introduction to signals and systems lecture #2 ... · in “state-space”, dynamical systems are...

Introduction to Signals and Systems Lecture #2 - Mathematical Representations of Dynamical Systems

Guillaume Drion Academic year 2017-2018

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Outline

Mathematical modeling: basics and illustrations

System dynamics: static gain, input integration and state interactions (feedback)

System output: update function vs output function

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A simple modeling example: the wheat and chessboard problem

If you start by putting one grain on the first square and double the number for each consecutive square: (i) how many grains would be put on the nth square? (ii) how many grains do you need to fill the chessboard?

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Mathematical model of the problem

What is ? What values can take? And ? What does the first equation describes? What does the second equation describes?

x[n] n x[n]

x[0] = 1,

x[n+ 1] = 2x[n]


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What is ? What values can take? And ?x[n] n x[n]

is the number of wheat grains on the nth square. is the square number.

can take values in the range .

can be any positive integer.

x[n] n

n

x[n]

Such function is called signal.

x[0] = 1,

x[n+ 1] = 2x[n]

[0, 63]


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x[n] n

n

x[n]


This defines the domain of the signal .x[n]

x[0] = 1,

x[n+ 1] = 2x[n]

[0, 63]


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x[n] n

n

x[n]


This defines the domain of the signal .x[n]

This defines the image of the signal .x[n]

x[0] = 1,

x[n+ 1] = 2x[n]

[0, 63]

Signals: domain and image

Systems treat signals.

Signals (x) are defined by their domain (X) and their image (Y):

Domain (for us): (discrete signal) or (continuous signal).

Image: set of values that the signal can take (ex: probabilities/binary signals).

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Signals

Inputs, outputs, variables, etc. are signals. Examples: “Hi!” on a computer is “01001000 01101001 00100001” (Ascii) => Discrete, binary signal x[n].

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Signals

Inputs, outputs, variables, etc. are signals. Examples: “Hi!” on a computer is “01001000 01101001 00100001” (Ascii) => Discrete, binary signal x[n]. Electrical signaling of a neuron: => Continuous signal x(t), larger than VK and smaller than VNa.

400 ms20 mV

VK

VNa

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Inputs, outputs, variables, etc. are signals. Examples: “Hi!” on a computer is “01001000 01101001 00100001” (Ascii) => Discrete, binary signal x[n]. Electrical signaling of a neuron: => Continuous signal x(t), larger than VK and smaller than VNa.

400 ms20 mV

Domain Image

Domain Image (+⊂ℝ)

VK

VNa

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A continuous signal can have a somehow “discrete image”... Ex: current flowing through an ion channel over time:

... and a discrete signal can have a “continuous image”. Ex: many digitalized signals.

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Why should we care about the domain of a signal?

Time-shift

Time-reversal

Time-scaling

Comparing continuous and discrete signals:

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Time-shift

Time-reversal

Time-scaling

Why should we care about the domain of a signal?

Some values of α lead to problems

Comparing continuous and discrete signals:

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Why should we care about the image of a signal?

In theory, you can design any system using mathematical modeling.

Ex: we want to design a system that controls the movement of a robot arm for weight lifting, and proudly come up with this system that works perfectly:

But in real-life, the values that the signals can take are constrained. You have to take this limitations of the signal image into account in your model!

Ex: in the robot, the voltage signal cannot exceed some saturation voltage. With your design, the robot can barely lift his arm, let alone any additional weight...

Input voltage

Lifted weight ( relative to arm weight)

123456

15

Why should we care about the image of a signal?

In theory, you can design any system using mathematical modeling.

Ex: we want to design a system that controls the movement of a robot arm for weight lifting, and proudly come up with this system that works perfectly:

But in real-life, the values that the signals can take (image) are constrained. You have to take this limitation into account in your model!

Ex: in the robot, the voltage signal cannot exceed some saturation voltage. With our design, the robot can barely lift his arm, let alone any additional weight...

Input voltage

Lifted weight ( relative to arm weight)

123456

Voltage saturation

16

Back to the wheat and chessboard problem


17

Mathematical model of the problem

What is ? What values can take? And ? What does the first equation describes? What does the second equation describes?

x[n] n x[n]

x[0] = 1,

x[n+ 1] = 2x[n]


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What does the first equation describes?

is the initial condition of my system.

