Ga algorithm fitness value not improving

Hello users,

This is the first time I'm using genetic algorithms. I am attempting to tune PID gains. As I observe the results I find that the fitness value is not improving, infact it's not changing at all. I am wondering what I am doing wrong. Before running the code I first initized k1=[1 1 1]. Below is my script written as clearly as I could.

% Inner Loop Controller

%Population range
InitialPopulationRange1 = [50 50 10;inf inf inf];

var_num1=3; % number of variables
lb1=[50 50 10]; % solution lower bound
ub1=[inf inf inf]; % solution upper bound

%Initial population
PopulationSize = 10;  %Population size
InitialPopulationMatrix1=rand(PopulationSize,3);

MaxGenerations=10;  % max no. of generations
MaxStallGenerations=5; %max generations when solution doesn't change

% ga options

ga_opt1 = optimoptions(@ga,'MaxGenerations',MaxGenerations,'MaxStallGenerations',MaxStallGenerations,'PopulationSize',PopulationSize,'SelectionFcn',{@selectiontournament,4},'FitnessScalingFcn',@fitscalingprop,'MutationFcn',{@mutationadaptfeasible},'PlotFcn',@gaplotbestf);

% Reproduction options
ga_opt1.EliteCount = 0;
ga_opt1.CrossoverFraction=0.8;

obj_fn=@(k) optimization_PID(k1);

% ga Command

[k1,best]=ga((obj_fn),var_num1,[],[],[],[],lb1,ub1,[],ga_opt1)

Answer 1

[Sam Chak]

Hi @Phumelele Magagula

To observe convergence, you should consider increasing the values of 'PopulationSize' and 'MaxGenerations'. In the example provided below, it's evident from the GA Best plot that convergence takes place within 150 generations.

%Population range
InitialPopulationRange1 = [50 50 10; inf inf inf];

var_num1 = 3; % number of variables
lb1      = [50 50 10]; % solution lower bound
ub1      = [inf inf inf]; % solution upper bound

%Initial population
PopulationSize = 25;  %Population size
InitialPopulationMatrix1 = rand(PopulationSize, 3);

MaxGenerations = 150;  % max no. of generations
MaxStallGenerations = 5; %max generations when solution doesn't change

% ga options

ga_opt1 = optimoptions(@ga,'MaxGenerations',MaxGenerations,'MaxStallGenerations',MaxStallGenerations,'PopulationSize',PopulationSize,'SelectionFcn',{@selectiontournament,4},'FitnessScalingFcn',@fitscalingprop,'MutationFcn',{@mutationadaptfeasible},'PlotFcn',@gaplotbestf);

% Reproduction options
ga_opt1.EliteCount = 0;
ga_opt1.CrossoverFraction=0.8;

obj_fn = @optimization_PID;

% ga Command

[k1, best] = ga((obj_fn), var_num1, [], [], [], [], lb1, ub1, [], ga_opt1)

Optimization terminated: maximum number of generations exceeded.

k1 = 1×3

60.0037 69.9995 80.0101

best = 1.1521e-04

% Test multivariate convex function
function J = optimization_PID(k1)
    J = (k1(1) - 60)^2 + (k1(2) - 70)^2 + (k1(3) - 80)^2;
end

[Phumelele Magagula]

Thank you Sam, I will do that

[Phumelele Magagula]

Hello Sam. So I have increased the population and number of generations, but now I am facing a new problem. I get a best solution whose cost function value seems reasonable but when I run the corresponding gains displayed as 'k' the value of the cost function is different, indicating that this k value is not actually the optimal solution. I don't know what's going on or how to fix it. Please help.

[Walter Roberson]

@Phumelele Magagula

You are copying the k values as text and passing those into the functions. But with the default MATLAB output format, you are getting at most 4 digits of precision in the values... and you need higher precision in order to get accurate cost function values.

Instead of displaying k1 the way you are now, use

fprintf("%.17g %.17g %.17g\n", k1);

There are some IEEE 754 Double Precision numbers that need 17 digits to exactly reproduce.

Using the default format short will always be too few digits for accurate calculations with nonlinear functions.

Using format long g is better but only displays one few digit than is required for accurate reproduction in most cases... two fewer in some cases.

[Sam Chak]

Hi @Phumelele Magagula

See my comment below.

rng default

%Population range
InitialPopulationRange1 = [50 50 10; inf inf inf];

var_num1 = 3; % number of variables
lb1      = [50 50 10]; % solution lower bound
ub1      = [inf inf inf]; % solution upper bound

%Initial population
PopulationSize = 25;  %Population size
InitialPopulationMatrix1 = rand(PopulationSize, 3);

MaxGenerations = 150;  % max no. of generations
MaxStallGenerations = 5; %max generations when solution doesn't change

% ga options

ga_opt1 = optimoptions(@ga,'MaxGenerations',MaxGenerations,'MaxStallGenerations',MaxStallGenerations,'PopulationSize',PopulationSize,'SelectionFcn',{@selectiontournament,4},'FitnessScalingFcn',@fitscalingprop,'MutationFcn',{@mutationadaptfeasible},'PlotFcn',@gaplotbestf);

% Reproduction options
ga_opt1.EliteCount = 0;
ga_opt1.CrossoverFraction=0.8;

obj_fn = @optimization_PID;

% ga Command

[k1, best] = ga((obj_fn), var_num1, [], [], [], [], lb1, ub1, [], ga_opt1)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

k1 = 1×3

60.0001 70.0017 80.0020

best = 7.0672e-06

If you want to assess the precision of the PID gains returned by the Genetic Algorithm, follow @Walter Roberson's advice.

fprintf("%.17g %.17g %.17g\n", k1)

60.000087400197792 70.001732581595888 80.002014388590453

If you directly plug the corresponding gains into the same cost function, 'optimization_PID()', you will obtain the same fitness value as the 'best' value.

J1 = optimization_PID(k1)

J1 = 7.0672e-06

However, if you merely use the displayed 4-digit precision values, the computed fitness value () will not match the 'best' value (). If the error between the computed fitness value and the 'best' value is not significant, and if , then you can be sure that the displayed 4-digit precision values are indeed 'better' gains. This is because they are much closer to the true optimal values [60, 70, 80]."

J2 = optimization_PID([60.0001 70.0017 80.0020])

J2 = 6.9000e-06

% Test multivariate convex function
function J = optimization_PID(k1)
    J = (k1(1) - 60)^2 + (k1(2) - 70)^2 + (k1(3) - 80)^2;
end