When a multivariable process with several inputs and outputs is controlled with decentralized single loops, this tool decides which input to pair with which output. Enter a 2x2 gain matrix and see Bristol's Relative Gain Array, the recommended pairing and the strength of the loop interaction update in real time.
Parameters
Elements of the 2x2 steady-state process gain matrix K. g_ij is the gain from input u_j to output y_i.
Gain g11 (u1 → y1)
Gain g12 (u2 → y1)
Cross-interaction gain from input 2 to output 1
Gain g21 (u1 → y2)
Cross-interaction gain from input 1 to output 2
Gain g22 (u2 → y2)
Results
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RGA element λ11
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RGA element λ12 (=1−λ11)
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Recommended pairing
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Interaction strength
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Niederlinski index NI
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Controllability verdict
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Process structure and RGA map
Four gain paths connect the two inputs (u1, u2) on the left to the two outputs (y1, y2) on the right; line thickness is proportional to |g_ij|. On the right is the RGA matrix map (green = good pair near 1, red = negative, yellow = severe interaction near 0.5). The recommended pairing paths pulse.
The Relative Gain Array of a 2x2 process. g_ij are the elements of the gain matrix K. Every row and column of the RGA sums to 1, and pairings on a negative element must be avoided.
$$\det K = g_{11}g_{22}-g_{12}g_{21},\qquad \text{NI}=\frac{\det K}{g_{11}g_{22}}$$
The determinant detK of the gain matrix and the Niederlinski index NI for the diagonal pairing. When |detK| is near 0 the matrix is singular (un-controllable). A negative NI means the diagonal pairing is structurally unstable.
What is the Relative Gain Array (RGA)?
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"Relative Gain Array" sounds intimidating — what does this thing actually compute?
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Put simply, it is a matrix that tells you "which knob should be in charge of which meter". Take a distillation column with two valves (u1, u2) and two things you want to measure, say a temperature and a composition (y1, y2). Normally you want u1 to control y1 and u2 to control y2 with separate PID loops. The trouble is that moving u1 changes not just y1 but y2 as well. The Relative Gain Array — RGA for short — invented by Edgar Bristol, turns that "loop interaction" into numbers you can read.
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I see. But couldn't I just look at the gain matrix K to see whether there is interaction? Why bother converting it to an RGA?
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Good question. The raw gain matrix K has a trap: just changing the units of an input or output makes the numbers swing wildly. Measure temperature in degrees Celsius or kelvin, flow in L/min or m3/h, and K looks completely different. So saying "the interaction is strong" has no objective meaning. The beautiful thing about the RGA is that each element is defined as a ratio between a loop's own gain and its gain when the other loops are closed, so its value does not change no matter how you scale things. That is why you can use it very early in design, even before the instruments are chosen.
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Once I have the RGA numbers, how exactly do I read them? When I move g12 on the left, lambda11 keeps changing.
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Reading it is simple. An RGA element near 1 means "that pair barely interacts — a clean pairing". Near 0.5 means "the two loops pull on each other half and half — severe interaction". And once an element goes negative, that is a danger signal. An element well above 1 is also "ill-conditioned" and becomes sensitive to small modelling errors. The rule is "pair the input and output at the element that is positive and closest to 1". For the default matrix lambda11 is about 1.09, so the diagonal pairing (y1-u1, y2-u2) is recommended.
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Why is a "negative" RGA element so bad?
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This is the single most important warning the RGA gives. A negative element means the gain of that pair changes sign between "the other loop open" and "the other loop closed". Say you design the controller while the other loop is on manual (open) and conclude "increasing u1 raises y1", so you build positive feedback. The moment you switch the other loop to automatic (closed), increasing u1 starts to lower y1 instead. The control system runs away. That is why a pairing on a negative RGA element must be avoided no matter how hard you tune.
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The "Niederlinski index" also shows up — is that something separate from the RGA?
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They complement each other. The RGA tells you "which pair to choose", but even after you choose the diagonal pairing, whether it can actually be stabilised with all loops closed needs a separate check. That is what the Niederlinski index NI = detK/(g11*g22) is for. If NI is negative, a PID controller with integral action can never be made stable for that pairing — a structural NG verdict. In practice the workflow is "pick a positive-element pairing with the RGA" and "confirm the NI is positive", and only when both pass do you move forward with that loop configuration.
Frequently Asked Questions
The Relative Gain Array (RGA), introduced by Edgar Bristol, is a matrix that measures the interaction between loops in a multi-input multivariable process. For a 2x2 system it is computed as lambda11 = (g11*g22)/(g11*g22-g12*g21), and the full RGA is [[l11,1-l11],[1-l11,l11]]. A key feature of the RGA is that it is independent of the units and scaling of the inputs and outputs. This tool computes the RGA from a gain matrix and tells you which input to pair with which output.
The rule is to pair each output with the input whose RGA element is positive and closest to 1. For a 2x2 system, if lambda11 >= 0.5 choose the diagonal pairing (y1-u1, y2-u2); if lambda11 < 0.5 choose the off-diagonal pairing (y1-u2, y2-u1). Never pair on a negative RGA element: closing a loop on a negative element makes its gain change sign between the other loop being open and closed, which drives the system unstable.