A system/model can have a very different output for different initial conditions. Example:

vs

x[0] = 1,

x[n+ 1] = 2x[n]

x[0] = 1,

x[n+ 1] = 2x[n]

x[0] = 1

x[0] = 0,

x[n+ 1] = 2x[n]


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What does the second equation describes?

describes the “future” evolution of my signal .

This equation contains the dynamics of my system.

The signal is a variable of my mathematical model.

x[n+ 1] = 2x[n]x[n]

x[n]

x[0] = 1,

x[n+ 1] = 2x[n]


If you start by putting one grain on the first square and double the number for each consecutive square: (i) how many grains would be put on the nth square?

20

x[0] = 1,

x[n+ 1] = 2x[n]

x[0] = 1 = 20,

x[1] = 2 ⇤ 1 = 2 = 21,

x[0] = 2 ⇤ 2 = 4 = 22,

...


If you start by putting one grain on the first square and double the number for each consecutive square: (i) how many grains would be put on the nth square?

21

x[0] = 1,

x[n+ 1] = 2x[n]

x[0] = 1 = 20,

x[1] = 2 ⇤ 1 = 2 = 21,

x[0] = 2 ⇤ 2 = 4 = 22,

...

x[n] = 2n

This function describes the state of my system at “time” .n

In “state-space”, dynamical systems are modeled using difference equations (discrete domain) or differential equations (continuous domain).

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, are variables of the systems (the model describes their evolution). A system can have many variables (dimension).x[n]

is a parameter of the system. A system can have many parameters.a

x =dx

dt

(where )x[n+ 1] = ax[n]

! x[n] = a

n

x = ax

! x(t) = e

at

Discrete domain Continuous domain

x(t)

In “state-space”, dynamical systems are modeled using difference equations (discrete domain) or differential equations (continuous domain).

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is a parameter of the system. A system can have many parameters.a

x =dx

dt

(where )x[n+ 1] = ax[n]

! x[n] = a

n

x = ax

! x(t) = e

at

Discrete domain Continuous domain

a > 0a < 0


If you start by putting one grain on the first square and double the number for each consecutive square: (ii) how many grains do you need to fill the chessboard?

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x[0] = 1,

x[n+ 1] = 2x[n]

This function describes the output of my system at “time” .n

y[n] =nX

i=0

x[i]

Modeling example: predator-prey model

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H[n+ 1] = H[n] + bH[n]� aL[n]H[n]

L[n+ 1] = L[n] + cL[n]H[n]� dL[k]


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H[n+ 1] = H[n] + bH[n]� aL[n]H[n]

L[n+ 1] = L[n] + cL[n]H[n]� dL[k]

1. If Hare or Lynx do not die or the food supply does not increase, the populations remain stable.


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H[n+ 1] = H[n] + bH[n]� aL[n]H[n]

L[n+ 1] = L[n] + cL[n]H[n]� dL[k]

2. If Hares get more food (parameter ), their population increases. If Lynxes get more Hares to eat, their population increases with a rate .

bc


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H[n+ 1] = H[n] + bH[n]� aL[n]H[n]

L[n+ 1] = L[n] + cL[n]H[n]� dL[k]

3. If Hares are eaten by lynxes, their population decreases with a rate . If Lynxes die, their population decreases with a rate .

ad


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H[n+ 1] = H[n] + bH[n]� aL[n]H[n]

L[n+ 1] = L[n] + cL[n]H[n]� dL[k]

Observations/data

Model simulation

Modeling example: neuron firing

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Equivalent scheme


31Equivalent scheme

Mathematical model


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Experimental data

Model simulation

Examples of problems requiring mathematical modeling

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Design of mass-damper Design of a robot arm

Design of a DC/DC converter

Outline




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System dynamics

Modeling and analysis of systems: open loop. “Observing and analyzing the environment”

SYSTEMInput Output

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System dynamics: static gain

Example #1: Applying a force on a spring.

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kF

xF = kx () x =

1

k

F

F (Input)0

1

x (Output)

0

1/k


Example #1: Applying a force on a spring. The output is solely shaped by the input at the present time: static system.

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kF

xF = kx () x =

1

k

F

F (Input)0

1

x (Output)

0

1/kstatic gain


Example #2: A current through a resistive wire.

