Multiple Input, Multiple Output I: Numerical Decoupling By Peter Woolf University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons

Inputs and Outputs Up to now, we have relied on our intuition about a process to decide how to connect sensors to actuators, but there are a few problems that can arise: (1)Sometimes these parings can be difficult to determine (2)Sometimes it is difficult to distinguish between pairings (3)Sometimes the parings we think will work don’t work (4)Sometimes there are no pairings that will work Questions: Are there other, data driven ways to see if a system can be decoupled? How can we evaluate the best pairing in a more objective way?

Four terms: 1) SISO: Single input, single output. Simplest to design, uses data from one sensor to control one thing. PID controller TC v1 2) SIMO: Single input, multiple output. Uses data from one sensor to control multiple things. #1 PID controller TC v1 #2 PID controller TC v2 3) MISO: Multiple input, single output. More complex as it uses data from multiple sensors to control one thing. E.g. cascade control #1 PID controller TC v1 FC #2 PID controller Set point

4) MIMO: Multiple Input, Multiple Output. Hardest to design as it integrates multiple sensor data to coordinate multiple actuators MIMO controller TC v2 FC v1 v3 General strategy: MIMO controllers are more complex, and as such designers often try to avoid them. Process design attempts to minimize cross-talk between sensors and actuators if possible. Even if there is significant cross talk, can we get away with a simpler controller by finding approximate pairings? Note: MIMO controllers are generally not PID, and as such are often designed for each particular case.

Example Process Schematic Process C AO Q CATCAT Simplified Process schematic Simplified Process Model

gain array Elements of the array: Note that this is not the jacobian.. Experimental alternative Change each single input at a time and observe outputs Calculate local change in outputs and these are entries in the gain array

Intuitive evaluation of gain arrays: Good or bad?

Intuitive evaluation of gain arrays: Best case because each manipulated variable exactly controls only one output. Here Cao controls Ca and Q controls T. Good or bad?

Intuitive evaluation of gain arrays: Good or bad?

Intuitive evaluation of gain arrays: Good or bad? Worst case. Manipulated variables can’t individually control outputs. System is fully coupled. How can we measure the degree of coupling?

Condition number (CN) Tells how near to a linearly dependent system we are based on a single value decomposition (SVD)

Condition number (CN) Tells how near to a linearly dependent system is based on a singular value decomposition (SVD) of the gain array Ratio of largest to smallest value=CN =5.46/0.36= 15.1 In Mathematica Terms Map[MatrixForm,{u, w, v}=SingularValueDecomposition[G]]

CN limits

The rule of thumb is that CN>50 is too hard to decouple.. What does CN=50 mean in this matrix above?

CN limits The rule of thumb is that CN>50 is too hard to decouple.. What does CN=50 mean in this matrix above?

In mathematica terms In matrix terms Intuitively and mathematically RGA is a normalized form of the gain matrix that describes the impact of each control variable on the output relative to the control variables impact on other variables.

A few features of an RGA matrix: The nearer all of the entries are to 1 the more decoupled the system is. The best pairing is found by taking the max of the RGA matrix for each row. Each row and each column of the RGA sum to 1

Example: Gain array from a system where the pressure and temperature of a reactor can be controlled by two valves v1 and v2. CN=402 Interpretation: Poor decoupling The same as Interpretation: All values far from 1, so poor decoupling. However, best coupling would link v2 to T and v1 to P. Interpretation: v2 influences T a little bit more than P, but both v1 and v2 strongly influence T and P.

What pairing would work best here? Therefore can be decoupled. Best pairings Note that the pairings are easier to identify using the RGA vs. G matrix.

Material from previous lectures

Take Home Messages It is possible to numerically determine if a system can be decoupled using the condition number RGA helps determine optimal pairings Not all systems can be effectively decoupled Even if a system can be decoupled, it may or may not perform well