Control Systems I - ETH Z · Karl J. ˚Astrom and Richard M. Murray ... (ETH) Lecture 1: Control...

Control Systems I

Lecture 1: Introduction

Suggested Readings: Astrom & Murray Ch. 1, Guzzella Ch. 1

Emilio Frazzoli

Institute for Dynamic Systems and Control

D-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 disaster

strikes


https://www.youtube.com/watch?v=f2at-cqaJMM

https://www.youtube.com/watch?v=iP_lAjIfZwU

https://www.youtube.com/watch?v=faB5bIdksi8


Outline

1 Overview

2 Logistics

3 Signals and Systems


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 e↵ectively using mathematical tools.

Analysis: understand the basic characteristics of a system (e.g., stability,controllability, observability), and how the input a↵ects the output.

Synthesis: figure out how to change a system in such a way that it behaves ina desirable way.



In particular, we will concentrate on systems that can be modeled byOrdinary Di↵erential 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.


Your new best friend


https://www.youtube.com/watch?v=-o6-vU7RsTM&feature=youtu.be



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 e↵ort to emphasize the connection between the physicalworld (and real numbers) and the Laplace/frequency domain (and complexnumbers).

. . . if all else fails . . .


Outline

1 Overview

2 Logistics



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. O�ce hours TBA.


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


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


Tentative schedule

# Date Topic1 Sept. 22 Introduction, Signals and Systems2 Sept. 29 Modeling, Linearization3 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

condition10 Nov. 24 Specifications for feedback systems11 Dec. 1 Loop Shaping12 Dec. 8 PID control13 Dec. 15 Implementation issues14 Dec. 22 Robustness


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 di↵erent interconnections of systems

Distinguish between MIMO and SISO


Outline

1 Overview

2 Logistics



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]


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 di↵erent in terms of sources and characteristics.More on this later in the course.


Systems: Input-Output models 2/2

An input-output model is a map ⌃ from an input signal u : t 7! u(t) to anoutput signal y : t 7! y(t),

y = ⌃u,

that is,

y(t) = (⌃u)(t), 8t 2 T.

⌃u y

Block diagram representation


Your new best friend, revisited


Memoryless (or static) systems

An input-output system ⌃ is memoryless (or static) if there exists a functionS : W ! W such that, for all t 2 T,

y(t) = (⌃u)(t) = S(u(t)).

Examples:

y(t) = 3u(t),

y(t) =p

sin(u(t)2).

Not a static system: y(t) =R t�1 u(⌧) d⌧ .

This system (an integrator) remembers everything that happened in the past.


Time invariance

Let the time-shift operator �⌧ be defined as follows, for any signal u:

(�⌧u)(t) = u(t � ⌧), 8t 2 T.

An input-output system ⌃ is time-invariant if it commutes with the time-shiftoperator, i.e., if

⌃�⌧u = �⌧⌃u = �⌧y 8⌧ 2 T.

Examples:

Time-invariant: y(t) = 3sin(u(t)),

Time-varying: y(t) = 3sin(t)u(t).


Linearity

An input-output system ⌃ is linear if, for all input signals ua, ub, and scalars↵,� 2 R,

⌃(↵ua + �ub) = ↵(⌃ua) + �(⌃ub) = ↵ya + �yb.

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.


Causality

An input-output system ⌃ is causal if, for any t 2 T, the output at time tdepends only on the values of the input on (�1, 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 ⌃ is causal if

PT⌃PT = PT⌃, 8T 2 T.

An input-output system ⌃ is strictly causal if, for any t 2 T, the output attime t depends only on the values of the input on (�1, t).


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.


Interconnections

Control/dynamical systems can be interconnected in various ways:

Serial interconnection:

⌃ = ⌃2⌃1

Parallel interconnection:

⌃ = ⌃1 + ⌃2

(Negative) Feedbackinterconnection:

⌃ = (I + ⌃2⌃1)�1⌃1

⌃1 ⌃2

u y

⌃1

⌃2

u y

⌃1

⌃2

u y

�


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.


Basic control architectures

F Pr yu

Feed-forward

C Pr e u y

�

Feedback

C

F

Pr e u y

�

Two degrees of freedom


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.


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 di↵erent interconnections of systems

Distinguish between MIMO and SISO


Control Systems I - ETH Z · Karl J. ˚Astrom and Richard M. Murray ... (ETH) Lecture 1: Control...

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