Reference values and interpretation of Cronbach’s alpha (KR20).
\r\n\tThis book intends to provide the reader with a comprehensive overview of the current state-of-the-art novel imaging techniques by focusing on the most important evidence-based developments in this area.
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He has authored or co-authored peer-reviewed articles and book chapters in the field of cardiac pacing, defibrillation, electrophysiological study, and catheter ablation.",coeditorOneBiosketch:"Raluca Tomoaia is an MD, Ph.D. in novel techniques in Echocardiography at the University of Medicine and Pharmacy in Cluj-Napoca, Romania., assistant professor, and a researcher in echocardiography and cardiovascular imaging.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"191888",title:"Dr.",name:"Gabriel",middleName:null,surname:"Cismaru",slug:"gabriel-cismaru",fullName:"Gabriel Cismaru",profilePictureURL:"https://mts.intechopen.com/storage/users/191888/images/system/191888.png",biography:"Dr. Cismaru Gabriel is an assistant professor at the Cluj-Napoca University of Medicine and Pharmacy, Romania, where he has been qualified in cardiology since 2011. 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It is defined as “the systematic collection and analysis of information to improve student learning” [1]. Test (exam) is a part of student assessment and should be “An objective and standardized measure of a sample of behavior” [2]. Item analysis is a post-examination evaluation and can provide information about the quality of tests.
Item analysis is a statistical analysis of the student’s responses on a test. Collection and summarization of students’ responses can provide quantitative objective information that is useful in deciding the quality of the test items and increasing the assessment’s efficiency [3, 4]. Also, Item analysis “investigates the performance of items considered individually either in relation to some external criterion or the remaining items on the test” [5].
Any educational test should measure students’ achievement in content material. Also, it leads to an overall assessment of students’ development to decide their academic status [6, 7].
The importance of item analysis is determined by the objective of the assessment [8]. In summative assessment, the assessment results should be reliable and valid because incorrect decisions about the academic status will lead to negative consequences [9]. While for the formative evaluation where the target is students learning, the item analysis has no much importance in giving feedback about items construction to test composers.
In literature, many reasons were reported for the conduction of item analysis, including examining if the item is functioning as intended, did it assess the required concepts (content)?, did it discriminate between those who master the content material and those who were not? was it within the acceptable level of difficulty?, whether the distracters are functioning or not? [10, 11].
Many factors can affect item analysis and hence its interpretation [8]. Difficulty and discrimination indices were constantly changing per administration and influenced by the ability and number of the examinee, the number of items, and the quality of instructions [8, 12].
Whatever the exam or test blueprinting (item selection) method, exam items remain a sample of the needed content material. The number of items (item sampling) carries excellent importance because one cannot ask about all contents. With a too-small number of items, the results may not be enough to reflect true student ability [8]. Technical item flaws are divided into two major types, test wiseness, and irrelevant difficulty. Test wiseness flaws can result in more easy items. Faults related to irrelevant difficulty can result in more challenging items unrelated to the content under assessment. It was reported that item analysis of exam with 200 examinees is stable, and with fewer than 100 examinees should be interpreted with caution (item difficulty or item discrimination index). While Downing and Yudkowsky described that even for a small number of the examinee (e.g., 30) still, the item analysis can provide a piece of a helpful information to improve item [13, 14].
The item or psychometric analysis parameters include difficulty index, reliability, discrimination index, distractor efficiency [2]. The descriptive statistics of the exam are important and can provide helpful generalized information [2]. The descriptive statistics include scores frequency, the mean, the mode, the median, and the standard deviation.
Cronbach’s alpha (KR20) is widely accepted and used estimate of test reliability (the internal consistency) and reported to be superior to the split-half estimate [15, 16]. Although validity and reliability are closely associated, the reliability of an assessment does not depend on its validity [16, 17]. Coefficient alpha is known to be equal to Kr-20 if the item has a single answer, such as in the case of type A MCQs or binary [18, 19, 20, 21].
