamic system reacts to input commands from
the driver such as steering gas and break and how it reacts to disturbances such as wind,
road surface and different vehicle loads.
The effects of the inputs and disturbances on the states such as velocity and
rotation rate of the vehicle are defined
by the kinematic and dynamic models we developed.
The role of the controller then is to
regulate some of these states of the vehicle by sensing
the current state variables and then generating
actuator signals to satisfy the commands provided.
For longitudinal control, the controller sensing the vehicle speed and adjust
the throttle and break commands to match
the desired speed set by the autonomous motion planning system.
Let's take a look at a typical feedback control loop.
The plant or process model takes the actuator signals as
the input and generates the output or state variables of the system.
These outputs are measured by sensors and
estimators are used to fuse measurements into accurate output estimates.
The output estimates are compared to
the desired or reference output variables
and the difference or error is passed to the controller.
The controller can be seen as
a mathematical algorithm that generates actuator signals so that
the error signal is minimized and
the plant state variables approach the desired state variables.
The plant model be it linear or nonlinear can be represented in several ways.
Two of the most common ways are state-space form
which tracks the evolution of an internal state to connect
the input to the output and transfer function form
which models the input to output relation directly.
Note that for transfer functions,
the system must be linear and time-invariant.
A transfer function G is a relation between inputs U and outputs Y of
the system defined in the Laplace domain as a function of S a complex variable.
We use the Laplace transform to go from
the time domain to the S domain because it allows for
easier analysis of an input-output relation
and is useful in understanding control performance.
When working with the transfer functions,
the numerator and denominator roots provide
powerful insight into the response of a system to input functions.
The zeros of a system are the roots of
the numerator and the poles of the system are the roots of its denominator.
Control algorithm design can vary from simple such as constant gain multiplication,
lookup tables and linear equations to more detailed methods based
on non-linear functions and optimization over finite prediction horizons.
Some of the basic and classic controllers include
lead-lag controllers and proportional integral and derivative or PID controllers.
In the rest of this video,
we will go into more detail on
the PID control combination as a useful starting point for longitudinal control.
More involved control design is also possible
and it's particularly useful for non-linear system models,
time-varying models, or models with constraints that limit output selection.
Nonlinear methods such as feedback linearization,
back stepping and sliding mode control are beyond the scope of
this course but can certainly be applied to self-driving vehicle control problem.
Optimization-based methods are heavily used in autonomous driving and so we'll look
at model predictive control as an example of
this group of controllers later on in the course.
PID control is mathematically formulated by
adding three terms dependent on the error function.
A proportional term directly proportional to the error E,
an integral term proportional to the integral of the error,
and a derivative term proportional to the derivative of the error.
The constants Kp, Ki,
and Kd are called
the proportional integral and derivative gains and govern the response so
the PID controller which is denoted U of t as it is the input to the plant model.
Taking the Laplace transform of the PID control yields
the transfer function Gc of S. Multiplying by
S in the Laplace domain is equivalent to taking a derivative in
the time domain and dividing by S is equivalent to taking the integral.
By adding these three terms of the PID controller together,
we get a single transfer function for PID control.
Note that not all gains need to be used for all systems.
If one or more of the PID gains are set to zero,
the controller can be referred to as P, Pd or Pi.
The PID transfer function contains
a single pole at the origin which comes from the integral term.
It also contains a second-order numerator with two zeros that can be
placed anywhere in the complex plane by selecting appropriate values for the gains.
PID control design therefore,
boils down to selecting zero locations to achieve
the desired output or performance based on the model for the plant.
There are also several algorithms to tune PID gains,
among them, Ziegler Nichols is one of the most popular.
Closed loop response denotes the response of a system when
the controller decides the inputs to apply to the plant model.
For a step input on the reference signal we can define
the rise time as the time it takes to reach 90 percent of the reference value.
The overshoot as the maximum percentage the output exceeds this reference.
