Instrumentation and Process Control

Feedback and feedforward control structures

Feedback control measures the controlled variable, compares it with the set point, and changes the manipulated variable to remove the error. It acts only after a disturbance has already moved the output, so it is a reactive scheme. Feedforward control measures a disturbance directly and computes a corrective action before the disturbance reaches the controlled variable. Because the correction is calculated from the measurement, feedforward is possible only when the disturbance can be measured by a sensor. An unmeasured disturbance can be handled only by feedback. In practice feedforward is usually combined with feedback, because feedback removes the residual error caused by model mismatch.

A feedback loop must manipulate a variable that directly changes the quantity being controlled. To hold the liquid volume in a vessel, the level is measured and a stream that changes the holdup, such as the outlet flow, is manipulated. A temperature measurement or a utility stream that changes only heat transfer does not control the volume.

For a disturbance d with disturbance transfer function G_d and manipulated-variable transfer function G_p, the ideal feedforward compensator cancels the disturbance effect: G_{ff} = -G_d/G_p. The minus sign means the correction opposes the disturbance. Example: if G_d = 1/(2s+1) and G_p = 2/(s+1), then G_{ff} = -\dfrac{s+1}{2(2s+1)}. If the ratio contains a dead-time advance e^{+\theta s}, it cannot be realized, so only the delay-free part is kept. A quick check is the steady-state gain, since G_{ff}(0) = -G_d(0)/G_p(0) must hold for complete rejection of a sustained disturbance.

Closed-loop stability criteria

A closed-loop system is stable when every root of the characteristic equation 1 + G_c G_p = 0 has a negative real part. Several conditions follow from this. For a polynomial characteristic equation, all coefficients must be positive when the system is stable, and the first column of the Routh array must have no sign change. For the Bode criterion, an open-loop stable process with feedback is stable only if the amplitude ratio is less than 1 at the phase crossover (critical) frequency, where the phase lag is -180^\circ. These are necessary conditions.

The Routh test for a cubic characteristic equation a_3 s^3 + a_2 s^2 + a_1 s + a_0 = 0 with all coefficients positive is a_2 a_1 > a_3 a_0. Example: for s^3 + 4s^2 + 3s + K_c = 0, stability requires 12 > K_c.

Dead time e^{-\theta s} is not a polynomial, so the Routh test cannot be applied to it directly. A Pade approximation can convert it to a rational function, and Nyquist analysis can be used instead. A closed loop with dead time may be stable, so dead time does not by itself make a system unstable. Sign conventions matter in multi-loop structures, because a positive feedback path changes the sign of the loop term in the characteristic equation.

For an open-loop unstable process controlled by a PD controller, derivative action can stabilize the loop, but a derivative time that is too large destabilizes it. The Routh inequalities give an upper bound on the derivative time for a given process, which shows the trade-off between stabilization and derivative action. In a cascade structure, the inner and outer loops combine into one characteristic polynomial, and the Routh criterion gives the largest outer controller gain for which the cascade stays stable.

Control valve inherent characteristics

The inherent characteristic of a control valve is the relation between flow and percent lift when the pressure drop across the valve is held constant. The installed characteristic is the actual relation in the plant, where the pressure drop changes with flow.

• Linear valve — flow changes in proportion to lift, so equal lift increments give equal flow increments.
• Equal percentage valve — equal lift increments give equal percentage changes in flow, so the curve is exponential and the valve has high gain at high lift.
• Quick-opening valve — most of the flow change occurs at small lift, and the curve levels off near full lift.
• Hyperbolic valve — an intermediate form, often written as Q/Q_{max} = 1/[1 + (R-1)(1-x)], where R is the turndown ratio and x is the fractional opening.

A valve curve that is steep at low lift and flattens near full open is quick opening, and a curve that stays low and then rises steeply is equal percentage.

The turndown (rangeability) of a valve is the ratio of maximum flow to minimum controllable flow. For a hyperbolic valve, knowing the flow at one opening and the turndown ratio fixes the maximum flow, and the flow at any other opening follows from the same formula. Example: with R = 20 and a flow of 2 m^3/h at 50% opening, the flow at 25% opening is found by first computing Q_{max} and then substituting x = 0.25.

P&ID instrumentation symbols

A piping and instrumentation diagram (P&ID) uses standard symbols for equipment, piping, valves and instruments. Equipment such as vessels, columns, pumps and heat exchangers each have a fixed graphic form, and the reader identifies them by shape. Instruments are shown as circles, and their tag letters give the measured variable and the function.

In an instrument tag, the first letter names the measured variable (for example P for pressure, T for temperature, L for level, F for flow), and the following letters name the function. Common function letters are I for indicator, T for transmitter, C for controller, Y for relay or computing function, and the alarm and limit letters A (alarm), H (high) and L (low). Example: LAL denotes a low-level alarm.

An alarm element only signals an abnormal condition and does not regulate the process. A controller both measures and acts on a valve. For this reason, a high-pressure alarm tag is never the function of a throttling or pressure-control element.

