a brief introduction to feedback control · a brief introduction to feedback control ......

A Brief Introduction to Feedback Control

VO Embedded Systems Engineering Benedikt Huber

WS 2010/11

Open and Closed Loop Systems

System 1 System 2 r u y

System 1 System 2

r

u y

Example: Glucose Concentration

Pancreas

Liver, Body Cells

90 mg / dl

Glucose level too high

Glucose level too low

Beta cells emit more insulin

Alpha cells emit glucagon

Liver releases/produces glucose

Cells take glucose from blood

Glucose concentration influences pancreas cells

[Idea: AstromMurray08, Pictures: wikipedia.org]

Computer Controlled Systems

Actuators System

Controller Filter

Observed Output

Sensors Noise Noise

External Input

External disturbances

Clock

r(t)

z(t)

u(t)

y(t)

Actuation Command

Engineered Control Systems

Server Admission

Control

Simple Feedback Control

•  Use control error e(t) -  e(t) = r(t) – y(t)

•  On/Off Control •  u = umax if e > 0 •  u = umin if e ≤ 0 •  Dead Zone •  Bang-Bang (Hysteresis)

e

u

u

e

Why Feedback Control?

Advantages of Feedback Control •  Control without exact system model •  Robustness: Less sensitive to disturbances •  Performance: Reach setpoint more quickly •  Feedback can be used to linearize system

Disadvantages •  Risk of instability (e.g.: Microphone Feedback) •  Additional components for providing the feedback signal

are required (measurement).

Introduction Modeling Dynamic Systems Designing PID Controllers Fuzzy Control

Simple Example Spring Mass System

–  d

–  p ... posi?on –  m ... mass of the vehicle –  c ... fric?on coeffficient –  k ... spring constant –  u ... applied force

System Dynamics: m p‘‘ + c p‘ + k p = u

Continous and Discrete Models

Continous •  Differential Equations •  Examples: Spring Mass, Cruise Control, DC Motor, ...

Discrete •  Difference Equations •  Example Email-Server: y[k+1] = a y[k] + b u[k] - y ... MaxUsers: deviation from operating point - u ... Remote Invocation Requests: deviation from op. Point

Linear Time-Invariant Systems

•  LTI Systems •  Time-Invariance: x(t-t0) → y(t-t0) •  Linearity: a·x1(t) +b·x2(t) → a·y1(t) +b·y2(t) •  Superposition principle: On piecewise constant input,

output is superposition of step responses •  Linearity

•  Most systems have non-linear behavior •  LTI Systems are often a good (local) approximation •  Feedback can be used to linearize systems •  Beware of finite actuator ranges (clipping)

Transfer Function (1) •  For Transfer Functions, the effect of a model component

is described in the Laplace Domain •  Laplace Transform L: f(t) F(s) •  Laplace Domain: Frequency domain (complex numbers) •  Step function: u(s) = 1/s

Time Domain Transforma-on

f(t), g(t) is transformed to

F(s),G(s) is transformed to

Linear term a f(t) + b g(t) a F(t) + b G(t)

Convolu?on f(t) * g(t) = F(s) G(s)

Differenta?on f‘(t) s F(s) – f(0)

Integra?on = (u*f) t F(s) / s f (! )d!0

t

!

f (! )g(t !! )d!0

t

"

Introduc?on to Control Theory 13

The Transfer Function (2)

•  The transfer func?on G(s) is defined as the ra?o of the Laplace transformed output signal Y(s) to the input signal X(s): G(s) = Y(s) / X(s)

•  In the ?me domain, the transfer func?on is a map x(t) → y(t)

g(t) y(t) x(t) G(s) Y(s) X(s)

Transfer Function Building Blocks (1)

Name Transfer Func-on Modeling

Gain (Propor+onal Elem.) G(s) = k y(t) = k x(t)

Integra+ng Element G(s) = s y(t) = x‘(t)

Differien+al Element G(s) = 1/s y‘(t) = x(t)

First order delay (PT1) G(s) = KP /(1 + T0∙s) T0∙y’(t) + y(t) = KP∙x(t)

