Modeling and Analysis of Systems General Informationsguilldrion/Files/SYST0002-2016-Lecture1.pdf · Modeling and Analysis of Systems General Informations ... Systems modeling in three

Modeling and Analysis of SystemsGeneral Informations

Guillaume DrionAcademic year 2015-2016

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 main lessons - Monday 10:45 to 12:45 9 tutorial sessions split in 4 groups - Monday 8:30 to 10:30 (room 1.97 of B28) - Monday 16:00 to 18:00 - Wednesday 9:00 to 11:00 - Friday 14:00 to 16:00

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]




Tutorial groups: student list

Schedule of the year

A1, A2, A3 and A4 correspond to 4 facultative assignments that could give you some bonus points.

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

Guillaume DrionAcademic year 2015-2016

So far, you have acquired most of the basic technical skills of an Engineer

Mathematical analysis

Ordinary differential equationsSeriesFourier transformConvolution Linear algebra

Matrix algebraDifference equationsPhysics

Laws of mechanicsLaws of electricity and electromagnetism

Informatics

AlgorithmsProgramming

Chemistry

Chemical reactionsOrganic chemistryThermodynamics

Numerical analysis

Numerical methodsOptimization

How will you apply these technical skills in a working environment?

Any project that you will be working on will require to combine the technical skills learned from various “disciplines”.

Linear algebra

Matrix algebraDifference equations

Informatics


Numerical analysis



Ordinary differential equationsSeriesFourier transformConvolution

Physics


Chemistry


The main goal of this course is to provide a general framework for the modeling, analysis, design and implementation of “systems”

SYSTEMS MODELING

AnalysisDesignImplementation

Linear algebra


Informatics


Numerical analysis




Physics


Chemistry


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

Open loop systems vs closed loop systems

Poikilotherm (“open loop”) vs homeotherm (“closed loop”)

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

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


Informatics


Numerical analysis




Physics


Chemistry


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


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

Left ventricle

Aortic valve (r)

Aorta


Peripheryvessels

(R1, R2, ..., Rn)

r

RCaP(t)

u(t)

PCa(t)Pr(t)



Left ventricle

Aortic valve (r)

Aorta


Peripheryvessels

(R1, R2, ..., Rn)

r

RCaP(t)

u(t)

PCa(t)Pr(t)



Left ventricle

Aortic valve (r)

Aorta


Peripheryvessels

(R1, R2, ..., Rn)

r

RCaP(t)

u(t)

PCa(t)Pr(t)



r

RCaP(t)

u(t)

PCa(t)Pr(t)



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

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 influence 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 influence the output (y = Du) (feedthrough matrix).

Why is the state-space representation important?

Isolation of the internal states (dimensionality of the system)

Analysis of key features of the system: stability, reachability, observability, etc.

Construction of a block diagram

Analysis in the frequency domain

y = Cx + Du

x = Ax + Bu

Block diagrams

INT

Display Control

Graphics Engine

TFT LCD Display

Proc & Mem I/F

SD/MMC

I/F

Audio Codec Camera

WLAN

Power Conv.

Battery

Keyboard/ Touchpad

USB PHY

DDR3512 MB

CMOS Camera

I/FI2S Audio

I/F

SD Card Slot

SPI I/F

Display Control

USB Ports

ARM

ARMADA610

TCON

IO3731

Security CPU

ARM

SDRAM

CPU

SDIOI/F

I2C I/F

SD/MMC

I/F

NAND FlashInternal SD

Accelerometer

USB Hub

32b

18b

8bI2C

Input Buttons GPIO

OneWire

FlashROM

FlashROM

SPI

PS2 (x2)

SPI

Open Firmware

XO-1.75 Block Diagram7/04/11

SPI I/FSPI

TWSI2

TWSI1

I2S

SSP1

SSP3

CCIC1

MMC2

MMC1

MMC3

I2C I/FGPIO

CMDACK

INT

INT

Always powered

Powered in idle and run

Optionally powered

EC

NAND Flash4GB eMMC

OR

USB

SPI I/FGPIO

Battery Charger

DC Input

Wikipedia: A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks.

https://en.wikipedia.org/wiki/Diagram

https://en.wikipedia.org/wiki/Diagram

https://en.wikipedia.org/wiki/System

https://en.wikipedia.org/wiki/System

Block diagrams

y = Cx + Du

x = Ax + Bu

The 3-Element Windkessel model - Block diagram


y = Cx + Du

x = Ax + Bu

Block diagram representation (ex: implementation in Simulink)

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

A = −

1

RCa

,B =1

Ca

,C = 1,D = r

+u(t) 1Ca

X(t) X(t)+

-1RCa

r

y(t)


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 mathematical analysis.

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).


You have seen the Fourier transform in mathematical analysis.

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.



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

Example: a 5Hz sine wave


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


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







Highlights of the day

1. Signals: continuous, discrete, domain, image.

2. Systems: open loop, closed loop, variables, parameters, inputs, outputs, states, modeling (ODEs or Difference Equations), simulation.

3. State-space representation: linear time-invariant (LTI) systems, “A, B, C, D” representation, block diagrams.

4. Frequency domain: Transforms, transfer functions.

5. Modeling scheme

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Modeling and Analysis of Systems General Informationsguilldrion/Files/SYST0002-2016-Lecture1.pdf · Modeling and Analysis of Systems General Informations ... Systems modeling in three