introduction to signals and systems general informationsguilldrion/files/syst0002-2017-18... ·...

Introduction to Signals and Systems General Informations

Guillaume Drion Academic year 2017-2018

SYST0002 - General informations

Website: http://sites.google.com/site/gdrion25/teaching/syst0002

Contacts: Guillaume Drion - [email protected] Marie Wehenkel (teaching assistant) - [email protected]

Organization: 11 or 12 main lessons - Wednesdays 13:30 or 13:30? 10 tutorial sessions split in 7 groups (B5b, see website)

Theory and exercises follow the textbooks provided on the website (in French). The textbooks are the same as last year!

http://sites.google.com/site/gdrion25/teaching/syst0002

mailto:[email protected]

mailto:[email protected]

Schedule of the year

Tutorials will start on Wednesday, October 4!

Goals of the course and evaluation

Goals of the course:

Lessons: intuition! The main goal of this course is to provide a general (and simple) framework for the analysis of possibly complex systems.

Tutorials: develop your technical skills, methods.

Leve

l of c

ompl

exity

Time

Past

Future

SYST0002

Goals of the course and evaluation

Goals of the course:

Lessons: intuition! The main goal of this course is to provide a general (and simple) framework for the analysis of possibly complex systems.

Tutorials: develop your technical skills, methods.

Evaluation:

2 short assignments (matlab).

Exam: 2-3 questions to test your technical skills (tutorial style). 1-2 questions to test your basic knowledge and intuition.

What changes this year?

The course absorbs the basic signal processing course:

Addition of the concept of Fourier series, increased focus on Fourier transforms from a signal processing viewpoint.

Addition of 1-2 courses on signal processing (sampling, windowing).

On the other hand, some concepts regarding the analytical solutions of state-space equations will be removed:

This part has already been taught in the calculus course (linear differential equations with constant coefficients).

Introduction to Signals and Systems Lecture #1 - Introduction to Systems Theory

Guillaume Drion Academic year 2017-2018

Introduction to Signals and Systems

Neville Hogan (MIT): The Paradox of Human Performance


Neville Hogan (MIT): The Paradox of Human Performance

As engineers, your task will not only be to use existing tools to design complex systems, but also to develop novel tools to shape tomorrow’s technology.


Example: can you accurately describe the motion of this simple pendulum…


Example: can you accurately describe the motion of this simple pendulum…

… without using Newton’s laws of motion?


What was mechanics like before Newton?How much did Newton’s laws changed our understanding of mechanics?

Mathematical modeling can turn any problem into an engineering problem

Real life problems


Engineering problems

Real life problems


Engineering problems

SYSTEMS MODELING

AnalysisDesignImplementation

“Applied mathematics”

Real life problems

Example: drones flying in formation. ithout using Newton’s laws of motion?

Systems modeling is a key method for the development of novel engineering tools

Example: drones flying in formation. Source of inspiration? ithout using Newton’s laws of motion?


Example: neuroscience and deep learning.


NeuromodulationSynaptic plasticity


Contemporary examples:

analysis and design of next generation materials (graphene).

engineering in life sciences (cardiovascular physiology, neuroscience).

Systems modeling in three courses

SYST0002: Introduction to signals and systems: open loop. “Observing and analyzing the environment”

SYST0003: Linear control systems: closed loop. “Interacting with the environment”

SYST0017: Advanced topics in systems and control: goes further. (nonlinear systems, chaos, etc.)

SYSTEMInput Output

SYSTEMInput Output

CONTROLLER

What is the value of a mathematical model?


And what about after Einstein?

v = v1 + v2

v1 v2

v

Galilean relativity




v = v1 + v2

v1 v2

v

Galilean relativity Special relativity

v =v1 + v21 + v1v2

c2

Which one is correct, which one is wrong?




v = v1 + v2

v1 v2

v

Galilean relativity Special relativity

v =v1 + v21 + v1v2

c2

Which one is correct, which one is wrong? Which one is useful?




All models are wrong, some are useful.

George E. P. Box


Contemporary examples: Graphene

Theory showed that “2D crystals are unstable” ( L. D. Landau, and E. M. Lifshitz, 1980)




Andre Geim and Konstantin Novoselov did it anyway… (2004)

The micromechanical cleavage technique (‘Scotch tape’ method) for producing graphene

https://fr.wikipedia.org/wiki/Andre_Geim




Andre Geim and Konstantin Novoselov did it anyway… (2004)


Contemporary examples: Standard model in physics (“theory of everything”).

Correctly deducts the existence of the Higgs Boson.


Contemporary examples: Standard model in physics (“theory of everything”).

Correctly deducts the existence of the Higgs Boson.

