Feedback Control of Dynamic Systems

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Feedback Control of Dynamic Systems

Yves Briere

yves.briere@isae.fr

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I. Introduction

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9/23/2009 I. Introduction 3

Introduction

• Aim of the course

– Give a general overview of classical and modern control theory

– Give a general overview of modern control tools

• Prerequisites

– Mathematics : complex numbers, linear algebra

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Introduction

• Tools

– Matlab / Simulink

• Book

– « Feedback Control of Dynamics Systems », Franklin, Powell, Amami-Naeini, Addison-Wessley Pub Co

– Many many books, websites and free references...

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Introduction

270 BC : the clepsydra and other hydraulically regulated devices for time measurement (Ktesibios)

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Introduction

1136-1206 : Ibn al-Razzaz al- Jazari

“The Book of Knowledge of Ingenious Mechanical Devices”

crank mechanism, connecting rod,

programmable automaton, humanoid robot, reciprocating piston engine, suction pipe, suction pump, double-acting pump, valve, combination lock, cam, camshaft, segmental gear, the first mechanical clocks driven by water and weights, and especially the crankshaft, which is considered the most important mechanical invention in history after the wheel

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Introduction

1600-1900 : pre-industrial revolution

Thermostatic regulators (Cornelius Drebbel 1572 - 1633)

Windmill speed

regulation.

1588 : mill hoper ; 1745 : fantail by Lee ; 1780 : speed regulation by Mead

Water level regulation (flush toilet, steam machine)

Steam engine pressure

regulation (D.

Papin 1707)

Centrifugal mechanical governor ( James Watt, 1788)

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Introduction

1800-1935 : mathematics, basis for control theory

Differential equations first analysis and proofs of stability condition for feedback systems (Lagrange, Hamilton, Poncelet, Airy-1840, Hermite-1854, Maxwell-1868, Routh-1877, Vyshnegradsky-1877, Hurwitz-1895, Lyapunov-1892)

Frequency domain approach (Minorsky-1922, Black-1927, Nyquist-1932, Hazen-1934)

1940-1960 : classical period

Frequency domain theory : (Hall-1940, Nichols-194, Bode-1938)

Stochastic approach (Kolmogorov-1941, Wiener and Bigelow-1942)

Information theory (Shannon-1948) and cybernetics (Wiener-1949)

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Introduction

1960-1980 : modern period, aeronautics and spatial industry

Non linear and time varying problems (Hamel-1949, Tsypkin-1955, Popov- 1961, Yakubovich-1962, Sandberg-1964, Narendra-1964, Desoer-1965,Zames- 1966)

Optimal control and Estimation theory (Bellman-1957, Pontryagin-1958,Kalman- 1960)

Control by computer, discrete systems theory : (Shannon-1950, Jury-1960, Ragazzini and Zadeh-1952, Ragazzini and Franklin-1958,(Kuo-1963, Aström-1970) 1980-... : simulation, computers, etc...

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Introduction

What is automatic control ?

Basic idea is to enhance open loop control with feedback control This seemingly idea is tremendously powerfull

Feedback is a key idea in control

Open loop Controler

Process

Input Output

Input reference

Perturbation

Closed loop Controler

Process

Input Output

Input reference

Measurement

Perturbation

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Introduction

Example : the feedback amplifier Harold Black, 1927

A R2 R1

V1 + V2

- Amplifier A has a high gain (say 40dB)

1 R

2 R 1

R 2 1 R

A 1 1

1 1

R 2 R 1

V 2

V ≈ −



 



 + +

⋅

−

=

Resulting gain is determined by passive components ! amplification is linear

reduced delay noise reduction

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Introduction

Use of block diagrams

Capture the essence of behaviour standard drawing

abstraction

information hiding

points similarities between systems

Same tools for :

generation and transmission of energy transmission of informaiton

transportation (cars, aerospace, etc...) industrial processes, manufacturing mechatronics, instrumentation

Biology, medicine, finance, economy...

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Introduction

Basic properties of feedback (1)

V1 A

k - V2

+

( )

k 1 k A 1 1

1 k

1 1 V

2 V

1 V A k

A 1 2 V

2 V k 1 V A 2 V

≈ + ⋅

⋅

=

⋅

=

⋅ +

⋅

−

⋅

=

Resulting gain is determined by feedback !

