Fractional Order Controls A Beginner’s Point of View

Zhuo Li Zhuo.li@aggiemail.usu.edu 5/15/2012

Outlines • Key Techniques • Backgrounds • Distributed Order • Approximation and Discretization • Tools in Matlab • IOPID, FOPI, FO[PI], DOPI and DO[PI] • My ECE5320 MAD Report • My Views on the 3 Tuning Equations • Proposed Approach • Some Results

2

Weird Things to Me!

• For me, these are the barriers between a beginner and TD, FO, DO.

• 𝑒𝑒−𝑠𝑠𝑠𝑠How can the “s” climb up to e? (9/2011)

• 𝐷𝐷0.5

𝐷𝐷𝑡𝑡0.5 = 𝑠𝑠0.5 = (𝑗𝑗𝜔𝜔)0.5First time in my life heard about FOC. (10/2011)

• ∫ 1𝑠𝑠𝛼𝛼𝑑𝑑𝛼𝛼 = 𝑠𝑠−𝑎𝑎−𝑠𝑠−𝑏𝑏

ln (𝑠𝑠)𝑏𝑏𝑎𝑎 What is this? (5/2012)

• ln (𝑗𝑗𝜔𝜔)Complex logarithm. Never heard! (5/2012)

3

Time Delay

4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

10 dB

-20 dB

-10 dB

-6 dB

-4 dB-2 dB

20 dB

10 dB

6 dB

4 dB2 dB

Nyquist Diagram

Real Axis

Imag

inar

y Ax

is

NyquistCurve of 𝐺𝐺 𝑠𝑠 = 𝑒𝑒−𝑠𝑠

-1

-0.5

0

0.5

1

Mag

nitu

de (d

B)

10-1

100

101

102

-5760-5040-4320-3600-2880-2160-1440-720

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)

Bode Plot of 𝐺𝐺 𝑠𝑠 = 𝑒𝑒−𝑠𝑠

Key techniques

• Recall Euler's formula 𝑒𝑒𝑗𝑗𝜔𝜔 = cos 𝜔𝜔 + 𝑗𝑗𝑠𝑠𝑗𝑗𝑗𝑗(𝜔𝜔)

• 𝑒𝑒𝑠𝑠 = 𝑒𝑒𝑗𝑗𝜔𝜔 Now, we can deal with it.

• 𝑗𝑗 = cos 𝜋𝜋2

+ 𝑗𝑗𝑠𝑠𝑗𝑗𝑗𝑗 𝜋𝜋2

= 𝑒𝑒𝜋𝜋2𝑗𝑗

• (𝑗𝑗)𝛼𝛼≜ 𝑒𝑒𝜋𝜋2𝑗𝑗𝛼𝛼 = cos 𝜋𝜋

2𝛼𝛼 + 𝑗𝑗𝑠𝑠𝑗𝑗𝑗𝑗 𝜋𝜋

2𝛼𝛼 (5/2012)

• ln 𝑗𝑗𝜔𝜔 = ln 𝜔𝜔 + ln (𝑒𝑒𝜋𝜋2𝑗𝑗)(5/2012)

5

Control Engineer’s View

• With these basic techniques, we can further deal with the analysis in frequency domain.

• IOPID • FO𝑃𝑃𝑃𝑃𝛼𝛼, FO[𝑃𝑃𝑃𝑃]𝛼𝛼 • DO𝑃𝑃𝑃𝑃𝛼𝛼, DO[𝑃𝑃𝑃𝑃]𝛼𝛼

6 Board writing by Dr. Y.Q. Chen

Background

7 Slide from: Dr. Yangquan Chen and Dr. Igor Podlubny.

Fact

• FOC depicts most physical phenomenon more precisely.

• Superior to Integer order in a lot of cases

8

Three Common Definitions

• Riemann-Liouville derivative

• Grünwald-Letnikov derivative

• Caputo fractional derivative

9

Two Important Functions

Source, “Engineering Education and Research Using MATLAB”, INTECK, editor Ali H. Assi, Chapter10, Ivo Petráš, 2011.

