State and Output Function of a nonlinear dynamics to use is NMPC

Hello,

So far I simulated a dynamic equation based on real data inputs in Simullink. I also estimated my coefficients to getting as close as possible to the real data.

My dynamic equation belongs to an axial-flow pump intended to use for pacients with heart desease is as below:

a,b,c,d, and L are coefficients which I tuned.

and omega(refrence speed), H(head pressure) and Q(flow) are inputs from test data.

My intention is to build a nonlinear MPC for this dynamics. but the problem is I don't know how to write State and Output functions for it.

On this dynamics, I can contorl Omega and H and the output is flow. I want to control the flow.

Thank you for your time in Advance!

Best Regarsd

Answer 1

[Sam Chak]

Hi @seyyed Erfan ghoreyshipour

The axial-flow pump system is governed by this equation:

.

So, the State variable in this equation is Q, and the Output that you want is the flow, which is also Q.

The State function refers to the uncontrolled dynamics of the pump system. That means you don't need to supply the any math equation to the inputs ω and H because I think the NMPC calculate for you without requiring you to understand the behavior of the system at all. For example, here is the code for myStateFunction, where x , u(1) , and u(2) .

function z = myStateFunction(x, u)
    a = 276.73;
    b = -0.2703;
    c = 17.5260;
    d = -1.5806e-06;
    L = -11.7480;
    z = b/L*u(1) + c/L*u(2) + (d/L*u(1)^2 + a/L)*x;
end

If the output is also the state, then the code for myOutputFunction is given by

function y = myOutputFunction(x, u)
    y = x;
end

NMPC-free method:

If you can manipulate the angular speed ω and the pressure head H, then you can probably control the flow Q without using the NMPC. Here is an example:

tspan  = [0 10];
initv  = 1;
[t, Q] = ode45(@odefcn, tspan, initv);
plot(t, Q, 'linewidth', 1.5), grid on, xlabel('t'), ylabel('Q'), ylim([0.5 2.5])

function dQdt = odefcn(t, Q)
    a     = 276.73;
    b     = -0.2703;
    c     = 17.5260;
    d     = -1.5806e-06;
    L     = -11.7480;
    k     = 1;                                          % affects the rate of convergence
    Qref  = 2;                                          % desired flowrate
    omega = - (L/b)*k*(Q - Qref);  % <--- correction    % angular Speed 
    H     = - (L/c)*((d/L*omega^2 + a/L)*Q);            % pressure Head
    dQdt  = b/L*omega + c/L*H + (d/L*omega^2 + a/L)*Q;  % ODE
    endf

[seyyed Erfan ghoreyshipour]

Dear @Sam Chak,

Thank you so much. I tried to replace my test data in omega and H and also I replaced the coefficients with my data as well as the tspan. However my desired flow rate (Q_ref) is not a single number.below you can see

So, should I replace my refrence flow along with othe coefficients and variables or not? if yes, How?

span  = timevals;
initv  = 1;
[t, Q] = ode45(@odefcn, tspan, initv);
plot(t, Q), grid on, xlabel('t'), ylabel('Q'), 
% ylim([0 1])
function dQdt = odefcn(t, Q)
    a     = 276.73;
    b     = -0.2703;
    c     = 17.5260;
    d     = -1.5806e-06;
    L     = -11.7480;
    k     = 1;
    Qref  = 2;                                        % desired flowrate
    v_meas_sim = - k*(Q - Qref);                             % angular Speed
    H     = - (L/c)*((d/L*v_meas_sim^2 + a/L)*Q);            % pressure Head
    dQdt  = b/L*v_meas_sim + c/L*H + (d/L*v_meas_sim^2 + a/L)*Q;  % ODE
end

The result is good. So I can be hopefull to design my controller.

So, this function would be my state function. right?

do you have any idea about my output function?

[Sam Chak]

Hi @seyyed Erfan ghoreyshipour

I have made a minor correction on this line:

omega = - (L/b)*k*(Q - Qref);  % <--- correction    % angular Speed 

If everything works well, then the system becomes (after making all substitutions and eliminations)

and the output function for the flow is given by

[seyyed Erfan ghoreyshipour]

Dear @Sam Chak,

Thank you so much for your response.

state function causes an error that I can not understand how to fix it!

As stated in the mathworks Prediction Model Specification, state function must contain states and inputs. while here I think it does not contain!

when I want to run my simulink containing nlmpc object that I made by your help this error pups up; Execution of script sputnikStFcn as a function is not supported

Do you know How can I fix it?

I am eagerly waiting for your response.

Best Wishes

[Sam Chak]

Hi @seyyed Erfan ghoreyshipour, I updated to include myStateFunction and the myOutputFunction

[seyyed Erfan ghoreyshipour]

Dear @Sam Chak,

Sorry for the delay. I am progressing with jacobian functions.

Thank you for your help. This helped me a lot!

I am currently writing state and output jacobian functions. If you have any idea about that please let me know.

Best Regards

Answer 2

[Robbie]

Thanks for the solution. That works!

@Sam Chak wordle unlimited

[seyyed Erfan ghoreyshipour]

Dear @Robbie,

Can you please tell me how did you implement the model?

I am still strugling with it! :)

Answer 3

[Anna]

Thank you so much. It worked for my issue well. I appreciate that a lot Godskin Apostle

Answer 4

[ann lily]

The article about is very good, thank you for sharing! cuphead

Answer 5

[emily]

Interesting Simulink simulation for your axial-flow pump! For your MPC state-space representation, consider defining states as functions of Q (flow) and its derivatives. Your output function would simply be y = Q, since you're controlling flow. Have you explored using a simpler model, like a linearized version, initially for MPC development? This sounds like navigating challenging terrain, like riding a virtual Snow Rider !