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R

i

V = Ri () i =1

RV

static gain

i (Input)0

1

V (Output)

0

R


Mathematical representation: functions. 1 input, 1 output: where is the static gain.

More than one input and/or output: matrix representation:

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y(t) = Ku(t) K

y(t) = k1u1(t) + k2u2(t) + · · ·+ knun(t)

=�k1 k2 · · · kn

�

0

BB@

u1

u2

· · ·un

1

CCA = Ku(t)

System dynamics: input integration

Example #1bis: Applying a force on a damper.

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F

x

cF = cv = cx

wherex =

dx

dt

=) x =1

c

F

F (Input)0

1

x (Output)

0

1/c


Example #1bis: Applying a force on a damper. The output depends on past values of input: energy storage (memory). The energy from input F is stored in position x. x is a state of the system.

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F

x

cF = cv = cx

wherex =

dx

dt

=) x =1

c

F

F (Input)0

1

x (Output)

0

1/c


Example #2bis: A current “through” a capacitor. The energy from input i is stored in voltage V. V is a state of the system.

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i

C i = CV () V =1

Ci

i (Input)0

1

V (Output)

0

1/C


Integrators store past values of the input into states.

Mathematical representation: ordinary differential equations (continuous). or difference equations (discrete).

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Input

State

x = Bu

x[n+ 1] = Bu[n]or

System dynamics: state interactions

Example #1ter: Applying a force on a spring and a damper in parallel.

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k

c

x

F

F = kx+ cx

Behavior?


Example #1ter: Applying a force on a spring and a damper in parallel. (a) : integration of the input

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k

c

x

F

F = kx+ cx

=) x =1

c

[F � kx]

x =1

c

F ⌘ 1

c

u


Example #1ter: Applying a force on a spring and a damper in parallel. (a) : integration of the input (b) : does not depend on input - Internal dynamics!

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k

c

x

F

F = kx+ cx

=) x =1

c

[F � kx]

x =1

c

F ⌘ 1

c

u

x = �k

c

x


Example #1ter: Applying a force on a spring and a damper in parallel. (b) : Internal dynamics. An increase in the position induces a decrease in the speed . Adding the spring adds negative feedback to the system.

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k

c

x

F

F = kx+ cx

=) x =1

c

[F � kx]

x = �k

c

x

x

x


Example #1ter: Applying a force on a spring and a damper in parallel.

48

k

c

x

F

F = kx+ cx

=) x =1

c

[F � kx]

F (Input)0

1

x (Output)

0

?


Example #2ter: Applying an input voltage on an RC circuit. (a) : integration of the input (b) : Internal dynamics.

49

R

i

VC

i

i = iR + iC =V

R+ CV

=) V =1

C

i� V

R

�

V =1

Ci

V = � 1

RCV


Example #2ter: Applying an input voltage on an RC circuit.

50

R

i

VC

i

i = iR + iC =V

R+ CV

=) V =1

C

i� V

R

�

i (Input)0

1

V (Output)

0

?


Mechanical system vs electrical system.

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=) V =1

C

i� V

R

�=) x =

1

c

[F � kx]

Mechanical system Electrical system


Mechanical system vs electrical system.

52

=) V =1

C

i� V

R

�=) x =

1

c

[F � kx]


F 1/c x

-k/c

y i 1/C V

-1/RC

y

Block diagram representation


Static gain of the mechanical and electrical systems?

53

=) V =1

C

i� V

R

�=) x =

1

c

[F � kx]


Input0

1

Output

0

?



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=) V =1

C

i� V

R

�=) x =

1

c

[F � kx]


Steady-state: x = 0

x =F

k

1

c

[F � kx] = 0

=)



The damper (resp. capacitor) only plays a dynamical role (no static effect).

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=) V =1

C

i� V

R

�=) x =

1

c

[F � kx]


Steady-state: Steady-state: x = 0 V = 0

V = Ri

1

C

i� V

R

�= 0

x =F

k

1

c

[F � kx] = 0

=) =)

System dynamics: mathematical representation

The dynamics of a system are represented by ordinary differential equations (continuous) or difference equation (discrete) of the form

For linear systems, it writeswhere and are vectors, and and are matrices (dimensionality of the system).