In Matlab, y=gamma(x)

[e]=mlf(alf,bet,c,fi) f=gml_fun(a,b,c,x,eps0)

10

Gamma function

Mittag-Leffler function

Approximating FO using IO

• Discretization z-transform

11 [♠] Y.Q. Chen, Ivo Petr´aˇs and D.Y. Xue, “Fractional Order Control - A Tutorial”, ACC, 2009.

[♠]

Discrete Approximation - PSE • Generating function

𝜔𝜔 𝑧𝑧−1 = 21 − 𝑧𝑧−1

1 + 𝑧𝑧−1

The forward method is not suitable for cause problems

Grünwald-Letnikov method

12

Discrete Approximation - CFE

13 User and programmer manual, Ninteger v. 2.3 Fractional control toolbox for MatLab. Universidade Técnica De Lisboa Instituto Superior Técnico, 2005

Discrete Approximation – irid

• Impulse-response invariant discretization (IRID) method

• The approximated discrete time model matches the impulse response of the DO model precisely at every sampling point.

14

• FIR, IIR • e.g.

Realization

15 [*] Y.Q. Chen, Ivo Petr´aˇs and D.Y. Xue, “Fractional Order Control - A Tutorial”, ACC, 2009.

Figure from [*]

• Definition of Distributed Order FO Integrator/differentiator [*]

• Analytical Solution [**]

Distributed Order

16

[*] H. Sheng, Y.Q. Chen, T.S. Qiu, “Fractional Processes and Fractional-order Signal processing”, Springer, 2012. [**] F.Y. Zhou, Y. Zhao, Y. Li, Y.Q. Chen, “An Implementation of Distributed Order PI Controller and Its Applications to the Wheeled Service Robot”, ACC13 Submission.

�𝟏𝟏𝒔𝒔𝜶𝜶 𝒅𝒅𝜶𝜶 =

𝒔𝒔−𝒂𝒂 − 𝒔𝒔−𝒃𝒃

𝒍𝒍𝒍𝒍 (𝒔𝒔)

𝒃𝒃

𝒂𝒂

𝓛𝓛−𝟏𝟏 �𝟏𝟏𝒔𝒔𝜶𝜶 𝒅𝒅𝜶𝜶

𝒃𝒃

𝒂𝒂=

𝟏𝟏𝟐𝟐𝟐𝟐𝟐𝟐� 𝒆𝒆𝒔𝒔𝒔𝒔

𝝈𝝈+𝟐𝟐𝒊

𝝈𝝈−𝟐𝟐𝒊

𝒔𝒔−𝒂𝒂 − 𝒔𝒔−𝒃𝒃

𝒍𝒍𝒍𝒍 (𝒔𝒔) 𝒅𝒅𝒔𝒔

Available Tools in Matlab

17

Toolbox Names Authors

Ninteger Toolbox Duarte Valério

Crone Toolbox CRONE Team

@fotf Toolbox Dr. Dingyu Xue

Irid_doi(), gml_fun, FOCP, nilt (non-integer order inverse Laplace) ora_foc(), etc.

Dr. Yangquan Chen, Hu Sheng, and Christophe Tricaud, et al.

mlf(), etc. Dr. Igor Podlubny

dfod1(), dfod1(), dfod1(), DFOC, etc. Dr. Ivo Petras

fderiv() F. M. Bayat

etc. etc.

Demos

18

%% ninteger Toolbox G_nid = nid(1, 0.5, [1e-3 1e4], 10, 'crone') figure(5) step(G_nid) figure(6) bodeFr(G_nid)

10-6

10-4

10-2

100

102

104

106

-50

0

50Bode diagram

frequency / rad s-1ga

in /

dB

10-6

10-4

10-2

100

102

104

106

0

20

40

60

phas

e / ?

