Final Control Elements
A final control element is a device that receives the manipulated variable from the controller as input and takes action that influences the process in the desired manner. In the process industries valves and pumps are the most common final control elements, because of the necessity to adjust a fluid flow rate such as coolant, steam, or the main process stream.
Control Valves
The control valve is designed to be remotely positioned by the controller output, which is usually an air pressure signal. The valve design contains an opening with variable cross sectional area through which the
fluid can flow. A stem that travels up or down according to a change in the manipulated variable changes the area of the opening and thereby the flowrate. At the end of the stem is a plug of specific shape, which fits into a seat or ring on the perimeter of the valve port. This plug/seat combination is called the valve trim, and it determines the steady-state gain characteristics of the control valve. Based on safety considerations the control valve can be configured to be either air-to-open or air-to-close.
The inherent characteristics of control valves allow classification into three main groups, based on the relationship between valve flow and valve position under constant pressure: linear, equal-percentage, and quick opening (Seborg, Edgar, and Mellichamp, 2004; see Fig. 18.51). Usually in-plant testing is used to determine the actual valve characteristics because the dynamics of the valve can depend on other flow resistances in the process.
The response time of a pneumatic valve is related to the distance from the actuator to the valve, because the pneumatic signal has to travel the distance between the actuator and the valve. It is important for the design of the process that the response time of the valve is at least one order of magnitude faster than the process. Usually valve response times are several seconds, which is small compared to process time constants of minutes and hours. Pneumatic valve performance often suffers from nonideal behavior such as valve hysteresis. In order to achieve better reproducibility and a faster response a valve positioner can be installed to control the actual stem position rather than the diaphragm air pressure (Edgar, Smith, Shinskey, Gassman, Schafbuch, McAvoy, and Seborg, 1997).
Adjustable Speed Pumps
Instead of using fixed speed pumps and then throttling the process with a control valve, an adjustable speed pump can be used as the final control element. In these pumps speed can be varied by using variable- speed drivers such as turbines or electric motors. Adjustable speed pumps offer energy savings as well as performance advantages over throttling valves to offset their higher cost. One of these performance advantages is that, unlike a throttling valve, an adjustable pump does not contain dead time for small amplitude responses. Furthermore, nonlinearities associated with friction in the valve are not present in electronic adjustable speed pumps. However, adjustable speed pumps do not offer shutoff capability like control valves and extra flow check valves or automated on/off valves may be required.
Feedback Control Systems
Feedback control is a fundamental concept that is employed in PID controllers as well as in advanced process control techniques. Figure 18.52 shows a simplified instrumentation diagram for feedback control of the stirred tank discussed earlier, where the inlet flow is equal to the outlet flow (hence, no level control is needed) and outlet temperature is controlled by the steam pressure. Figure 18.53 shows a generic block diagram for a feedback control system. In feedback control the controlled variable is measured
and compared to the set point. The resulting error signal is then used by the controller to calculate the appropriate corrective action for the manipulated variable. The manipulated variable influences the controlled variable through the process, which is dynamic in nature.
In many industrial control problems, notably those involving temperature, pressure, and flow control, measurements of the controlled variable are available, and the manipulated variable is adjusted via a control valve. In feedback control, corrective action is taken regardless of the source of the disturbance. Its chief drawback is that no corrective action is taken until after the controlled variable deviates from the set point. Feedback control may also result in undesirable oscillations in the controlled variable if the controller is not tuned properly, that is, if the adjustable controller parameters are not set at appropriate values. The tuning of the controller can be aided by using a mathematical model of the dynamic process, although an experienced control engineer can use trial-and-error tuning to achieve satisfactory performance in many cases. Next, we discuss the two types of controllers used in most commercial applications.
On-Off Controllers
An on-off controller is the simplest type of feedback controller and is widely used in the thermostats of home heating systems and in domestic refrigerators because of its low cost. This type of controller is seldom used in industrial plants, however, because it causes continuous cycling of the controlled variable and excessive wear on the control valve.
In on-off control, the controller output has only two possible values:
Three Mode (PID) Controllers
Most applications of feedback control employ a controller with three modes: proportional (P), integral (I), and derivative (D). The ideal PID controller equation is
where p is the controller output, e is the error in the controlled variable, and p¯ is the bias, which is set at the desired controller output when the error signal is zero. The relative influence of each mode is determined by the parameters Kc , τI , and τD .
