Question

Write a MATLAB program to demonstrate the Ziegler-Nichols method of PID loop tuning. Also in no more than one A4 page justify how your MATLAB code demonstrates the PID loop tuning procedure.

Answer 1

• CLS( Wp,Wc ) :

function sys = CLS( Wp,Wc )
%CLS Closed loop system function

%% Parameters
% Wp : Plant transfer function
% Wc : Controller transfer function
% sys : Closed Loop transfer function with assuming unity feedback.

%% Function implementation
sys=feedback(series(Wp,Wc),1);
end

• CreatePlant( num,den ):

function [ Wp ] = CreatePlant( num,den )
%CreatePlant Creates plant transfer function.
%   The returned value is the system in numerator/denomerator format
%% Parameters
% num : Numerator vector (starting from highest order of coefficients)
% den : Denomerator vector (starting from highest order of coefficients)
% plant : Plant transfer function 
%% EXAMPLE
%    num=[1];
%    den=[1 0 1];
%    sys=CreatePlant(num,den)

%% Result is        
%             1
% sys= ---------------
%           S^2+1
%% Function implementation
syms s;
Wp=tf(num,den);
end

• example(varargin):

function varargout = example(varargin)
% EXAMPLE MATLAB code for example.fig
%      EXAMPLE, by itself, creates a new EXAMPLE or raises the existing
%      singleton*.
%
%      H = EXAMPLE returns the handle to a new EXAMPLE or the handle to
%      the existing singleton*.
%
%      EXAMPLE('CALLBACK',hObject,eventData,handles,...) calls the local
%      function named CALLBACK in EXAMPLE.M with the given input arguments.
%
%      EXAMPLE('Property','Value',...) creates a new EXAMPLE or raises
%      the existing singleton*.  Starting from the left, property value pairs are
%      applied to the GUI before example_OpeningFcn gets called.  An
%      unrecognized property name or invalid value makes property application
%      stop.  All inputs are passed to example_OpeningFcn via varargin.
%
%      *See GUI Options on GUIDE's Tools menu.  Choose "GUI allows only one
%      instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES

% Edit the above text to modify the response to help example

% Last Modified by GUIDE v2.5 01-Sep-2014 10:35:27

% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name',       mfilename, ...
                   'gui_Singleton',  gui_Singleton, ...
                   'gui_OpeningFcn', @example_OpeningFcn, ...
                   'gui_OutputFcn',  @example_OutputFcn, ...
                   'gui_LayoutFcn',  [] , ...
                   'gui_Callback',   []);
if nargin && ischar(varargin{1})
    gui_State.gui_Callback = str2func(varargin{1});
end

if nargout
    [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
    gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT

% --- Executes just before example is made visible.
function example_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)
% varargin   command line arguments to example (see VARARGIN)

% Choose default command line output for example
handles.output = hObject;

% Update handles structure
guidata(hObject, handles);

initialize_gui(hObject, handles, false);

% UIWAIT makes example wait for user response (see UIRESUME)
% uiwait(handles.figure1);


% --- Outputs from this function are returned to the command line.
function varargout = example_OutputFcn(hObject, eventdata, handles)
% varargout  cell array for returning output args (see VARARGOUT);
% hObject    handle to figure
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% Get default command line output from handles structure
varargout{1} = handles.output;


% --- Executes during object creation, after setting all properties.
function density_CreateFcn(hObject, eventdata, handles)
% hObject    handle to density (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called

% Hint: popupmenu controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end



function density_Callback(hObject, eventdata, handles)
% hObject    handle to density (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of density as text
%        str2double(get(hObject,'String')) returns contents of density as a double
density = str2double(get(hObject, 'String'));
if isnan(density)
    set(hObject, 'String', 0);
    errordlg('Input must be a number','Error');
end

% Save the new density value
handles.metricdata.density = density;
guidata(hObject,handles)

% --- Executes during object creation, after setting all properties.
function volume_CreateFcn(hObject, eventdata, handles)
% hObject    handle to volume (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    empty - handles not created until after all CreateFcns called

% Hint: popupmenu controls usually have a white background on Windows.
%       See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end



function volume_Callback(hObject, eventdata, handles)
% hObject    handle to volume (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of volume as text
%        str2double(get(hObject,'String')) returns contents of volume as a double
volume = str2double(get(hObject, 'String'));
if isnan(volume)
    set(hObject, 'String', 0);
    errordlg('Input must be a number','Error');
end

% Save the new volume value
handles.metricdata.volume = volume;
guidata(hObject,handles)

% --- Executes on button press in calculate.
function calculate_Callback(hObject, eventdata, handles)
% hObject    handle to calculate (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

mass = handles.metricdata.density * handles.metricdata.volume;
set(handles.mass, 'String', mass);

% --- Executes on button press in reset.
function reset_Callback(hObject, eventdata, handles)
% hObject    handle to reset (see GCBO)
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

initialize_gui(gcbf, handles, true);

% --- Executes when selected object changed in unitgroup.
function unitgroup_SelectionChangeFcn(hObject, eventdata, handles)
% hObject    handle to the selected object in unitgroup 
% eventdata  reserved - to be defined in a future version of MATLAB
% handles    structure with handles and user data (see GUIDATA)

if (hObject == handles.english)
    set(handles.text4, 'String', 'lb/cu.in');
    set(handles.text5, 'String', 'cu.in');
    set(handles.text6, 'String', 'lb');
else
    set(handles.text4, 'String', 'kg/cu.m');
    set(handles.text5, 'String', 'cu.m');
    set(handles.text6, 'String', 'kg');
end