Fails to explain the origin of gravitational forces (so far). Adding a “graviton”?

Why do we need a theory to analyse and design dynamical systems?

Dynamical systems can have counter-intuitive properties.

Example: Briggs-Rauscher reaction (color shows iodine concentration)


In 1838, Pierre-François Verhulst proposed a dynamical model for the growth of a population (N) depending on the intrinsic growth rate (r) and the maximum number of individuals the environment can support (K).

This equation is called the logistic equation.

Simple behavior: If r >> and N << K: the population grows fast. If N = K: the population does not grow anymore.

Example: mathematical modelling in ecology

The logistic equation

Simulation of the logistic equation for different growth rates and K=1.

1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1Logistic equation, r=2

1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1Logistic equation,r=4

time

N

N

A discrete equivalent of the logistic equation: the logistic map

In 1976, Robert May proposed a “discrete equivalent”

As opposed to the continuous system, the dynamics of the discrete system is extremely rich, and can be “chaotic” for certain values of α.

vs

Dynamical behavior of the logistic map

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=2

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=3.3

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=4

Dynamical behavior of the logistic map: chaos.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=2

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=3.3

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=4

Dynamical behavior of the logistic map: chaos.

In 1976, Robert May proposed a “discrete equivalent”

As opposed to the continuous system, the dynamics of the discrete system are extremely rich, and can be “chaotic” for certain values of α.

This dynamical richness comes from the nonlinearity of the system. But it highlights the fact that continuous and discrete systems are not always “equivalent”.

vs


Because every system is dynamical in nature…


Because every system is dynamical in nature…

Systems theory studies systems dynamical behavior:

Stability

Oscillations

Response speed

Overshoots

“Resonance”

etc.

Where is systems theory useful?

High levels of automation. Example: SpaceX automatic landing.


Engineering in life sciences. Example: cochlear implants.


Engineering in life sciences. Example: deep brain stimulation in Parkinson’s disease.


Civil engineering. How can we study the effect of a fire on a beam in a building? Construct a whole building in a laboratory?Possible solution: submit a real beam to a fire in a laboratory, measure the forces and displacements and feed them to a numerical model of the building.

Aerospace engineering. Will my satellite survive the launch to space?Identification of resonance peaks and nonlinearities using system identification.

What does systems theory consist of?

In this course, we will mainly focus on methods that have been developed in the case of linear, time-invariant (LTI) systems.

The course will introduce two main approaches:

The state-space approach (exhaustive description of the system)

The input-output approach (the system under study is a “black box”)

The course will introduce novel mathematical methods that will look complex at first sight but, when used wisely, can drastically simplify your engineering work.

What does systems theory consist of?

Example: design of an electrical circuit in the frequency domain.

The use of frequency domain methods has made the design of complex systems possible. But it first requires to master the concepts of Fourier transforms, Laplace transforms, etc.

… The order of the set of differential equations describing the typical negative feedback amplifier used in telephony is likely to be very much greater. As a matter of idle curiosity, I once counted to find out what the order of the set of equations in an amplifier I had just designed would have been, if I had worked with the differential equations directly. It turned out to be 55.

Hendrik Bode, 1960

A unified theory to study systems is possible because systems share a lot of properties

Illustration: what is the common point between a simple suspension, an electrical circuit and the mammalian cardiovascular system?



Answer: they all have very similar dynamical and input/output behaviors.




⌧ =m

B=

B

K

= RC =L

R

Stimulation ON Stimulation OFF

m = mass B = damping K = stiffness

R = resistance C = capacitance L = inductance




⌧ =m

B=

B

K

= RC =L

R

Stimulation ON Stimulation OFF

m = mass B = damping K = stiffness

R = resistance C = capacitance L = inductance

?

Open loop systems modeling: analyzing the environment

Case study: cardiovascular physiology. Our system: heart + vessels + blood.

Question: how can the blood flow be continuous knowing that the heart generates pulses?

vs

Modeling the cardiovascular system

Measurements: pressure in the left ventricle LV (input) and in the Aorta Ao(output).


Measurements: pressure in the left ventricle LV (input) and in the Aorta Ao(output).

LV: large variations. Ao: stays highly positive (between 80 and 120 mmHg).

InputOutput

LV and Ao pressure variations over time are signals.


Pathology: some patients have higher systolic pressure with lower diastolic pressure. Why? (It happens mostly in older patients).

Answering this question is very important because these patients are prone to heart failures. What can we do to “fix” the problem?

Modeling the cardiovascular system: Otto Frank.

In 1899, german physiologist Otto Frank came up with a first mathematical representation of the LV-Ao system: the Windkessel Model.