0 0.05 0.1 0.150

20

40

60

80

100

120Step Response

Time (seconds)

Ampl

itude

Half order differentiator 𝑠𝑠0.5

10dB/dec

Demos

19

%% ora_foc() by Christophe Tricaud sys=ora_foc(-0.5,3,0.01,100) figure(3) step(sys) figure(4) bode(sys)

-30

-20

-10

0

10

20

30

Mag

nitu

de (d

B)

10-4

10-2

100

102

104

-60

-30

0Ph

ase

(deg

)

Bode Diagram

Frequency (rad/s)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

5

10

15

20

25

30

35Step Response

Time (seconds)

Ampl

itude

Half order itegrator 1𝑠𝑠0.5

-10dB/dec

Demos

20

%% dfod1() by Dr. Ivo Petras sys=dfod1(50, 0.001, 1/3, 0.5) figure(1) bode(sys) sys_c=d2c(sys) figure(2) step(sys_c)

10

15

20

25

30

35

40

Mag

nitu

de (d

B)

100

101

102

103

104

0

30

60Ph

ase

(deg

)

Bode Diagram

Frequency (rad/s)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

5

10

15

20

25

30

35

40Step Response

Time (seconds)

Ampl

itude

Half order differentiator 𝑠𝑠0.5

Demos

21

%% dfod1(), Ivo's demo T=0.1; a=1/3; z=tf('z',T,'variable','z^-1'); Hz=((1+a)/T)*((1-z^-1)/(1+a*z^-1)); Gz_DCM=0.08/(Hz*(0.05*Hz+1)); % DC modtor plant Cz=0.625*dfod1(5,T,a,0.5)+12.5*dfod1(5,T,a,-0.5); % controller Gz_close=(Gz_DCM*Cz)/(1+(Gz_DCM*Cz)); figure(6); step(Gz_close,15);grid; Gz_open=(Gz_DCM*Cz); figure(7); bode(Gz_open); [Gm,Pm] = margin(Gz_open);

-50

0

50

Mag

nitu

de (d

B)

10-2

10-1

100

101

102

-180

-135

-90

-45

0

45

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (seconds)

Ampl

itude

Demos

22

%% mlf(), time domain solution using Mittag-leffler, Dr.Igor Podlubny clear all; close all; a=2; b=1; alpha=1.5; t=0:0.001:20; % for time step 0.001 and computational time 20 sec y=(1/a)*t.^(alpha).*mlf(alpha,alpha+1,((-b/a)*t.^(alpha))); figure(5) plot(t,y); xlabel('Time [sec]'); ylabel('y(t)');grid; 0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

y(t)

Demos

23

%% fotf() Dr.Dingyu Xue b = [-2 4]; a = [2 3.8 2.6 2.5 1.5]; nb = [0.63 0]; na =[3.5 2.4 1.8 1.3 0]; t = 0:0.5:28; G = fotf(a,na,b,nb) yi = step(G,t); figure(6); hold on; plot(t,yi,'g'); 0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5

3

3.5

Demos

24

%% FO derivative sine wave, using fderiv() by Bayat h=0.01; t=0:h:2*pi; y=sin(t); order=0:0.1:1; for i=1:length(order) yd(i,:)=fderiv(order(i),y,h); end [X, Y]=meshgrid(t,order); mesh(X,Y,yd) xlabel('t'); ylabel('\alpha'); zlabel('y')

02

46

8

0

0.5

1-1

-0.5

0

0.5

1

tα

y

Demos

25

%% irid_doi(), By Dr.Y Li, Dr.YQ Chen Impulse response invariant discretization of distributed order integrator doi=irid_doi(0.01,0.75,1,5,5);

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Time

Impu

lse

resp

onse

impulse response of ∫abs-αdα

approximated impulse response

10-1

100

101

102

103

104

-100

-50

0

50

Frequency (Hz)M

agni

tude

(dB

)

mag. Bode of ∫abs-αdα

approximated mag. Bode

10-1

100

101

102

103

104

-100

0

100

Frequency (Hz)

Pha

se (d

egre

es)

phase Bode of ∫abs-αdα

approximated phase Bode

�1𝑠𝑠𝛼𝛼

𝑏𝑏

𝑎𝑎𝑑𝑑𝛼𝛼,

a, b ∈ 0.5, 1 , 𝑎𝑎 < 𝑏𝑏

Three-parameter-tunable Controllers

• IOPID

• FOPI

• FO[PI]

• DOPI

• DO[PI]

26

C s = Kp +𝐾𝐾𝑖𝑖𝑠𝑠 + Kds

C s = Kp(1 +𝐾𝐾𝑖𝑖𝑠𝑠𝛼𝛼)

