CHEN90032 · Process Simulation and Control
Multivariable (MIMO) Control: RGA & Decoupling
Multivariable (MIMO) control is the closing module of University of Melbourne CHEN90032 Process Simulation and Control: it deals with plants that have several controlled and manipulated variables whose loops interact. You learn to score that interaction with the Relative Gain Array (RGA), use it to pair each output with the right manipulated variable, cancel the leftover coupling with ideal decouplers (checking they are physically realizable), and meet Model Predictive Control (MPC) as the modern alternative. It maps directly to the multivariable question on the written final.
What this chapter covers
- 01Why loops interact: in an n×n process each manipulated variable m_j can move several controlled variables c_i, so independent single-loop controllers can fight each other
- 02The steady-state gain matrix K and its diagonal (intended) vs off-diagonal (interaction) gains
- 03The Relative Gain Array λ_ij = K_ij·(K⁻¹)_ji — dimensionless, with every row and column summing to 1
- 04The 2×2 shortcut λ11 = 1/[1 − K12·K21/(K11·K22)] and how it fixes the whole array
- 05Loop-pairing rules: pair on the positive λ closest to 1; why λ < 0 is dangerous and must never be paired
- 06The RGA ignores dynamics — checking the pairing against time constants and the Niederlinski index
- 07Ideal decouplers D12 = −G12/G11 and D21 = −G21/G22 that turn a coupled 2×2 into two SISO loops
- 08Realizability: a decoupler must be proper and free of predictive (positive) dead time e^{+θs}
- 09The lead-lag approximation K_D(τ_a s + 1)/(τ_1 s + 1) for a decoupler that cannot be built exactly
- 10Model Predictive Control (MPC): predict → optimise-with-constraints → apply first move → receding horizon
RGA and loop pairing of a 2×2 process
- +1Form the gain ratio needed by the 2×2 shortcut: K12·K21/(K11·K22) = (+1.0)(−2.0)/[(+2.0)(+3.0)] = −2.0/6.0 = −0.333. Keep the sign of K21 (it is negative), because it decides which side of 0.5 λ11 lands on.
- +2Relative gain from the shortcut λ11 = 1/[1 − K12·K21/(K11·K22)] = 1/[1 − (−0.333)] = 1/1.333 = 0.75. It is dimensionless — the units of the two gains cancel.
- +1Complete the array using the row/column-sum-to-1 property: RGA = [[0.75, 0.25], [0.25, 0.75]]. Check each row and column: 0.75 + 0.25 = 1.00.
- +1Pair on the positive relative gain closest to 1. Here λ11 = 0.75 sits in the acceptable band 0.5 < λ < 1, so pair c1–m1 and c2–m2 — the 1–1 / 2–2 (diagonal) pairing.
- +1State the caveat: λ = 0.75 means mild but real interaction, so verify the choice against the dynamics and add decouplers if the coupling degrades the response. The RGA uses steady-state gains only.
Key terms
- Process interaction
- The situation in a multivariable plant where a manipulated variable affects more than one controlled variable, so single loops disturb one another. Quantified by the off-diagonal gains of the steady-state gain matrix K.
- Relative Gain Array (RGA)
- A matrix Λ = [λ_ij] computed from the steady-state gains as the element-by-element product of K with the transpose of its inverse. It is dimensionless and every row and column sums to 1.
- Relative gain λ_ij
- λ_ij = K_ij·(K⁻¹)_ji, the ratio of the open-loop gain (other loops open) to the gain with all other loops on perfect control. λ = 1 means no interaction on that pairing.
- Loop pairing
- Deciding which manipulated variable controls which controlled variable. The rule is to pair on the relative gain that is positive and closest to 1; a negative λ must never be paired because the effective gain sign flips when other loops close.
- Ideal decoupler
- An extra transfer-function block that cancels an interaction path. For a 2×2 process paired 1–1 / 2–2 the ideal decouplers are D12 = −G12/G11 and D21 = −G21/G22, which leave two independent single-loop problems.
- Realizability
- Whether a decoupler can be physically built. It must be proper (numerator order not greater than denominator order) and must not contain a predictive (positive) dead time e^{+θs}; otherwise it is approximated.
- Lead-lag unit
- A first-order compensator K_D(τ_a s + 1)/(τ_1 s + 1) used to approximate an unrealizable ideal decoupler: keep the static gain and the dominant lead and lag, and drop the un-implementable piece.
- Model Predictive Control (MPC)
- A control strategy that uses a model of the whole plant to predict future outputs, optimises a sequence of manipulated moves (subject to constraints), applies only the first move, and repeats each sample — a receding horizon that handles interaction and limits directly.
Multivariable (MIMO) Control: RGA & Decoupling FAQ
How do I know which loop to pair with which manipulated variable?
Compute the Relative Gain Array from the steady-state gains and pair each controlled variable with the manipulated variable whose relative gain λ is positive and closest to 1. A value between 0.5 and 1 is a good pairing with mild interaction; a value below 0.5 means the indirect path dominates; and a negative λ is dangerous because the effective gain changes sign when the other loops are switched to automatic, so it must never be paired.
What is the difference between decoupling and MPC?
Decoupling is a pairwise fix: after choosing pairings you add cross-elements D12 = −G12/G11 and D21 = −G21/G22 that cancel each interaction path so ordinary PID loops can be tuned independently. MPC takes the whole coupled plant at once — it uses one model to predict future outputs and optimises all the manipulated moves together, subject to constraints, so interactions are handled by construction rather than cancelled loop-by-loop. MPC also enforces hard limits that PID cannot.
Can AI help me with multivariable (MIMO) control and RGA in CHEN90032?
Yes — an AI tutor such as Sia can explain, step by step, how to build a steady-state gain matrix, compute the Relative Gain Array, apply the pairing rules, and derive and test ideal decouplers, then generate similar practice questions so you can check your own working. Use it to understand the method and rehearse; it is a study aid, not a source of ready-made answers, so always confirm assessment details on Canvas and do the derivations yourself for the closed-book exam.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Multivariable control is compact and high-yield, so drill the standard sequence until it is automatic. First, be able to build the steady-state gain matrix K from a set of transfer functions (the gains are the s→0 limits) and compute λ11 from the 2×2 shortcut, watching the signs of K12 and K21 — a sign slip is the usual lost mark and it can reverse your pairing advice. Second, memorise the pairing rules as a ladder: λ = 1 ideal, 0.5 < λ < 1 acceptable, λ < 0.5 poor, λ < 0 never, and always state that λ is dimensionless. Third, practise deriving both ideal decouplers D12 = −G12/G11 and D21 = −G21/G22, dividing the dead times to a single exponent and reading its sign: a negative exponent is a realizable delay, a positive one is a prediction and is not realizable, so quote the lead-lag fallback. Finish by rehearsing a two-sentence MPC description (predict, optimise with constraints, apply the first move, recede) so you can pick up the concept marks. With about 1.8 minutes per mark on the written final, a 15-mark multivariable question is worth roughly 27 minutes; leave enough of it for the realizability sentence, where marks are won or lost. Confirm the exam date on Canvas for the ~June end-of-Semester-1 exam period.