Coefficient alpha reflects the degree to which item response scores correlate with total test scores [15]. It also describes the degree to which items in the exam measure the same concept or construct [22]. Therefore, it is connected to the inter-relatedness and dimensionality of the items within the exam [16, 20]. Cronbach’s alpha is affected by exam time, the number and inter-relation of the items (dimensionality) and easy or hard, poorly written or confusing items, Variations in examinee responses, curriculum content not reflected in the test, Testing conditions, and Errors in recording or scoring [22, 23, 24]. The value of alpha is decreased in the exam with fewer items and increased if items assessing the same concept (unidimensionality of the exam) [16]. Other factors were reported to impact alpha value, such as item difficulty, number of the examinee, and student performance in the exam time. It was argued that very high alpha values could indicate lengthy exams, parallel items, or a narrow coverage of the content material [22].
The alpha value of the exam can be increased by increasing the number of items with a high p-value (difficulty index). It was reported that items with moderate difficulty could maximize alpha value and while those with zero difficulties or 100 can minimize it [15]. In the same way, deletion of faulty items can increase alpha value. It should be considered that repetition of items in the same exam or using items assessing the same concept can increase alpha value.
The interpretation of reliability is the correlation of the test with itself. When the estimate of reliability increases, the portion of a test score related to the error will decrease. Wise interpretation of alpha needs an understanding of the interrelatedness of items and whether the items measure a single latent trait or construct. Exam or test with different content materials such as integrated courses, for example, in the musculoskeletal system course, although is dominated by anatomy it contains other subjects of basic medical and clinical sciences that have different contains. Therefore, interpretation of such a course exam needs deep looks beyond the alpha figure. It was reported that KR20 of 0.7 is acceptable to short test (less than 50 items) and KR20 of 0.8 for an extended test (more than 50 item-test) [25]. Moreover, it was documented that a multidimensional exam does not have a lower (Table 1) alpha value than a unidimensional one [30].
Author | Interpretation of Cronbach’s alpha (KR20) |
---|---|
Robinson, Shaver et al. [26] | ≥0.80 Exemplary |
0.70–0.79 Extensive | |
0.60–0.69 Moderate | |
<0.60 Minimal | |
Cicchetti [27] | <0.70 Unacceptable |
0.70–0.80 Fair | |
0.80–0.90 Good | |
< 0.90 Excellent | |
Axelson and Kreiter [28] | >0.90 is needed for very high stakes tests (e.g., licensure, certification exams) |
0.80–0.89 is acceptable for moderate stakes tests (e.g., end-of-year summative exams in medical school, end-of-course exams) | |
0.70–0.79 would be acceptable for lower stakes assessments (e.g., formative or summative classroom-type assessments created and administered by local faculty | |
<0.70 might be useful as one component of an overall composite score. | |
Obon and Rey [12] | >0.90 Excellent reliability |
0.80–0.90 Very good for a classroom test | |
0.70–0.80 good for a classroom test | |
0.60–0.70 Somewhat low (The test needs to be supplemented by other measure) | |
0.50–0.60 Suggests need for revision of test (unless it is quite short, ten or fewer Items). | |
0.50 < Questionable reliability. | |
Hassan and Hod [29] | > 0.7 is excellent |
0.6–0.7 is acceptable | |
−0.5-0.6 is poor | |
< 0.5 is unacceptable | |
< 0.30 is unreliable |
Reference values and interpretation of Cronbach’s alpha (KR20).
A low alpha value can be due to a smaller number of items, reduced interrelatedness between items, or heterogeneous constructs [22]. A high value of alpha can suggest exam reliability, and some items are non-functional as they are testing the same content but in a different guise or repeated ones [16, 22]. Also, a high value indicates items with high interrelatedness, indicating a limited coverage of the content materials [22].
Adding new items with an acceptable difficulty index, high discrimination power and distractor efficiency can increase the test reliability [22, 31, 32]. In addition, deletion of faulty items or those with low or very high p-value can improve Cronbach’s alpha. Items with poor correlation or are not related should be revised or discarded from the exam.
