# Fundamentals of linear control theory.

In the following presentation of linear feedback control theory, the author discusses basic concepts, time constants, system analysis, and stability.

BASIC CONCEPTS

There are three topics to consider under basic concepts: (1) systems, (2) transients, and (3) feedback.

Systems

It is necessary to consider the entire control system as an entity. Every element has an influence on the output and must be considered in determining the overall system characteristics. Take, for example, the room temperature control system shown in Fig. 1. Changes in the load affect the room air temperature (output) in a manner determined by the room characteristics (controlled system). The temperature change is sensed by the bimetal (sensing element). The resulting motion activates the valve top (control elements), resulting in a change in valve position. This change in valve position modulates the output of the convector (mechanical equipment), thereby changing the room air temperature. An essential point is that each component takes time to respond, thus contributing to the controllability of the system and thus making it necessary to include time as a variable in the system analysis.

Transients

The consideration of time in the analysis implies that conditions are changing with time. This consideration is fundamental to the understanding of control theory. In the above example, the load is influenced by atmospheric conditions such as wind, temperature, solar radiation, etc. Because these factors change continuously, the system is always in a transient state; thus, the transient behavior is fundamental.

A quantity which is solely dependent on time is called a function of time and is mathematically written as f(t) or g(t). In this article, we will deal with such functions in systems of linear devices. A device is defined as linear if its output to input ratio is a function of time and not dependent on the magnitude of the input signal. For example, if the output of a device is f(t) for an input of g(t), then the output will be 2f(t) for an input of 2g(t).

A system is linear if all the devices in that system are linear. Strictly speaking, if hysteresis, saturation, or dead zone is present, the system is not linear. If the effects of these non-linearities are small, however, or if they fall outside the range of operation, the general behavior can be described by linear analysis.

Feedback

The third basic concept is feedback, and we will see how the system concept and the transient concept are brought together with the concept of feedback. The significance of feedback is that it permits the continuous adjustment of the system so as to minimize the difference between the actual output and a desired output (set point). When a system has feedback, it is called a closed loop system.

[FIGURE 1 OMITTED]

The system shown in Fig. 1 is a closed loop system. When a system does not have feedback, it is an open loop system. In open loops the corrective effort is determined by the set point alone. Room temperature controlled by an outside temperature controller is an example of an open loop system. In general, better control is obtained by a closed loop system.

So far we have considered only system feedback; however, there is also a secondary feedback when the controller itself has a feedback loop, this is frequently referred to as internal feedback. The benefit of feedback within an instrument lies in the ability to use larger signals and then reduce the signal by feedback. This generally develops superior performance without need for precision manufacturing.

TIME CONSTANTS

There are two topics that should be considered under the concept of time constants: (1) response time (or lag) and (2) dead time (or transportation lag). When we discussed systems it was indicated that each component takes time to respond to a change in input. This response time, T, is related to the physical properties of the component; therefore, it is predictable and independent of the change in input.

The two properties necessary to establish T are resistance, R, and capacitance, C. These are usually thought of as electrical terms, but actually can be thermal or hydraulic terms also. As in electrical terminology, resistance is the opposition to flow, and capacitance is the change in quantity per unit change in potential.

Potential can be given in units of electrical strength, temperature, or pressure. The product of resistance and capacitance determines the response time, written T = RC. The units of T are time units (seconds, minutes, etc.).

If a system has only one resistance and one capacitance, it is called a first-order system, and the value of T explicitly determines the time response of the system. If, however, there are two or more resistances and capacitances in a system, there is no single time constant as such, and the response is related to the products of the individual resistances and capacitances. The number of these products that exist determines the order of the system.

Let us consider an example of a stem thermometer in thermal equilibrium suddenly being immersed in a bath at a higher temperature. The thermal resistance retards the heat flow from the bath into the thermometer and the capacitance (mass and specific heat) determines the rate of temperature rise. As the thermometer warms, the rate of heat flow diminishes, due to the reduced thermal potential, and finally reaches zero when the bath and thermometer are at the same temperature. At any instant the thermometer temperature is E = I (l - [e.sup.-t/T]), where I is the initial temperature difference and t is the time after immersion. If T is known, the entire time response curve can be computed for this first-order system. A typical curve for a first-order system is shown in Fig. 2.

However, if the thermometer were placed in a well, then the resistance and capacitance of the well would enter into the response curve. If it is assumed that [T.sub.l] represents the response time of the thermometer and [T.sub.2] the well, then the above example would be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (Note: A and B are constants determined by [T.sub.1] and [T.sub.2].) This is a second-order system, and no single value such as T suffices to completely explain the system response.

