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Control Systems ILecture 1: Introduction
Suggested Readings: Astrom & Murray Ch. 1, Guzzella Ch. 1
Emilio Frazzoli
Institute for Dynamic Systems and ControlD-MAVT
ETH Zurich
September 22, 2017
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 1 / 31
The hidden technology [Karl Astrom]
Widely used
Very successful
Seldom talked about
Except when disasterstrikes
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 2 / 31
https://www.youtube.com/watch?v=f2at-cqaJMM
https://www.youtube.com/watch?v=iP_lAjIfZwU
https://www.youtube.com/watch?v=faB5bIdksi8
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 3 / 31
Outline
1 Overview
2 Logistics
3 Signals and Systems
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 4 / 31
Course Objectives 1/3
The course addresses dynamic control systems, i.e., systems that
evolve over time,
have inputs and outputs.
We have three main objectives:
Modeling: learn how to represent a dynamic control system in a way that itcan be treated effectively using mathematical tools.
Analysis: understand the basic characteristics of a system (e.g., stability,controllability, observability), and how the input affects the output.
Synthesis: figure out how to change a system in such a way that it behaves ina desirable way.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 5 / 31
Course Objectives 2/3
In particular, we will concentrate on systems that can be modeled byOrdinary Differential Equations (ODEs), and that satisfy certain linearityand time-invariance conditions. In this course, we will focus on systemswith a single input and a single output (SISO).
This will allow us to use ”classical control” tools that are very powerful andeasy to use (i.e., mostly graphical), and which are really laying the foundationof any followup work on more challenging control problems.
We will analyze the response of these systems to inputs and initial conditions:for example, stability and performance issues will be addressed. It is ofparticular interest to analyze systems obtained as interconnections (e.g.,feedback) of two or more other systems.
We will learn how to design (control) systems that ensure desirableproperties (e.g., stability, performance) of the interconnection with a givendynamic system.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 6 / 31
Your new best friend
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 7 / 31
https://www.youtube.com/watch?v=-o6-vU7RsTM&feature=youtu.be
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 8 / 31
Course Objectives 3/3
A large part of the course will require us to work in the Laplace and in thefrequency domain and complex numbers, rather than something “physical”like time and real numbers. This requires a big leap of faith, making thelearning curve quite steep for many students.
We will make every effort to emphasize the connection between the physicalworld (and real numbers) and the Laplace/frequency domain (and complexnumbers).
. . . if all else fails . . .
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 9 / 31
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 10 / 31
Outline
1 Overview
2 Logistics
3 Signals and Systems
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 11 / 31
Course Information
Instructor Prof. Emilio Frazzoli <[email protected]>, Room ML K 32.1
Lead TA Julian Zilly <[email protected]>, Room ML K 42.3
Admin Assistant Ms. Annina Fattor <+41 44 632 87 96>, Room ML K32.2
Lectures F 10-12, Room HG F 5, 7.
Exercises F 13-15, Various rooms (arranged by groups, refer to Julian).
Prof. Office hours TBA.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 12 / 31
Reading material
Lecture and exercise notes will be posted on the course web site.
A nice introductory book on feedback control, available online for free:
Feedback Systems: An Introduction for Scientists and EngineersKarl J. Astrom and Richard M. Murray
http://www.cds.caltech.edu/~murray/amwiki/index.php/First_Edition
Online discussion forum: https://piazza.com/, sign up with your ETHaccount for ”151-0591-00L: Control Systems I” as a student.Detailed instructions on the course homepage:
http://www.idsc.ethz.ch/education/lectures/control-systems-i.html
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 13 / 31
Another suggested (not required) textbook
Von Herr Prof. Dr. E. Frazzoli empfohlen
TitelAnalysis and Synthesis
ISBN9783728133861
AutorLino Guzzella
StudentenpreisCHF 30.00
Preis NormalCHF 37.50
Erhältlich in den Filialen:
ETH Store PolyterrasseOffen Mo-Fr 9-18 Uhr
ETH Store HönggerbergOffen Mo-Fr 8-18 UhrSa 11-16 Uhr
Online bestellen:www.eth-store.ch
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 14 / 31
Tentative schedule
# Date Topic
1 Sept. 22 Introduction, Signals and Systems2 Sept. 29 Modeling, Linearization
3 Oct. 6 Analysis 1: Time response, Stability4 Oct. 13 Analysis 2: Diagonalization, Modal coordinates5 Oct. 20 Transfer functions 1: Definition and properties6 Oct. 27 Transfer functions 2: Poles and Zeros7 Nov. 3 Analysis of feedback systems: internal stability,
root locus8 Nov. 10 Frequency response9 Nov. 17 Analysis of feedback systems 2: the Nyquist
condition
10 Nov. 24 Specifications for feedback systems11 Dec. 1 Loop Shaping12 Dec. 8 PID control13 Dec. 15 Implementation issues14 Dec. 22 Robustness
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 15 / 31
Today’s learning objectives
After today’s lecture, you should be able to:
Understand control systems in terms of input and output signals.
