Digital Control Systems: Difference between revisions

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===Links===
===Links===
*Textbooks
====Textbooks====
**[http://www.me.unm.edu/~starr/teaching/me581/textbook.pdf Introduction to Applied Digital Control, Greg Starr, University of New Mexico]
**[http://www.me.unm.edu/~starr/teaching/me581/textbook.pdf Introduction to Applied Digital Control, Greg Starr, University of New Mexico]
***[http://www.me.unm.edu/~starr/teaching/me581.html Greg's ME481/ME581 web pages] contain solutions to the book problems and other things.
**[http://www.control.utoronto.ca/people/profs/francis/sd_book.pdf Optimal Sampled-Data Control Systems, Chen & Francis]
**[http://www.control.utoronto.ca/people/profs/francis/sd_book.pdf Optimal Sampled-Data Control Systems, Chen & Francis]
**[http://faculty.ksu.edu.sa/hedjar/Documents/CEN455/Digital%20Control%20Systems.pdf Digital Control Systems Notes] Irfan Ahmed
**[http://www.ee.nchu.edu.tw/Pic/CourseItem/4559_digital_control_engineering_2nd_edition.pdf Digital Control Engineering] Fadali & Visioli
**[http://een.iust.ac.ir/profs/jahed/digital%20controll/e%20book/discrete-time_control_systems.pdf Discrete Time Control Systems] Ogata
**[http://een.iust.ac.ir/profs/jahed/digital%20controll/e%20book/discrete-time_control_systems.pdf Control System Engineering, Chapter 13 (Digital Control)] Norman Nise
**[http://uav.ece.nus.edu.sg/~bmchen/courses/ee3304.pdf Digital Control Systems Notes] Ben Chen
**[https://www.springer.com/cda/content/document/cda_downloaddocument/9781846280559-c1.pdf?SGWID=0-0-45-166313-p52482958 Computer Control Systems]
**[http://santcas.com/uploads/2/8/1/3/2813995/digital_control_system_analysis_and_design_3rd_edition_phillips.pdf Digital Control System Analysis and Design, 3rd Ed., Philips]
**[http://en.wikibooks.org/wiki/Control_Systems Control Systems and Control Engineering]
**[http://www.ece.rutgers.edu/~orfanidi/intro2sp/orfanidis-i2sp.pdf Introduction to Signal Processing]
**[http://eceweb1.rutgers.edu/~gajic/psfiles/ Zoran Gajic's Book on Control Systems] (Continuous & Discrete) and some other documents. Look for the documents entitled Chapter N.
**[http://nptel.ac.in/courses/108103008/25# NPTEL Course on Digital Control]
====Lectures====
*2016
**[http://people.wallawalla.edu/~Rob.Frohne/ClassHandouts/Digital_Control/LQR%20Outline.pdf LQR Derivation Outline from Class]
**[http://people.wallawalla.edu/~Rob.Frohne/ClassHandouts/Digital_Control/Kalman%20Filters%20Derivation.pdf Kalman Filters Derivation Outline from Class]
***[http://people.wallawalla.edu/~Rob.Frohne/ClassHandouts/Digital_Control/Kalman%20Filter%20Summary.pdf Kalman Filter Summary from Class]