C s = (Kp +𝐾𝐾𝑖𝑖𝑠𝑠 )𝛼𝛼

C s = � Kp(1 +1𝑇𝑇𝑖𝑖𝑠𝑠

)𝛼𝛼𝑏𝑏

0𝑑𝑑𝛼𝛼

C s = Kp + K𝑖𝑖 �1𝑠𝑠𝛼𝛼

𝑏𝑏

0𝑑𝑑𝛼𝛼 = Kp + K𝑖𝑖

1 − 𝑠𝑠−𝑏𝑏

ln (𝑠𝑠)

Tuning Rules

• Idea: given ϕm, ω𝑐𝑐 , obtain flat phase at ω𝑐𝑐

• ∠G jω = ∠C jωc + ∠P jωc = −π + ϕm

• G jω = C jω𝑐𝑐 P jω𝑐𝑐 = 0𝑑𝑑𝑑𝑑

• 𝑑𝑑∠G jω𝑑𝑑ω

�ω=ω𝑐𝑐

= 0

27

Procedure

• Separate the real and imaginary part of P j𝜔𝜔 andC j𝜔𝜔 , (for each controller C j𝜔𝜔 ) – using the techniques on slide 4

• Calculate the gain and phase • Solve the 3 equations numerically

– Get Kp, Ki, 𝛼𝛼

• Digital simulation and implementation – Using the tools on slide 16.

28

My ECE5320 MAD Report

29

My ECE5320 MAD Report

30

My ECE5320 MAD Report

31

My ECE5320 MAD Report

32

My ECE5320 MAD Report

33

My ECE5320 MAD Report

34

0 0.5 1 1.5 2 2.5 3 3.5

x 105

25

30

35

40

45

50

55

60

65

70

75

0 0.5 1 1.5 2 2.5

x 105

25

30

35

40

45

50

55

60

65

70

PID

FOPI

FO[PI]

My views on the 3 Eq. • Allowing users to specify the Gain cross-over frequency 𝜔𝜔𝑐𝑐 gives the user

too much freedom. Barely using anyone of the above five controllers cannot always satisfy this strong design constraint. More specifically, in most cases, for sure there is an 𝜔𝜔𝑐𝑐, and there is an 𝜔𝜔𝑑𝑑 at which the phase plot is flat.

𝑑𝑑∠𝐺𝐺(𝑗𝑗𝜔𝜔)𝑑𝑑𝜔𝜔

= 0. By adjusting the 3 tunable parameters, these two frequencies can be aligned, meaning that they match each other. However, they cannot be aligned at arbitrary frequency. As we can see from the Bode plot of the plant, the time delay pulls down the phase plot too quickly at high frequency, which gives us less time to regulate the plant. At the meanwhile, all of the 5 controllers provide limit phase compensation at high frequency, which is powerless to bring the phase back from −∞. This can be seen clearly from the Bode plot of the controllers. Hence, if the desired 𝜔𝜔𝑐𝑐 is assigned "too late", obtaining a "flat phase" at 𝜔𝜔𝑐𝑐 becomes impossible.

35

My views on the 3 Eq.

36

• The above statement has shown that freely defining 𝜔𝜔𝑐𝑐 by user is already a “luxurious service”. In spite of this, allowing users to specify the phase margin 𝜙𝜙𝑚𝑚 is another strong requirement. The point where 𝜔𝜔𝑐𝑐 and 𝜔𝜔𝑑𝑑 are aligned is not guaranteed to provide the desired phase margin. The Bode plot of a given plant is fixed. The shape of the phase plot of a given controller is also determined. As an example, let's see fig 1 and fig 2. For 𝜆𝜆 ∈ 0~1 , the FOPI is able to provide −90° phase compensation at the most. If the 𝜙𝜙𝑚𝑚 is assigned to be 50° at 𝜔𝜔 <0.01 𝑟𝑟𝑎𝑎𝑑𝑑/𝑠𝑠, then, it is obvious the demand cannot be meet, unless using 𝜆𝜆 > 1.

My views on the 3 Eq.

37

• Either one of the above 2 design specifications may make the 3 tuning equations unsolvable. Combining them together, the 3 tuning equations is possible to become contradictory.

• For some combinations of plant and controller, there may not exist a "flat phase".

