---- Forward QSS1 ----
(t, y0, ty, r0, tr, d0, td, e)
---- Forward d(t + tau(t)) ----
Forward d(t + tau(t)) = d0
Forward d(t + tau(t)) = d0
expression = d0 == y0
---- coefficients:----
do = y0
---- Forward QSS2 ----
(t, y0, y1, ty, r0, r1, tr, d0, d1, td, e)
---- Forward d(t + tau(t)) ----
Forward d(t + tau(t)) = (r1*(t - tr) + r0 + t - td)*d1 + d0
Forward d(t + tau(t)) = -d1*r1*tr + d1*r0 + (d1*r1 + d1)*t - d1*td + d0
expression = -d1*r1*tr + d1*r0 + (d1*r1 + d1)*t - d1*td + d0 == (t - ty)*y1 + y0
---- coefficients:----
do = d1*r1*tr - d1*r0 + d1*td - ty*y1 + y0
d1 = y1/(r1 + 1)
---- When new tau arrives:----
do = (tr - ty)*y1 + y0 (same as the definition of y(tr)=(tr - ty)*y1 + y0)
---- When new y arrives:----
do = y0 (same as the definition of y(ty)=y0)
---- Forward QSS3 ----
(t, T_y, T_r, T_d, y_0, y_1, y_2, r_0, r_1, r_2, d_0, d_1, d_2)
---- Forward d(t + tau(t)) ----
Forward d(t + tau(t)) = ((T_r - t)^2*r_2 - (T_r - t)*r_1 - T_d + r_0 + t)^2*d_2 + ((T_r - t)^2*r_2 - (T_r - t)*r_1 - T_d + r_0 + t)*d_1 + d_0
Forward d(t + tau(t)) simplified = T_r^4*d_2*r_2^2 + d_2*r_2^2*t^4 + T_r^2*d_2*r_1^2 - 2*(2*T_r*d_2*r_2^2 - (d_2*r_1 + d_2)*r_2)*t^3 + T_d^2*d_2 + d_2*r_0^2 + (6*T_r^2*d_2*r_2^2 + d_2*r_1^2 + 2*d_2*r_1 - (6*T_r*d_2*r_1 + 2*(T_d + 2*T_r)*d_2 - 2*d_2*r_0 - d_1)*r_2 + d_2)*t^2 - T_d*d_1 - (2*T_d*d_2 - d_1)*r_0 + (2*T_d*T_r*d_2 - 2*T_r*d_2*r_0 - T_r*d_1)*r_1 - (2*T_r^3*d_2*r_1 + 2*T_d*T_r^2*d_2 - 2*T_r^2*d_2*r_0 - T_r^2*d_1)*r_2 - (4*T_r^3*d_2*r_2^2 + 2*T_r*d_2*r_1^2 + 2*T_d*d_2 - 2*d_2*r_0 + (2*(T_d + T_r)*d_2 - 2*d_2*r_0 - d_1)*r_1 - 2*(3*T_r^2*d_2*r_1 - 2*T_r*d_2*r_0 - T_r*d_1 + (2*T_d*T_r + T_r^2)*d_2)*r_2 - d_1)*t + d_0
---- Removing 3 and 4 order ----
d(t + tau(t)) = T_r^4*d_2*r_2^2 + T_r^2*d_2*r_1^2 + T_d^2*d_2 + d_2*r_0^2 + (6*T_r^2*d_2*r_2^2 + d_2*r_1^2 + 2*d_2*r_1 - (6*T_r*d_2*r_1 + 2*(T_d + 2*T_r)*d_2 - 2*d_2*r_0 - d_1)*r_2 + d_2)*t^2 - T_d*d_1 - (2*T_d*d_2 - d_1)*r_0 + (2*T_d*T_r*d_2 - 2*T_r*d_2*r_0 - T_r*d_1)*r_1 - (2*T_r^3*d_2*r_1 + 2*T_d*T_r^2*d_2 - 2*T_r^2*d_2*r_0 - T_r^2*d_1)*r_2 - (4*T_r^3*d_2*r_2^2 + 2*T_r*d_2*r_1^2 + 2*T_d*d_2 - 2*d_2*r_0 + (2*(T_d + T_r)*d_2 - 2*d_2*r_0 - d_1)*r_1 - 2*(3*T_r^2*d_2*r_1 - 2*T_r*d_2*r_0 - T_r*d_1 + (2*T_d*T_r + T_r^2)*d_2)*r_2 - d_1)*t + d_0
y(t) = (T_y - t)^2*y_2 - (T_y - t)*y_1 + y_0
expression = T_y^2*y_2 - T_y*y_1 + y_0 == T_r^4*d_2*r_2^2 + T_r^2*d_2*r_1^2 + T_d^2*d_2 + d_2*r_0^2 - T_d*d_1 - (2*T_d*d_2 - d_1)*r_0 + (2*T_d*T_r*d_2 - 2*T_r*d_2*r_0 - T_r*d_1)*r_1 - (2*T_r^3*d_2*r_1 + 2*T_d*T_r^2*d_2 - 2*T_r^2*d_2*r_0 - T_r^2*d_1)*r_2 + d_0