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Introduction

Basic properties of feedback (2) : static properties

k_p r

-

k_c

e u

d

y k_p

k_c

e u

d

y r : reference e : error

d : disturbance y : output

k_c : control gain K_p : process gain

Open loop control : y = k_p ⋅k_c ⋅e+ k_p ⋅d

Closed loop control : ^p ^c ^p

p c p c

k k k

y r d

1 k k 1 k k

= ⋅ ⋅ + ⋅

+ ⋅ + ⋅

If k_c is big enough y tend to r and d is rejected

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Introduction

Basic properties of feedback (2) : dynamics properties Closed loop control can :

enhance system dynamics stabilize an unstable system

make unstable a stable system !

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Introduction

The On-Off or bang-bang controller : u = {u_max , u_min}

e u

The proportional controller : u=k_c.(r – y)

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Introduction

The proportional derivative controller

( ) ( ) ( )

dt t k de

t e k t

u = _p ⋅ + _d ⋅

Gives an idea of future : phase advance The proportional derivative controller

( ) ( ) ( )

dt t k de

t e k t

u = _p ⋅ + _d ⋅

The proportional integral controller

( )

t ⁼ kp ^⋅e

( )

t ⁺ ki ^⋅

_∫

₀^te

( )

^τ ^⋅d^τ u

e(t) tends to zero !

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9/23/2009 II. A first controller design 18

II. A first controller design

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A first control design

• Use of block diagrams

• Compare feedback and feedforward control

• Insight feedback properties :

– Reduce effect of disturbances

– Make system insensitive to variations – Stabilize unstable system

– Create well defined relationship between output and reference – Risk of unstability

• PID controler :

( )

⁼ p ^⋅

( )

⁺ d ^⋅

( )

⁺ ki ^⋅

_∫

₀^te

( )

^τ ^⋅d^τ dt

t k de

t e k t

u

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Cruise control

mg

F

θθθθ

A cruise control problem :

• Process input : gas pedal u

• Process output : velocity v

• Reference : desired velocity v_r

• Disturbance : slope θ Construct a block diagram

• Understand how the system works

• Identify the major components and the relevant signals

• Key questions are :

– Where is the essential dynamics ?

– What are the appropriate abstractions ?

• Describe the dynamics of the blocks

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Cruise control

We made the assumptions :

• Essential dynamics relates velocity to force

• The force respond instantly to a change in the throttle

• Relations are linear

v_r Body -

Engine F

ext. force Controller

v Throttle

We can now draw the process equations

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Cruise control

Process linear equations :

( )

₊ _⋅ ₌ ₋ _⋅ _⋅_θ

⋅ k v F m g

dt t m dv

Reasonable parameters according to experience :

( )

₊ ₀_.₀₂_⋅_v ₌ _u ₋₁₀_⋅_θ

dt t dv Where :

• v in m.s^-1

• u : normalized throttle 0 < u < 1

• θ slope in rad

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Cruise control

Process linear equations :

PI controller :

( )

₊ ₀_.₀₂_⋅_v ₌ _u ₋₁₀_⋅_θ

dt t dv

( )

t ⁼ k^⋅

(

vr ⁻ v

( )

t

)

⁺ ki ^⋅

_∫

₀^t

(

vr ⁻ v

( )

^τ

)

^⋅d^τ u

Combining equations leads to :

( ) ( ) ( ) ( ) ( )

dt t t d

e dt k

t k de

dt t e d

i

⋅ θ

=

⋅ +

+ 0 . 02 10

2 2

Integral action

Steady state and θ = 0 e = 0 !

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Cruise control

Now we can tune k and k_i in order to achieve a given dynamics

( ) ( ) ( ) ( ) ( )

dt t 10 d

t e dt k

t k de

02 . dt 0

t e d

i

2 + + ⋅ + ⋅ = ⋅ θ

( ) ( )

^x

( )

^t ⁰

dt t 2 dx

dt t x

d ₂

0 0

2 + ⋅σ⋅ω ⋅ + ω ⋅ =

How to choose ω₀ and σ ?

