Proof of RGA Negative Number Theorem

The Theorem takes the form: "If the RGA value corresponding to an array element is negative (and the corresponding Process Transfer Function Matrix element does not itself have a right half plane zero*), then the closed loop transfer function seen by any controller corresponding to that element will contain either a (real) right half plane zero or pole."


The proof depends on considering the general impact of the Initial and Final Value theorems: The low s (or frequency) behavior of the closed loop transfer function comes from the closed loop gain (the gain of the pij), whereas the high s (frequency) behavior comes from the open loop transfer function φij. Thus, with the RGA element μ < 0, the low and high frequency transfer sections are on opposite sides of the x-axis. The only way that they can meet is if the transfer function crosses the x-axis, with a RHPZ, or if it crosses through ∞, with a RHPP.

* This condition usually does not limit the result even if in theory it should.