The PID controller input and output signals are continuous signals which are either pneumatic or electrical. The standard range for pneumatic signals is 3–15 psig, whereas several ranges are available for electronic controllers including 4–20 mA and 1–5 V. Thus, the controllers are designed to be compatible with conventional transmitters and control valves. If the PID controller is implemented as part of a digital control system, the discrete form of the PID equation is used,
where Λt is the sampling period for the control calculations and n denotes the current sampling time. Digital signals are scaled from 0 to 10 V DC.
The principle of proportional action requires that the amount of change in the manipulated variable (e.g., opening or closing of the control valve) vary directly with the size of the error. The controller gain Kc affects the sensitivity of the corrective action and is usually selected after the controller has been installed. The actual input-output behavior of a proportional controller has upper and lower bounds; the controller output saturates when the limits of 3 or 15 psig are reached. The sign of Kc depends on whether the controller is direct acting or reverse acting, that is, whether the control valve is air-to-open or air-to-close.
Integral action in the controller brings the controlled variable back to the set point in the presence of a sustained upset or disturbance. Because the elimination of offset (steady-state error) is an important control objective, integral action is normally employed, even though it may cause more oscillation. A potential difficulty associated with integral action is a phenomenon known as reset windup. If a sustained error occurs, then the integral term in Eq. (18.9) becomes quite large and the controller output eventually saturates. Reset windup commonly occurs during the start up of a batch process or after a large set-point change. Reset windup also occurs as a consequence of a large sustained load disturbance that is beyond the range of the manipulated variable. In this situation the physical limitations on the manipulated variable (e.g., control valve fully open or completely shut) prevent the controller from reducing the error signal to zero. Fortunately, many commercial controllers provide antireset windup by disabling the integral mode when the controller output is at a saturation limit.
Derivative control action is also referred to as rate action, preact, or anticipatory control. Its function is to anticipate the future behavior of the error signal by computing its rate of change; thus, the shape of the error signal influences the controller output. Derivative action is never used alone, but in conjunction with proportional and integral control. Derivative control is used to improve the dynamic response of the controlled variable by decreasing the process response time. If the process measurement is noisy, however, derivative action will amplify the noise unless the measurement is filtered. Consequently, derivative action is seldom used in flow controllers because flow control loops respond quickly and flow measurements tend to be noisy. In the chemical industry, there are more PI control loops than PID.
To illustrate the influence of each controller mode, consider the control system regulatory responses shown in Fig. 18.54. These curves illustrate the typical response of a controlled process for different types of feedback control after the process experiences a sustained disturbance. Without control the process slowly reaches a new steady state, which differs from the original steady state. The effect of proportional control is to speed up the process response and reduce the error from the set point at steady state, or offset. The addition of integral control eliminates offset but tends to make the response more oscillatory.
Adding derivative action reduces the degree of oscillation and the response time, that is, the time it takes the process to reach steady state. Although there are exceptions to Fig. 18.54, the responses shown are typical of what occurs in practice.
Stability Considerations
An important consequence of feedback control is that it can cause oscillations in closed-loop systems. If the oscillations damp out quickly, then the control system performance is generally considered to be acceptable. In some situations, however, the oscillations may persist or their amplitudes may increasewith time until a physical bound is reached, such as a control valve being fully open or completely shut. In the latter situation, the closed-loop system is said to be unstable. If the closed-loop response is unstable or too oscillatory, this undesirable behavior can usually be eliminated by proper adjustment of the PID controller constants, K , τ , and τ . Consider Kc 2 Kc 2 > Kc1 c I D the closed-loop response to a unit step change in setpoint for various values of controller gain Kc . For small values of Kc , say Kc 1 and Kc 2, typical closed-loop responses are shown in Fig. 18.55. If the controller gain is increased to a value of Kc 3, the sustained oscillation of Fig. 18.56 results; a larger gain, Kc 4 produces the unstable response shown in Fig. 18.56. Note that the amplitude of the unstable oscillation does not continue to grow indefinitely because a physical limit will eventually be reached, as previously noted. In general as the controller gain increases, the closed-loop response typically becomes more oscillatory and, for large values of Kc , can become unstable.