Another complication arises if the thermometer signal is transmitted sufficiently far to cause a measurable delay. For example, assume that the thermometer is replaced by an instrument called a temperature transmitter, and that this transmitter is placed some 500 ft away from the receiver. The 500 ft will be connected in this example by a copper tube that will transmit a pneumatic signal. The signal (in this case a pressure) will now be restricted in the tube and the capacitance of the receiver will add a time constant term to the response equation (remember the transmitter will also have a response time,).

In addition to these two response times, the signal will take a certain amount of time to travel the 500 ft. This lag is called transportation lag, or dead time. This delay is not serious in an indicating system, but can be significant and worth considering in a controlling system. Fig. 2 illustrates three typical response curves for the examples that have been discussed.

It can be shown mathematically that a single-order system cannot overshoot; hence, it is always stable. A second-order system may oscillate but is always stable from a mathematical standpoint. A third or higher order system may be truly unstable. Most systems encountered in the field are much higher than third order but, since many of their characteristics can be neglected, it is possible to approximate the response of the system by using an equivalent third or even second-order system for the purpose of analysis.

[FIGURE 2 OMITTED]

SYSTEM ANALYSIS

Armed with these concepts and tools, we are now ready to consider system analysis. The acceptability of a control system usually can be described by two interdependent criteria: (1) steady state and (2) dynamic performance. Generally, one must be compromised somewhat by the other. We will now try to see how these two criteria are related, and also to see their relative importance in describing the complete system.

As indicated previously, an actual system never truly reaches equilibrium, however, for all practical purposes systems do settle out so that the output falls within the offset band as shown in Fig. 3. The width of this band is specified and is a yardstick with which to measure control accuracy.

One way to reduce offset is to increase controller gain, another is to use reset (more properly called integral action). An increase in gain increases the chances of instability and the introduction of reset action decreases the speed of response, in other words, each has its disadvantage and the improvement of the steady-state performance is made at the expense of dynamic performance.

Dynamic Criteria

There are two criteria for system performance in the dynamic state. (1) transient response and (2) frequency response. Transient response refers to the response of a system to a single, or non-recurring, change of input, as for example, a step change. The output signal is recorded as a function of time. The equation of this timeoutput plot is used to determine a transfer function. This mathematical expression enables the engineer to predict and describe the complete system performance.

Frequency response refers to the response of a system to a sinusoidally changing input. Again, the output is recorded as a function of time. From this plot, a ratio of output to input amplitude is made and the phase shift of output and input sinusoids is determined. These terms are then used to derive the transfer function of the system. In air-conditioning systems, response times are quite long and, as a result, frequency response techniques are seldom feasible on an actual system.

Use of Transient Response

Transient response data can be expressed by two indexes--overshoot and settling time. The proper combination of these determines the acceptability of the dynamic performance, just as offset determines the acceptability of the steady-state performance. Improvements in these indexes can be made by adjusting the controller gain or by using additional modes, such as rate action, and/or reset action.

The purpose of system analysis is to enable the designer or operator to adjust the system so as to achieve the new steady state in as short a time as is consistent with the limitations of the equipment and in such a way as to maintain specified steady-state performance.

STABILITY

The fourth and last part of this paper deals with the most important characteristic of a control system--stability. Typical response curves as given in Fig. 2 were indicative of stable systems. Fig. 4, however, illustrates an unstable system. The significant difference in these two illustrations can be described in terms of the way the final value is reached after an input disturbance.

Additionally, however, we have a system, as illustrated in Fig. 3, that for practical purposes can also be considered a stable system and that reaches a final value in still a third manner. Let us consider these three classes of systems. They are overdamped, underdamped, and unstable. Overdamped systems, as illustrated in Fig. 2, gradually approach the final value, and at no time have an output signal that exceeds the desired final value. Underdamped systems, sometimes called oscillatory systems, are illustrated in Fig. 3. These systems oscillate about the final value several times, but with decreasing overshoot, until finally the amplitude of the oscillations falls within the offset band.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Truly unstable systems, as illustrated in Fig. 4, oscillate about the final value indefinitely with increasing amplitudes until one or more of the components in the system saturates, at which time the amplitude of overshoot remains constant.