*2014
**[http://www.stanford.edu/class/archive/ee/ee392m/ee392m.1056/Lecture5_Digital.pdf Sampled Time Control]
**[http://unitedthc.com/CTL/dig_ctl.pdf Duy-Ky Nguyen's Concise Digital Control Notes] I like this presentation.
**[http://eceweb1.rutgers.edu/~gajic/psfiles/observers.pdf Zoran Gajic's notes on Observers.] This is a very short set of lecture notes that explains the why's of observers, with several interesting derivations.
**[http://cse.lab.imtlucca.it/~bemporad/automatic_control_course.html Professor Bemporad's Automatic Control Class Notes] These are very good.
***[http://cse.lab.imtlucca.it/~bemporad/teaching/ac/pdf/07-output_feedback.pdf Professor Bemporad's] presentation on the separation principle of state space regulator design (showing that the poles of the control setting are preserved even when using a full order observer).
**[http://ocw.mit.edu/courses/mechanical-engineering/2-154-maneuvering-and-control-of-surface-and-underwater-vehicles-13-49-fall-2004/lecture-notes/ MIT Class on Underwater Vehicle Control]
***[http://ocw.mit.edu/courses/mechanical-engineering/2-154-maneuvering-and-control-of-surface-and-underwater-vehicles-13-49-fall-2004/lecture-notes/lec19.pdf Linear Quadratic Regulators (Mathematically Deep)]
***[http://ocw.mit.edu/courses/mechanical-engineering/2-154-maneuvering-and-control-of-surface-and-underwater-vehicles-13-49-fall-2004/lecture-notes/lec20.pdf A Mathematically Involved, but Insightful Look at Kalman Filters.] It has some interesting remarks about the duality between the Kalman-Bucy Filter for continuous systems and the LQR algorithm for continuous systems. I expect the discrete versions are also duals.
**[http://www.cl.cam.ac.uk/~rmf25/papers/Understanding%20the%20Basis%20of%20the%20Kalman%20Filter.pdf Kalman Filter Applied to Cart in One Dimensional Motion]
**[https://www.youtube.com/watch?v=18TKA-YWhX0 Intuitive Lecture on Practical Kalman Filter given by Francis Grovers III]
**[http://greg.czerniak.info/guides/kalman1/ The first 80% of what you need to know about Kalman Filters from Greg Czerniak.]
**[http://www.academia.edu/1512888/Introduction_to_the_Kalman_Filter_and_its_Derivation Introduction the the Kalman Filter and its Derivation, Brent Perreault]
**[http://people.wallawalla.edu/~Rob.Frohne/ClassHandouts/Digital_Control/Kalman_1.pdf A Simple Introduction to the Kalman Filter] (This reference from Duy-Ky Nguyen has some excellent examples and very readable derivations.)
*Papers
**[http://docphdpersonalstuff.googlecode.com/svn/trunk/literature/an%20introduction%20to%20observers.pdf David Luenberger's Introduction to Observers]
**F.L. Lewis' LQR Explanations
***[http://www.uta.edu/utari/acs/Lectures/lqr.pdf F.L. Lewis Linear Quadratic Regulator Derivation and Examples]
***[http://www.uta.edu/utari/acs/ee5321/2013%20notes/2%20lqr%20dt%20and%20sampling.pdf F.L. Lewis Discrete LQR Derivation and Examples] This goes nicely with the above paper doing it for continuous time.
**[http://www.youtube.com/watch?v=St5L-ekOKGA Jake Abbott's lecture on the Linear Quadratic Regulator]
**[http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf Greg Welch & Gary Bishop's Introduction to Kalman Filters]
**[http://users.isr.ist.utl.pt/~mir/pub/kalman.pdf Maria Isabel Ribeiro paper on Kalman Filters]
**[https://www.control.isy.liu.se/research/reports/LicentiateThesis/Lic1085.pdf The Descriptor State Space Model]
**[https://nms.kcl.ac.uk/simon.salamon/P/dedm.pdf Relation Between Continuous Differential Equations and Discrete Difference Equations]
*2016
**[http://ocw.mit.edu/courses/mechanical-engineering/2-004-systems-modeling-and-control-ii-fall-2007/lecture-notes/lecture05.pdf Equations of Motion for a DC Motor]
**[https://courses.engr.illinois.edu/ece486/documents/lecture_notes/state_space_realizations.pdf Controllable and Modal Canonical Forms of the State Space Representation]

