Multiloop feedback control system find closed loop gain

I want to find the closed loop gain of this block diagram:

I remember a rule that since these 2 loops collide with each other we ignore them so is the closed loop gain just equal to $$\frac{H_{1}(s)}{1+H_{1}(s)H_{2}(s)}$$?

• Homework? What have you done to try to solve this so far? Jun 13, 2023 at 23:58
• @TimWescott I am preparing for a exam.And I have already posted a guess the answer should be Y/N. Jun 14, 2023 at 4:31
• Is there a source for this block diagram ? A signal line where signal flows in two directions is not usually seen in text books. This diagram may simply be wrong. Please provide a source.
– AJN
Jun 14, 2023 at 16:14
• I didn't consider that making a wild guess (and a wrong one, at that) shows enough effort. Something like the answer below, with a math error in the middle -- that counts as effort, and worthy of effort on my part. Jun 14, 2023 at 23:22

We can then find an expression for $$Y(s)/U(s)$$ as follows: \begin{align} Y(s) &= X_1(s) - X_2(s) \\ &= H_1(s)X_3(s) - H_2(s)Y(s) \\ &= H_1(s)(U(s) - X_2(s)) - H_2(s)Y(s) \\ &= H_1(s)U(s) - H_1(s)X_2(s) - H_2(s)Y(s) \\ &= H_1(s)U(s) - H_1(s)(H_2(s)Y(s)) - H_2(s)Y(s) \\ &= H_1(s)U(s) - H_1(s)H_2(s)Y(s) - H_2(s)Y(s) \\ Y(s) + H_1(s)H_2(s)Y(s) + H_2(s)Y(s) &= H_1(s)U(s) \\ (1 + H_1(s)H_2(s) + H_2(s))Y(s) &= H_1(s)U(s) \\ \frac{Y(s)}{U(s)} &= \frac{H_1(s)}{1 + H_1(s)H_2(s) + H_2(s)} \\ \end{align} The general strategy for dealing with block diagrams is to make sure that all inputs and outputs of summing junctions are labelled, and then we can express all labelled signals as a function of either $$Y(s)$$ or $$U(s)$$. We then solve for $$Y(s)/U(s)$$.