i need code for the example1 model equation using pso

i need code using pso ,without data flitering using model equation for ex-1,is it possible to give code for me to generate simulation results.

[Torsten]

If you have a licence for the optimization toolbox, you can use

https://uk.mathworks.com/help/gads/particleswarm.html

Answer 1

[Sam Chak]

Hi @Sirisha

Properly modeling the fractional-order Wiener–Hammerstein system and the adaptive fuzzy system structure in Example 1 in MATLAB would require additional coding time or tools from MATLAB File Exchange.

If you’d like to see an example of using PSO to identify the parameters of a nonlinear dynamic system, here is a basic code using particleswarm() for identifying parameters {a, b, c} in this system:

% noise-free data
tin     = linspace(0, 10, 51)';
xout    = tanh(tin);    % a known solution for dx/dt = 1 - x(t)² with initial value x(0) = 0

% identification via PSO
fun     = @(p) costfcn(p, tin, xout);   % find p vector only in J
nvars   = 3;                            % 3 unknown variable in p
lb      = [0.9, 0.9, 1.9];              % lower bound (requires user guess)
ub      = [1.1, 1.1, 2.1];              % upper bound (requires user guess)
[p, fval, exitflag, output] = particleswarm(fun, nvars, lb, ub)

Optimization ended: relative change in the objective value 
over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.

p = 1×3

1.0003 1.0003 1.9972

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fval = 2.1429e-04

exitflag = 1

output = struct with fields:
      rngstate: [1×1 struct]
    iterations: 80
     funccount: 2430
       message: 'Optimization ended: relative change in the objective value ↵over the last OPTIONS.MaxStallIterations iterations is less than OPTIONS.FunctionTolerance.'
    hybridflag: []

% comparison
[t, x]  = ode45(@(t, x) odefcn(t, x, p), tin, xout(1));
plot(t, x), hold on
plot(tin, xout, '.'), hold off
grid on
xlabel('Time, t')
ylabel('Amplitude, x(t)')
title('Comparison')
legend('Identified system', 'True solution (data)')

% cost function, J, (to minimize the error between x and xout)
function J = costfcn(p, tin, xout)
    tspan  = tin;
    x0     = 0;
    [t, x] = ode45(@(t, x) odefcn(t, x, p), tspan, x0);
    J      = norm(x - xout);
end

% nonlinear system, dx = f(x(t)), only parameters are unknown, but the structure is known
function dx = odefcn(t, x, p)
    % dx    = 1 - x^2;              % known system
    dx      = p(1) - p(2)*x^p(3);   % unknown system
end

Answer 2

[Sam Chak]

Hi @Sirisha

You appear to be an expert in PSO. I used your standalone PSO code, with minor tweaks, from another thread to identify the parameters of the same nonlinear dynamic system. The results are impressive as well.

%% ==========================================
% DATASET
%% =========================================
g           = linspace(0, 10, 51)';
x           = tanh(g);

%% ==========================================
% PSO PARAMETERS
%% ==========================================
dim         = 3;
nParticles  = 100;
maxIter     = 500;
lb          = [0.9, 0.9, 1.9];      % lower bound (requires user guess)
ub          = [1.1, 1.1, 2.1];      % upper bound (requires user guess)

%% ==========================================
% INITIALIZATION
%% ==========================================
particle = struct;
for i = 1:nParticles
    particle(i).position        = lb + rand(1,dim).*(ub-lb);
    particle(i).velocity        = zeros(1,dim);
    particle(i).cost            = fitness_functionh(particle(i).position,x,g);
    particle(i).best.position   = particle(i).position;
    particle(i).best.cost       = particle(i).cost;
end

costs       = [particle.cost];
[~, idx]    = min(costs);
global_best = particle(idx).best;

%% ==========================================
% PSO Constants
%% ==========================================
w               = 0.6;
c1              = 1.5;
c2              = 1.5;
BestCost        = zeros(maxIter, 1);
theta_history   = zeros(maxIter, dim);
mse_history     = zeros(maxIter, 1);