Name examples and describe what states, input and output of a system is
Describe the benefits of using control systems to another student
Know how to classify signals/systems as linear/nonlinear, causal/acausal,time invariant/variant, memoryless (static) / dynamic
Distinguish and calculate different interconnections of systems
Distinguish between MIMO and SISO
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 16 / 31
Outline
1 Overview
2 Logistics
3 Signals and Systems
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 17 / 31
Signals
Signals: maps from a set T to a set W.
Time axis T, for us this will be the real line, i.e., T = R. One could alsoconsider, e.g., the set of natural numbers (discrete-time systems).
Signal space W: for us this will be the real line too, W = R. One could alsoconsider vector-valued signals, for which W = Rn for some fixed integer n.
t
y(t)
k
y [k]
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 18 / 31
Systems: Input-Output models 1/2
In this course, we will consider a system a map between signals, i.e.,something that transforms some input signal into some output signal.
It is convenient to define an input signal (typically this is the signal that canbe manipulated by the designer), and an output signal (which captures howthe system performs a certain task).
Other signals that are of interest include disturbances and noise. Both areexogenous inputs, but are different in terms of sources and characteristics.More on this later in the course.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 19 / 31
Systems: Input-Output models 2/2
An input-output model is a map Σ from an input signal u : t 7→ u(t) to anoutput signal y : t 7→ y(t),
y = Σu,
that is,
y(t) = (Σu)(t), ∀t ∈ T.
Σu y
Block diagram representation
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 20 / 31
Your new best friend, revisited
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 21 / 31
Memoryless (or static) systems
An input-output system Σ is memoryless (or static) if there exists a functionS : W→W such that, for all t ∈ T,
y(t) = (Σu)(t) = S(u(t)).
Examples:
y(t) = 3u(t),
y(t) =√
sin(u(t)2).
Not a static system: y(t) =∫ t
−∞ u(τ) dτ .This system (an integrator) remembers everything that happened in the past.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 22 / 31
Time invariance
Let the time-shift operator στ be defined as follows, for any signal u:
(στu)(t) = u(t − τ), ∀t ∈ T.
An input-output system Σ is time-invariant if it commutes with the time-shiftoperator, i.e., if
Σστu = στΣu = στy ∀τ ∈ T.
Examples:
a
b
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 23 / 31
Linearity
An input-output system Σ is linear if, for all input signals ua, ub, and scalarsα, β ∈ R,
Σ(αua + βub) = α(Σua) + β(Σub) = αya + βyb.
The key property of linearity is superposition. In other words, if I know that
if I apply ua I get ya, and
if I apply ub I get yb,
then I also know that
if I apply ua + ub, I get ya + yb.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 24 / 31
Causality
An input-output system Σ is causal if, for any t ∈ T, the output at time tdepends only on the values of the input on (−∞, t].
In other words, define the truncation operator P as
(PTu)(t) =
{u(t) for t ≤ T0 for t > T .
Then an input-output system Σ is causal if
PTΣPT = PTΣ, ∀T ∈ T.
An input-output system Σ is strictly causal if, for any t ∈ T, the output attime t depends only on the values of the input on (−∞, t).
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 25 / 31
Scope of the course
In this course, we will consider only LTI SISO systems
Single Input, Single Output (i.e., W = R)
Linear
Time invariant
Causal.
This is a very restrictive class; in fact, most systems are NOT LTI. On theother hand, many systems are approximated very well by LTI models. This isa key idea.
As long as we are mindful of the errors induced by the LTI approximation, themethods discussed in the class are very powerful.
Indeed, most control systems in operation are designed according to theprinciples that will be covered in the course.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 26 / 31
Interconnections
Control/dynamical systems can be interconnected in various ways:
Serial interconnection:
Σ = Σ2Σ1
Parallel interconnection:
Σ = Σ1 + Σ2
(Negative) Feedbackinterconnection:
Σ = (I + Σ2Σ1)−1Σ1
Σ1 Σ2u y
Σ1
Σ2
u y
Σ1
Σ2
u y
−
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 27 / 31
Control objectives
Stabilization: make sure the system does not “blow up.”
Regulation: Maintain a desired operating point in spite of disturbances
Tracking: follow the reference trajectory as closely as possible.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 28 / 31
Basic control architectures
F Pr yu
Feed-forward
C Pr e u y
−
Feedback
C
F
Pr e u y
−
Two degrees of freedom
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 29 / 31
benefits/dangers of feedback
Feed-forward control relies on a precise knowledge of the plant, and does notchange its dynamics.
Feedback control allows one to
Stabilize an unstable system;
Handle uncertainties in the system;
Reject external disturbances.
However, feedback can
introduce instability, even in an otherwise stable system!
feed sensor noise into the system.
Two degrees of freedom (feedforward + feedback) allow better transientbehavior, e.g., can yield good tracking of rapidly-changing reference inputs.
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 30 / 31
Today’s learning objectives
After today’s lecture, you should be able to:
Understand control systems in terms of input and output signals.
Name examples and describe what states, input and output of a system is
Describe the benefits of using control systems to another student
Know how to classify signals/systems as linear/nonlinear, causal/acausal,time invariant/variant, memoryless (static) / dynamic
Distinguish and calculate different interconnections of systems
Distinguish between MIMO and SISO
E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/2017 31 / 31