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Cruise control

Compare open loop and closed loop

Open loop

Closed loop

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Cruise control

Compare different damping σ (ω₀ = 0.1)

σ = 0.5 σ = 1 σ = 2

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Cruise control

Compare different natural frequencies ω₀ (σ= 1)

ω₀ = 0.05 ω₀ = 0.1 ω₀ = 0.2

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Cruise control

Control tools and methods help to :

• Derive equations from the system

• Manipulate the equations

• Understand the equations (standard model)

– Qualitative understanding concepts – Insight

– Standard form – Computations

• Find controller parameters

• Validate the results by simulation END 1

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Standard models

Standard models are foundations of the “control language”

Important to :

Learn to deal with standard models Transform problems to standard model

The standard model deals with Linear Time Invariant process (LTI), modelized with Ordinary Differential Equations (ODE) :

( ) ( ) ( ) ( )

^... ^b ^u

( )

^t

dt t u b d

t y a

...

dt t y a d

dt t y d

1 n n 1 n 1 1 n

n 1 n n 1

n + ⋅ ⁻ ₋ + ⋅ = ⋅ ⁻ ₋ + ⋅

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Standard models

Example (fundamental) : the first order equation

( ) ( )

^t ^y ⁰ ^e ^a ^t

y

0 t

y dt a

t dy

⋅

⋅ −

⇒ =

=

⋅ +

( ) ( ) ( )

( ) ( )

⁼ ^⋅ ⁺ ^⋅ ^{( )} ^⋅

( )

^τ ^⋅ ^τ

⇒

⋅

=

⋅ +

∫

⁻ ^⋅ ⁻^τ

⋅

− b e u d

e 0 y t

y

t u b t

y dt a

t dy

t 0

t a t

a

Input signal Initial conditions

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Standard models

A higher degree model is not so different :

( ) ( )

^... ^a ^y

( )

^t ⁰

dt t y a d

dt t y d

1 n n 1 n n 1

n + ⋅ ⁻ ₋ + ⋅ =

Characteristic polynomial is :

( )

n

1 n 1

n a s ... a

s s

A = + ⋅ ⁻ +

If polynomial has n distinct roots α_k then the time solution is :

( ) ∑

=

⋅

⋅ α

= ⁿ

1 k

t k e ^k C

t y

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Standard models

Real α_k roots gives first order responses :

Complex α_k=σ±i.ω roots gives second order responses :

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Standard models

General case (input u) :

( ) ( ) ( ) ( ) ( )

( )

⁼

∑ ( )

^⋅ ⁺

_∫ ( )

⁻^τ ^⋅ ^τ

⇒

⋅ +

⋅

=

⋅ +

=

⋅ α

−

t 0 n

1 k

t k

1 n n 1 n 1 1 n

n 1 n n 1

n

d t

g e

t C t

y

t u b ...

dt t u b d

t y a

...

dt t y a d

dt t y d

k

Where :

• C_k(t) are polynomials of t

•

( ) ∑ ( )

=

⋅

⋅ α

= ⁿ

1 k

k t

' _k

e t C t

g

A system is stable if all poles have negative real parts

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Standard models

Transfer function

without knowing anything about Laplace transform it can be useful to store a_k and b_k coefficients in a convenient way, the transfer function :

( ) ( ) ( )

₁ ⁿ ¹ _n ⁿ

1 n 1 n

b ...

s b

a ...

s a s

s A

s s B

F ⋅ +

+

⋅

= +

= ₋ ⁻

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9/23/2009 III. Laplace transforms 35

III. The Laplace transform

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• We assume the system to be LINEAR and TIME INVARIANT

The output (y) of the the system is related to the input (u) by the convolution :

System

u y

Laplace transform (1) : convolution

d τ τ ) h(t

) u( τ

y(t) = ∫

−^+∞∞

⋅ − ⋅

• Example : u(t) is an impulsion (0 everywhere except in t = 0)

h(t) y(t) =

h(t) is called the impulse response, h(t) describes completely the system

• Causality : h(t) = 0 if t < 0

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Time space

u(t) h(t) y(t)

Laplace space

x(t)

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

Laplace transform (1) : definition

∫

^+∞ ^⋅ ⁻ ^⋅ ^⋅

= 0

t

s dt

e ) t ( x )

s ( X

t : real (time) s : complex (frequency)

∫

₋⁻_⋅_∞^⋅^∞

^⋅

⋅ ⋅ π

= ⋅

^c ^j

j c

t

s

ds

e ) s ( j X

2 ) 1 t (

x

X(s)

d τ τ ) h(t )

u( τ

y(t) = ∫

−^+∞∞

⋅ − ⋅

Y(s) = H(s) . U(s)

☺ Mathematical formulas are never used !