%% ==========================================
% PSO MAIN LOOP
%% ==========================================
for iter = 1:maxIter
    
    for i = 1:nParticles
        
        particle(i).velocity = w*particle(i).velocity + c1*rand*(particle(i).best.position-particle(i).position) + c2*rand*(global_best.position-particle(i).position);
        particle(i).position = particle(i).position + particle(i).velocity;
        particle(i).position = max(particle(i).position,lb);
        particle(i).position = min(particle(i).position,ub);
        particle(i).cost     = fitness_functionh(particle(i).position, x, g);
        
        if particle(i).cost < particle(i).best.cost
            particle(i).best.position = particle(i).position;
            particle(i).best.cost = particle(i).cost;
        end
        
        if particle(i).best.cost < global_best.cost
            global_best = particle(i).best;
        end
        
    end
    
    theta_history(iter,:)   = global_best.position;
    mse_history(iter)       = global_best.cost;
    BestCost(iter)          = global_best.cost;
    fprintf('Iteration %d   Best Cost = %e\n', iter, global_best.cost);
end

Iteration 1   Best Cost = 1.726705e-07
Iteration 2   Best Cost = 1.710162e-07
Iteration 3   Best Cost = 8.333953e-08
Iteration 4   Best Cost = 7.501296e-08
Iteration 5   Best Cost = 7.501296e-08
Iteration 6   Best Cost = 5.437119e-08
Iteration 7   Best Cost = 4.906860e-08
Iteration 8   Best Cost = 4.906860e-08
Iteration 9   Best Cost = 4.440833e-08
Iteration 10   Best Cost = 4.440833e-08
Iteration 11   Best Cost = 3.798649e-08
Iteration 12   Best Cost = 3.786442e-08
Iteration 13   Best Cost = 3.327946e-08
Iteration 14   Best Cost = 2.740116e-08
Iteration 15   Best Cost = 1.256102e-08
Iteration 16   Best Cost = 3.227137e-09
Iteration 17   Best Cost = 1.424000e-09
Iteration 18   Best Cost = 1.424000e-09
Iteration 19   Best Cost = 1.172687e-09
Iteration 20   Best Cost = 1.134270e-09
Iteration 21   Best Cost = 1.134270e-09
Iteration 22   Best Cost = 9.610087e-10
Iteration 23   Best Cost = 9.610087e-10
Iteration 24   Best Cost = 9.610087e-10
Iteration 25   Best Cost = 9.248028e-10
Iteration 26   Best Cost = 9.235203e-10
Iteration 27   Best Cost = 9.235203e-10
Iteration 28   Best Cost = 9.228408e-10
Iteration 29   Best Cost = 9.206767e-10
Iteration 30   Best Cost = 9.073599e-10
Iteration 31   Best Cost = 9.014428e-10
Iteration 32   Best Cost = 8.990494e-10
Iteration 33   Best Cost = 8.974113e-10
Iteration 34   Best Cost = 8.974113e-10
Iteration 35   Best Cost = 8.973885e-10
Iteration 36   Best Cost = 8.973746e-10
Iteration 37   Best Cost = 8.973746e-10
Iteration 38   Best Cost = 8.973655e-10
Iteration 39   Best Cost = 8.973655e-10
Iteration 40   Best Cost = 8.973655e-10
Iteration 41   Best Cost = 8.973655e-10
Iteration 42   Best Cost = 8.973636e-10
Iteration 43   Best Cost = 8.973562e-10
Iteration 44   Best Cost = 8.973511e-10
Iteration 45   Best Cost = 8.973511e-10
Iteration 46   Best Cost = 8.973511e-10
Iteration 47   Best Cost = 8.973454e-10
Iteration 48   Best Cost = 8.973452e-10
Iteration 49   Best Cost = 8.973452e-10
Iteration 50   Best Cost = 8.973445e-10
Iteration 51   Best Cost = 8.973445e-10
Iteration 52   Best Cost = 8.973445e-10
Iteration 53   Best Cost = 8.973441e-10
Iteration 54   Best Cost = 8.973441e-10
Iteration 55   Best Cost = 8.973441e-10
Iteration 56   Best Cost = 8.973441e-10
Iteration 57   Best Cost = 8.973441e-10
Iteration 58   Best Cost = 8.973441e-10
Iteration 59   Best Cost = 8.973441e-10
Iteration 60   Best Cost = 8.973440e-10
Iteration 61   Best Cost = 8.973440e-10
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Iteration 70   Best Cost = 8.973440e-10
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Iteration 75   Best Cost = 8.973440e-10
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Iteration 80   Best Cost = 8.973440e-10
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Iteration 381   Best Cost = 8.973440e-10
Iteration 382   Best Cost = 8.973440e-10
Iteration 383   Best Cost = 8.973440e-10
Iteration 384   Best Cost = 8.973440e-10
Iteration 385   Best Cost = 8.973440e-10
Iteration 386   Best Cost = 8.973440e-10
Iteration 387   Best Cost = 8.973440e-10
Iteration 388   Best Cost = 8.973440e-10
Iteration 389   Best Cost = 8.973440e-10
Iteration 390   Best Cost = 8.973440e-10
Iteration 391   Best Cost = 8.973440e-10
Iteration 392   Best Cost = 8.973440e-10
