15.2 Feedback Control Loops, PID Tuning, and Control Hardware

Closed-Loop Control Systems

A closed-loop control system uses feedback to adjust a manipulated variable based on measurements of the process variable. The standard components of a feedback loop are:

1. Sensor/Transmitter ($H$): Measures the process variable (PV) and transmits a signal.
2. Comparator: Calculates the error signal, $E(s) = R(s) - Y_m(s)$, where $R(s)$ is the setpoint (SP) and $Y_m(s)$ is the measured PV.
3. Controller ($G_c$): Decides on corrective action based on the error.
4. Final Control Element / Valve ($G_v$): Adjusts the physical flow rate of the manipulated variable (MV).
5. Process ($G_p$): The system being controlled.
6. Disturbance ($G_d$): Uncontrolled variables that affect the process variable.

Closed-loop transfer functions represent the relationship between inputs and outputs. We differentiate between setpoint tracking (servo problem) and disturbance rejection (regulatory problem):

• Servo Transfer Function ($D(s) = 0$): $\frac{Y(s)}{R(s)} = \frac{G_c G_v G_p}{1 + G_c G_v G_p H}$
• Regulatory Transfer Function ($R(s) = 0$): $\frac{Y(s)}{D(s)} = \frac{G_d}{1 + G_c G_v G_p H}$

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Control Strategies

• Feedback Control: Measures the process variable and adjusts the manipulated variable. It requires no model of process disturbances, but it is purely reactive: control action is only initiated after the process variable deviates from the setpoint.
• Feedforward Control: Measures a disturbance directly and calculates the necessary adjustment to the manipulated variable to compensate before the disturbance affects the process variable. It is proactive and can theoretically achieve perfect control. However, it requires an accurate process model and cannot correct for unmeasured disturbances; therefore, it is almost always implemented in combination with a feedback loop.
• Cascade Control: Uses two nested control loops. The primary (outer) loop measures the primary variable (e.g., reactor temperature) and determines the setpoint for the secondary (inner) loop. The secondary loop measures and controls a faster intermediate variable (e.g., cooling water flow rate) to reject local disturbances (such as utility pressure fluctuations) before they can affect the primary variable. The secondary loop must respond significantly faster than the primary loop.
• Ratio Control: Standard configuration to maintain a fixed ratio between two flow streams (e.g., fuel and air in combustion, or reactants in a stoichiometric ratio). A "wild" stream is measured, and its flow rate is multiplied by the desired ratio to set the setpoint for the controlled stream's flow controller.

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PID Controller Equations

The Proportional-Integral-Derivative (PID) controller is the most common controller in the process industries. In the Laplace domain, its standard transfer function is: $G_c(s) = K_c \left(1 + \frac{1}{\tau_I s} + \tau_D s \right)$ where $K_c$ is the controller gain, $\tau_I$ is the integral time constant (reset time), and $\tau_D$ is the derivative time constant (rate time).

• Proportional Mode ($K_c$): Adjusts the output based on the current error. Increasing $K_c$ speeds up the response and reduces steady-state offset, but increases overshoot and destabilizes the system if too high. Proportional-only control results in a permanent steady-state offset for step changes in first- or second-order systems.
• Integral Mode ($\tau_I$): Eliminates steady-state offset by integrating the error over time. However, it adds phase lag, increases overshoot, and can cause integral windup. Windup occurs when a sustained error causes the integral term to accumulate to its limit, keeping the control valve saturated even after the process variable has crossed the setpoint.
• Derivative Mode ($\tau_D$): Reacts to the rate of change of the error, predicting future error. This provides a damping effect, reducing overshoot and settling time. However, derivative action cannot be used alone and is highly sensitive to measurement noise, which it amplifies.

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PID Tuning Methods

Tuning is the process of selecting the controller parameters ($K_c$, $\tau_I$, $\tau_D$) to achieve stable and fast control.

• Ziegler-Nichols Closed-Loop Method:
  1. Set the controller to proportional-only mode ($K_c > 0$, $\tau_I = \infty$, $\tau_D = 0$).
  2. Increase $K_c$ until the closed-loop system exhibits sustained, constant-amplitude oscillations. This value is the ultimate gain ($K_{cu}$), and the period of oscillation is the ultimate period ($P_u$).
  3. Calculate the tuning parameters using the following formulas:
     • P-control: $K_c = 0.5 K_{cu}$
     • PI-control: $K_c = 0.45 K_{cu}, \quad \tau_I = P_u / 1.2$
     • PID-control: $K_c = 0.6 K_{cu}, \quad \tau_I = 0.5 P_u, \quad \tau_D = 0.125 P_u$
• Ziegler-Nichols Open-Loop Method: Uses a step test on the open-loop process. The resulting process reaction curve is modeled as a first-order plus dead time (FOPDT) system: $G_p(s) = \frac{K_p e^{-\theta s}}{\tau_p s + 1}$ where $K_p$ is process gain, $\tau_p$ is time constant, and $\theta$ is dead time. Tuning coefficients are calculated from these three parameters.
• Cohen-Coon Method: An empirical method designed to achieve a quarter-amplitude decay response, particularly suited for systems with relatively large dead times ($\theta/\tau_p$).

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Control Hardware: Sensors, Valves, and DCS/PLC

Implementing control in a chemical plant requires robust instrumentation:

• Sensors:
  • Temperature: Thermocouples (generate voltage via the Seebeck effect at junctions of dissimilar metals) and RTDs (resistance changes linearly with temperature).
  • Flow: Orifice plates and Venturi tubes measure pressure drop ($\Delta P \propto F^2$), Coriolis meters measure mass flow directly, and rotameters measure volumetric flow.
  • Level: Differential pressure transmitters ($\Delta P = \rho g h$) and radar/ultrasonic level sensors.
• Control Valves: Regulate flow by varying the opening area. The valve flow capacity is characterized by the valve coefficient ($C_v$): $Q = C_v \sqrt{\frac{\Delta P}{SG}}$ where $Q$ is flow rate, $\Delta P$ is pressure drop across the valve, and $SG$ is specific gravity.
• Valve Characteristics: Flow capacity as a function of valve stem position:
  • Linear: Flow is directly proportional to valve travel.
  • Equal Percentage: Equal increments of valve travel produce equal percentage changes in flow. Used for pressure control and long piping lines.
  • Quick Opening: Large flow changes occur at low valve travel. Used for on-off service and safety shutdown.
• Valve Failure Positions: Selected based on safety considerations in the event of loss of power or instrument air:
  • Fail-Open (FO / Air-to-Close ATC): Valve opens when air pressure is lost. Used for cooling water lines to prevent reactor runaway, or pressure relief lines.
  • Fail-Closed (FC / Air-to-Open ATO): Valve closes when air pressure is lost. Used for fuel feed lines or reactant feed lines to isolate the unit and stop chemical reactions.
• DCS vs. PLC: Distributed Control Systems (DCS) are used for large continuous chemical processes requiring complex PID loops and central monitoring. Programmable Logic Controllers (PLCs) are used for high-speed discrete control, interlock systems, and emergency shutdown systems.