Second order delay (PT2) G(s) = KP /(a∙s2 + b∙s + 1)

a∙y’’(t) + b∙y’(t) + y(t) = KP∙x(t)

Transfer Function Building Blocks (2)

•  Serial Composition

•  Feedback

)()()( 21 sGsGsG ⋅=

01 2

1 2 0

G (s)G (s) G (s)G(s)

1 G (s) G (s) 1 G (s)⋅

= =+ ⋅ +

Spring Mass - System Model

Transfer Function •  Second order delay •  G(s)= Y(s) / U(s) = 1 / [m s^2 – k c s – m k]

[modeled using ScicosLab]

Step Response

[Image: wikipedia.org]

OVERSHOOT


Step Response (2)

  Proportional element y(t) = KP · x(t)

  Integrating element y(t) = KI · ∫ x(t) dt

Pictures denote the response to the

standard step signal

y(t)

t

K p

y(t)

t

1

1/K I


Step Response (3)

  Differential element y(t) = KD · dx(t)/dt

  First order delay T0·y’(t) + y(t) = KP·x(t)

Pictures denote the response to the

standard step signal

y(t)

t

inifinite amplitude not poss ible (clipping)

y(t)

t

K p

T 0


Step Response (4)

  Second order delay

Output depends on internal parameters

[image from wikiepdia.org]

Step Response: Spring Mass System

State Space Models

•  Matrix-oriented view on LTI models •  State dynamics: dx/dt = A x + B u

•  x ... State vector •  u ... Control vector •  A ... Dynamics matrix •  B ... Control matrix

•  Output dynamics: y = C x + D u •  y ... output vector •  C .,. Sensor matrix •  D ... Direct term (often 0)

LTI State Space Models

•  Solve differential equation by matrix exponentiation •  Given x’(t) = A x(t), x(0) = x0 •  We have: y(t) = eAt x0

•  Discretization •  Fix sampling interval T •  Given x’(t) = A x(t), x(0) = x0 •  We have: x((k+1) T) = eAT x(k T) + ((eAT - I) B A -1) u(k T) •  With fixed T, very simple discrete simulation (but discretization can

be expensive)

•  Advantages •  State Space Model directly corresponds to differential equations •  Convenient to model MIMO (multiple input, multiple output) systems

(cf. SISO)

Simple Example: State Space Model Spring Mass System •  State Space Equations in (p,p‘)

•  For improved readbility: Variables (p,v) with v = p‘ •  First row states that p‘ = v •  Second row is differential equation modeling system

!x

!t=

p '

v '

"

#$$

%

&''=

0 1

(k

m(c

m

"

#

$$$

%

&

'''

p v( )+0

1

m

"

#

$$$

%

&

'''F

y =1

0

!

"#

$

%&( p v )+

0

0

!

"#

$

%&F

Introduction Modeling Dynamic Systems Designing PID Controllers Beyond PID Controllers

Goals of Effective Control

S-A-S-O •  Stability

•  High Accuracy (Small Steady State Error)

•  Short Settle Time

•  Minimize Overshoot

Proportional Control

•  Idea: Amplify the error •  u(t) = kp e(t) •  Design Parameter: Proportional gain kp

•  Steady State Error (SSE) indirect prop. to kp •  Gain too small: large SSE, large settle time •  Gain too high: Overshoot, oscillation

•  Correcting SSE with feed forward term •  u(t) = kp e(t) + uff •  How to choose uff? •  Requires precise system model

Proportional Control Spring Mass System

kp = 50

kp = 15

Input Amplifier

PI Controller

•  Integral component eliminates SSE and attenuates disturbance

•  PI Controller: kp e(t) + ip ∫e (t) •  In a stable PI-controlled system, SSE becomes 0

eventually •  Integrating element (ip=kp) can be used to generate uff

•  Design tradeoffs for gain ip: •  Larger gain attenuates disturbances effectively •  Gain to large: oscillatory behavior, poor robustness,

possible instability

PI Controller

kp = 10, ki=3 (Long settlement time)

kp = 10, ki=0 (SSE)