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x(t) = f(x(t), u(t)) x[n+ 1] = f(x[n], u[n])

x = Ax+Bu

x[n+ 1] = Ax[n] +Bu[n]

BAx u


Classical example : Newton’s second law of motion. State #1: position ( )State #2: speed ( )

57

F = mx

x2 = x

x1 = x

x1

x2

�=

0 10 0

� x1

x2

�+

0

1/m

�F


Classical example : Newton’s second law of motion. State #1: position ( )State #2: speed ( )

58

F = mx

x2 = x

x1 = x

x1

x2

�=

0 10 0

� x1

x2

�+

0

1/m

�F

Force affects speed (Input integration)

Speed affects position (state interactions)

Outline




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States are not always outputs of the system

Example: a series RC circuit in a black box but we cannot measure ! is the update function: it describes systems dynamics

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R Vout

C

i

Vin

Vc =1

RC(Vin � Vc)

Vc

Vc =1

RC(Vin � Vc)


Example: a series RC circuit in a black box but we cannot measure ! Systems output? In terms of input and state: is the output function.

61

R Vout

C

i

Vin

Vc =1

RC(Vin � Vc)

Vc

y = VR = Vin � Vc

y = Vin � Vc


Example: a series RC circuit in a black box but we cannot measure ! update function output function

62

R Vout

C

i

Vin

Vc =1

RC(Vin � Vc)

Vc

Vc

=1

RC(V

in

� V c)

Vout

= Vin

� Vc

General form of dynamical systems: state-space representation

63

A dynamical system is represented by its update function and its output function

For linear systems, it writeswhere , and are vectors,

, , and are matrices (dimensionality of the system).

x = f(x, u)

y = g(x, u)

x = Ax+Bu

y = Cx+Du

BA

x u y

DC

x[n+ 1] = Ax[n] +Bu[n]

y[n] = Cx[n] +Du[n]

x[n+ 1] = f(x[n], u[n])

y[n] = g(x[n], u[n])

General form of dynamical systems: state-space representation

64

A dynamical system is represented by its update function and its output function

For linear systems, it writeswhere , and are vectors,

, , and are matrices (dimensionality of the system).

x = f(x, u)

y = g(x, u)

x = Ax+Bu

y = Cx+Du

BA

x u y

DC

x[n+ 1] = Ax[n] +Bu[n]

y[n] = Cx[n] +Du[n]

x[n+ 1] = f(x[n], u[n])

y[n] = g(x[n], u[n])

Linear systems: A,B,C,D representation

Linear, time-invariant (LTI) dynamical systems can be represented in the formwhere describes how the dynamics of the system evolve (dynamics matrix) describes how the input influences the states (input matrix) describes how the states “are seen” in the output (output matrix). describes how the input directly influences the output (feedthrough matrix).

The output is therefore a function of the input and the states. Simple but recurrent case: the output is one of the states.

65

x = Ax+Bu

y = Cx+Du

BA

DC

Linear systems: block diagram representation

66

General form of dynamical systems

In this course, we will develop tools to analyze the behavior of dynamical systems.

In particular: stability, static behavior, dynamical behavior.

Input 0

1

Output #1

0

Output #2

0

K

Why do we need general tools to analyze system dynamics? Illustration.

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.

Mathematical modeling is widely used 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

69

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

70

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

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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


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=4

72

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


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1Logistic map, a=4

73

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

74

How can general tools help us analyze the behavior of dynamical systems?

Illustration: systems stability. Back to the spring-damper system .

if (classical spring): the spring acts against the movement, giving a stable steady-state to the position. negative feedback

if : the further the system is, the faster is goes away. the system will go away until the “negative spring” saturates. positive feedback

=) x =1

c

[F � kx]

k > 0

k < 0


In a linear system: no saturation Any positive feedback brings instability!

How to assess stability of a general linear system (1D): System is stable if .

a < 0x = ax+ bu


In a linear system: no saturation Any positive feedback brings instability!

How to assess stability of a general linear system (1D): System is stable if .

How to assess stability of a general linear system (N-D): Multiple state-interactions: what are the feedbacks?N-feedback directions (eigenvectors of matrix ), each having its own amplitude (eigenvalues of ). The system is stable if (all feedbacks negative).

a < 0x = ax+ bu

x = Ax+Bu

R(eig(A)) < 0

AA

introduction to signals and systems lecture #2 ... · in “state-space”, dynamical systems are...

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