The settling time as the time to settle to within five percent of the reference
and the steady-state error as the error between
the output and the reference at steady-state.
The effects of each P,
I and D action are summarized in the following table.
For instance, an increase in Kp leads to a stronger reaction to
errors and therefore a decrease in
rise time in response to a step change in the reference signal.
Similarly, since Kd reacts to the rate of change of the error
an increased Kd leads to a decrease in overshoot or the rate of change of error is high.
It may simultaneously lead to a decrease in oscillations
about the reference and a decreased settling time as a result.
Finally, an increase in Ki can eliminate
steady-state errors but may lead to increased oscillation in the response.
Ultimately, the P, I and D gains must be selected with knowledge of
the interaction of their effects to adjust
the system response to get the right closed loop performance.
You'll get a chance to see these interactions as you develop
your own PID controller as part of the assessment for this course.
Now, let's take a look at
the well-known second-order spring-mass damper model as shown in the figure.
In this example, we'll first review the transfer function of
the proposed dynamic system and then design a PID controller for it.
The dynamics of the spring-mass damper system
were derived in an earlier video in this course.
The system is subjected to the input force F
and the output of the model is the displacement of the body x.
The mass M is connected to a rigid foundation by
a spring with spring constant K and a damper with damping coefficient b.
Now to transform the equation into the S domain or Laplace domain,
we use the Laplace transform and write the second-order equation as follows.
This relies on the fact that the derivative in
the time domain are multiplications by S in the Laplace domain.
Finally, the transfer function is formed which represents the relation
between the output x of s and the input F of S and is
defined as the plant transfer function G of s. This is
a second-order system with two poles defined by
the mass spring constant and damping coefficient.
To evaluate the system characteristics,
we excite the system by using a unit step input.
This is normally the first step to evaluate the dynamic characteristics of a plant.
For example, the system response x is plotted
here for the parameter values given as m equals one,
b equals 10 and k equals 20.
This type of response is easily generated with
scientific computing tools such as Matlab recite pi.
The input is the unit step F equals one and the output is once again x.
This response is called the open-loop response
since there is no controller applied to the system at this point.
If a controller is added to the plant and the output of the model
is measured and compared with the desired output or reference signal,
then the response of the system is called the closed loop response.
For unity feedback, the sensor transfer function is
assumed to be one and in general it could be any transfer function.
The closed loop transfer function given here can be performed from
the transfer functions of the controller and the plant.
For those of you who have studied classical feedback control,
you'll know that the poles of the open-loop system
define the characteristics of the closed-loop response.
You may have also seen root locus bodhi and Nyquist design techniques which can
be used to select controllers that meet specific output specifications.
We've left some links to appropriate resources for
those who'd like to learn more in the supplemental material.
Let's look at the step response for a few different PID controllers.
The dashed horizontal line represents the reference or desired output and
the controllers goal is to keep the actual output close to this reference.
In the first example,
the step responses for pure proportional control of the spring-mass damper system.
In the P controller response,
we see a fast rise time,
significant overshoot and prolonged oscillation leading to a long settling time.
Adding derivative control improves the step response in terms of
overshoot and settling time but slows down the rise time.
Adding the integral term instead maintains a short rise time and is
able to reduce oscillations and overshoot leading to a fast settling time as well.
The simple Pi control is an excellent design for the spring-mass damper system.
Including all three PID terms in the controller,
leads to even more flexibility in designing the step response.
By carefully tuning the controller gains,
we can use the benefits of all three to eliminate
overshoot and still maintain very short rise and settling times.
As can be seen in the plot,
the system approaches the reference at much more
quickly without any overshoot with PID control.
In this video, we've covered the concepts of
controller design and why we integrate controllers into a dynamic model.
We also reviewed the PID controller and learned how to control
the step response of a spring-mass damper system with PID control.
In the next video,
you will learn how to apply PID control to
regulate the speed of a self-driving car. See you there.