Input Amplifier

kp = 10, ki=1

PI Control - Spring Mass System

PI Controller: Windup

•  Real-world actuators have limited operating range •  What happens if control point out of actuator limits? •  Integrator windup: Integral term and control point can get

very large

•  Solution: Anti-Windup Component •  Measure Difference: actuator output - controller ouput •  Feed back into integrator component

PD Controller

•  Intuition: Error prediction using derivative •  u = kp e + kd (de/dt) = kp ( e + Td (de/dt) ) •  e + Td (de/dt): Prediction of error at time t + Td by linear

extrapolation, where Td = kd / kp

•  Design Tradeoffs for derivative gain •  Higher gain damps oscillations •  Performance decreases if gain is too high

PID Controller

[Image: wikipedia.org]

PID Control - Spring Mass System

Input Amplifier

More PID Controller Improvements

•  Low-pass filter for derivative •  Problem: Derivative component sensitive to high-

frequency noise •  Solution: Low-pass filter input of derivative component

•  Setpoint weighting •  Problem: Step in reference input causes peak of output

due to derivative term •  Solution: Weight the reference input‘s influence on the PD

components


Designing Control Systems using the Step Response

  Step response measurement for the control path •  test signal is the standard step signal:

  Step response provides the characteris?c values Tu, Ta and Ks

1

t

x(t)


Designing Control Systems using the Step Response (2)

y(t)

t

Ks

Tu Ta

  Tu …effec?ve delay ?me   Ta …effec?ve

compensa?on ?me   Ks …gain


Designing Control Systems using the Step Response (3)   Based on the values Tu, Ta and Ks the rules by [Chien, Hrones, and Reswick] can be used to find a first se]ng of the controller: •  supports P, PI, and PID controller types •  se]ngs for 0% and 20% allowed overshoo?ng •  depending on the control path, fine tuning may be required.

39

Rules from Chien, Hrones, and Reswick


Boundary conditions of Designing a Control System by Rules using the Step Response

  Requires time invariance and linearity of control path.

  The described method is for analog controllers.   Method gives an approximate starting configuration   Possible optimizations either manually or using

sophisticated algorithms (e.g. genetic algorithm)


Stability

•  Most important property of controlled system •  BIBO stability: Bounded Input only generates Bounded Output

•  Several Methods •  Frequency Domain: Nyquist Criterion, Bode Plot •  State Space Model: Lyapunov Stability Theorem

Introduction Modeling Dynamic Systems Designing PID Controllers Fuzzy Control


Fuzzy Logic – Fuzzy Control

  Multi-valued logic   Instead of true or false: degree of membership to a “Fuzzy

Set”   Membership function:

]1,0[:)( →XxAµ


Example: Fuzzy Variable “temp”

-  5 fuzzy sets (very cold, cold, …) -  µvery cold(x) = 0 -  µcold(x) = 0.14 -  µmoderate(x) = 0.33 -  µhot(x) = 0 -  µvery hot(x) = 0

0

1

T0 T 7 T 8 T 9T 6T 5T 4T 3T 2T 1

verycold cold moderate hot very

hot

0.140.33

x


Operators on Fuzzy Sets

(x)]µ (x),min[µ (x)µ BABA =∩

(x)µ1(x)µ AA −=¬

(x)]µ (x),max[µ (x)µ BABA =∪


Process flow of fuzzy systems

F uzzyficationR ule

E valuation(n R ules )

Aggregation Defuzzyfication

Input Variable(s )

Output Variable(s )