Commonly are formed of a stem with or without leading question and five or four alternatives (type A MCQs). Among item’s alternatives, only one is the key answer and others are called distractors [4]. Distractors should carry or convey a miss concept about the key answer and appear plausible. The distractors should appear similar to the key answer in terms of the used words, grammatical form, style, and length [19]. Distractor efficiency (DE) is the ability of incorrect answers to distract the students [12].
A functional distractor (FD) the distractor that is selected by 5% or more of the examinee [4, 33]. At the same time, those chosen by less than 5% of the examinee are considered non-functional (NFD) [4]. In comparison, other authors reported 1% of the examinee as the demarcation of functional distractors [34, 35]. Commonly items are categorized based on the numbers of NFDs in the item (Table 2) [12, 26, 36, 37].
Number of NFD | Percentage | Interpritation |
---|---|---|
3 | 0 | Poor |
2 | 33.3 | Moderate |
1 | 66.6 | Good |
0 | 100 | Excellent |
Classification of items according to thee number of the nonfunctional distractors (NFD).
The occurrence of NFD makes the item easier and reduces its discrimination power, while FD distractors are making it more difficult [36, 38]. It was reported that non-functional distractors are negatively correlating with reliability [38]. The presence of non-functional distractors can be related to two main causes. First is the training and construction ability of the item writer or composer. Second, the miss-match between the target content and the possible number of a distractor created. Thus, training and more effort in item writing and construction can decrease NFDs [36]. Other causes were related to NFDs, including the low cognitive level of the item, irrelevant or limited number of plausible distractors, or presence of logic cues [39]. Another possibility of NFDs is mastering the content material of the item, and students can identify the distractor as the wrong one. If no other cause (s) for NFDs, they should be removed or changed with a more plausible option because it has no contribution to the measurement of the test [12]. If a distractor is selected more frequently than the key answer by a higher-scoring examinee, this may indicate poor constructions or a misleading question or miss or double-keyed [12, 40]. In this, concerning the use of three options is more practical than four, does not affect reliability, and does not affect the discrimination index significantly [26, 35, 36, 37].
Furthermore, it was reported that there is no psychometric reason that all items in the exam should have the same number of distractors [26, 41]. The required number of options in an item should be considered according to the content material from which plausible distractors can be developed [33, 40, 42].
Reducing the number of options/distractors will result in other important benefits such as reducing the answering time of the test and safe time can be used to cover more content material, reduce the burden on item composers, and have items with more acceptable parameters [43, 44].
Puthiaparampil et al. reported a non-high significant negative and positive correlation between the number of functional distractors and difficulty and discrimination indices, respectively [34]. While a significant positive correlation was reported between the DIF and the number of NFDs [45].
Many authors concluded that no predictable relationship between DE and difficulty index and discrimination index [26, 31, 40, 46, 47]. In addition Licona-Chávez et al. did not find a parallel performance between DE and other parameters of item analysis including Cronbach alpha [46]. In contrast, some authors claimed that low DE decreases the difficulty index [47, 48].
Restoring the optimal DE of the item can be achieved by identifying flaws related to the NFDs and correcting them or removing the NFDs from the item [39].
The item difficulty (easiness, facility index, P-value) is the percentage of students who answered an item correctly [6, 40]. The difficulty index ranges from 0 to 100, whereas the higher the values indicate, the easier the question and the low value represents the difficulty of hard items. The ideal (optimal) difficulty levels for type A MCQs is varying according to the number of the options (Table 3) [49, 50]. The range of items difficulty can be categorized into difficult, moderate, and easy. Easy and difficult items were reported to have very little discrimination power [48]. Item difficulty is related to the item and the examinee that took the test in the given time [24]. Thus, reusing of the item depending on its difficulty index should be controlled. Some authors found that difficulty indices of items assessing high cognitive levels in Bloom’s taxonomy such as evaluation, explanation, analysis, and synthesis are lower than those assessing remembering, understanding, and applying [51, 52].
