ASHRAE Journal - August 2009 - 62

ASHRAE 19 50 59–2009 YE A R S JOURNAL 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 E = I (1 – Ae –t/ T1 + Be –t/T2); 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 Tl or T2 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 overdamped, 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 62 ASHRAE Journal SUMMARY The major points of interest in any control system are, according to a decreasing order of importance: 1. System stability; 2. Satisfactory steady-state performance; and 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. 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. a s h r a e. o rg August 2009

ASHRAE Journal - August 2009

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