% --------------------------------------------------------------------
function initialize_gui(fig_handle, handles, isreset)
% If the metricdata field is present and the reset flag is false, it means
% we are we are just re-initializing a GUI by calling it from the cmd line
% while it is up. So, bail out as we dont want to reset the data.
if isfield(handles, 'metricdata') && ~isreset
    return;
end

handles.metricdata.density = 0;
handles.metricdata.volume  = 0;

set(handles.density, 'String', handles.metricdata.density);
set(handles.volume,  'String', handles.metricdata.volume);
set(handles.mass, 'String', 0);

set(handles.unitgroup, 'SelectedObject', handles.english);

set(handles.text4, 'String', 'lb/cu.in');
set(handles.text5, 'String', 'cu.in');
set(handles.text6, 'String', 'lb');

% Update handles structure
guidata(handles.figure1, handles);

• PIDExperiment(varargin):

function varargout = PIDExperiment(varargin)
gui_Singleton = 1;
gui_State = struct('gui_Name',       mfilename, ...
                   'gui_Singleton',  gui_Singleton, ...
                   'gui_OpeningFcn', @PIDExperiment_OpeningFcn, ...
                   'gui_OutputFcn',  @PIDExperiment_OutputFcn, ...
                   'gui_LayoutFcn',  [] , ...
                   'gui_Callback',   []);
if nargin && ischar(varargin{1})
    gui_State.gui_Callback = str2func(varargin{1});
end

if nargout
    [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
    gui_mainfcn(gui_State, varargin{:});
end

function PIDExperiment_OpeningFcn(hObject, eventdata, handles, varargin)
% Choose default command line output for PIDExperiment
handles.output = hObject;
handles.tfdata.num=1;
handles.tfdata.den=[1000 300 30 1];
handles.tfdata.t='0:0.01:300';
% Update handles structure
guidata(hObject, handles);


% --- Outputs from this function are returned to the command line.
function varargout = PIDExperiment_OutputFcn(hObject, eventdata, handles) 
% Get default command line output from handles structure
varargout{1} = handles.output;



function num_Callback(hObject, eventdata, handles)
num=str2num(get(handles.num,'string'));
handles.tfdata.num=num;
guidata(hObject,handles);


% --- Executes during object creation, after setting all properties.
function num_CreateFcn(hObject, eventdata, handles)
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end



function den_Callback(hObject, eventdata, handles)
den=str2num(get(handles.den,'string'));
handles.tfdata.den=den;
guidata(hObject,handles);


% --- Executes during object creation, after setting all properties.
function den_CreateFcn(hObject, eventdata, handles)
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end


% --- Executes on button press in gettf.
function gettf_Callback(hObject, eventdata, handles)
Plant=CreatePlant(handles.tfdata.num,handles.tfdata.den);
handles.tfdata.plant=Plant;
guidata(hObject,handles);
sys=evalc('Plant');
set(handles.tf,'string',sys);


% --- Executes on button press in plotnyquist.
function plotnyquist_Callback(hObject, eventdata, handles)
axes(handles.nyquistaxis);
nyquist(handles.tfdata.plant);


% --- Executes on button press in findkc.
function findkc_Callback(hObject, eventdata, handles)
[Kc,Pm,Wc,Wm]=margin(handles.tfdata.plant);
pu=2*pi/Wc;
set(handles.kc,'string',num2str(Kc));
set(handles.pu,'string',num2str(pu));
handles.tfdata.kc=Kc;
handles.tfdata.pu=pu;
guidata(hObject,handles);


% --- Executes on button press in plotziegler.
function plotziegler_Callback(hObject, eventdata, handles)

%P-Controller
Kpp=handles.tfdata.kc*0.5;
Tip=100000;
Tdp=0;

%PI-Controller
Kppi=handles.tfdata.kc*0.45;
Tipi=handles.tfdata.pu*0.83;
Tdpi=0;

%PID-Controller
Kppid=handles.tfdata.kc*0.59;
Tipid=handles.tfdata.pu*0.5;
Tdpid=handles.tfdata.pu*0.12;

%Closed Loop transfer function
Wcp=ZieglerNicholasPID(Kpp,Tip,Tdp);
Wcpi=ZieglerNicholasPID(Kppi,Tipi,Tdpi);
Wcpid=ZieglerNicholasPID(Kppid,Tipid,Tdpid);

sysp=CLS(handles.tfdata.plant,Wcp);
syspi=CLS(handles.tfdata.plant,Wcpi);
syspid=CLS(handles.tfdata.plant,Wcpid);

zntable=[Kpp Tip Tdp;Kppi Tipi Tdpi;Kppid Tipid Tdpid ];
set(handles.table,'data',zntable);

%plotting
axes(handles.responseaxis);
t=str2num(handles.tfdata.t);
step(sysp,syspi,syspid,t),
legend('P','PI','PID');


function t_Callback(hObject, eventdata, handles)
t=num2str(get(handles.t,'string'));
handles.tfdata.t=t;
guidata(hObject,handles);


% --- Executes during object creation, after setting all properties.
function t_CreateFcn(hObject, eventdata, handles)
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor'))
    set(hObject,'BackgroundColor','white');
end

• ZieglerNicholasPID( Kc,Ti,Td ):

function Wc  = ZieglerNicholasPID( Kc,Ti,Td )
% ZieglerNicholasPID function to generate the PID controller transfer 
%% Parameters
% Kc : Critical gain
% Ti : Reset time (minutes)
% Td : Derivative time (minutes)
% Wc : The laplace representation of Z-N

%% EXAMPLE
%    Kc=10;
%    Ti=0.83;
%    Td=2.5
%    Wc=ZieglerNicholasPID( Kc,Ti,Td )
%% Result is        
%               1 
% Wc= 10*(1+ -------- + 2.5*s
%             0.83*s 

%% Function implementation 
s=tf('s');
Wc=Kc*(1+(1/(Ti*s))+Td*s);
end