The ideal (optimal) difficulty level (for tests with 100 items).
During item or exam construction, the constructor should aim for acceptably level of difficulty [6]. Sugianto reported that items within the exam could be distributed according to difficulty to moderate level (40%), easy and challenging levels (20%), and easier and more challenging levels (10%) [6]. Other authors reported that most items should be of moderate difficulty or 5% should be in the difficult range [50, 53]. Some authors found that difficulty indices of items assessing high cognitive levels in Bloom’s taxonomy such as evaluation, explanation, analysis, and synthesis are lower than those assessing remembering, understanding, and applying [51, 52]. Regarding the general arrangement of test or examination, easy items start first then are followed by difficult ones. At the same time, in the case of diagnostic assessment, the sequence of the learning material is more important [6, 7].
Easy and difficult items affect the item’s ability to discriminate between students and show low discrimination power. Some reports described a negative correlation between exam reliability and difficult and easy items [38]. Oermann et al. reported that educationalists must be careful in deleting items with poor DIF because the number of items has more effect on test validity [54]. It is recommended that difficult items should be reviewed for the possible technical and content causes [50]. Possible causes of low difficulty index include uncovered (taught) content material, challenging items, missed key or no correct answer among the item options [55]. Easy items (high
In literature including medical education, many ranges of difficulty indices were reported (Table 4).
Author | Difficulty index | Interpretation |
---|---|---|
Uddin et al. [50] | >80% | Easy |
30–80% | Moderate | |
<30% | Difficult | |
Kaur, Singla et al. [56] | >80 | Easy |
40–80 | Moderate | |
<39 | Difficult | |
Sugianto [6] | 90 | Easy |
50 | Moderate | |
10 | Difficult | |
Date, Borkar et al. [37] and Kumar, Jaipurkar et al. [36] | <30 | Too difficult |
>70% | Too easy | |
50–60% | Excellent/ideal | |
30–70% | Good/acceptable/average | |
Obon and Rey [12] | > 0.76 | Easy (Revise or Discard) |
0.26–0.75 | Right difficult (Retain) | |
0–0.25 | Difficult (Revise or Discard) | |
Bhat and Prasad [57] | >70% | Easy |
30–70% | Good | |
<30% | Difficult |
Reference values and interpretation of difficulty index (p-value).
Item discrimination (DI) is the ability of an item to discriminate between higher achiever (good) students and low ones. It was defined as “stated that item discrimination is a statistic that indicates the degree to which an item separates the students who performed well from those who did poorly on the test as a whole” [6]. The discrimination power of an item is calculated by categorizing the examinee into upper 27% and lower 27% according to their total test score. The difference between the upper and lower group is divided by the number of the examinee in the upper group or the larger group or by half of the total number of the examinee or even by the total number [4, 6, 58, 59]. Obon and Rey [12] calculated the discrimination index as the difference of difficulty index between the upper and lower groups [12]. In literature, both 25 and 27% were reported as possible percentages of examinee categorization [60, 61]. The 27% is commonly used to maximize differences in normal distributions and increase the number of examinees in each category. The discrimination index range from 1.0 to −1.0. The positive discrimination index indicates that high achievers answer the item correctly more than those in the lower ones, which is desirable. The negative discrimination index reflects that lower achiever examinees answer the item more correctly, while zero discrimination indicates equal numbers of students in the upper and lower groups [36, 37]. Negative discrimination is thought to be due to item flaws or inefficient distractors, miss keys, ambiguous wording, gray areas of opinion, and areas of controversy [12, 62]. Nevid and McClelland [52] reported that items assessing evaluation and explanation domains could discriminate between high and low performers, while Kim et al. [51] comments that items assessing remembering and understanding levels have low discrimination power [52, 54].