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Step fonction :

t<0 : x(t) = 0

t>0 : x(t) = 1 X(s) = 1/s

Impulse fonction

t=0 : x(t) = infinite

x(t)=0 X(s) = 1

Derivation :

) t ( dt x

y(t) = d

^Y⁽^s⁾ = ^s^.^X⁽^s⁾− ^x⁽⁰+⁾

Sinusoïdal fonction :

) t sin(

y(t) = ω ⋅

2 2

s ) 1 s (

Y = +ω

Laplace transform (2) : properties

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Delay :

) t t ( x

y(t) = −

_d

Y ( s ) = X ( s ) ⋅ e

⁻^t^d^⋅^s

Initial value theorem :

( ) ( ^s ^Y ^s )

lim )

0 y( =

s

⋅

∞ + →

Final value theorem (if limit exists) :

( ) ( ^s ^Y ^s )

lim )

y( + ∞ =

s 0

⋅

→

Laplace transform (3) : properties

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From t to s

Laplace transform (4) : tables

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From s to t

Laplace transform (4) : tables

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( ) ^t â ^x ( ) ^t â ^x ( ) ^t ^b û ( ) ^t ^b û ( ) ^t

x

a

_o

+

₁

+

₂

=

₀

+

₁

+

Theorem of differentiation

( ) ( ( ) ( ) ) ( ( ( ) ( ) ) ( ) ) ( ) ( ( ) ( )

+

)

+ +

+

−

⋅ +

=

−

⋅

⋅ +

−

⋅

⋅ +

0 u s

U s b s

U b

0 x 0

x s

X s s a 0

x s

X s a

s X a

1 0

2 1

o

Laplace transform and differential equations

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( ) ( ( ) ( ) ) ( ( ( ) ( ) ) ( ) ) ( ) ⁺ ( ^⋅ ( ) ( ) ⁻

⁺₊

)

⁺ ⁺

=

−

⋅

⋅ +

−

⋅

⋅ +

0 u s

U s b s

U b

0 x 0

x s

X s s a 0

x s

X s a

s X a

1 0

2 1

o

( ) ^{( ) (} ^{) ( )} ^{( )}

( ⁺ ^⋅ ) ( ) ^⋅ ⁻ ^⋅ ( )

₊ ⁺ ⁺

=

⋅

−

⋅

⋅ +

−

⋅

⋅ +

0 u b s

U s

b b

0 x a

0 x s a a

s X s

a s a a

1 1

0

2 2

1 2

2 1

o

( ) ( ) ( ) ( ) ( ) ( )

2 2 1

o

1 2

2 1

2 2 1

o

1 0

s a s a a

0 u b 0

x a

0 x s a s a

s U a

s a a

s b s b

X + ⋅ + ⋅

⋅

−

⋅ +

⋅

⋅ + +

⋅ +

⋅

= +

⁺

⁺ ⁺

Initial conditions Transfer function

Laplace transform and differential equations

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( ) ( ) ( ) ( ) ^s ^H ^s ^U ^s ^I ^s

X = ⋅ +

( ) ( ) ( ) ( ) ( ) ( )

2 2 1

o

1 2

2 1

2 2 1

o

1 0

s a s a a

0 u b 0

x a

0 x s a s a

s U a

s a a

s b s b

X + ⋅ + ⋅

⋅

−

⋅ +

⋅

⋅ + +

⋅ +

⋅

= +

⁺

⁺ ⁺

( ) ( ) ( ) ( ) ^t ^h ^t ^u ^t ⁱ ^t

x = ∗ +

☺ Initial conditions are generally assumed to be null !