Proposed Approach

38

• instead of specifying 𝜔𝜔𝑐𝑐, 𝜙𝜙𝑚𝑚, and exactly solving the 3 equations, we can try to align 𝜔𝜔𝑑𝑑 with 𝜔𝜔𝑐𝑐 first, then manually examine the phase margin.

10-3

10-2

10-1

100

101

102

-50

0

50

100

150Loop transfer function with FOPIλ

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

Some Results

39

10-3

10-2

10-1

100

101

102

-20

0

20

40

X: 26.13Y: 0.04891

FO model of the HFE plant

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

ωcFig1. The FO plant 𝑷𝑷 𝒔𝒔 = 𝑲𝑲

𝑻𝑻𝒔𝒔𝒂𝒂+𝟏𝟏𝒆𝒆−𝑳𝑳𝒔𝒔

K =66.16; T =12.73; L = 1.93; % time delay a=0.5; % alpha

10-3

10-2

10-1

100

101

102

-20

0

20

40

60FOPIλ controller

10-3

10-2

10-1

100

101

102

-100

-50

0

Fig2. FOPI controller with initially guessed parameters Kp_fopi=0.8812; Ki_fopi=0.3153; lambda = 1.082;

Some Results

40

These set of parameters are obtained without specifying phase margin. Kp = 0.2535 Ki = 0.3630 𝜆𝜆 = 1.5338 Pm_fopi = 22.9815

10-3

10-2

10-1

100

101

102

-50

0

50

100

150Loop transfer function with FOPIλ

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

Some Results

41

These set of parameters are obtained withspecifying phase margin to be 𝜙𝜙𝑚𝑚 = 50°. Kp = 0.0501 Ki = 0.3815 𝜆𝜆 = 1.0618 Pm_fopi = 50.0000°.

10-3

10-2

10-1

100

101

102

-50

0

50

100Loop transfer function with FOPIλ

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

Some Results

42

These set of parameters are obtained withspecifying phase margin to be 𝜙𝜙𝑚𝑚 = 50°. And aligning wc with phase_min. Kp = 0.0241 Ki = 0.3821 𝜆𝜆 =1.0614 Pm_fopi = 50.0000°.

10-3

10-2

10-1

100

101

102

-50

0

50

100Loop transfer function with FOPIλ

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

Some Results

43

Some Results

44

10-3

10-2

10-1

100

101

102

-20

0

20

40

60Loop transfer function with initial attampted DOPIα

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

Bode plot of loop transfer function with an arbitrary guessed DOPI Kp_dopi=1;

Ki_dopi=0.5; b_dopi=0.8;

10-3

10-2

10-1

100

101

102

-60

-40

-20

0

20Tuned DOPIα controller

10-3

10-2

10-1

100

101

102

0

50

100

150

Some Results

45

10-3

10-2

10-1

100

101

102

-100

-50

0

50Loop transfer function with DOPI

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

100

The tuning fails by using DOPI. This confirms that for some plant, this tuning rule has no solution x_dopi = 0.0075 -0.0180 1.1039

10-3

10-2

10-1

100

101

102

0

10

20

30DOPIα controller with initial guessed parameters

10-3

10-2

10-1

100

101

102

-60

-40

-20

0

10-3

10-2

10-1

100

101

102

-20

0

20

40

X: 26.13Y: 0.04891

FO model of the HFE plant

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

ωc

Some Results

46

Bode plot of loop transfer function with tuned IOPID x_pid = 0.0048 0.0102 0.8440

10-3

10-2

10-1

100

101

102

-50

0

50

100Loop transfer function with IOPID

10-3

10-2

10-1

100

101

102

-300

-200

-100

0

100Phase margin is: 211.1609

error = abs(wc-wd(end)) + abs(Pm-50)

Some Results

47

Bode plot of loop transfer function with tuned IOPID x_pid = 0.0048 0.0102 0.8440

error = abs(wc-wd(end)) + abs(Pm-50)

Future Work

48

• Mathematically show for what type of plants the 3 tuning equations are not applicable

• Implementing the 5 controllers on varieties of plant models.

Acknowledgements

49

• Dr. Yangquan Chen • Hadi Malek • Dr. Luo Ying • Calvin Coopmans

Q&A

50

I

O P I

[PI]