It was reported that discrimination indices are positively associated with difficulty index and distractor efficiency [39, 63]. The discrimination power of the item is reduced by the increased number of non-functional distractors [36].
A test with poor discriminating power will not provide a reliable interpretation of the examinee’s actual ability [6, 64]. In addition, discrimination power will not indicate item validity, and deletion of items with poor discrimination power negatively impacts validity due to a decrease in the item number [65].
Discrimination coefficients can evaluate item discrimination. The discrimination coefficients include point biserial correlation, biserial correlation, and phi coefficient. Although point biserial correlation is used interchangeably with the discrimination index, discrimination coefficients are considered superior to the discrimination index [24]. The superiority came from the fact that discrimination coefficients are calculated using all examinees’ responses in the item rather than only 54% of the examinees such as in the discrimination index.
The difference between Point-biserial correlation (rBP) and discrimination indexes is that rBP is the correlation between an item in the exam and the overall student score [2, 66]. In cases of highly discriminating items, the examinees who responded to the item correctly also did well on the test. In general, the examinees who responded to the item incorrectly also tended to perform poorly on the overall test. It was suggested that point biserial can express the predictive validity better than Biserial correlation coefficients [61, 67].
Discrimination power of items more than 0.15 was reported as evidence of item validity [50, 53]. While any item with less than 0.15 or negative should be reviewed [50] (Table 5).
Author | Discrimination power | Interpretation |
---|---|---|
Elfaki, Bahamdan et al. [53] | ≥0.35 | Excellent |
0.25–0.34 | Good | |
0.21–0.24 | Acceptable | |
≤ 0.20 | Poor | |
Obon and Rey [12] | ≥ 0.50 | Very Good Item (Definitely Retain) |
0.40–0.49 | Good Item (Very Usable) | |
0.30–0.39 | Fair Quality (Usable Item) | |
0.20–0.29 | Potentially Poor Item (Consider Revising) | |
≤ 0.20 | Potentially Very Poor (Possibly Revise Substantially or Discard) | |
Bhat and Prasad [57] | > 0.35 | Excellent |
0.2–0.35 | Good | |
< 0.2 | Poor | |
Sugianto [6] | >0.40 | Very good |
0.30–0.39 | Reasonably good possibly need to improvement | |
0.20–0.29 | Marginal item usually needing and being to improvement | |
<0.19 | Poor item rejected or improved by revision | |
Aljehani, Pullishery et al. [66] and Sharma [4] | ≥ 0.40 | Very discriminating, very good item(Keep) |
0.30–0.39 | Discriminating item, good item (Keep) | |
0.20–0.29 | Moderately discriminating, fair item (Keep) | |
< 0.20 | Not discriminating item, marginal item (Revise/Discard) | |
Negative | Worst/ defective item (Definitely Discard) | |
Ramzan, Imran et al. [63] | > 0.30 | Excellent discrimination |
0.20–0.29 | Good discrimination | |
0–0.19 | Poor discrimination | |
00 | Defective | |
Uddin et al. [50] | ≥ 0.35 | Excellent |
0.25–0.34 | Good | |
0.21–0.24 | Acceptable | |
< 0.20 | Poor |
Reference values and interpretation of discrimination index (power).
When interpreting the discrimination power of an item to decide about, especial consideration should be related to its difficulty. Items with a high difficulty index (most of the examinee answer it right) and those with low difficulty index (most of the examinee answer it wrong) commonly have low discrimination power [35, 63]. In both cases, such items will not discriminate examines as the majority are on one side. Thus items with a moderate difficulty index are more likely to have good discrimination power.
The common causes of poor discrimination power of item include technical or writing flaws, untaught or not well covered content material, ambiguous wording, gray areas of opinion and controversy, and wrong keys [12, 50, 62, 66].
In general, the statistical data obtained from item analysis can help item constructors and exam composers to detect defective items. The decision to revise an item or distractors must be based on the difficulty index, discrimination index, and distractor efficiency. Revision of items can lead to modification in the teaching method or the content material [68].
