control systems i - eth z€¦ · course objectives 1/3 this course is about control of dynamic...

Control Systems ILecture 1: Introduction

Suggested Readings: Astrom & Murray Ch. 1

Jacopo Tani

Institute for Dynamic Systems and ControlD-MAVT

ETH Zurich

September 21, 2018

J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 1 / 30

Figure: Space-X Falcon IX landing

Figure: nuTonomy self driving car in action


https://www.youtube.com/watch?v=1sJlFzUQVmY

https://vimeo.com/291048066

Examples of control systems fields of application

Thermostat

Power generation and transmission

Transportation networks

Aerospace

Robotics

Biological systems


Outline

1 Overview and course objectives

2 Logistics

3 Signals and Systems


The hidden technology [Karl Astrom]

Widely used

Very successful

Seldom talked about

Except when disasterstrikes


Course Objectives 1/3

This course is about control of dynamic systems, i.e., systems that

evolve over time,

have inputs and outputs.

The control problem is finding the right input sequence, over time, such thatthe system’s output follows a reference signal.

Or, in other words: make a system behave like the user wants to, and not likeit would naturally behave.

We learn how to control systems by achieving three 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, con-trollability, 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.



In particular, we will concentrate on systems that can be modeled by OrdinaryDifferential Equations (ODEs), and that satisfy certain linearity and time-invariance conditions. In this course, we will focus on systems with a singleinput 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 of partic-ular 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 desirable proper-ties (e.g., stability, performance) of the interconnection with a given dynamicsystem.



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.

Efforts will be made to emphasize the connection between the physical world(and real numbers) and the Laplace/frequency domain (and complexnumbers).

. . . if all else fails . . .


Outline


2 Logistics



Course Information

Instructor Dr. Jacopo Tani <[email protected]>, Room ML K 37.3.

Lead Teaching Assistant Dr. Shima Mousavi <[email protected]>, RoomML K 37.4.

Admin Assistants Julian Zilly <[email protected]>, Annina Fattor <+41 44632 87 96>, Room ML K32.2.

Lectures Friday 10-12, Lecture room HG F 7, with video transmission in HGF 5.

Exercises Friday 13-15, Various rooms (see course catalouge).

Study center Wednesday 13-15 in room ETZ E 8 (starting from week 3).

Instructor office hours Tuesday 14-15 in room ML K 37.3. (Excluding13.11, 4.12. I will be out of town those weeks. Dr. Mousavi will substituteme.)


http://www.vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheit.view?semkez=2018W&ansicht=LEHRVERANSTALTUNGEN&lerneinheitId=123501&lang=en

Smile, you are on camera!

All lectures are recorded and will be publicly available online.


CS I Staff

Lead Teaching Assistant Dr. Shima Mousavi <[email protected]>

Teaching Assistant Email contactBiagosch Carl Philipp <[email protected]>Friederich Rockenbauer <[email protected]>Giuseppe Rizzi <[email protected]>Jasan Zughaibi <[email protected]>Luna Meeusen <[email protected]>Marc Leibundgut <[email protected]>Moritz Reinders <[email protected]>Yannik Schnider <[email protected]>


Reading material

Lecture slides 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 ETH ac-count 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


http://www.cds.caltech.edu/~murray/amwiki/index.php/First_Edition

https://piazza.com/

http://www.idsc.ethz.ch/education/lectures/control-systems-i.html

Tentative schedule

# Date Topic

1 Sept. 21 Introduction, Signals and Systems2 Sept. 28 Modeling, Linearization

3 Oct. 5 Analysis 1: Time response, Stability4 Oct. 12 Analysis 2: Diagonalization, Modal coordinates5 Oct. 19 Transfer functions 1: Definition and properties6 Oct. 26 Transfer functions 2: Poles and Zeros7 Nov. 2 Analysis of feedback systems: internal stability,

root locus8 Nov. 9 Frequency response9 Nov. 16 Analysis of feedback systems 2: the Nyquist

condition

10 Nov. 23 Specifications for feedback systems11 Nov. 30 Loop Shaping12 Dec. 7 PID control13 Dec. 14 State feedback and Luenberger observers14 Dec. 21 On Robustness and Implementation challenges