Laplace transform and differential equations

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System

u(t) y(t)

What is the output y(t) from a given input u(t) ?

u(t)

Table of transform

U(s)

Y(s) = H(s) . U(s)

Y(s) y(t)

Table of transform

Finding output response with Laplace transform

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System

u(t) y(t)

What is the final output y(inf) from a given input u(t) ?

u(t)

Table of transform

U(s)

Y(s) = H(s) . U(s)

Y(s)

Theorem of final value

( ) (

^s ^Y ^s

)

lim )

y(+ ∞ = s 0 ⋅

→

Finding final value with Laplace transform

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Poles and zeros

H(s) Y

U

Transfer function is a ratio of polynomials : ...

s a s a s a a

...

s b s b b

) s ( D

) s ( ) N s ( ) H s ( U

) s ( Y

3 3 2

2 1

0

2 2 1

0

+ +

= +

=

Poles and zeros :

( ) ( )

(

⁻

) (

⁻^⋅ ⁻^⋅ ⁻

) (

^⋅ ⁻^⋅ ^⋅⋅

)

^⋅ ^⋅⋅

⋅

=

3 2

1

2 1

0 0

p s p

s p s

z s z s a

) b s ( ) H s ( U

) s ( Y

zero : z₁, z₂, …

poles : p₁, p₂, p₃…

p₂

p₁ z₁

Re Im

p₃

z₂

• Poles and zeros are either into the left plane ore into the right plane

• Complex poles and zeros have a conjugate

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• Poles are the roots of Transfer function denominator

– Real values or conjugate complex pairs

• Poles are also the eigenvalues of matrix A

• Poles = modes

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Poles and zeros : decomposition

H(s) Y

U

Transfer function can be expansed into a sum of elementary terms : ...

s a s

a s a a

...

s b s b ) b

s ( ) H s ( U

) s ( Y

3 3 2 2 1

0

2 2 1

0

+ +

= +

=

p ...

s p

s p ) s

s ( ) H s ( U

) s ( Y

3 3 2

2 1

1 +

− + α

−

= α

=

p₁ and p₂ are conjugate : ⁼ ⁻^ω ^⋅ ^⋅^θ ⁼ ⁻^ω0 ^⋅ ⁻^j^⋅^θ j

0

1 e ,p2 e

p

p ...

s s cos 2

) s s ( ) H s ( U

) s ( Y

3 3 2

0 0

2

2 ,

1 +

− + α ω +

⋅ θ

⋅ ω

⋅ +

= α

=

First orders Second orders

☺ Complex system response is the sum of first order and second order systems responses

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Dynamic response of first order systems

U Y

p1

s ) 1 s (

H = +

( )

^U

( )

^s

p s s 1 Y

+ 1

=

Example 1 : impulse response

u(t) is an impulsion (0 everywhere, except in 0 : ∞)

u(t)

Table of transform

U(s)=1

Table of transform

( )

^s ^H

( ) ( )

^s ^U ^s

Y = ⋅

( )

p1

s s 1

Y = +

t p₁

e ) t (

y = ⁻ ^⋅

(51)

Dynamic response of first order systems

U Y

p1

s ) 1 s (

H = +

( )

^U

( )

^s

p s s 1 Y

+ 1

=

Example 2 : step response u(t) is a step

u(t)

Table of transform Table of transform

( )

^s ^H

( ) ( )

^s ^U ^s

Y = ⋅

( )

s

1 p s s 1 Y

1

+ ⋅

=

(

^p ^t

)

1

e 1

p 1 ) 1 t (

y = − ⁻ ^⋅

( )

s s 1 U =

(52)

Properties of first order systems

U Y

p1

s ) 1 s (

H = +

Step response

t₁= 1/p₁ is the time constant of the system :

after t = t₁, 63% of the final value is obtained

( )

^U

( )

^s

p s s 1 Y

+ 1

=

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2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω

Dynamic response of second order systems

U Y

u(t)

Table of transform

U(s)=1

Table of transform

( )

^s ^H

( ) ( )

^s ^U ^s

Y = ⋅

(

¹ ^t

)

sin e

1 ) 1

t (

y ^t ₀ ²

2 0

à ⋅ ω −σ ⋅

σ ⋅

−

= ω ⁻^σ^⋅^ω ^⋅

2 0 0

2 2 s

s ) 1 s (

Y = + ⋅σ⋅ω ⋅ + ω

(54)

