1) The ZN works by finding the proportional ultimate gain coefficient Kp which produces a stable oscillation, then using the oscillation characteristics to set the P/I coefficients.
2) You are unable to find Kp.
3) Therefor you are unable to use the ZN method.
The sharp drop is because the error is high. The proportional term is going to go way down, and since you have no integral built up, of course the output will drop.
I'm not exactly keen on the "lookup table" approach, I'd rather see the loop better tuned, but if you are okay with the lookup table, just force that value to the integral term, suppressing P and D until the time expires, then enabling the full PID with the integral preset.
With regard to the comment about increasing the proportional gain, I'd advise against that as increasing the proportional gain would probably drive that system into oscillation; and increasing it would also cause a much sharper drop when switched from "Startup" to "Run" mode.
Ultimately, you probably do want to pre-load the integral term with your pre-calculated output value and just let it run. Tuning PID's by just playing with the gains (or gain/reset/rate, depending where your come from) usually doesn't end up with a stable system.
There are many methods of tuning PID loops, but if you must "Wing" it, typically reduce the I and D terms to zero, run P up until the system oscillates, and back P off by about half. That should give you a steady state error. Next, increase I slowly, until the error is eliminated, and the system doesn't hunt. D should normally only be used slightly in most loops, but might be necessary in very dynamic controls.
Best Answer
The figure below shows the steps in order to find the \$K_{cr}\$(or \$K_u\$) and \$P_{cr}\$ (or \$P_u)\$, by changing the proportional gain only (with \$T_d=0\$ and \$T_i=\infty\$) - an example for temperature control:
The time unit to be used should be consistent with its response curve. The relationship with the sample period \$\Delta t\$ can be obtained, after the discretization of the PID controller. Using the standard form, in contrast with other implementations (for example, when the derivative term is taken from output):
$$u(t)=K_pe(t)+\frac{K_p}{T_i}\int_0^t{e(\tau)d\tau+K_pT_d\frac{de(t)}{dt}}$$
Taking the derivative of \$u(t)\$: $$u'(t)=K_pe'(t)+\frac{K_p}{T_i}e(t)+K_pT_de''(t)$$ A possible approach: To approximate the first and second derivatives using finite differences (eg backward), where \$k\$ is the sample id: $$x'(t)\approx\frac{x_k-x_{k-1}}{\Delta t}$$ $$x''(t)\approx\frac{x_k-2x_{k-1}+x_{k-2}}{\Delta t^2}$$ So, the discrete PID controller takes the form (velocity algorithm): $$u_k=u_{k-1}+K_p[(1+\frac{\Delta t}{T_i}+\frac{T_d}{\Delta t})e_k+(-1-\frac{2T_d}{\Delta t})e_{k-1}+\frac{T_d}{\Delta t}e_{k-2}]$$
Alternative definitions can include \$K_i=\frac{K_p}{T_i}\$ and \$K_d=K_pT_d\$. Also, the derivative term can be modified in order to reduce issues with high frequency noise - eg a low pass filter. Other discretization methods exist as well, such as Tustin, ZOH.
FURTHER EXPLANATION:
Set \$K_p\$ to some low value (with \$T_i=\infty\$ and \$T_d=0\$ at this stage). So, the above equation is simplified to: $$u_k=u_{k-1}+K_p(e_k-e_{k-1})$$
Implement the previous equation (a P controller) in your digital system along with that suitable \$\Delta t\$, testing \$K_p\$ to see if it causes continuum oscillation (marginally stable). If the oscillations decay, keep increasing \$K_p\$. If the oscillations increase in amplitude (unstable system), reduce \$K_p\$. Do this until the system is marginally stable. When you arrive at this point, you have found \$K_{cr}=K_p\$ and \$P_{cr}=\$oscillation period (see the figure above).
Using the table you have provided (also above), determine the \$K_p\$, \$T_i\$ and \$T_d\$ values from the \$K_{cr}\$ and \$P_{cr}\$ ones.
Implement the complete PID controller: $$u_k=u_{k-1}+K_p[(1+\frac{\Delta t}{T_i}+\frac{T_d}{\Delta t})e_k+(-1-\frac{2T_d}{\Delta t})e_{k-1}+\frac{T_d}{\Delta t}e_{k-2}]$$