===MATLAB/Octave/Scilab===
====Gnu Octave====
Gnu Octave is a free open source clone of MATLAB. I use it all the time in lieu of MATLAB, and it works well on my Ubuntu Linux OS.
*[http://laris.fesb.hr/digitalno_vodjenje/octave/doc/octave_30.html Octave Control Systems Toolbox] This is not the same thing that is on [http://octave.sourceforge.net/control/overview.html Octave Forge here].
*[[Digital Control Octave Scripts]]
====MATLAB/Simulink====
*[http://ctms.engin.umich.edu/CTMS/index.php?aux=Activities_IOpack This page shows some neat experiments you can do with the ArduinoIO package which allows you to use an Arduino from Simulink/MATLAB.] You can also control an Arduino from any terminal program. It's license from Mathworks forbids its use with other software like Octave or Scilab, and probably from controlling an Arduino from your own program, unless it is for use with Mathworks software.

====Scilab/Xcos====
Scilab is a MATLAB like program. It isn't as compatible with MATLAB as octave is, but it has Xcos, which is a GUI block manipulation tool very similar to Scilab.
*[https://atoms.scilab.org/toolboxes/arduino The Xcos ATOM toolbox] is very similar to ArduinoIO, but for the open source Scilab/Xcos environment. It works only on Windows at this time, but looking at the source leads me to believe that it would be easy to port it to Linux or OS X and it is fully open source.

====Pendulum Specific Scripts====
=====Single Pendulum=====
<nowiki>
% Double Pendulum Parameters (Tentative: There are two pendulum setups, each with different parameters. I'm not sure which these go to.)
% This script is for balancing only the long rod.

% Run parameters
%f = input('Control Frequency (Hz) = ');
%Trun = input('Run Time (s) = ');
%f=130;
f=1000;
Trun=30;

kmax = round(f*Trun);
T = 1/f;
Maxpos = 0.25; % Max carriage travel +- 0.25 m
Maxangle = 0.175; % Max rod angle -- 10 deg
Maxvoltage = 20; % Max motor voltage, V
pstart = 0.005; % Carriage position starting limit, m
astart = 1*pi/180; % Angle starting limit, rad

g = 9.81; % m/s^2 Gravitational constant

% SYSTEM PARAMETERS
% Measured Mechanical Parameters
d1 = 0.323; % m Length of pendulum 1 (long)
d2 = 0.079; % m Length of pendulum 2 (short)
%mp1 = 0.0208; % kg Mass of pendulum 1
mp1 = 0.0318;
%mp2 = 0.0050; % kg Mass of pendulum 2
mp2 = 0.0085;
%m=.5;
m = 0.3163; % kg Mass of carriage
rd = 0.0254/2; % m Drive pulley radius
md = 0.0375; % kg Mass of drive pulley (cylinder)
%mc1 = 0.0036; % kg Mass of clamp 1*
%mc2 = 0.0036; % kg Mass of clamp 2*
mc1 = 0.0085;
mc2 = mc1;

% *Clamp Dimensions
% Rectangular 0.0254 x 0.01143 m
% The pivot shaft is 0.00714 m from the end

% Motor Parameters (Data Sheet)
Im = 43e-7; % kg m^2/rad Rotor moment of inertia
R = 4.09; % ohms Resistance
kt = 0.0351; % Nm/A Torque constant
ke = 0.0351; % Vs/rad Back emf constant

% Derived Mechanical Parameters

% kg m^2/rad Moment of inertia, clamp 1
%Ic1 = mc1*(0.01143^2 + 0.0254^2)/12 + mc1*(0.0127-0.00714)^2;
Ic1 = mc1*(0.0098^2 + 0.0379^2)/12;
Ic2 = Ic1; % kg m^2/rad Moment of inertia, clamp 2
Id = md*(rd^2)/2; % kg m^2/rad Moment of inertia, drive pulley
Imd = Im + Id; % kg m^2/rad Moment of inertia, combined

J1 = Ic1 + mp1*(d1^2)/3; % Total moment of inertia, pendulum 1 (long)
J2 = Ic2 + mp2*(d2^2)/3; % Total moment of inertia, pendulum 2 (short)
Jd = Im + Id; % Total moment of inertia, motor drive
Mc = m + mc1 + mc2; % Total carriage mass