We will use this example to introduce the different ways to model a system

1. Find an equivalent representation of the system under study (Ch2, Ch3)

2. Put system into equations (Ordinary Differential Equations or Difference Equations)

• State-space representation (Ch2-3-4)

3. Extract system input/output properties (Laplace/Fourier or z-transform) (Ch 5-6)

• Transfer function (Ch7)

• System analysis (effects of changes in parameters?) (Ch8-9-10)

Modeling scheme

1. Find an equivalent representation of the system under study


• State-space representation

3. Extract system input/output properties (Laplace/Fourier transform or z-transform)

• Transfer function

• System analysis (effects of changes in parameters?)

Find an equivalent representation of the system under study

Linear algebra

Matrix algebraDifference equations

Informatics

AlgorithmsProgramming

Numerical analysis

Numerical methodsOptimization

Mathematical analysis

Ordinary differential equationsSeriesFourier transformConvolution

Physics

Laws of mechanicsLaws of electricity and electromagnetism

Chemistry

Chemical reactionsOrganic chemistryThermodynamics

Equivalent representation of the left ventricle-aorta (LV-Ao) system

Otto Frank took advantage of the water circuit analogy to electric circuit.

Left ventricle

Aortic valve (r)

Aorta

Arterial compliance (Ca)

Peripheryvessels

(R1, R2, ..., Rn)

r

RCaP(t)

u(t)

PCa(t)Pr(t)

Equivalent representation of the left ventricle-aorta (LV-Ao) system

Otto Frank took advantage of the water circuit analogy to electric circuit.

Left ventricle

Aortic valve (r)

Aorta

Arterial compliance (Ca)

Peripheryvessels

(R1, R2, ..., Rn)

r

RCaP(t)

u(t)

PCa(t)Pr(t)

The 3-Element Windkessel model - Circuit diagram

1. Equivalent circuit of the LV-Ao system

r

RCaP(t)

u(t)

PCa(t)Pr(t)


1. Equivalent circuit of the LV-Ao system

Modeling scheme








2. Mathematical description of the dynamical system: ordinary differential equations r

RCaP(t)

u(t)

PCa(t)Pr(t)

The 3-Element Windkessel model - ODE’s

2. Mathematical description of the dynamical system: ordinary differential equations

Kirchhoff’s voltage law:

r

RCaP(t)

u(t)

PCa(t)Pr(t)

P(t) = Pr (t) + PCa(t) = ru(t) + PCa

(t)


2. Mathematical description of the dynamical system: ordinary differential equations

Kirchhoff’s current law:

Kirchhoff’s voltage law:

r

RCaP(t)

u(t)

PCa(t)Pr(t)

P(t) = Pr (t) + PCa(t) = ru(t) + PCa

(t)

u(t) = iCa(t) + ir (t) = Ca

dPCa(t)

dt+

PCa(t)

R


2. Mathematical description of the dynamical system: ordinary differential equations r

RCaP(t)

u(t)

PCa(t)Pr(t)

P(t) = ru(t) + PCa(t)

Ca

dPCa(t)

dt+

PCa(t)

R= u(t)


r

RCaP(t)

u(t)

PCa(t)Pr(t)


Ca

dPCa(t)

dt+

PCa(t)

R= u(t)

2. Mathematical description of the dynamical system: ordinary differential equations Elements that vary over time are called

variables. Ex: PCa (t)

Elements that are fixed are called parameters. Ex: r, R, Ca

u(t) is the input (commonly used), P(t) is the output.

The 3-Element Windkessel model - Simulation

2A. Validation of the model: simulation (ex: matlab)

u(t)

P(t)

r

RCaP(t)

u(t)

PCa(t)Pr(t)


Ca

dPCa(t)

dt+

PCa(t)

R= u(t)

The 3-Element Windkessel model - State-space

2B. State-space canonical representation

Linear, time-invariant (LTI) dynamical systems can be represented in the form

y = Cx + Du

x = Ax + Bu

where A is the dynamics matrix,

B is the input matrix,

C the output matrix,

D the feedthrough matrix.

Linear: the output is a linear function of the input (not y = x2 for instance)

Time-invariant: parameters do not change over time. (A, B, C and D does not depend on time). Here: the values of r, R and Ca are fixed.



y = Cx + Du

x = Ax + Bu





P(t) = PCa(t) + ru(t)

dPCa(t)

dt= −

1

RCa

PCa(t) +

1

Ca

u(t)






y = Cx + Du

x = Ax + Bu





P(t) = PCa(t) + ru(t)

x(t) = PCa(t), y(t) = P(t)

A = −

1

RCa

,B =1

Ca

,C = 1,D = r

dPCa(t)

dt= −

1

RCa

PCa(t) +

1

Ca

u(t)




Why is the state-space representation important?