‐

controller

Control Path


Example: Steam Turbine

0

1

T0 T 7 T 8 T 9T 6T 5T 4T 3T 2T 1


hot

0

1

P 0 P 7 P 8 P 9P 6P 5P 4P 3P 2P 1

weak low OK strong high

Input Variables: temperature and pressure

Output Variable: throttle setting

0

1

min max0

negative zero pos itive


Example: Fuzzyfication

0

1

T0 T 7 T 8 T 9T 6T 5T 4T 3T 2T 1


hot

0.6

x

1)  Temperature Input x: µcold(x) = 0.6 µvery cold(x) = 0 … µvery_hot(x) = 0

2)  Pressure Input y: µlow(y) = 0.37 µok(y) = 0.13 µweak(y) = 0 … µhigh(y) = 0 0

1

P 0 P 7 P 8 P 9P 6P 5P 4P 3P 2P 1

weak low OK strong high

0.13

0.37


Example: Rule Base

1)  IF temperature IS cold AND pressure IS low THEN

throttle IS positive

2)  IF temperature IS cold AND pressure IS ok THEN throttle IS zero

3)  IF temperature IS cold AND pressure IS strong THEN

throttle IS negative

4)  …


1)  IF temperature IS cold AND pressure IS low THEN throttle IS positive

2)  IF temperature IS cold AND pressure IS ok THEN throttle IS zero

3)  IF temperature IS cold AND pressure IS strong THEN throttle IS negative

Example: Rule Evaluation (Max/Min)

0

1

min max0

pos itive

0.37

µcold(x) = 0.6 µlow(y) = 0.37

µcold(x) = 0.6 µok(y) = 0.13

0

1

min max0

zero

0.13


Example: Aggregation (Max/Min)

0

1

min max0

pos itive

0.37

0

1

min max0

zero

0.13

0

1

min max0

zero pos itive


Example: Defuzzyfication

0

1

min max0 z

•  Generation of crisp output value by use of “center of gravity” method

center of gravity

References / Materials

•  Exploring Classical Control (Online Material) http://www.facstaff.bucknell.edu/mastascu/eControlHTML/CourseIndex.html

•  Feedback Systems: An Introduction for Scientists and Engineers.. Karl J. Åström and Richard M. Murray (2008).

•  Modeling a DC motor using MATLAB/Simulink http://www.mathworks.com/help/toolbox/control/getstart/f1-1010549.html

•  http://www.scicoslab.org/ •  www.gnu.org/software/octave/

ENDE

Danke für die Aufmerksamkeit!

Additional / Unused Material

Laplace Transformation Designing a Digital Controller Stability criteria


Laplace Transformation

Calculation in the Laplace domain (s).

∫

∫∞+

∞−

−

∞−

⋅⋅⋅

==

⋅==

+=

j

j

st

st

dsesXj

sXLtx

dtetxtxLsX

js

σ

σπ

ωσ

)(21)}({)(

)()}({)(

1

0


Laplace Transformation (2)

Important Laplace Transformations

x(t) X(t) = L{x(t)}

δ(t) 1

1 or σ(t) 1 / s

t 1 / s2

sin(b t) b / (s2 + b2)

ea· t

1 / (s – a)


Designing a Digital Controller (PID)

  Position algorithm:

  Velocity algorithm:

  Sampling time Ts must be sufficiently small: Ts ≤ 0.1 · Ta Ts ≤ 0.25 · Tu

s

nnD

n

isiInPn T

eeKTeKeKu 1−−++= ∑

s

nnnDsnInnPn T

)ee(eKTeK)e(eKΔu 211

2 −−−

−−++−=


Stability: Nyquist Criterion

  Plo]ng the transfer func?on of the opened control system as Nyquist Plot (separate axis for real and imaginary part)

  Cri?cal point of F(ω): [‐1,0i]   Amplitude passage ωD1 : |F(ωD1)| = 1   Phase passage ωD2 : arc(F(ωD2)) = π   Stable behavior: ωD1 < ωD2


Stability: Nyquist Criterium (2)

-1

ω >

Im

Re

ω=0 ω=∞ critical point ωD1

ωD2


Stability: Bode Plot |G (jω)|

ω

ω

‐90°

‐180°

‐270°

10

1

|F | > 1 … instabil|F | = 1|F | < 1 … s tabil

Φ > ‐180° … s tabilΦ = ‐180°Φ < ‐180° … instabil

Φ

Quality of Stability: •  Amplitude margin

•  Phase margin

a brief introduction to feedback control · a brief introduction to feedback control ......

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