% Friction Test Data
% Carriage Slope = 19 deg; Terminal Velocity xdotss = 0.312 m/s; From
% twincarriage.m; formula b = m g sin(theta)/xdotss
% Pendulum 1 (long) Exponent a1 = 0.0756 1/s; From longfit.m
% Pendulum 2 (short) Exponent a2 = 0.2922 1/s; From shortfit.m
% formula b = 2 a J

%alpha = 19;
alpha = 12.2;
%xdotss = 0.312;
xdotss = 0.4852;
%a1 = 0.0756;
%a2 = 0.2922;
a1 = 0.0185;
a2 = 0.012;
% Ns/m Viscous friction of carriage system
b = (Mc + mp1 + mp2)*g*sin(alpha*pi/180)/xdotss;
b1 = 2*a1*J1; % Nms/rad Viscous friction of pendulum 1 (rotational)
b2 = 2*a2*J2; % Nms/rad Viscous friction of pendulum 2 (rotational)

scale = [rd*2*pi/4096 2*pi/4096 -0.05/250];


T = 1/f;

% The data above comes from the fweb wiki.

M=Mc; %mass of cart
m=mp1; %mass of pendulum 1
b=b; %friction
l=d1/2; %length of pendulum
I=J1; %inertia of pendulum
%q=(M+m)*(l+m*l^2)-(m*l)^2;
%num=[m*l,0]; %numerator for transfer function
%den=[q,b*(l+m*l^2),-m*g*l*(M+m),-b*m*g*l]; %denominator for transfer function
%[A,B,C,D]=tf2ss(num,den)
%A,B,C,D matricies for the state space model
% x_vec is [x, x_dot, theta, theta_dot]'
% See the web site: https://www.library.cmu.edu/ctms/ctms/examples/pend/invpen.htm
A=[ 0 1 0 0;
0 ((-(I+m*l^2)*b)/(I*(M+m)+M*m*l^2)) ((m^2*g*l^2)/(I*(M+m)+M*m*l^2)) 0;
0 0 0 1;
0 ((-m*l*b)/(I*(M+m)+M*m*l^2)) ((m*g*l*(M+m))/(I*(M+m)+M*m*l^2)) 0];
B=[ 0;
((I+m*l^2)/(I*(M+m)+M*m*l^2));
0;
((m*l)/(I*(M+m)+M*m*l^2))];
C=[ 1 0 0 0;
0 0 1 0];
D=[ 0;
0];
cont_sys = ss(A,B,C,D)
rank_ctrb = rank(ctrb(A,B))
original_poles = eig(A) %poles for our system
</nowiki>

These Pendulum [[Parameters]] and control scripts are for our hardware.

===Z Transforms===
*[https://en.wikipedia.org/wiki/Z_transform#Relationship_to_Laplace_transform Relationship between the Laplace and Z transforms]
*Convolution and Z Transforms
**[http://www.ualberta.ca/~msacchi/GEOPH426/chapter2.pdf Z Transforms and Convolution]
**[http://www.jhu.edu/signals/convolve/index.html Here is an animation of convolution with continuous signals.] Look at it so you understand what is happening with the mechanics of convolution. To convolve <math>x(t)</math> with <math>h(t)</math>, you flip shift <math>h(t)</math> into <math>h(t-t_0)</math> on the <math>t_0</math> axis, then you multiply it by <math>x(t_0)</math> to get <math>x(t_0)h(t-t_0)</math>, then you integrate with respect to <math>t_0</math>, so that the convolution is: <math>x(t) * h(t) = \int_{-\infty}^\infty x(t_0)h(t-t_0) dt_0</math>. The first animation shows this happening with sampled waveforms: <math>x_s(t) = x(t) \sum_{n=0}^\infty \delta (t-nT) = \sum_{n=0}^\infty x(nT) \delta (t-nT)</math> and <math>h_s(t) = h(t) \sum_{n=0}^\infty \delta (t-nT)= \sum_{n=0}^\infty h(nT) \delta (t-nT)</math>.
***Discrete convolution animations
****[http://www-rohan.sdsu.edu/~jiracek/DAGSAW/4.3.html Here is an animation of discrete convolution to illustrate the above equations.]
****[http://www.jhu.edu/signals/discreteconv2/index.html This one lets you try several functions.]
****[http://www.cse.yorku.ca/~asif/spc/ConvolutionSum_Final3.swf This one shows what happens if your signals happen prior to time zero.]
**Notice that this is really the same as [http://www.mathworks.com/help/matlab/ref/conv.html Polynomial Multiplication].
===Discretization===
*[https://en.wikipedia.org/wiki/Discretization This is how the c2d zero order hold works in MATLAB/octave.] It uses the "exact" solution to the discretization problem.
* Also check Chapter 7.3 of [http://www.me.unm.edu/~starr/teaching/me581/textbook.pdf Greg Starr's book].