A lambda11 near 0.5 means the two loops interact strongly with each other. Moving one input changes both outputs by a similar amount, so tuning one loop on its own disturbs the other and undoes the change. Decentralized single-loop control becomes very difficult: settling is slow and oscillations may not die out. The remedies are to add a decoupler (an interaction-compensating controller) or to switch to multivariable control such as model predictive control.
The Niederlinski index NI = detK/(g11*g22) is a structural stability check for the diagonal pairing with all loops closed. If NI is negative, that diagonal pairing is unstable for any tuning as long as the controllers contain integral action (the Niederlinski instability condition). A positive NI is a necessary but not sufficient condition for stability. A pairing is structurally sound only when the RGA has no negative element and the NI is positive.
Real-World Applications
Distillation column control: The most classic use of the RGA is the distillation column. Two manipulated variables — reflux rate and reboiler heat duty — hold two controlled variables, the top and bottom compositions. The two interact strongly, and for the typical LV configuration the RGA elements are known to take large positive or negative values. The sign and magnitude of the RGA tell you, early in design, which valve to assign to composition control and whether decoupling is needed.
HVAC and air-conditioning systems: When a single air handler controls temperature and humidity at the same time, the cooling-coil capacity and the reheat or humidification action interact. Cooling lowers temperature while also dehumidifying, so pairing the temperature and humidity loops wrongly leaves the two chasing each other forever. The RGA decides quantitatively which manipulated variable should be assigned to which measurement.
Chemical reactors and blending processes: Many processes have manipulated variables that are coupled geometrically or thermodynamically — reactor temperature and pressure, or the ratio and total flow of a blending tank. The RGA is used to assign the reactor jacket temperature and feed flow, or to select the acid and base dosing loops of a pH neutralisation process, giving an opportunity to revise the structure before interaction degrades controllability.
Deciding when to move to multivariable control: The RGA also helps decide whether to keep single-loop control or move to multivariable control such as model predictive control (MPC). If the RGA elements stay near 1, decentralized PID is enough; but for an "ill-conditioned" process with values near 0.5, negative values, or extremely large values, investing in decouplers or MPC is justified. It is the first screening tool for deciding the control architecture of an entire plant.
Common Misconceptions and Pitfalls
The most common mistake is assuming the RGA alone fully determines the pairing. The RGA is based only on the steady-state (zero-frequency) gains and reflects none of the process dynamics — no time constants, no dead time, no transfer-function phase. A pairing that looks good on the steady-state RGA can interact strongly dynamically if one loop is much faster than the other. A serious design uses the frequency-dependent RGA evaluated near the crossover frequency. Treat the steady-state RGA in this tool as a first screening for structure selection only.
Next, thinking that a positive Niederlinski index means stability. A positive NI is only a necessary condition for stability, not a sufficient one. If NI is negative you can be certain that the diagonal pairing is always unstable with integral control, but even with a positive NI the system can still be unstable. The NI is a sieve that rules out structurally impossible pairings. Final stability must be confirmed with the controller actually designed, using a Nyquist test or simulation.
Finally, simply assuming that a larger RGA element means stronger interaction. Interaction is in fact most severe when lambda11 is near 0.5. When lambda11 is near 1, or near 0, the interaction is weak and the loops separate cleanly. When lambda11 is well above 1, or negative, the interaction is "strong / ill-conditioned" — a state with extreme sensitivity to modelling error. RGA elements must be read by both "how far from 1" and "sign"; judging by absolute magnitude alone is wrong. This tool classifies the interaction strength using both |lambda11-0.5| and the sign.
How to Use
Enter the 2×2 process gain matrix elements (g11, g12, g21, g22) representing steady-state gains between two inputs and two outputs in your multivariable system (e.g., distillation column: reflux-to-overhead composition, reboiler-to-bottoms composition).
The simulator calculates the Relative Gain Array (RGA) by computing λij = gij·(G⁻¹)ji / det(G) for pairing analysis and the Niederlinski index NI = det(G)·Πλii to assess closed-loop stability risk.
Review the recommended pairing (1-1/2-2 or 1-2/2-1), interaction strength percentage, and controllability verdict to decide decentralized PI controller assignment.
Worked Example
A 2×2 distillation process has gains: g11=2.0 (reflux→overhead composition %/%), g12=−1.5 (reboiler→overhead), g21=−0.8 (reflux→bottoms), g22=1.8 (reboiler→bottoms). The RGA yields λ11≈0.63, λ12≈0.37. Since λ11 >0.5, pairing output 1 with input 1 and output 2 with input 2 is preferred. Interaction strength ≈37% indicates moderate coupling; Niederlinski index NI=0.84 confirms acceptable closed-loop stability margin for decentralized control.
Practical Notes
RGA values near 0.5 signal high interaction; prioritize input-output pairings where λij approaches 1.0 or 0.0 for minimal loop coupling in refineries and paper mills.
Negative Niederlinski index (NI <0) flags potential instability regardless of pairing—consider MIMO control or valve rearrangement before commissioning.
Off-diagonal elements (g12, g21) represent cross-coupling; gains >50% of diagonal magnitude typically demand 2×2 decoupler or model-predictive control rather than simple SISO loops.