Iteration 393   Best Cost = 8.973440e-10
Iteration 394   Best Cost = 8.973440e-10
Iteration 395   Best Cost = 8.973440e-10
Iteration 396   Best Cost = 8.973440e-10
Iteration 397   Best Cost = 8.973440e-10
Iteration 398   Best Cost = 8.973440e-10
Iteration 399   Best Cost = 8.973440e-10
Iteration 400   Best Cost = 8.973440e-10
Iteration 401   Best Cost = 8.973440e-10
Iteration 402   Best Cost = 8.973440e-10
Iteration 403   Best Cost = 8.973440e-10
Iteration 404   Best Cost = 8.973440e-10
Iteration 405   Best Cost = 8.973440e-10
Iteration 406   Best Cost = 8.973440e-10
Iteration 407   Best Cost = 8.973440e-10
Iteration 408   Best Cost = 8.973440e-10
Iteration 409   Best Cost = 8.973440e-10
Iteration 410   Best Cost = 8.973440e-10
Iteration 411   Best Cost = 8.973440e-10
Iteration 412   Best Cost = 8.973440e-10
Iteration 413   Best Cost = 8.973440e-10
Iteration 414   Best Cost = 8.973440e-10
Iteration 415   Best Cost = 8.973440e-10
Iteration 416   Best Cost = 8.973440e-10
Iteration 417   Best Cost = 8.973440e-10
Iteration 418   Best Cost = 8.973440e-10
Iteration 419   Best Cost = 8.973440e-10
Iteration 420   Best Cost = 8.973440e-10
Iteration 421   Best Cost = 8.973440e-10
Iteration 422   Best Cost = 8.973440e-10
Iteration 423   Best Cost = 8.973440e-10
Iteration 424   Best Cost = 8.973440e-10
Iteration 425   Best Cost = 8.973440e-10
Iteration 426   Best Cost = 8.973440e-10
Iteration 427   Best Cost = 8.973440e-10
Iteration 428   Best Cost = 8.973440e-10
Iteration 429   Best Cost = 8.973440e-10
Iteration 430   Best Cost = 8.973440e-10
Iteration 431   Best Cost = 8.973440e-10
Iteration 432   Best Cost = 8.973440e-10
Iteration 433   Best Cost = 8.973440e-10
Iteration 434   Best Cost = 8.973440e-10
Iteration 435   Best Cost = 8.973440e-10
Iteration 436   Best Cost = 8.973440e-10
Iteration 437   Best Cost = 8.973440e-10
Iteration 438   Best Cost = 8.973440e-10
Iteration 439   Best Cost = 8.973440e-10
Iteration 440   Best Cost = 8.973440e-10
Iteration 441   Best Cost = 8.973440e-10
Iteration 442   Best Cost = 8.973440e-10
Iteration 443   Best Cost = 8.973440e-10
Iteration 444   Best Cost = 8.973440e-10
Iteration 445   Best Cost = 8.973440e-10
Iteration 446   Best Cost = 8.973440e-10
Iteration 447   Best Cost = 8.973440e-10
Iteration 448   Best Cost = 8.973440e-10
Iteration 449   Best Cost = 8.973440e-10
Iteration 450   Best Cost = 8.973440e-10
Iteration 451   Best Cost = 8.973440e-10
Iteration 452   Best Cost = 8.973440e-10
Iteration 453   Best Cost = 8.973440e-10
Iteration 454   Best Cost = 8.973440e-10
Iteration 455   Best Cost = 8.973440e-10
Iteration 456   Best Cost = 8.973440e-10
Iteration 457   Best Cost = 8.973440e-10
Iteration 458   Best Cost = 8.973440e-10
Iteration 459   Best Cost = 8.973440e-10
Iteration 460   Best Cost = 8.973440e-10
Iteration 461   Best Cost = 8.973440e-10
Iteration 462   Best Cost = 8.973440e-10
Iteration 463   Best Cost = 8.973440e-10
Iteration 464   Best Cost = 8.973440e-10
Iteration 465   Best Cost = 8.973440e-10
Iteration 466   Best Cost = 8.973440e-10
Iteration 467   Best Cost = 8.973440e-10
Iteration 468   Best Cost = 8.973440e-10
Iteration 469   Best Cost = 8.973440e-10
Iteration 470   Best Cost = 8.973440e-10
Iteration 471   Best Cost = 8.973440e-10
Iteration 472   Best Cost = 8.973440e-10
Iteration 473   Best Cost = 8.973440e-10
Iteration 474   Best Cost = 8.973440e-10
Iteration 475   Best Cost = 8.973440e-10
Iteration 476   Best Cost = 8.973440e-10
Iteration 477   Best Cost = 8.973440e-10
Iteration 478   Best Cost = 8.973440e-10
Iteration 479   Best Cost = 8.973440e-10
Iteration 480   Best Cost = 8.973440e-10
Iteration 481   Best Cost = 8.973440e-10
Iteration 482   Best Cost = 8.973440e-10
Iteration 483   Best Cost = 8.973440e-10
Iteration 484   Best Cost = 8.973440e-10
Iteration 485   Best Cost = 8.973440e-10
Iteration 486   Best Cost = 8.973440e-10
Iteration 487   Best Cost = 8.973440e-10
Iteration 488   Best Cost = 8.973440e-10
Iteration 489   Best Cost = 8.973440e-10
Iteration 490   Best Cost = 8.973440e-10
Iteration 491   Best Cost = 8.973440e-10
Iteration 492   Best Cost = 8.973440e-10
Iteration 493   Best Cost = 8.973440e-10
Iteration 494   Best Cost = 8.973440e-10
Iteration 495   Best Cost = 8.973440e-10
Iteration 496   Best Cost = 8.973440e-10
Iteration 497   Best Cost = 8.973440e-10
Iteration 498   Best Cost = 8.973440e-10
Iteration 499   Best Cost = 8.973440e-10
Iteration 500   Best Cost = 8.973440e-10

