Mathematically speaking, the order of a continuous process equals the number of differentiation operators required to construct its governing equation. In layman's terms, that describes how dramatically the process variable can change when a control effort is applied to the process. Higher order processes demonstrate more complicated behavior and are commensurately harder to control.
At A Glance
Basic concepts defined
Fundamental formulas explained
Simple examples
Variations on the theme
Mathematically speaking, the order of a continuous process equals the number of differentiation operators required to construct its governing equation .In layman’s terms, that describes how dramatically the process variable can change when a control effort is applied to the process. Higher order processes demonstrate more complicated behavior and are commensurately harder to control.Consider, for example, the well-insulated kettle of beer brewing in Figure 1. An adjustable burner determines the amount of heat that enters the beer, and a thermocouple measures its temperature. As shown in Figure 2, the rate at which the beer heats up (T process ) is proportional to the difference between the temperature of the burner (T control ) and the temperature of the beer (T process ). The higher that difference, the faster the beer temperature changes.
The beer in this industrial kettle is heated by the burner below. By varying the gas flow, the temperature of the burner (Tcontrol ) can be manipulated so as to control the temperature of the beer (Tprocess ).
Figure 2 Governing equation
This first order relationship governs how the temperature of the beer (Tprocess ) will respond to the temperature of the burner (Tcontrol ). Tprocess is the time derivative of Tprocess. The value of the fixed time constant t depends on the thermodynamic properties of the process. A lower value of t represents a more rapid transfer of heat from the burner into the beer.
The relationship shown in Figure 2 is the governing equation of the brewing process. It is a first order relationship since it contains just one differentiation operator. The process itself is also described as being first order.Like most governing equations, this one can be solved for the process variable (T process ) by eliminating the differentiation operator through a series of convoluted mathematical manipulations. Fortunately, the end result is relatively straightforward in the special case of a control effort (T control ) that undergoes a step change as shown in Figure 3.Initially, the temperature of the beer begins to rise rapidly, but as the beer gets warmer and warmer, its temperature rises slower and slower until it reaches the temperature of the burner. The temperature of the beer can never exceed the temperature of the burner, so the process variable can never oscillate unless the control effort should happen to start oscillating.
A more complex example
Figure 4 shows a mechanical process that is capable of oscillating without being forced. It consists of a weight hanging from a spring. The weight’s vertical position is determined by the downward pull of gravity, the upward pull of the spring, and the force of friction that opposes the weight’s motion in either direction.The spring force is proportional to the weight’s position (X process ), the frictional force is proportional to the weight’s velocity (X process ), and the force of gravity is proportional to the weight’s acceleration (X process ). Combining these with the displacement of the handle (X control ) yields the governing equation shown in Figure 5. This is a second order relationship because it contains both a single differentiation operator and a double differentiation operator.
The brewing process will respond with an exponentially decaying rise in the process variable (Tprocess ) when the control effort (Tcontrol ) changes abruptly from zero to DT at time zero.
Unlike the first order process in the previous example, this second order process will sometimes oscillate even if the control effort does not. Specifically, if the friction is low enough, the spring is stiff enough, and the weight is heavy enough, the process will be underdamped and a simple tug will cause the weight to bounce as shown in Figure 6. Conversely, an overdamped second order process will respond without oscillations as shown in Figure 7.
Other processes
The relationships shown in Figures 3, 6, and 7 are not unique to brewing beer and bouncing springs. They apply to all first and second order processes—just about everything that can be controlled with a traditional PID (proportional-integral-derivative) loop. Only the values ofτ, ζ, and ω n change from one process to the next. All first order step responses look like Figure 3, and all second order step responses look like either Figure 6 or Figure 7. Only the scale changes.In the first order case, the time constantτ determines the duration of the step response by defining exactly when the process variable will reach 63% of its final value (which is to say, T process = 0.63ΔX when t = τ). The smaller the value of τ, the sooner that 63% point will be reached, and conversely.Similarly, the values of the damping ratioζ and the natural frequency ω n completely determine the duration and amplitude of an underdamped second order step response. As shown in Figure 6, the inverse ofζ ω n serves as the time constant for the decaying exponential term e-ζωnτ. The productω n √1-ζ2(that is,αω n ) serves as the frequency of the sinusoidal term sin(αω n t+φ). The phase of that sinusoid is φ and its amplitude is 1/α, both of which are determined by the magnitude of ζ.