===Scilab/Xcos===
*[https://www.scilab.org/content/download/1107/10095/file/Xcos_beginners.pdf Xcos Beginners Guide]
*[http://www.iitg.ernet.in/trivedi/Downloads/scilab/book.pdf Scilab/Scicos Book]

Latest revision as of 15:11, 3 June 2016

Links

Textbooks

Lectures

MATLAB/Octave/Scilab

Gnu Octave

Gnu Octave is a free open source clone of MATLAB. I use it all the time in lieu of MATLAB, and it works well on my Ubuntu Linux OS.

MATLAB/Simulink

Scilab/Xcos

Scilab is a MATLAB like program. It isn't as compatible with MATLAB as octave is, but it has Xcos, which is a GUI block manipulation tool very similar to Scilab.

  • The Xcos ATOM toolbox is very similar to ArduinoIO, but for the open source Scilab/Xcos environment. It works only on Windows at this time, but looking at the source leads me to believe that it would be easy to port it to Linux or OS X and it is fully open source.

Pendulum Specific Scripts

Single Pendulum
% Double Pendulum Parameters (Tentative:  There are two pendulum setups, each with different parameters.  I'm not sure which these go to.)
% This script is for balancing only the long rod.

% Run parameters
%f = input('Control Frequency (Hz) = ');
%Trun = input('Run Time (s) = ');
%f=130;
f=1000;
Trun=30;

kmax = round(f*Trun);
T = 1/f;
Maxpos = 0.25;              % Max carriage travel +- 0.25 m
Maxangle = 0.175;           % Max rod angle -- 10 deg
Maxvoltage = 20;            % Max motor voltage, V
pstart = 0.005;             % Carriage position starting limit, m
astart = 1*pi/180;          % Angle starting limit, rad

g = 9.81;                   % m/s^2     Gravitational constant

% SYSTEM PARAMETERS
% Measured Mechanical Parameters
d1 = 0.323;                 % m         Length of pendulum 1 (long)
d2 = 0.079;                 % m         Length of pendulum 2 (short)
%mp1 = 0.0208;              % kg        Mass of pendulum 1
mp1 = 0.0318;
%mp2 = 0.0050;              % kg        Mass of pendulum 2
mp2 = 0.0085;
%m=.5;
m = 0.3163;                 % kg        Mass of carriage
rd = 0.0254/2;              % m         Drive pulley radius
md = 0.0375;                % kg        Mass of drive pulley (cylinder)
%mc1 = 0.0036;              % kg        Mass of clamp 1*
%mc2 = 0.0036;              % kg        Mass of clamp 2*
mc1 = 0.0085;
mc2 = mc1;

% *Clamp Dimensions
%  Rectangular 0.0254 x 0.01143 m
%  The pivot shaft is 0.00714 m from the end

