Digital Control Systems

Links

Textbooks

• • Introduction to Applied Digital Control, Greg Starr, University of New Mexico
    • Greg's ME481/ME581 web pages contain solutions to the book problems and other things.
  • Optimal Sampled-Data Control Systems, Chen & Francis
  • Digital Control Systems Notes Irfan Ahmed
  • Digital Control Engineering Fadali & Visioli
  • Discrete Time Control Systems Ogata
  • Control System Engineering, Chapter 13 (Digital Control) Norman Nise
  • Digital Control Systems Notes Ben Chen
  • Computer Control Systems
  • Digital Control System Analysis and Design, 3rd Ed., Philips
  • Control Systems and Control Engineering
  • Introduction to Signal Processing
  • Zoran Gajic's Book on Control Systems (Continuous & Discrete) and some other documents. Look for the documents entitled Chapter N.
  • NPTEL Course on Digital Control

Lectures

• 2016
  • LQR Derivation Outline from Class
  • Kalman Filters Derivation Outline from Class
    • Kalman Filter Summary from Class

• 2014
  • Sampled Time Control
  • Duy-Ky Nguyen's Concise Digital Control Notes I like this presentation.
  • 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.
  • Professor Bemporad's Automatic Control Class Notes These are very good.
    • 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).
  • MIT Class on Underwater Vehicle Control
    • Linear Quadratic Regulators (Mathematically Deep)
    • 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.
  • Kalman Filter Applied to Cart in One Dimensional Motion
  • Intuitive Lecture on Practical Kalman Filter given by Francis Grovers III
  • The first 80% of what you need to know about Kalman Filters from Greg Czerniak.
  • Introduction the the Kalman Filter and its Derivation, Brent Perreault
  • A Simple Introduction to the Kalman Filter (This reference from Duy-Ky Nguyen has some excellent examples and very readable derivations.)
• Papers
  • David Luenberger's Introduction to Observers
  • F.L. Lewis' LQR Explanations
    • F.L. Lewis Linear Quadratic Regulator Derivation and Examples
    • F.L. Lewis Discrete LQR Derivation and Examples This goes nicely with the above paper doing it for continuous time.
  • Jake Abbott's lecture on the Linear Quadratic Regulator
  • Greg Welch & Gary Bishop's Introduction to Kalman Filters
  • Maria Isabel Ribeiro paper on Kalman Filters
  • The Descriptor State Space Model
  • Relation Between Continuous Differential Equations and Discrete Difference Equations
• 2016
  • Equations of Motion for a DC Motor
  • 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.

• Octave Control Systems Toolbox This is not the same thing that is on Octave Forge here.
• Digital Control Octave Scripts

MATLAB/Simulink

• 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.

• 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

• Relationship between the Laplace and Z transforms
• Convolution and Z Transforms
  • Z Transforms and Convolution
  • 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 $x(t)$ with $h(t)$, you flip shift $h(t)$ into $h(t-t_{0})$ on the $t_{0}$ axis, then you multiply it by $x(t_{0})$ to get $x(t_{0})h(t-t_{0})$, then you integrate with respect to $t_{0}$, so that the convolution is: $x(t)*h(t)=\int _{-\infty }^{\infty }x(t_{0})h(t-t_{0})dt_{0}$. The first animation shows this happening with sampled waveforms: $x_{s}(t)=x(t)\sum _{n=0}^{\infty }\delta (t-nT)=\sum _{n=0}^{\infty }x(nT)\delta (t-nT)$ and $h_{s}(t)=h(t)\sum _{n=0}^{\infty }\delta (t-nT)=\sum _{n=0}^{\infty }h(nT)\delta (t-nT)$.
    • Discrete convolution animations
      • Here is an animation of discrete convolution to illustrate the above equations.
      • This one lets you try several functions.
      • This one shows what happens if your signals happen prior to time zero.
  • Notice that this is really the same as Polynomial Multiplication.

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 Greg Starr's book.

Scilab/Xcos

• Xcos Beginners Guide
• Scilab/Scicos Book