Standard item analysis of mid-course examination. The total number of items is 40, and the total number of the examinee is 21. The KR20 is 0.82. Pt.Biserial: Point biserial correlation, Disc Index: discrimination index, Correct: number and percentage of the correct answer (difficulty index), Pct. Incorrect: percentage of an incorrect answer.
In this Example 1.
The number of examinees was 21.
The number of test items (Total possible) is 40.
The highest and lowest scores were 38 and 14 respectively.
The class average (mean) (30.3) is more than the class median (30) which represents a positively skewed distribution of examinee scores. Despite this, examinee scores may show normal ball-shape distribution. If the median is larger than the average (mean), the examinee scores will be negatively skewed distribution. Average equals median, the examinees’ scores are symmetrically (zero skewed) and normally distributed with ball-shaped.
The KR20 (Cronbach’s alpha) is 0.82 which is an acceptable value for most of the authors. Such value of internal consistency of exam allows deciding pass/fail. Lower values put the exam in questionable status.
Item 1: the difficulty index is 85.7% (easy). Although it has high discrimination power (DE = 0.6, Pbiserial = 0.58), two distractors are non-functional (B, C).
Comment: the item needs reediting. Distractors B and C need to be revised or changed by more plausible ones before being re-used.
Item 2: the difficulty index is 100% (easy). It has low discrimination power (DE = 00, Pbiserial = 00), all distractors are non-functional.
Comment: the item needs major revision or rewriting. This item is absolutely easy with no difficulty or discrimination index. Such items should be removed from the question bank and removal from the exam is considered valid.
Item 6: the difficulty index is 66.7% (moderate). It has high discrimination power (DE = 0.6, Pbiserial = 0.43) and all the distractors are functional.
Comment: The item has acceptable indices. Such items can be saved in the question bank for further use. The distractors need to be updated to have more efficiency.
Item 7: the difficulty index is 28.6% (difficult). Although it has high discrimination power (DE = 0.67, Pbiserial = 0.39), all the distractors are functional.
Comment: The item has acceptable indices. Such items can be saved in the question bank for further use. The distractors need to be updated to have more efficiency.
Item 8: the difficulty index is 76.2% (moderate). This item has a negative discrimination index (−0.33) and poor Pbiserial (0.04). Only one distractor is functional (C). The negative discrimination index is caused by the increased number of students in the lower account (27%) than those in the upper account (27%).
Comment: although the item has a moderate difficulty index, but is poorly discriminating. Such an item needs major revision.
Standard item analysis of Mid-course examination. The total number of items is 40, and the total number of examinee is 25. The KR20 is 0.74. Pt.Biserial: Point biserial correlation, Disc Index: discrimination index, Correct: number and percentage of the correct answer (difficulty index), Pct. Incorrect: percentage of an incorrect answer.
In this Example 2.
The number of examinees was 25.
The number of test items is 40.
The highest and lowest scores were 33 and 13 respectively.
The class average (mean) (24.6) is more than the class median (25), distribution of examinee scores is skewed to the left. Despite this, examinee scores may show normal ball shape distribution.
The KR20 (Cronbach’s alpha) is 0.74 which is an acceptable value for most of the authors. Such a value of internal consistency is suitable for class tests.
Item 8: the difficulty index is 4.0% (difficult). It has negative discrimination power (DE = -0.17, Pbiserial = −0.06), one distractors is non-functional (C).
Comment: the correct answer is (A) while most of the examinees chose (B). According to distractor analysis, this item is miss-keyed rather than an implausible distractor.
Item 9: the difficulty index is 20% (difficult). It has low discrimination power (DE = 0.17, Pbiserial = 0.09), all distractors are functional.
Comment: distractor analysis show option number (A) and (B) are more selected by examinees. This can be due to implausible. The presence of implausible can affect the item difficulty index. Distractors in this item should be revised or changed with plausible ones.
