modeling and analysis of systems general informationsguilldrion/files/syst0002-2016-17... ·...

Modeling and Analysis of SystemsGeneral Informations

Guillaume DrionAcademic year 2016-2017

SYST0002 - General informations

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

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

Organization: 9 or 10 main lessons - Tuesday 10:45 to 12:45 9 tutorial sessions split in 4 groups - Monday 10:45 to 12:45 (rooms to be determined) - Thursday 8:30 to 10:30

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

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

mailto:[email protected]




Schedule of the year

Tutorials will start on Monday, 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.

Evaluation:

Project: throughout the year (cut in pieces with several deadlines).

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

Modeling and Analysis of SystemsLecture #1 - Introduction to Systems Theory

Guillaume DrionAcademic year 2016-2017

Modeling and analysis of systems

Real problems


Real problems

Engineering problems


Real problems

Engineering problems

SYSTEMS MODELING

AnalysisDesignImplementation

“Applied mathematics”


Historical example: mechanics before and after Newton.


Contemporary examples:

analysis and design of next generation materials (graphene).


Contemporary examples:

analysis and design of next generation materials (graphene).

engineering in life sciences (cardiovascular physiology, neuroscience).

Systems modeling in three courses

SYST0002: Modeling and analysis of 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


Systems theory:

translates a possibly complex problem in mathematics.

develops and uses specific set of tools to solve these problems.

more general/abstract = more powerful.


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 = massB = dampingK = stiffness

R = resistanceC = capacitanceL = inductance




⌧ =m

B=

B

K

= RC =L

R

Stimulation ON Stimulation OFF

m = massB = dampingK = stiffness

R = resistanceC = capacitanceL = 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


Kirchhoff’s voltage law:

r

RCaP(t)

u(t)

PCa(t)Pr(t)

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

(t)



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



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)


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)





A - the dynamics matrix (“states feed back on themselves”)

y = Cx + Du

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

A = −

1

RCa

,B =1

Ca

,C = 1,D = r

u is the input of the system, y is the output. What is x?

x is called a state. A state describes the internal dynamics of the system.Hence the name state-space representation.

The matrix A describes how the states influence themselves (x = Ax), and therefore describes how the dynamics of the system evolve (dynamics matrix).


B - the input matrix

y = Cx + Du


A = −

1

RCa

,B =1

Ca

,C = 1,D = r

The input not only directly affects the output of the system, but also affects the internal states.

The matrix B describes how the input influences the states (x = Bu) (input matrix).


C - the output matrix

y = Cx + Du


A = −

1

RCa

,B =1

Ca

,C = 1,D = r

The output of a system is mostly shaped by its internal states.

The matrix C describes how the states are seen in the output (y = Cx) (output matrix).


D - the feedthrough matrix

y = Cx + Du


A = −

1

RCa

,B =1

Ca

,C = 1,D = r

Some part of the input might also affect the output without “being affected” by the dynamics of the system.

The matrix D describes how the input directly influences the output (y = Du) (feedthrough matrix).

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


A = −

1

RCa

,B =1

Ca

,C = 1,D = r

Modeling scheme







Frequency domain: introduction

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 gives

and 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







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