# Closed Loop transfer Function

Hi Guys wanted some help for breaking down a block diagram which can be seen below. I know that typically with feedback systems be it negative or positive you would employ using the following:-

Close Loop = $$C.L.T = \frac{open loop} {closed loop}$$

However the signal $$q_{i-1}$$ is confusing me typically without it, the closed loop would be

$$\frac{q_i}{d_r,i} = \frac{PDG_i(s)}{1-PDG_{i}(s)}$$ as a result of positive feedback I'm not sure how to treat that $$q_{i-1}$$ signal in the system can anyone help help derive the transfer function. Suppose we acknowledge $$q_{i-1}$$ as an independent signal, then using the block diagram, the following equations can be derived: $$e_i = d_i - d_{r,i} = - q_i+q_{i-1} - d_{r,i}$$ $$q_i = G_i(s)u_i = PDG_i(s)e_i$$ substitute the first in the latter: $$q_i = PDG_i(s)(- q_i+q_{i-1} - d_{r,i})$$ $$(1+PDG_i(s))q_i = PDG_i(s)(q_{i-1} - d_{r,i})$$ $$q_i = \frac{PDG_i(s)}{1+PDG_i(s)}(q_{i-1} - d_{r,i})$$ Which represents a transferfunction with $$(q_{i-1} - d_{r,i})$$ as input and $$q_i$$ as output. The troublesome part here is this $$q_{i-1}$$, of which I do not know its meaning (is $$i$$ a discrete time index, or is $$q$$ a set of systems where $$q_i$$ is an part of it). Even thought one could substitute $$q_{i-1} = \frac{PDG_{i-1}(s)}{1+PDG_{i-1}(s)}(q_{i-2} - d_{r,i-1})$$ into the equation, the problem is only shifted to $$q_{i-2}$$ and so on. As such, the complete transfer matrix (where the input is every instance of $$d_{r,i-n}$$ and the output is every instance of $$q_{i-n}$$) will be the full definition of your system. However, if $$i$$ is a discrete time index, you could try to convert the model and controller to the Z domain and describe $$q_{i-1} = z^{-1}q_i$$
• Hi @Petrus sorry for the late response $q_{i-1}$ represents the position of a vehicle in front while $q_i$ represents the position of a vehicle in the back so therefore q is a set of systems in which $q_i$ is part of it. Feb 13, 2021 at 2:22