The conditions under which a feedback control system becomes unstable can be determined theoretically using a number of different techniques (Seborg, Edgar, and Mellichamp, 2004).
Manual/Automatic Control Modes
In certain situations the plant operator may wish to override the automatic mode and adjust the controller output manually. In this case there is no feedback loop. This manual mode of operation is very useful during plant start-up, shut-down, or in emergency situations. Testing of a process to obtain a mathematical model is also carried out in the manual mode. Commercial controllers have a manual automatic switch for switching from automatic mode to manual mode and vice versa. Bumpless transfers, which do not upset the process, can be achieved with commercial controllers.
Tuning of PID Controllers
Before tuning a controller, one needs to ask several questions about the process control system:
✁ Are there any potential safety problems with the nature of the closed-loop response?
✁ Are oscillations permissible?
✁ What types of disturbances or set-point changes are expected?
There are several approaches that can be used to tune PID controllers, including model-based correlations, response specification, and frequency response (Seborg, Edgar, and Mellichamp, 2004; Ogunnaike and Ray, 1995). One approach that has recently received much attention is model-based controller design. Model-based control requires that a dynamic model of the process is available. The dynamic model can be empirical, such as the popular first-order plus time delay type of model, or it can be of higher order.
For most processes it is possible to find a range of controller paramters (Kc , τI , τD ) that give closed loop stability; in fact, most design methods provide ∆y a reasonable margin of safety for stability and the controller parameters are chosen in order to meet dynamic performance criteria in addition to guaranteeing stability.
One well-known technique for determining an empirical model of a process is the process reaction curve (PRC) method. In the PRC technique, the actual process is operated under manual control, and a step change in the controller output (Λp) is carried 0 θ out. The size of the step is typically 5% of the span, depending on noise levels for the process variables.
One should be careful to take data when other plant fluctuations are minimized. For many processes, the response of the system Λy (change in the measured value of the controlled variable using a sensor) follows the curve shown in Fig. 18.57 (see case 6 in Fig. 18.50). Also shown is the graphical fit bya first order plus time delay model, which is described by three parameters:
1. the process gain K
2. the time delay θ
3. the dominant time constant τ
While nonlinear regression can be employed to find K , θ, and τ from step response data, graphical analysis of the step response can provide good estimates of θ and τ . In Fig. 18.57 the process gain is a steady-state characteristic of the process and is simply the ratio Λy/Λp. The time delay θ is the time elapsed before Λy deviates from zero. The time to reach 63% of the final response is equal to θ + τ .
A variety of different tuning rules exists which are based on the assumption that the model can be accurately approximated by a first-order plus time delay model. One popular approach, which is not
restricted to this type of model but nevertheless achieves good performance for it, is internal model control (IMC) (Rivera, Morari, and Skogestad, 1986; Bequette, 2003; Seborg, Edgar, Mellichamp, 2004). Controller tuning using IMC requires only choosing the value of the desired time constant of the closed- loop system, τc , in order to compute the values of the three controller tuning parameters, KC , τI , τD . The value of τc can be further adjusted based on simulation or experimental behavior. The tuning relations for a PI and PID controller for a process which is described by a first-order plus time delay model are shown in Table 18.5 (Chien and Fruehauf, 1990). Tuning relations for other types of process models can be found in Seborg, Edgar, Mellichamp (2004).
Another popular tuning method uses integral error criteria such as the time-weighted integral absolute error (ITAE). The controller parameters are selected in order to minimize the integral of the error. ITAE weights errors at later times more heavily, giving a faster response. Seborg, Edgar, and Mellichamp (2004) have tabulated power law correlations for these PID controller settings for a range of first-order model parameters K, τ , and θ .
Other plant testing and controller design approaches such as frequency response can also be used for more complicated models; refer to Seborg, Edgar, and Mellichamp (2004) for more details.
On-Line Tuning
The model-based tuning of PID controllers presumes that the model is accurate; thus these settings are only a first estimate of the best parameters for an actual operating process. Minor on-line adjustments of Kc , τI , and τD are usually required. Another technique, called continuous cycling, actually operates the loop at the point of instability, but this approach has obvious safety and operational problems, so it is not used much today. A recent improvement on continuous cycling uses a relay feedback configuration, which puts the loop into a controlled oscillation. The amplitude and period of the oscillation can be used to set proportional and integral settings.