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω

Dynamic response of second order systems

U Y

Example 2 : step response u(t) is an step

u(t)

Table of transform

U(s)=1/s

Table of transform

( )

^s ^H

( ) ( )

^s ^U ^s

Y = ⋅

(

^ω ⁻^σ ^⋅ ⁺

( )

^σ

)

⋅ σ ⋅

−

− ω

= e⁻^σ^⋅^ω ^⋅ sin 1 t arcos 1

1 1 ) t (

y ^t ₀ ²

2 0

à

s 1 s

2 s

) 1 s (

Y ₂

0 0

2 ⋅

ω +

⋅ ω

⋅ σ

⋅

= +

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Properties of second order systems

Step response :

σ is the damping factor ω₀ is the natural frequency

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω Y U

(

^ω ⁻^σ ^⋅ ⁺

( )

^σ

)

⋅ σ ⋅

−

− ω

1 1 ) t (

y ^t ₀ ²

2 0

à

2 0 1−σ

ω is the pseudo-frequency 5% of the final value is obtained after :

0

% 5

t 3

ω

⋅

≈ σ

overshoot

Overshoot increases as σ decreases

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Properties of second order systems

Step response (continued) :

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω Y U

poles : p₁_,₂ = −ω₀ ⋅e^±^θ cos(θ)=σ

θ ω0

Re Im

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Stability

Re Im

Unstable

Stable

unstable pole, deverges like exp(t)

stable pole, decays like exp(-4.t)

Any pole with positive real part is unstable

Any input (even small) will lead to instability

See animation

(58)

« fast poles » vs « slow poles »

Re Im

slow

fast

slow pole, decays like exp(-t)

constant time : t₁ = 1s fast pole, decays like

exp(-4.t)

constant time : t₁ = 4s Fast poles can be neglected

See animation

(59)

Effect of zeros

See animation

Re Im

Zeros modify the transient response

• Fast zero : neglected

• Slow zero : transient response affected

• Positive zero : non minimal phase system, step response start out in the wrong

direction

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Ex. analysis of a feedback system

Process model :

( )

₊₀_.₀₂_⋅_v ₌ _u ₋₁₀_⋅_θ

dt t dv

Transfer function

s

:

( ) ( ) ( )





θ

⋅

−

=

⋅ +

⋅

=

⋅ +

⋅

s 10

s V 02 . 0 s

V s

s U s

V 02 . 0 s

V s

( ) ( ) ( ) ( ) ( )









− + θ =

= +

=

⇒

s 02 . 0

10 s

s V

s 02 . 0 s 1

s F U

s V

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Ex. analysis of a feedback system

Transfer function of the controller (PID) :

( ) ( ) ( ) ( )

( ) ( )

^t ^k ^k ^s ^k ¹^s

E t U

d t e dt k

t k de

t e k t

u

i d

t i 0 d

⋅ +

⇒ =

τ

⋅

⋅ +

⋅

=

∫

(62)

Ex. analysis of a feedback system

We can now combine transfer functions :

v_r -

F

θ

PID v

u

e -10

( )

^s ¹ ^F

( )

^s ¹^PID

( ) ( )

^s ^V ^s ¹ ^F

( )

^s ¹⁰^PID

( ) ( )

^s ^E ^s

V _r ⋅

⋅ +

+ −

⋅ ⋅

= +

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9/23/2009 IV. Design of simple feedbacks 63

IV. Design of simple feedback

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P(s)

u y

Introduction

Standard problems are often first orders or second orders

• Standard problem standard solution

C(s)

d r

-

( )

s a s b

P = +

( )

2 1

2

2 1

a s a s

b s s b

P + ⋅ +

= +

(65)

Control of a first order system

Most physical problems can be modeled as first order systems

( )

s a s b

P = +

Step 1 : transform your problem in a first order problem :

( )

s

k k s

C = + ⁱ Step 2 : choose a PI controller

Step 3 : combine equations and tune k and in k_iin order to achieve the desired closed loop behavior (mass-spring damper analogy)

( ) ( ) ( ) ( ) ( )