General representation! At this stage, we use the same tools, whether the system is a car suspension, an electrical circuit, a chemical reaction, the cardiovascular system, etc.

Four matrices summarize the behavior of any LTI system, regardless of its complexity.

We can use this representation to analyze the key features of the system: stability, reachability, observability, etc.

Very important for system realization: still contains the real system parameters.

y = Cx + Du

x = Ax + Bu

Back to our case study

Questions:

How can the blood flow be continuous knowing that the heart generates pulses?

Some patients have higher systolic pressure with lower diastolic pressure. Why?

y = Cx + Du

x = Ax + Bu x(t) = PCa(t), y(t) = P(t)

A = −

1

RCa

,B =1

Ca

,C = 1,D = r

Modeling scheme







Frequency domain: introduction

(Some of) you have seen the Fourier transform in calculus.

In this course, we will use the Fourier transform, and others such as the Laplace transform (continuous time) and z-transform (discrete time) to move from the time domain to the frequency domain. where ω is an angular frequency (rad/s).


The idea is to decompose a signal into the frequencies that compose it and analyze how a system transmit/transform these frequencies.


The idea is to decompose a signal into the frequencies that compose it and analyze how a system transmit/transform these frequencies.

Low frequency High frequency

Frequency domain: Fourier transform vs Laplace transform

Fourier transform where ω is an angular frequency (rad/s).

Laplace transform where s is the complex frequency s = σ + jω.

Why working in the frequency domain?

Many advantages, here is one of them:

Time domain Frequency domain

The 3-Element Windkessel model - Transfer function

3. Input/output properties: transfer function (frequency domain via Laplace transform)

Idea: describe the system through a simple function that characterizes the way it affects an input U(s)

“s” is the complex number frequency (s = σ+jω). If σ=0: Fourier transform!

U(s) H(s) Y(s) and



U(s) H(s) Y(s)

Idea: describe the system through a simple function that characterizes the way it affects an input U(s)

“s” is the complex number frequency (s = σ+jω). If σ=0: Fourier transform!

There are different ways to compute the transfer function of a system. However, it is convenient to start from the canonical state-space representation (if available)

y = Cx + Du

x = Ax + Bu

which gives

H(s) =Y (s)

U(s)= C (sI − A)−1

B + D (see next slide)

and


Transfer function from state-space representation: which givesand therefore

(1)

(2)

(1)

(1) (2)



Transfer function of the 3-Element Windkessel model ( )

H(s) =Y (s)

U(s)= C (sI − A)−1

B + D

A = −

1

RCa

,B =1

Ca

,C = 1,D = r


H(s) =Y (s)

U(s)= C (sI − A)−1

B + D


Transfer function of the 3-Element Windkessel model ( )A = −

1

RCa

,B =1

Ca

,C = 1,D = r

= 1(s +1

RCa

)−11

Ca

+ r


H(s) =Y (s)

U(s)= C (sI − A)−1

B + D



1

RCa

,B =1

Ca

,C = 1,D = r

= 1(s +1

RCa

)−11

Ca

+ r

=R

RCas + 1+ r


H(s) =Y (s)

U(s)= C (sI − A)−1

B + D

≡

K

τs + 1+ r => Low pass filter! (r<< physiologically)

K=R: gain τ=RCa: time constant ωc=1/τ: cutoff frequency



1

RCa

,B =1

Ca

,C = 1,D = r

=R

RCas + 1+ r

= 1(s +1

RCa

)−11

Ca

+ r


≡

K

τs + 1+ rLow-pass filter

The transfer function of the Windkessel model helps making predictions on the potential effects of physiological and/or pathological conditions on blood pressure

Low frequency High frequency

Low pass filter

τ = RCa

H(s) =R

RCas + 1+ r



Loss of low-pass filtering properties



Loss of low-pass filtering properties

Low pass filter

τ = RCa

H(s) =R

RCas + 1+ r


Atherosclerosis: loss of arterial compliance => Ca decreases => τ=RCa decreases

τ = RCa

Low pass filter

H(s) =R

RCas + 1+ r

Modeling the cardiovascular system: conclusion

The vascular system acts as a low-pass filter, following slow heart movements but filtering fast heart movements.

This allows to maintain a rather constant blood flow in the system.

InputOutput

Modeling scheme







Systems theory: state-space vs input-output approaches


Ca

dPCa(t)

dt+

PCa(t)

R= u(t)

H(s) =R

RCas + 1+ r

State-space approach

Potentially complex and high-dimensionalNon unique Closely relates to physics/biology

Input-output approach

Low dimensional, simple to interpret UniqueMore abstract

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