Cascade, ratio, feedforward and split-range control

Why it matters

A single feedback loop only acts after the controlled variable has already moved. Cascade, ratio, feedforward and split-range schemes are the standard ways plants deal with fast disturbances, blending, measurable upsets and processes needing more than one final element. Almost every P&ID you will read in industry contains at least one of them.

Key ideas

Cascade control. Two feedback controllers in series. The primary (master, outer) controller measures the variable you really care about and its output becomes the set point of the secondary (slave, inner) controller, which manipulates the valve.

• Example: reactor temperature (primary) → jacket temperature or steam flow (secondary) → valve.
• Benefit: disturbances that enter the inner loop (steam supply pressure changes, valve stiction, cooling-water temperature) are corrected by the fast inner loop before they upset the primary variable. The inner loop also linearises the valve.
• Rules: the inner loop must be clearly faster than the outer one (a factor of about 3 to 5 or more in response time); the secondary variable must be measurable and must respond to the valve; the inner loop is tuned first with the outer loop in manual, then the outer loop. The inner controller runs in "remote set point" (cascade) mode.

Ratio control. Keeps the flow of one stream (controlled flow) in a fixed proportion to another uncontrolled ("wild") flow — fuel/air in a burner, reactant feeds, dilution and blending. Two implementations:

• Multiply the wild-flow measurement by the desired ratio R and use the result as the set point of the controlled-flow loop. Preferred: the loop gain stays constant.
• Divide the two measured flows and control the computed ratio. Simpler to display but the gain of the divider changes with the wild flow, so it is non-linear. Orifice (differential-pressure) signals are proportional to flow squared, so take square roots before ratioing, or remember that a flow ratio R corresponds to a DP ratio R².

Feedforward control. Measure a major disturbance and act on the manipulated variable before the controlled variable moves. Ideal compensator G_ff = −G_d/G_p, where G_d is the disturbance-to-output transfer function and G_p the manipulated-to-output one. In practice it is implemented as a static gain, often with a lead-lag and a dead time. It is only as good as the model and does nothing for unmeasured disturbances, so it is almost always combined with feedback trim, which removes the remaining error. The compensator is physically realisable only if the dead time of G_d is at least that of G_p.

Split-range control. One controller output drives two (or more) final elements over different parts of its range. Examples:

• Heating and cooling of a reactor jacket: 0–50 % output moves the cooling valve, 50–100 % the heating valve, with both closed near 50 %.
• Pressure in a nitrogen-blanketed tank: low output opens the nitrogen make-up valve, high output opens the vent.
• Large and small valves in parallel for wide rangeability. Valves are calibrated (positioner ranges) so that each gets a full stroke over its part of the signal. A small dead band at the split point prevents both valves fighting.

Related schemes. Override (selective) control uses high/low selectors so a constraint controller takes over the valve when a limit is approached; auctioneering selects the highest of several measurements (hot-spot temperature). Both need anti-windup on the controller that is not selected.

Formulas

F_B,sp = R · F_A

• F_A: wild flow (kg/s or m³/h); R: desired ratio (dimensionless when both flows are in the same unit); F_B,sp: set point of the controlled flow.

ΔP_B / ΔP_A = R² (orifice signals without square-root extraction, same meter factors)

G_ff(s) = − G_d(s) / G_p(s)

• Feedforward compensator from disturbance measurement to controller output. Static version: K_ff = − K_d / K_p.

valve opening (%) = (u − u_start)/(u_end − u_start) × 100

• u: controller output (%); u_start, u_end: the signal range assigned to that valve. Reverse the expression for a valve that closes as the signal rises.

K_cl = K_c2·K_2 / (1 + K_c2·K_2), τ_cl = τ_2 / (1 + K_c2·K_2)

• Inner loop of a cascade with P-only secondary control of a first-order secondary process (gain K_2, time constant τ_2): the closed inner loop is faster by the factor (1 + K_c2·K_2).

Worked examples

Example 1 (standard) — split range and ratio. (a) A jacket uses a cooling valve over 0–50 % controller output (fully open at 0 %, closed at 50 %) and a heating valve over 50–100 % (closed at 50 %, fully open at 100 %). Find the valve openings at outputs of 30 % and 80 %. (b) A burner needs an air/fuel mass ratio of 15. Fuel flow is 0.2 kg/s. What is the air-flow set point?

1. At u = 30 % the cooling valve is in range: opening = (50 − 30)/(50 − 0) × 100 = 40 % open; heating valve closed.
2. At u = 80 % the heating valve is in range: opening = (80 − 50)/(100 − 50) × 100 = 60 % open; cooling valve closed.
3. (b) F_air,sp = R·F_fuel = 15 × 0.2 = 3.0 kg/s.

Example 2 (GATE level) — feedforward and cascade. (a) G_p(s) = 2e^(−2s)/(10s + 1) and G_d(s) = 0.5e^(−4s)/(5s + 1) (time in min). Design the ideal feedforward compensator and find the controller output change, immediately after the compensator's delay and at steady state, for a disturbance step of +4 units. (b) The secondary process of a cascade is 1/(4s + 1) and the secondary controller is P-only with K_c2 = 4. Find the closed inner-loop gain and time constant.

1. G_ff = −G_d/G_p = −(0.5/2)·[(10s + 1)/(5s + 1)]·e^(−(4 − 2)s) = −0.25·(10s + 1)/(5s + 1)·e^(−2s).
2. It is realisable: positive delay of 2 min, lead time 10 min, lag time 5 min.
3. Steady-state change: −0.25 × 4 = −1.0 unit.
4. Initial change (after 2 min): a lead-lag step response starts at (lead/lag) × gain = −0.25 × (10/5) × 4 = −2.0 units, then decays to −1.0 with time constant 5 min.
5. (b) K_c2·K_2 = 4 × 1 = 4. K_cl = 4/5 = 0.8; τ_cl = 4/5 = 0.8 min — five times faster than the open inner process, which is why cascade suppresses inner-loop disturbances so well.

Common mistakes

• Tuning the outer loop first, or making the inner loop slower than the outer one; cascade then gives no benefit and may oscillate.
• Forgetting to put the secondary controller in cascade (remote set-point) mode, so the primary output goes nowhere.
• Ratioing raw DP signals as if they were flows; the ratio of DP signals is the square of the flow ratio.
• Using feedforward alone and expecting zero offset; model error always leaves some, which feedback trim must remove.
• Getting the sign of the feedforward gain wrong — the compensator must oppose the disturbance effect.
• Overlapping split ranges with no dead band, so heating and cooling run at the same time.

For GATE IN

Typical items: draw or identify the block diagram of cascade or feedforward-plus-feedback control; derive the closed-loop transfer function of a cascade loop; design G_ff = −G_d/G_p and comment on realisability; compute the set point in a ratio loop, including square-root effects; and map a controller output to valve positions in split range. Practise block-diagram reduction for nested loops.

Quick check

1. In cascade control, which loop must be faster and which is tuned first?
2. Air/fuel ratio 12, fuel 0.5 kg/s. What is the air set point?
3. G_d = 3/(2s + 1), G_p = 1.5/(2s + 1). What is the ideal feedforward compensator?
4. In a 0–50 % / 50–100 % split range, how far open is the second valve at 65 % output?

Answers: 1. The inner (secondary) loop is faster and is tuned first. 2. 6 kg/s. 3. G_ff = −2 (a pure static gain). 4. 30 % open.