Feedback control
The object of this child’s toy is to bounce the weight up and down. The position of the weight (Xprocess ) can be controlled by manipulating the position of the handle (Xcontrol ).
Figure 5 Governing equation
This second order relationship governs how the position of the weight (Xprocess ) will respond to the position of the handle (Xcontrol ). Xprocess is the time derivative of Xprocess and Xprocess is its second derivative. The fixed values of the damping ratio z and the natural frequency vn depend on the physical properties of the process—the mass of the weight, the viscosity of the frictional force, and the strength of the spring.
These relationships are particularly convenient for feedback controller design. They allow a controller to predict the step response of any first order process with a known time constant or any second order process with a known damping ratio and natural frequency. The controller can then be configured to apply just the right series of steps required to drive the process variable towards the desired setpoint along a desired trajectory.Just how the values ofÏ„, ζ, and ω n can be translated into appropriate controller parameters, such as P, I, and D, is often a matter of significant technical complexity. Fortunately, there are formulas available for many of the simplest cases. See, for example, the famous Ziegler-Nichols tuning rules for first order processes in “Loop Tuning Fundamentals,” Control Engineering , July 2003, at www.controleng.com .Determining the values ofÏ„, ζ, and ω n that best represent the behavior of the process in question can also be a challenge. First principles analysis uses the laws of chemistry, physics, electricity, and thermodynamics to deduce their values analytically. This tends to be the preferred approach for academic control problems.
If the damping ratio (z) is between zero and one, the process is underdamped and its process variable (Xprocess ) demonstrates a decaying sinusoidal response when the control effort (Xcontrol ) changes abruptly from zero to DX at time zero. The constants a and f are determined by z according to:
If the damping ratio (z) is greater than one, the process is overdamped and its process variable (Xprocess ) demonstrates a step response very similar to a first order step response. The constants b, t1, and t2 are determined by z and vn according to:
Empirical analysis , on the other hand, is considerably simpler and generally more practical for most industrial applications. An operator simply forces a step change in the control effort and compares the results to a collection of charts like Figures 6 and 7, each plotted with different values ofζ and ω n . The values ofζ and ω n that best represent the behavior of the process can then be read off the chart that most resembles the step response recorded by the operator. Or, if it turns out that the process behaves in more of a first order fashion, a collection of first order step responses can be consulted for the value of t that best fits the experimental results.
Variations
It is also possible to find that no first or second order step response will match the experimental results. The process could have an order greater than two, especially if it is comprised of multiple elements. For example, if a spring-loaded valve were used to regulate the flow of gas into the brew kettle’s burner, the combined process would have a third order governing equation since a process of order Y in series with a process of order Z yields a process of order Y+Z. Fortunately, higher order processes can often be well approximated with first and second order governing equations.
A more common variation on the basic first or second order process is deadtime . Deadtime is the interval that elapses between the application of a control effort and its first effect on the process variable. The process variable remains constant until the deadtime has elapsed. This has the effect of delaying the start of a step response, but does not otherwise change its shape. Deadtime often occurs in fluid flow processes where measurements are taken downstream from the point where the control efforts are applied.
A related phenomenon—known as sag or droop —can also delay the start of the step response, but not by holding the process variable constant. Instead, the process variable actually starts to decrease when the control effort increases. It then changes course and starts to rise in the right direction. Both first and second order processes can demonstrate sag, making them extremely difficult to control. Fortunately, sag is a fairly rare phenomenon.
Equally difficult to control are processes that demonstrate negative damping ; that is to say, negative values ofζ or τ. Rather than diminishing over time, the exponential term in the step response grows larger and larger. A feedback controller can be designed to actively compensate for such unstable behavior, but even the slightest mistake in the choice of a particular control effort can have disastrous consequences.
Finally, there are some first and second order processes that are non-linear by virtue of having values forÏ„, ζ, and ω n that vary over time. If such variations are slow enough and the values ofÏ„, ζ, and ω n can be automatically deduced while the controller is in operation, the controller can be designed to compensate by continuously redesigning its control strategy on-line. For more on such adaptive controllers , see “Techniques for Adaptive Control,” available from Control Engineering online at www.controleng.com/process .