do = -T_r^4*d_2*r_2^2 - T_r^2*d_2*r_1^2 - T_d^2*d_2 - d_2*r_0^2 + T_y^2*y_2 + T_d*d_1 + (2*T_d*d_2 - d_1)*r_0 - (2*T_d*T_r*d_2 - 2*T_r*d_2*r_0 - T_r*d_1)*r_1 + (2*T_r^3*d_2*r_1 + 2*T_d*T_r^2*d_2 - 2*T_r^2*d_2*r_0 - T_r^2*d_1)*r_2 - T_y*y_1 + y_0
d1 = -(4*T_r^3*d_2*r_2^2 + 2*T_r*d_2*r_1^2 + 2*T_d*d_2 - 2*d_2*r_0 + 2*((T_d + T_r)*d_2 - d_2*r_0)*r_1 - 2*(3*T_r^2*d_2*r_1 - 2*T_r*d_2*r_0 + (2*T_d*T_r + T_r^2)*d_2)*r_2 + 2*T_y*y_2 - y_1)/(2*T_r*r_2 - r_1 - 1)
d2 = -(r_2*y_1 - (2*(T_r + T_y)*r_2 - r_1 - 1)*y_2)/(8*T_r^3*r_2^3 - r_1^3 - 12*(T_r^2*r_1 + T_r^2)*r_2^2 - 3*r_1^2 + 6*(T_r*r_1^2 + 2*T_r*r_1 + T_r)*r_2 - 3*r_1 - 1)
---- When new tau arrives:----
do = (16*T_r^7*d_2*r_2^5 - 2*T_r^2*d_2*r_1^5 + 2*(T_r*d_2*r_0 - (T_d*T_r + 4*T_r^2)*d_2)*r_1^4 - 8*(7*T_r^6*d_2*r_1 - 2*T_r^5*d_2*r_0 + (2*T_d*T_r^5 + 5*T_r^6)*d_2)*r_2^4 + 4*(2*T_r*d_2*r_0 - (2*T_d*T_r + 3*T_r^2)*d_2)*r_1^3 + 4*(19*T_r^5*d_2*r_1^2 - 10*T_r^4*d_2*r_0 + (10*T_d*T_r^4 + 9*T_r^5)*d_2 - 2*(5*T_r^4*d_2*r_0 - (5*T_d*T_r^4 + 14*T_r^5)*d_2)*r_1)*r_2^3 - 2*T_d*T_r*d_2 + 2*T_r*d_2*r_0 + 4*(3*T_r*d_2*r_0 - (3*T_d*T_r + 2*T_r^2)*d_2)*r_1^2 - 2*(25*T_r^4*d_2*r_1^3 - 18*T_r^3*d_2*r_0 - 3*(6*T_r^3*d_2*r_0 - (6*T_d*T_r^3 + 19*T_r^4)*d_2)*r_1^2 + (18*T_d*T_r^3 + 7*T_r^4)*d_2 - 3*(12*T_r^3*d_2*r_0 - (12*T_d*T_r^3 + 13*T_r^4)*d_2)*r_1)*r_2^2 + 2*(4*T_r*d_2*r_0 - (4*T_d*T_r + T_r^2)*d_2)*r_1 + 2*(8*T_r^3*d_2*r_1^4 - 7*T_r^2*d_2*r_0 - (7*T_r^2*d_2*r_0 - (7*T_d*T_r^2 + 25*T_r^3)*d_2)*r_1^3 - 3*(7*T_r^2*d_2*r_0 - (7*T_d*T_r^2 + 9*T_r^3)*d_2)*r_1^2 + (7*T_d*T_r^2 + T_r^3)*d_2 - (21*T_r^2*d_2*r_0 - (21*T_d*T_r^2 + 11*T_r^3)*d_2)*r_1)*r_2 + (8*T_r^3*r_2^3 - r_1^3 - 12*(T_r^2*r_1 + T_r^2)*r_2^2 - 3*r_1^2 + 6*(T_r*r_1^2 + 2*T_r*r_1 + T_r)*r_2 - 3*r_1 - 1)*y_0 + ((T_r + T_y)*r_1^3 - (3*T_r^4 + 8*T_r^3*T_y)*r_2^3 + 3*(T_r + T_y)*r_1^2 + 6*(T_r^3 + 2*T_r^2*T_y + (T_r^3 + 2*T_r^2*T_y)*r_1)*r_2^2 + 3*(T_r + T_y)*r_1 - 2*((2*T_r^2 + 3*T_r*T_y)*r_1^2 + 2*T_r^2 + 3*T_r*T_y + 2*(2*T_r^2 + 3*T_r*T_y)*r_1)*r_2 + T_r + T_y)*y_1 + ((T_r^2 - 2*T_r*T_y - T_y^2)*r_1^3 - 2*(T_r^5 - 3*T_r^4*T_y - 4*T_r^3*T_y^2)*r_2^3 + 3*(T_r^2 - 2*T_r*T_y - T_y^2)*r_1^2 + (5*T_r^4 - 12*T_r^3*T_y - 12*T_r^2*T_y^2 + (5*T_r^4 - 12*T_r^3*T_y - 12*T_r^2*T_y^2)*r_1)*r_2^2 + T_r^2 - 2*T_r*T_y - T_y^2 + 3*(T_r^2 - 2*T_r*T_y - T_y^2)*r_1 - 2*(2*T_r^3 - 4*T_r^2*T_y - 3*T_r*T_y^2 + (2*T_r^3 - 4*T_r^2*T_y - 3*T_r*T_y^2)*r_1^2 + 2*(2*T_r^3 - 4*T_r^2*T_y - 3*T_r*T_y^2)*r_1)*r_2)*y_2)/(8*T_r^3*r_2^3 - r_1^3 - 12*(T_r^2*r_1 + T_r^2)*r_2^2 - 3*r_1^2 + 6*(T_r*r_1^2 + 2*T_r*r_1 + T_r)*r_2 - 3*r_1 - 1) )
y(T_r) = (T_r - T_y)*y_1 + (T_r^2 - 2*T_r*T_y + T_y^2)*y_2 + y_0