Resetting only a section of my state during event driven ode45 simulation

Hello,

I'm trying to write a code that simulates a closed loop event driven LTI system (with a poleplacement controller u(t) = -K*x(t)). I'm using the following event formulation:

where the ξ stands for the actual state and the ϵ stands for the error between the current state and the state measured during an event trigger. So:

So I want to trigger an event the moment the value becomes 0.

Now, the problem I'm having is that I want the ϵ value (so the error) to reset to 0 upon an event but the ξ part of the state should not reset. I'm having trouble coming up with a way to exactly program this. I'm currently just simulating the system dynamics for the augmented system, so for . For now I have the following in matlab:

a = 4;
b = 6;
c = 9;
poles = [-1+2i; -1-2i];

A = [0.3+a-b, 0.5-c; 0, 1];
B = [0; 1];
C = eye(2);
D = [0; 0];
K = place(A, B, poles);

A_aug = [A-B*K -B*K;
        -A+B*K B*K];%Defining augmented CL A-matrix for augmented state x_augmented = [x;error]//[xi;epsilon]
tspan = [0,10];
x0 = [1;1;0;0];
Q = eye(2);
P = 2*eye(2);
sig = 0.1;

options = odeset('Events', @(t,x) myEventsFcn(t,x,sig,Q,P,B,K));

[t,y,te,ye,ie] = ode45(@(t,x) LTIaug_fun(t,x,A_aug),tspan,x0,options)

t = 25x1

1.0e+00 * 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002

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y = 25x4

1.0000 1.0000 0 0 0.9999 1.0000 0.0001 -0.0000 0.9999 1.0000 0.0001 -0.0000 0.9998 1.0000 0.0002 -0.0000 0.9998 1.0000 0.0002 -0.0000 0.9995 1.0000 0.0005 -0.0000 0.9993 1.0000 0.0007 -0.0000 0.9990 1.0000 0.0010 -0.0000 0.9988 1.0000 0.0012 -0.0000 0.9975 1.0001 0.0025 -0.0001

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te = 0.0514

ye = 1x4

0.4953 1.0120 0.5047 -0.0120

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ie = 1

Where the LTIaug_fun and myEventsFcn are defined as follows:

function dxdt = LTIaug_fun(~,x,A_aug)
dxdt = A_aug*x;
end

function [value,isterminal,direction] = myEventsFcn(t,x,sig,Q,P,B,K)
Th = [(1-sig)*Q P*B*K;
      (B*K)'*P zeros(2,2)];
Th_e = x'*Th*x; 

value = Th_e; 
isterminal = 1;
direction = 0;
end

So how do I set my ϵ part of my augmented state to 0 after/during an event trigger?

Answer 1

[Torsten]

After the event has been triggered, control is returned to the calling program.

If you want to continue integration with the epsilon part set to 0, you can just call ode45 anew with the initial x0-vector set to x0 = [ye(1:2),0,0].