% Motor Parameters (Data Sheet)
Im = 43e-7;                 % kg m^2/rad    Rotor moment of inertia
R = 4.09;                   % ohms            Resistance
kt = 0.0351;                % Nm/A            Torque constant
ke = 0.0351;                % Vs/rad        Back emf constant

% Derived Mechanical Parameters

                            % kg m^2/rad    Moment of inertia, clamp 1
%Ic1 = mc1*(0.01143^2 + 0.0254^2)/12 + mc1*(0.0127-0.00714)^2;
Ic1 = mc1*(0.0098^2 + 0.0379^2)/12;
Ic2 = Ic1;                  % kg m^2/rad    Moment of inertia, clamp 2
Id = md*(rd^2)/2;           % kg m^2/rad    Moment of inertia, drive pulley
Imd = Im + Id;              % kg m^2/rad    Moment of inertia, combined

J1 = Ic1 + mp1*(d1^2)/3;    % Total moment of inertia, pendulum 1 (long)
J2 = Ic2 + mp2*(d2^2)/3;    % Total moment of inertia, pendulum 2 (short)
Jd = Im + Id;               % Total moment of inertia, motor drive
Mc = m + mc1 + mc2;         % Total carriage mass

% Friction Test Data
%   Carriage Slope = 19 deg;  Terminal Velocity xdotss = 0.312 m/s; From
%       twincarriage.m; formula b = m g sin(theta)/xdotss
%   Pendulum 1 (long) Exponent a1 = 0.0756 1/s;  From longfit.m
%   Pendulum 2 (short) Exponent a2 = 0.2922 1/s; From shortfit.m
%        formula b = 2 a J

%alpha = 19;
alpha = 12.2;
%xdotss = 0.312;
xdotss = 0.4852;
%a1 = 0.0756;
%a2 = 0.2922;
a1 = 0.0185;
a2 = 0.012;
                            % Ns/m    Viscous friction of carriage system
b = (Mc + mp1 + mp2)*g*sin(alpha*pi/180)/xdotss;
b1 = 2*a1*J1;               % Nms/rad    Viscous friction of pendulum 1 (rotational)
b2 = 2*a2*J2;               % Nms/rad    Viscous friction of pendulum 2 (rotational)

scale = [rd*2*pi/4096  2*pi/4096 -0.05/250];


T = 1/f;

% The data above comes from the fweb wiki.

M=Mc;                       %mass of cart
m=mp1;                      %mass of pendulum 1
b=b;                        %friction
l=d1/2;                     %length of pendulum
I=J1;                       %inertia of pendulum
%q=(M+m)*(l+m*l^2)-(m*l)^2;
%num=[m*l,0];                                %numerator for transfer function
%den=[q,b*(l+m*l^2),-m*g*l*(M+m),-b*m*g*l];  %denominator for transfer function
%[A,B,C,D]=tf2ss(num,den)
%A,B,C,D matricies for the state space model
% x_vec is [x,  x_dot, theta, theta_dot]'
% See the web site:  https://www.library.cmu.edu/ctms/ctms/examples/pend/invpen.htm
A=[ 0   1                                                  0                                   0;
    0   ((-(I+m*l^2)*b)/(I*(M+m)+M*m*l^2))  ((m^2*g*l^2)/(I*(M+m)+M*m*l^2))     0;
    0   0                                   0                                   1;
    0   ((-m*l*b)/(I*(M+m)+M*m*l^2))        ((m*g*l*(M+m))/(I*(M+m)+M*m*l^2))   0];
B=[ 0;
    ((I+m*l^2)/(I*(M+m)+M*m*l^2));
    0;
    ((m*l)/(I*(M+m)+M*m*l^2))];
C=[ 1   0   0   0;
    0   0   1   0];
D=[ 0;
    0];
cont_sys = ss(A,B,C,D)
rank_ctrb = rank(ctrb(A,B))
original_poles = eig(A) %poles for our system

These Pendulum Parameters and control scripts are for our hardware.

Z Transforms

Discretization

Scilab/Xcos