Today’s learning objectives

After today’s lecture, you should be able to:

Understand the approach of control systems in terms of “systems” with inputand output signals

Name examples and describe what input, output and states of a system are

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: series,parallel, feedback

Distinguish between MIMO and SISO systems


Outline


2 Logistics



Signals

Signals are maps from a set T to a set W. They receive a number as inputand produce a number as output.

Think of T as the time axis. It will be the real line, i.e., T = R, when talkingabout continuous-time systems. This is how “things work in nature”.

Or it could be the set of natural numbers: T = N, when talking aboutdiscrete-time systems. This is how things work on a computer.

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

Systems: in this course we will consider a system as a map between signals,i.e., something that transforms some input signal into an output signal.

Input signals can be thought of as something that can be manipulated by theuser.

Output signals instead capture how the system responds to a certain input.

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.


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

Depending on how Σ affects u, we classify the system in different ways. Forexample:

Static (memoryless) vs. Dynamic

Linear vs. Nonlinear

Causal vs. Acausal (not to be confused with casual!)

Time invariant vs. Time variant.

In this course, we will study how to control dynamic systems that are linear, andtime invariant; or LTI systems.


Memoryless (or static) systems

An input-output system Σ is memoryless (or static) if for all t ∈ T, y(t) is afunction of u(t).

In a static (memoryless) system: the output at the present time depends onlyon the value of input at the present time; not on the value of input in thepast or the future time.


Memoryless (or static) systems: Examples

Static systems:

y(t) = 3u(t),

y(t) = 2−(t+1)u(t),

y(t) =√

sin(u2(t)),

y(t) = <[u(t)] = u(t)+u∗(t)2

.

Dynamic system:

y(t) =∫ t

−∞ u(τ) dτ ,This system (an integrator) remembers everything that happened in the past.

y(t) = u(t),

y(t) = u(t2),

y(t) = u(t − a), ∀a 6= 0.


Causal systems

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: a system is causal if and only if the future input does not affectthe present output.

All practically realizable systems are causal. (It is impossible to implementa non causal system in “the real world”.)


Causal systems: truncation operator PT

We can express causality mathematically by introducing a truncation operatorPT :

(PTu)(t) =

{u(t) for t ≤ T0 for t > T .

PTu PTu


Causal systems

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).


Causal vs. non-causal systems: Examples

Causal systems (all practically/physically realizable systems are causal):

y(t) = u(t),

y(t) = u(t − τ), ∀τ > 0 (systems with delay),

y(t) = cos(3t + 1)u(t − 1) (don’t make the +1 fool you)

y(t) =∫ t

−∞ u(τ) dτ

Non-causal system:

y(t) = u(t − a), ∀a < 0, (actually anti-causal)

y(t) = u(t + 1) + u(t) + u(t − 1), (non causal)

y(t) = u(bt), ∀b > 0,

y(t) =∫ t+1

−∞ u(τ) dτ .

Q: Is y(t) = u(t) causal or non-causal? Furthermore, is it realizable or not?

A: It is causal but not realizable.


Causal vs. non-causal systems: Examples

Causal systems (all practically/physically realizable systems are causal):

y(t) = u(t),

y(t) = u(t − τ), ∀τ > 0 (systems with delay),

y(t) = cos(3t + 1)u(t − 1) (don’t make the +1 fool you)

y(t) =∫ t

−∞ u(τ) dτ

Non-causal system:

y(t) = u(t − a), ∀a < 0, (actually anti-causal)

y(t) = u(t + 1) + u(t) + u(t − 1), (non causal)

y(t) = u(bt), ∀b > 0,

y(t) =∫ t+1

−∞ u(τ) dτ .

