Cascade, feedforward and ratio control

Why it matters

Single feedback loops act only after the controlled variable has already moved. Cascade, feedforward and ratio control are the three standard enhancements that catch disturbances earlier or keep streams in proportion; they appear on almost every reactor, furnace, column and blending P&ID, and GATE tests both their block diagrams and their design equations.

Key ideas

Cascade control. Two feedback controllers are nested. The primary (master, outer) controller measures the variable that really matters (for example, reactor temperature) and its output becomes the set point of the secondary (slave, inner) controller, which measures an intermediate variable that responds quickly to the manipulated variable and to certain disturbances (for example, coolant flow or jacket temperature).

• The inner loop corrects disturbances that enter it (supply-pressure changes, valve non-linearity, sticking) before they reach the primary variable.
• It also speeds up the part of the process inside it, so the outer loop sees a smaller effective lag and can use a higher gain.
• Requirements: the secondary variable must be measurable, must respond to the manipulated variable, and the inner loop should be at least 3–5 times faster than the outer loop.
• Tuning order: tune the inner loop first (often P or PI with high gain; offset there does not matter because the outer loop has integral action), then the outer loop with the inner loop closed.
• Typical examples: temperature-to-flow (heat exchanger steam flow), reactor-temperature-to-jacket-temperature, level-to-flow.

Feedforward control. A disturbance D is measured and the manipulated variable is changed so that the disturbance effect is cancelled before the controlled variable moves. With C = G_d·D + G_p·M and feedforward M = G_f·D, perfect cancellation requires G_f = −G_d/G_p (including valve and transmitter transfer functions where present).

• Static (steady-state) feedforward uses only the gains: G_f = −K_d/K_p.
• Dynamic feedforward adds a lead–lag (and dead time, if the disturbance path is slower than the manipulated path). If G_p has more dead time than G_d, perfect compensation is impossible (it would need prediction).
• Feedforward needs a good model and handles only measured disturbances, so it is almost always combined with feedback trim (feedforward–feedback control). Feedforward does not change the characteristic equation of the feedback loop, so it does not affect stability.
• Examples: boiler drum level (steam flow as disturbance), heat exchanger (process-fluid flow and inlet temperature), distillation feed flow.

Ratio control. Keeps the flow of one stream (the controlled, or manipulated, stream) at a set ratio R to a measured wild stream that is not controlled. Preferred implementation: multiply the measured wild flow by R in a ratio station to get the set point of the controlled-stream flow controller. Dividing the two flows and controlling the quotient is avoided because the loop gain then varies with the wild flow. Ratio control is a form of feedforward. Examples: fuel–air ratio in burners, reflux-to-feed ratio, blending, reactant feeds.

Related schemes. Split-range control (one controller drives two valves over different signal ranges, for example heating and cooling), override/selective control (high or low selectors protect constraints), and inferential control (estimate a composition from temperatures).

Formulas

G_f(s) = −G_d(s)/G_p(s)

• Ideal feedforward controller; G_d disturbance-to-output, G_p manipulated-to-output transfer functions.

G_f = −K_d/K_p

• Static feedforward gain.

G_f(s) = −(K_d/K_p)·(τ_p·s + 1)/(τ_d·s + 1)·e^(−(θ_d − θ_p)·s)

• For first-order-plus-dead-time paths; realisable only if θ_d ≥ θ_p.

G_inner = G_c2·G_p2/(1 + G_c2·G_p2·G_m2)

• Closed inner loop in cascade (it replaces G_p2 in the outer loop).

F_c,sp = R·F_w

• Ratio station: set point of controlled flow from measured wild flow F_w (same units).

Worked examples

Example 1 (standard): designing a feedforward controller. For a heat exchanger, the outlet-temperature response to process-flow disturbance is G_d = 2·e^(−3s)/(5s + 1) and to steam valve signal is G_p = 0.5·e^(−s)/(2s + 1) (time in min). Design the ideal feedforward controller and its static version.

1. G_f = −G_d/G_p = −(2/0.5)·[(2s + 1)/(5s + 1)]·e^(−(3 − 1)s).
2. G_f = −4·(2s + 1)/(5s + 1)·e^(−2s): a lead–lag with lead 2 min, lag 5 min and a 2 min delay.
3. Realisable because θ_d = 3 min ≥ θ_p = 1 min.
4. Static feedforward: G_f = −K_d/K_p = −2/0.5 = −4.

G_f = −4·(2s + 1)·e^(−2s)/(5s + 1); static gain −4

Example 2 (GATE level): benefit of cascade. A process consists of three lags 1/((s + 1)(2s + 1)(4s + 1)) (min) under P control; the first lag (τ = 1 min) is the coolant-flow loop. Without cascade, K_cu = 11.25. A secondary flow controller with K_c2 = 9 (P only, unity gains) closes the inner loop around the 1 min lag. Find the new ultimate gain of the primary controller.

1. Inner closed loop: 9/(s + 1 + 9) = 0.9/(0.1s + 1).
2. Outer open loop: K_c1·0.9/((0.1s + 1)(2s + 1)(4s + 1)).
3. Expand: (0.1s + 1)(8s² + 6s + 1) = 0.8s³ + 8.6s² + 6.1s + 1.
4. Characteristic equation: 0.8s³ + 8.6s² + 6.1s + 1 + 0.9K_c1 = 0.
5. Cubic condition: 8.6·6.1 > 0.8·(1 + 0.9K_c1), so 52.46 > 0.8 + 0.72K_c1, giving K_c1 < 71.75.
6. ω_u = √(6.1/0.8) = 2.76 rad/min; P_u = 2π/2.76 = 2.28 min (was 6.72 min).

K_cu rises from 11.25 to 71.75 and P_u falls from 6.72 to 2.28 min: the outer loop can be tuned far tighter, and coolant-supply disturbances are corrected by the inner loop.

Example 3 (short): ratio station. Fuel (wild) flow is 120 kg/h and the required air-to-fuel mass ratio is 15. The air-flow set point is 15·120 = 1800 kg/h.

Common mistakes

• Making the inner loop slower than, or as slow as, the outer loop; cascade then gives little benefit and can oscillate.
• Tuning the outer loop before the inner loop, or with the inner loop in manual.
• Writing G_f = G_d/G_p without the minus sign, which doubles the disturbance effect.
• Expecting feedforward alone to remove offset caused by model error or unmeasured disturbances.
• Implementing ratio control by dividing the two flow signals, which makes the loop gain vary with the wild flow.

For GATE CH

Expect: deriving G_f = −G_d/G_p (often with dead times and lead–lag terms), closed-loop transfer functions of cascade schemes, the ultimate gain with and without cascade, ratio-station set points, and conceptual questions on which scheme suits a described disturbance. Draw the block diagram first, then reduce the inner loop before the outer one.

Quick check

1. G_d = 3/(4s + 1), G_p = 1.5/(s + 1). What is the ideal feedforward controller?
2. Which loop in a cascade is tuned first?
3. Does adding feedforward change the stability of the feedback loop?
4. A blend needs 0.2 kg additive per kg base. The base flow is 450 kg/h. What is the additive set point?

Answers: 1. G_f = −2·(s + 1)/(4s + 1). 2. The inner (secondary) loop. 3. No, the characteristic equation is unchanged. 4. 90 kg/h.