There are mathematical ways to describe the degree of stability, and these expressions are often used in industry. Stability in a linear feedback system exists only when the exponents of the output equation are negative. If the exponents were positive, the output would be as shown in Fig. 4. Note that when we examined the thermometer system above, the response equation was [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; the exponents were determined solely by the time constants.

Since the time constants for the real systems are, of course, positive, this system will always have negative exponents. (A positive exponent would require a negative value for [T.sub.1] or [T.sub.2] which is, of course, impossible.) It is for this reason that all open loop systems are stable. When feedback is employed, the exponents of the transient equations are no longer solely dependent on the component time constants, but are also dependent on the system gain. It is now possible for the interrelation of time constants and gain to create positive exponents in the transient equation, resulting in an unstable system.

The amount or degree of damping of a system illustrated by Fig. 3 depends upon the magnitude of the damping ratio. When this ratio is less than one, the system oscillates; when it is greater than one, the system is over-damped, that is, its response is represented by Fig. 2.

Strictly speaking, the damping ratio should be called a damping factor when it is equal to or greater than one. For practical systems, a good value for the damping ratio is about 0.6. Lower values cause over-sensitive systems that produce excessive oscillations, while higher values cause sluggish systems. Both of these systems require excessive settling time.

Instability does not result from the undesirable properties of any one component alone, but rather from the collection of properties, both good and bad, of all the components. It is necessary, therefore, to realize that no matter how insignificant component properties might seem, when viewed alone, they may contribute a significant part to the instability of the system.

When an actual control system is unstable, there are a few standard cures; the first is usually to lower the controller gain until the maximum permissible offset is reached. If this is not sufficient to stabilize the system, the gain is lowered still further until the system is stable and then integral action is added to reduce the offset. Either of these methods will work, but they certainly are not the only methods that can be employed. It might also be possible to alter the properties of one or more components in order to stabilize the system without increasing the offset band and without resorting to reset action. Also, it is true that in actual systems non-linearities affect stability. Fortunately, however, some non-linearities are helpful under certain conditions. Friction in a valve will help dampen a highly oscillatory system.

SUMMARY

The major points of interest in any control system are, according to a decreasing order of importance:

1. System stability;

3. Satisfactory response to transient conditions.

If a system is unstable, it is usually of no value whatsoever; therefore, the question of stability is of paramount importance. If the steady-state performance is not satisfactory, then adjustments or changes must be made before transient conditions are checked.

Finally, if the system is stable and has satisfactory steady-state performance, transient information is required to determine the system response time and overshoot.

It should be emphasized that it is absolutely essential that we understand and can define the transient and dynamic characteristics of all the mechanical equipment in an air-conditioning, refrigeration, heating or ventilating system.

Unless these characteristics are known, we cannot truly design or operate with optimum performance. Fortunately, ASHRAE has embarked on a research program called, Air Conditioning System Control, and the results of this program will give us some indication of system requirements.

It is imperative that more research of a similar nature be carried out by ASHRAE until it becomes standard practice for manufacturers to publish the transient, as well as the steady-state, characteristics of their equipment.

WILLIAM P. CHAPMAN, P.E., PRESIDENTIAL MEMBER/FELLOW ASHRAE

William P. Chapman, P.E., Presidential Member/Fellow ASHRAE, served as president of ASHRAE in 1976-77 and was the director of research and development at Johnson Service Company, Milwaukee.

After joining ASHVE in 1947, Chapman served on many Society committees. His service included the Presidential Ad Hoc International Standards Committee, the Research Administrative Council, a task group on air-conditioning system control, a task group on dynamic response, the Research and Technical Committee, chairing the Joint Expo Policy Committee, the Scholarship Trust Fund, the Global Warming Position Paper Committee, the Nominating Committee, the Publications Committee and the Guide Book Committee.

He served on the Historical Committee and TC 6.1, Hydronic and Steam Equipment Systems. He was elevated to the grade of Fellow in 1969 and received the Distinguished Service Award in 1969, the F. Paul Anderson Award in 1983, the ASHRAE-ALCO Medal for Distinguished Public Service in 1993, and the Distinguished 50-Year Member Award in 1998.

When presenting Chapman with the Anderson award, Presidential Member Clinton Phillips recognized him as "an engineer, researcher and author who has shaped the technical evolution of building system controls, benefiting mankind through the improvement of living conditions."

Chapman retired from Johnson Controls in 1984. He died in April 2002 at age 82.

BY WILLIAM P. CHAPMAN, P.E., PRESIDENTIAL MEMBER/FELLOW ASHRAE
COPYRIGHT 2009 American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc.
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