Q: Is y(t) = u(t) causal or non-causal? Furthermore, is it realizable or not?A: It is causal but not realizable.


Time-invariant vs. time-variant systems

A time invariant system is a time dependant map between input and outputsignals that is the same at any point in time.In other words, the system manipulates the input in a way that does notdepend on when the system is used.

More formally: consider the time-shift operator στ :

(στu)(t) = u(t − τ), ∀t ∈ T.

στu στu


Time-invariant systems

A system is time-invariant if:

Σστu = στΣu = στy ∀τ ∈ T.


Time-invariant vs. time-variant systems: Examples

Time-invariant:

y(t) = ku(t) + c, ∀c, k,

y(t) = 3sin(u(t)),

y(t) =∫ t

−∞ u(τ)dτ ,

y(t) = u(t − 1) + u(t + 2).

Time-variant:

y(t) = u(at), a 6= 0 (time scaling),

y(t) = 3u(sin(t)) (always time scaling),

y(t2) = u(t) (always time scaling, but on the output),

y(t) = cos tu(t) (if a coefficient is time dependent),

y(t) = u(t) + t (if any summed term - except input/output - is timedependent),

y(t) =∫ t

−∞ u(2τ)dτ .


Linear systems

An input-output system is linear if it is additive and homogeneous.Additivity: Σ(u1 + u2) = Σu1 + Σu2Homogeneity: Σ(ku) = kΣu.

In other words, Σ is linear if, for all input signals ua, ub, and scalars α, β ∈ R,

Σ(αua + βub) = α(Σua) + β(Σub) = αya + βyb.

Σua ya

Σub yb

Σαua αya

The key idea is superposition:

Σαua + βub αya + βyb


Linear vs. nonlinear: Examples

Linear:

y(t) = u(sin t),

y(t) = u(t2) (time scaling doesn’t make a system nonlinear),

y(t) = au(t),∀a,

y(t) = cos(t)u(t) (coefficients don’t influence linearity),

y(t) = u(t + a) + u(t − b) (time shift does not affect linearity)

y(t) =∫ t

∞ u(τ)dτ ,

y(t) = u(t) (differential and integral operators are linear).

Nonlinear:

y(t) = u2(t),

y(t) = u(t) + a, (summed terms - except input or output - make a systemnonlinear)

y(t) = sin(u(t)),

y(t) = <(u(t)).


Which systems will we treat in this course, and why?

In this course, we will consider only LTI SISO systems:

Single Input, Single Output ,

Linear,

Time invariant,

Causal.

This is a very restrictive class of systems; in fact, most systems are NOTLTI. On the other hand, many systems are approximated very well by LTImodels. This is a 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 of systems

Control/dynamical systems can be interconnected in various ways:

Serial interconnection:

Σ = Σ2Σ1

Parallel interconnection:

Σ = Σ1 + Σ2

(Negative) Feedbackinterconnection:

Σ = (I + Σ1Σ2)−1Σ1

Σ1 Σ2u y

Σ1

Σ2

u y

Σ1

Σ2

u y

−


What are the objectives of a control system?

Stabilization: make sure the system does not “blow up.”

Regulation: maintain a desired operating point in spite of disturbances.

Tracking: follow the reference/desired trajectory/behavior as closely aspossible.

Robustness: the controller that satisfies the above works even if the systemis “slightly” different than we expected.

Robustness is a more advanced topic which will be treated in Control Systems II.We will introduce it in the last class of this course.


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 is error based, compensates for unexpected / unmodeled phenomena(disturbances, noise, model uncertainty).

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 transient be-havior, 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 of systems.

Know how to classify signals/systems as linear/nonlinear, causal/acausal,time invariant/variant, memoryless (static) / dynamic.

Distinguish and calculate different interconnections of systems.

Explain the acronyms MIMO, SISO, LTI.

Describe the benefits of using control systems to another student.


control systems i - eth z€¦ · course objectives 1/3 this course is about control of dynamic...

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