%% ==========================================
% Estimated Parameters
%% ==========================================
theta = global_best.position

theta = 1×3

1.0003 1.0003 1.9970

<mw-icon class=""></mw-icon>

<mw-icon class=""></mw-icon>

%% ==========================================
% Comparison
%% ==========================================
[tsim, xsim]  = ode45(@(t, x) odefcn(t, x, theta), g, x(1));
plot(tsim, xsim), hold on
plot(g, x, '.'), hold off
grid on
xlabel('Time, t')
ylabel('Amplitude, x(t)')
title('Comparison')
legend('Identified system', 'True solution (data)')

%% ==========================================
% Cost function called "Fitness"
%% ==========================================
function j = fitness_functionh(xext, x, g)

    %% ==========================================
    % Column vectors
    %% ==========================================
    x = x(:);
    g = g(:);
    
    %% ==========================================
    % Parameters
    %% ==========================================
    p(1) = xext(1);
    p(2) = xext(2);
    p(3) = xext(3);
       
    %% ==========================================
    % Free Run Simulation
    %% ==========================================
    tspan       = g;
    x0          = x(1);
    [t, xhat]   = ode45(@(t, x) odefcn(t, x, p), tspan, x0);
    
    %% ==========================================
    % Mean Square Error
    %% ==========================================
    j = mean((x - xhat).^2);
    if isnan(j) || isinf(j)
        j = 1e20;
    end
    
end

%% ==========================================
% Nonlinear system
%% ==========================================
function dx = odefcn(t, x, p)
    % dx    = 1 - x^2;              % known system
    dx      = p(1) - p(2)*x^p(3);   % unknown system
end