5.6. PERFORMANCE INDICES
Modern complex control systems usually require more sophisticated performance criteria than those presented so far. As seen in the previous section, error and the time at which it occurs are very important factors that usually must be considered simultaneously. A performance index is a single measure of a system’s performance that emphasizes those characteristics of the response that are deemed to be important. The notion of a performance index is very important in estimator design using linear-state-variable feedback, which is presented in Sections 8.1 through 8.6, and in optimal control theory, where the system is designed to optimize this performance index given certain constraints. This subject will be discussed fully in Chapter 11.
Refrences 1 and 2 discuss an entire class of performance indices that are various functions of error and/or time. This section reviews those that are useful in the design of practical linear control systems.
A fairly useful performance index is the integral of the absolute magnitude of the error (IAE) criterion, which is
By utilizing the magnitude of the error, this integral expression increases for either positive or negative error, and results in a fairly good underdamped system. For a second-order system, S1 has a minimum for a damping ratio of approximately 0.7.
Another useful performance index is the integral of the square of the error (ISE) criterion, which is
By focusing on the square of the error function, it penalizes both positive and negative values of the error. For a second-order system, S2 has a minimum for a damping ratio of approximately 0.5.
A very useful criterion that penalizes long-duration transients is known as the integral of time multiplied by the absolute value of error (ITAE). It is given by
This performance index is much more selective than the IAE or the ISE. The minimum value of its integral is much more definable as the system parameters are varied. For a second-order system, S3 has a minimum for a damping ratio of 0.7.
Other figures of merit which have been proposed are the integral of time multiplied by the squared error (ITSE), the integral of squared time multiplied by the absolute value of error (IST AE), and the integral of squared time multiplied by square error (ISTSE). These performance indices are
The performance indices S4, S5, and S6 have not been applied to any great extent in practice because of the increased difficulty in handling them.
A comparison of this array of performance indices is very interesting. The ISE criterion is not very sensitive to parameter variations, because the minimum is usually broad [3]. In addition, the ISE criterion has the advantage of being easy to deal with mathematically. The IAE criterion gives a slightly better sensitivity than the ISE criterion. The ITAE criterion generally produces smaller overshoots and oscillations than the IAE and ISE criteria. In addition, it is the most sensitive of the three, and sometimes too sensitive—slight parameter variation degrades system performance [1, 2].
In practice, a relatively insensitive criterion may be more useful in those systems where the parameters may not be known very accurately. In addition, even though one tries to optimize a performance criterion, one may also have other performance characteristics in mind. Therefore, it is desirable in some applications to permit moderate deviation from the “optimum” setting of the parameters in order that these other performance characteristics can be achieved without appreciably increasing the performance index. Based on this logic, the ISE criterion may be the most desirable performance index in some practical applications and is considered further in Chapter 6 of the accompanying volume when optimal control theory is discussed. The reader is also referred to Chapter 2 of Reference 3 on the ISE criterion.
A paper by Graham and Lathrop [2] created a great deal of interest in the ITAE criterion. An important aspect of this paper was the detailed discussion and presentation of results of a comparison for various performance criterion with the ITAE. It is more sensitive than the IAE and ISE criteria, and is useful in those practical applications that require a very sensitive criterion. Because the ITAE criterion has practical value and is interesting academically, this performance criterion is studied further. First, however, we are interested in determining the form of the closed-loop system transfer function in order that zero error results for various kinds of inputs. This is necessary in order to determine the relationship of the various coefficients in the numerator and denominator of the closed-loop system transfer function in order to achieve zero error.