Electronic – Can the PID controller have negative p value and positive i value for a PID logic

It's not a complete answer but I hope that it could be of some help.

You could rewrite the first system as

$$ \begin{cases} P(n) = K_P E(n) \\ I(n) = I(n-1) + \frac{K_P}{T_I} E(n) \Delta t \\ D(n) = K_P T_D \frac{E(n) - E(n-1)}{\Delta t} \end{cases} $$

Where \$E(n) = G(n) - target(n)\$ and \$\Delta t\$ is your sampling interval. Note that \$T_D\$ and \$T_I\$ are not defined as gains. \$K_I = \frac{K_P}{T_I}\$ and \$K_D = K_P T_I\$ are respectively the integral gain and the derivative gain.

Now you can rewrite the system as a single function of the error.

$$ PID(n) = P(n) + I(n) + D(n) $$

$$ I(n-1) = PID(n-1) - P(n-1) - D(n-1) \\ = PID(n-1) - K_P E(n-1) - K_P T_D \frac{E(n-1) - E(n-2)}{\Delta t} $$

$$ PID(n) = K_P E(n) + PID(n-1) - K_P E(n-1) - K_P T_D \frac{E(n-1) - E(n-2)}{\Delta t} + \frac{K_P}{T_I} E(n) \Delta t + K_P T_D \frac{E(n) - E(n-1)}{\Delta t} \\ = PID(n-1) + K_P \left(\left(1 + \frac{\Delta t}{T_I} + \frac{T_D}{\Delta t} \right)E(n) - \left(1 + 2\frac{T_D}{\Delta t} \right)E(n-1) + \frac{T_D}{\Delta t} E(n-2) \right) $$

The second one is a bit more complex to rewrite as a single equation but you can do it in a similar way. The result should be

$$ R(n) = K_1 R(n-1) - (\gamma K_0 + K_2) R(n-2) + (1+\gamma) (PID(n) - K_1 PID(n-1) + K_2 PID(n-2)) $$

Now you only need to substitute the equation of the PID in order to obtain the equation of the regulator as function of the error.

Some comments to your question:

(1) A frequency response analysis can describe the systems behaviour only if the system is LINEAR (which is not the case for the thermostat system).

(2) For linear systems, it is best to open the feedback loop (however, for finding the conditions and the correct place for the opening requires some deeper knowledge). Then, you can analyze the loop gain response and apply one of the stability crireria (Nyquist, Bode). This method allows to determine the degree (quality) of stability (phase and/or gain margin)

(3) There is a rough method to evaluate the closed-system response (stable yes/no): The magnitude will exhibit a peaking in the "critical" frequency region where the system could oscillate. In this case, the SLOPE of the phase function at the peaking frequency must be NEGATIVE for a stable system (and will be positive in case of instability).

(4) A circuit involving thermostat devices should be analyzed in the time domain only.