2 0

i i

s s 1 2

s ' b K 1

s k k

a s 1 b

s k k

a s

b s

C s P 1

s C s s P

CL

+ ω ω ⋅

σ + ⋅

⋅

⋅ +

=



 



 + + ⋅

+



 



 + + ⋅

⋅ = +

= ⋅

(66)

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω

Dynamic response of second order systems

U Y

u(t)

Table of transform

U(s)=1

Table of transform

( )

^s ^H

( ) ( )

^s ^U ^s

Y = ⋅

(

¹ ^t

)

sin e

1 ) 1

t (

y ^t ₀ ²

2 0

à ⋅ ω −σ ⋅

σ ⋅

−

= ω ⁻^σ^⋅^ω ^⋅

2 0 0

2 2 s

s ) 1 s (

Y = + ⋅σ⋅ω ⋅ + ω

(67)

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω

Dynamic response of second order systems

U Y

Example 2 : step response u(t) is an step

u(t)

U(s)=1/s

Table of transform

( )

^s ^H

( ) ( )

^s ^U ^s

Y = ⋅

(

^ω ⁻^σ ^⋅ ⁺

( )

^σ

)

⋅ σ ⋅

−

− ω

1 1 ) t (

y ^t ₀ ²

2 0

à

s 1 s

2 s

) 1 s (

Y ₂

0 0

2 ⋅

ω +

⋅ ω

⋅ σ

⋅

= + Table of transform

(68)

Properties of second order systems

Step response :

σ is the damping factor ω₀ is the natural frequency

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω Y U

(

^ω ⁻^σ ^⋅ ⁺

( )

^σ

)

⋅ σ ⋅

−

− ω

1 1 ) t (

y ^t ₀ ²

2 0

à

2

0 1−σ

ω is the pseudo-frequency 5% of the final value is obtained after :

0

% 5

t 3

ω

⋅

≈ σ

overshoot

Overshoot increases as σ decreases

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Properties of second order systems

Step response (continued) :

2 0 0

2 2 s

s ) 1 s (

H = + ⋅σ⋅ω ⋅ + ω Y U

poles : p₁_,₂ = −ω₀ ⋅e^±^θ cos(θ)=σ

θ ω0

Re Im

(70)

Control of a second order system

Step 1 , step 2 : idem (PI controller)

Step 3 : Transfer function is now third order

( ) ( ) ( )

_^











+ ω ω ⋅

σ + ⋅

⋅ +

⋅

⋅ +

⋅ = +

= ⋅

2 0

s s 1 2

s a 1

s ' b K 1

s C s P 1

s C s s P

CL

2 dof (k and k_i) : the full dynamics (order 3) cannot be totally chosen

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Simulation tools

Matlab or Scilab

Transfer function is a Matlab object

Adapted to transfer function algebra (addition, multiplication…) Simulation, time domain analysis

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9/23/2009 VI. Design of simple feedbacks (Ctd) 72

Conclusion

Laplace Transform +

Simulation tools

Design of simple feedbacks

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9/23/2009 V. Frequency response 73

V. Frequency response

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Introduction

Frequency response :

• One way to view dynamics

• Heritage of electrical engineering (Bode)

• Fits well block diagrams

• Deals with systems having large order

– electronic feedback amplifier have order 50-100 !

• input output dynamics, black box models, external description

• Adapted to experimental determination of dynamics

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The idea of black box

The system is a black box : forget about the internal details and focus only on the input-output behavior

Frequency response makes a “giant table” of possible inputs-outputs pairs

Test entries are enough to fully describe LTI systems ☺

- Step response - Impulse response - sinusoids

System

u y

(76)

What is a LTI system

A Linear Time Invariant System is :

• Linear

If (u₁,y₁) and (u₂,y₂) are input-output pairs then (a.u₁+ b.u₂ , a.y₁+ b.y₂) is an input-output pair : Theorem of superposition

• Time Invariant

(u₁(t),y₁(t)) is an input-output pair then (u₁(t-T),y₁(t-T)) is an input- output pair

The “giant table” is drastically simplified :

( ) ( ) ( ) ( ) ( )

^s ^H ^s ^U ^s

Y

dτ u τ

t τ h y(t)

⋅

⇒ =

⋅

−

=

∫

₋^+∞_∞

(77)