Recognizing the equipment symbol is the first step in reading a P&ID. A pump, a packed column and a vessel are drawn with distinct shapes, and only symbols that appear in the diagram standard are used.

Second-order process step response overshoot

A second-order process with gain K is written as G(s) = \dfrac{K}{\tau^2 s^2 + 2\zeta\tau s + 1}, where \zeta is the damping ratio. If \zeta < 1, the response is underdamped and overshoots the new steady state. If \zeta \ge 1, the response is overdamped or critically damped and does not overshoot.

For an underdamped step response, the fractional overshoot is \exp\left(-\dfrac{\zeta\pi}{\sqrt{1-\zeta^2}}\right), which depends only on \zeta. The peak deviation is the step size times (1 + \text{overshoot}). When the input change is given in deviation variables, the new steady-state output equals the initial output plus K times the input change, and the overshoot is measured from that new steady state.

When a process has two real lags, the step response is monotonic and has no overshoot. To find the time of the maximum of a response that rises and then falls, write the step response as a sum of exponentials, differentiate it, and set the derivative to zero. Example: for y(t) = 1 + e^{-t} - 2e^{-t/2}, the derivative is always positive for t > 0, so there is no peak. For a response with a true peak, the root of dy/dt = 0 gives the time of the maximum.

Process measurement devices and measured variables

Each measuring device works on a specific physical principle and therefore measures one quantity. Knowing the principle identifies the variable.

• Bourdon gauge — an elastic tube that straightens under pressure, used to measure pressure.
• Pirani gauge — a thermal-conductivity gauge that measures low (vacuum) pressure from the heat loss of a heated filament.
• Orifice plate meter — a differential-pressure device that gives the flow rate from the pressure drop across a restriction.
• Pyrometer — a radiation device that measures high temperature from thermal radiation.
• Colorimeter — measures the concentration of a coloured solution from its light absorption.
• Thermocouple and resistance thermometer — measure temperature from an electrical signal.
• Float and differential-pressure level meters — measure liquid level.

A Bourdon gauge indicates pressure, and an orifice plate indicates flow rate, not pressure. A vortex shedding meter places a bluff body in the flow, and the frequency of the shed vortices is proportional to the mean velocity, f = St\,v/d, so the meter is used primarily to measure fluid flow rate.

Cascade control disturbance response

A cascade structure has two loops: a fast inner (secondary) loop that controls an intermediate variable, and a slower outer (primary) loop whose output is the set point of the inner controller. A disturbance that enters the inner loop is rejected by the inner loop before it reaches the primary output. The effect on the output is reduced by roughly the factor 1/(1 + G_{c2}G_{p2}) at the frequencies where the inner loop has high gain.

To find the disturbance transfer function, write the signal at each junction, eliminate the internal signals, and collect the output terms. The closed-loop denominator then contains the combined loop gains of both loops. A disturbance that enters the secondary loop appears in the output through a numerator that contains the outer controller only when the outer loop is closed.

For a block diagram with two loops, the general method is to set the set point to zero, write each output in terms of the controller and process transfer functions, and solve for the output in terms of the disturbance. For the disturbance d entering a secondary point, the denominator takes the form 1 + G_{c2}G_{p2} + G_{c1}G_{c2}G_{p1}, where each term is one loop gain, and the numerator is the product of the path gains from d to the output.

Process transfer function identification from step response

A transfer function can be identified from the shape of a step response. Each feature of the response corresponds to one element of the model.

• Dead time — no change in the output for a short interval after the step, modelled by e^{-\theta s}.
• Sigmoidal (S-shaped) rise — several lags in series, modelled by several first-order factors in the denominator.
• Initial overshoot followed by a slow tail — a lead term in the numerator, modelled by (\alpha s + 1).
• Initial change opposite to the final change — a right-half-plane zero (see inverse response).

The number of lags in the denominator is related to the number of inflection points in the response, and the dead time is read from the start of the rise.

The maximum slope method fits a first-order-plus-dead-time (FOPDT) model K e^{-\theta s}/(\tau s + 1) to a step response. Draw the tangent at the point of maximum slope S. Its intercept with the time axis gives \theta, and its intercept with the final value K gives \theta + \tau. The time constant is \tau = K/S, and the ratio \tau/\theta is a common measure of how easily a process can be controlled.

Instrument signal transmission standards

Measurement and control signals are sent between instruments and controllers in standard ranges so that devices from different makers can be connected.

• 4–20 mA DC — the most common current-loop standard, with 4 mA as the live zero.
• 3–15 psig — the classical pneumatic signal standard.
• 1–5 V DC — a standard voltage signal.
• 0–100 % — a percentage of the instrument span, used for scaling the signal in the controller, not a physical transmission standard.

A percentage scale describes the position of a signal within its range, while 4–20 mA, 3–15 psig and 1–5 V DC are the physical signal ranges that are actually transmitted.