Combine two controllers in ODE45 environment

Hi,

Please I need idea on how to implement the following problem on MATLAB.

My aim is to stabilize a plant using two dynamic controllers. Only one controller is implemented at a time. The controller to be implemented is depended on a logic variable q = 1 or q = 0.

If q = 0, controller 1 is implemented and if q = 1, controller 2 is implemented.

The controller output is given by:

u = q* K2(x) + (1 - q) * K1(x).

K1 and K2 are dynamic controllers. so they are solved using ODE45.

By default controller 1 will be triggered at the start of the program until output variable is less than 5. and the second controller is triggered.

The main challenge:

The last output of controller 1 (The output just before controller 2 is triggered) should be used as the initial condition of controller 2 when controller 2 is triggered.

[Sam Chak]

Hi @Telema Harry

"The last output of controller 1 should be used as the initial condition of controller 2 when controller 2 is triggered."

This implies that your K2 dynamic controller has function like this: . Please clarify.

[Davide Masiello]

It is possible to solve this using event location functions, but you need to share all the details of you problem.

[Telema Harry]

@Sam Chak. Excellent question. We have two options.

1. Both controllers depending on the state. i.e. if the feedback error is large use controller K1. and implement controller K2 if the feeback error is small. With this formulation, K2 does not have to use the last value of K1 in its computation.
2. Your description of the problem is correct. Here, we want to maintain the last K1 control output in the same neighbourhood using K2.

[Telema Harry]

@Davide Masiello thank you for the feedback. I will read about event location functions. I will also create a simplified problem.

[Sam Chak]

Yes @Telema Harry

Thanks for clarifying the matter. I think I understand that the dynamics is something like , where , in order to preserve the continuity of the control action. Actually, I'm thinking, "if this is a state-space, where do I put ?"

[Telema Harry]

@Sam Chak. Sorry, that I did not write the controller dynamics. Please see below a simplified version of the problem. Please let me know if it is not clear.

I am currently trying to code the problem using Euler's numerical technique. In the past, I used if statement in the ODE environment. But it did not work as intended.

Controller 2:

when q = 1,

when q = 0

u = K1

State Space Representation:

Assuming a linear system

Answer 1

[Sam Chak]

Hi @Telema Harry

Since you are unable to provide more info, here is a very simple example, which I'm not sure if this is what you want.

System: , where is a sinusoidal disturbance.

function dxdt = odefcn(t, x)

dxdt = zeros(2, 1);

h = 0.2; % activation if x < |h|

q = (sign(x(1) + h) - sign(x(1) - h))/2; % trigger condition for K2

u1 = sign(x(1)); % K1

u2 = x(1)/h; % K2

u = (1 - q)*u1 + q*u2;

d = 0.75*cos(t*2*pi/10) + 0.5*sin(t*2*pi/5); % disturbance

dxdt(1) = d + x(2);

dxdt(2) = (- u - x(2))/0.01;

end

Solving ODE

tspan = linspace(0, 20, 2001)';
x0 = [1 -1];
[t, x] = ode45(@odefcn, tspan, x0);

Plotting the results

plot(t, x, 'linewidth', 1.5)

[Telema Harry]

@Sam Chak Thank you so much for your help. Though I did not provide enough information (model) of my problem as it is my graduate school research project. You still managed to provide great help. I will say, your solution gave me another insight.

Thank you.