Transfer function where some of the coefficients are dependent on the output Y(S)

I have a problem to solve where I have a transfer function where the coefficients are a function of the system's position. This causes algebraic loops that I can't seem to put differently.

The system is an elevator where the weight and cable stiffness changes as the lift moves upwards in the following ways:

function M = fcn(y)

M = 1010+y;
end

And the stiffness is as follows:

function K = fcn(Mass, y)

K = (0.2*Mass)/(y-(y/1.0016));
end

I can neglect the mass function but the stiffness is important.

Here is the transfer function:

Y(S) = (U(S)*(k))/(M(Y)*s^3+b*s^2+k(M,Y)*s)

I have tried to implement this with the following block diagram:

Which returns an algebraic loop error.

I have verified this block diagram with constant coefficients so the only problem is the varying coefficients.

Is there any way to prevent this error from occuring without first estimating the height as a function of time and chaging the functions to be functions of time instead of "y"?

[Sam Chak]

Hi @Wynand

Can you show the mathematical model of the Elevator (preferably in Ordinary Differential Equation form)? It helps to cross-check whether the system is correctly modeled using the Simulink blocks. The elevator should be well-damped, or else, it will oscillate 'violently'. Please check if your system is consistent with my proposed ODE:

[Wynand]

Hi @Sam Chak. You actually helped me a while ago with a previous step in the problem

https://www.mathworks.com/matlabcentral/answers/2144394-using-ss2tf-when-your-matrices-are-variables-without-values-yet?s_tid=srchtitle

The ODE was as follows:

I just changed the output to be position instead of velocity.

The mass is just a function of components of the system and the stiffness was derived from various information given about the cable and has been verified to be correct.

Thank you for taking the time to assist

[Sam Chak]

Hi @Wynand

You need to be careful when designing the nonlinear function for the stiffness k because your function causes the cable to become infinitely stiff (resulting in division by zero) when the elevator reaches the ground floor (). What is the reasonable value for the cable stiffness when the elevator is at the ground floor?

Additionally, what is a reasonable value for the cable stiffness when the elevator is at the top floor?

With these two values, you can effectively interpolate the stiffness between the highest and lowest stiffness using either a linear or nonlinear function. You may create a sketch, and I will suggest an appropriate function for performing the interpolation.

M   = 100;
fplot(@(y) (0.2*M)./(y - y/1.0016)), grid on, axis([-2 10, 0 1e5])
xlabel('height, y'), ylabel('stiffness, k')
title('Your nonlinear stiffness, k(y)')

[Wynand]

Yes I have noticed this issue. The height (y) is taken from the reference point at the top of the shaft and increases as the lift lowers. I assumed when the cables is essentially as short as possible it will be incredibly stiff. I hoped this is a reasonable assumption to make.

If I could maybe point the discussion back towards the original question. Even if I can find a much nicer, linear eqaution for the stiffness. I still need to be able to express a closed loop feedback system where the output y, is a part of the function for the coefficients. I was hoping to use the variable coefficient transfer function component but it warns precisely against my use case.

[Sam Chak]

Hi @Wynand

It's up to you to decide on the stiffness function, as you are 100% fully aware that division by zero will absolutely mess up the simulation.

Regarding your Simulink model, since the elevator's motion is described by a single second-order ODE, you should only use one double integrator block. I recommend using the Second-Order Integrator:

[Mathieu NOE]

hi

even at the top position, the cable length is never zero , there is usually a minimum length of at least 1.5 or 2 m between cabin roof and the machinery on the buiding roof.

even the cabin structure has some flex in it.

[Wynand]

Good morning @Sam Chak

I have revised the model and come up with these constants.

M = 1018

B = 100

K = 11041666.6666667

I know this is not on the topic of the question anymore but I am having massive difficulties in designing a controller for the system.

This is what I have tried so far

overshoot = 10;
Zeta = (-log(overshoot/100))/sqrt(pi^2+(log(overshoot/100)^2))

Zeta = 0.5912

Omega_n = 5/(4*Zeta)

Omega_n = 2.1145

% Kp = (2*Zeta*Omega_n-1)/Kprocess
% Ki = (Omega_n^2)/(Kprocess)

R = 0.8

R = 0.8000

Td = 0.1

Td = 0.1000

Kp = 0.9/(R*Td)

Kp = 11.2500

Ki = 0.27/(R*Td^2)

Ki = 33.7500

However this produces massive oscillations and really unbelievable amplitudes.

You have helped me on quite a few of my posts. Can I ask for your assistance in this?

This was for a PI controller but the type of controller is not set in stone

[Sam Chak]

Hi @Wynand

Why not consider posting a new control problem complete with the following components?

• The elevator model (state-space or transfer function),
• Performance requirements (percent overshoot and settling time),
• The MATLAB code & simulation that produces significant oscillations.

[Wynand]

Thank you I will do so

[Wynand]

I am not sure why but I seem to be unable to type anything in the questions description when I try and post new questions

Answer 1

[Kothuri]

Hi Wynand,

I understand that you are trying to implement the elevator system where the weight and cable stiffness changes as the lift moves upwards, and the coefficients of the transfer function are a function of the system's position which is causing algebraic loops.

Here are some of the steps you can try to avoid algebraic loops:

• By introducing a “Unit Delay” block or a “Memory” block in the feedback loop for stiffness helps to break the algebraic loop by providing the previous value of the signal. This avoids the direct dependency on the current value of “y”.

You can refer the below links for more info on “Unit Delay” and “Memory” blocks

https://www.mathworks.com/help/simulink/slref/unitdelay.html

https://www.mathworks.com/help/simulink/slref/memory.html

• Use an iterative solver that can handle algebraic loops. Some solvers in Simulink are better suited for systems with algebraic loops. You can find these settings under the Solver Configuration block.
• Use the delayed or stored position values in your transfer function calculations to avoid direct algebraic dependency. This approach uses a persistent variable to store the previous value of y, thus breaking the direct dependency.

% Example of introducing a delay in stiffness calculation
function K = fcn_delayed(Mass, y)
    persistent previous_y;
    if isempty(previous_y)
        previous_y = y;
    end
    K = (0.2 * Mass) / (previous_y - (previous_y / 1.0016));
    previous_y = y;  % Update for next iteration
end

You can refer the below documentation on “Varying Transfer Function” for better understanding https://www.mathworks.com/help/control/ref/varyingtransferfunction.html

[Wynand]

HI, @Kothuri.

Thank you so much for your answer. This was what I was looking for. I ended up doing 'snapshots' of the system at different points and just extrapolating from there but this would have been a great implementation as well.