Item 11: the difficulty index is 44.0% (moderate). It has low discrimination power (DE = 0.0, Pbiserial = 0.01) and only one the distractors is non-functional.
Comment: The item has an acceptable difficulty index. Distractor (D) is more selected by upper examinee such as the key answer. Such a situation can favor missed key or implausible distractors. The distractors need to be updated to have more efficiency.
As it is emphasized in [1, 2], the tasks of analysis and synthesis of control processes occurring in dynamic systems of different physical nature, operating in conditions of substantial plant parametric uncertainty, including the engineering ones, are currently the most urgent and challenging within the framework of the control theory. Among these tasks, one could mention the problem of flux control for the electric motor vector control systems operating in uncertainty because the flux control quality strongly affects the electromagnetic torque and speed control quality, and thus the drive power efficiency. For this reason, of great importance are the tasks of stability investigation and parametric synthesis of robust control systems (their characteristic polynomials) for the plants which parameters vary within the given or unknown intervals of values.
\nIn the area of investigation and synthesis of dynamic system characteristic polynomials, there exists a lot of approaches and methods. For the first time, the necessary and sufficient conditions for systems up to the 3-rd order were formulated by James Maxwell in 1868. Later appeared the stability criteria of Routh–Hurwitz, Mikhajlov, Nyquist, and Bode, which made it possible to check stability of the systems of order
The methods for analysis and synthesis of polynomial families represent a separate group. One of the most effective solutions for the task of interval polynomial family investigation within the algebraic approach has been proposed by Kharitonov [4], where in the general case, the task of polynomial stability analysis is reduced to consideration of only four specific polynomials of the whole family with constant coefficients. In [3, 5], the frequency criteria of Hurwitz robust stability are considered, which allow to define the coefficient perturbation sweep for the nominally stable polynomial and various types of uncertainties. Hurwitz robust stability is also investigated in [6, 7, 8, 9, 10]. In [6], the maximal deviation intervals of perturbed Hurwitz polynomial coefficients assuring strict Hurwitz property are determined on the basis of the algebraic method worked out using Kharitonov’s polynomials [4]. The similar task is solved in [7] but using the Hermite-Biler theorem, which allows to reduce twice the power of investigated polynomial. The way for calculation of perturbed polynomial coefficients’ maximal limit values that guarantee sector stability is given in [8]. The linear dependence of coefficient perturbation is considered by Bartlett, et al. for a class of polynomial families generated by convex polytopes in the coefficient space [9]. Here the so-called edge theorem was proved assuring derivation of the stability analysis task to investigation of root location for the finite number of the parametric families. The edge theorem allows to analyze both stability and quality characteristics of the family. A combination of the stochastic and worst-case approaches to the problem of uncertainty is proposed in [10]. It certainly widens the scope of types of treatable uncertainties and reduces conservatism. However, it works properly only in the cases permitting an arbitrarily small probability of specification violation. Thus, to the specific extent, it still bears the drawbacks of the stochastic approach to control, which guarantees only the “average” performance.
\nAn analog of Kharitonov theorem [11] was formulated for the unstable interval polynomials’ homogeneous classes of equivalence. Criteria of existence of such classes of equivalence were obtained. Based on the new interval polynomial stability criterion and Lyapunov theorem, a robust optimal proportional-integral-derivative (PID) controller is proposed in paper [12] to carry out design for different plants that contain perturbations of multiple parameters. A new stability criterion of the interval polynomial is presented to determine whether the interval polynomial belongs to Hurwitz polynomial or not. Time-delay systems involving multiple imaginary roots (MIRs) and their stability analysis, which becomes much more complicated than that in the case with only simple imaginary roots, are treated in [13]. For a class of time-delay systems, it was proved that the invariance between the multiple imaginary roots and the simple imaginary roots holds for any multiplicity as well as for the degenerate cases. In paper [14], monic complex polynomials are identified with the sets of their roots instead of being identified with the vectors of their coefficients. A proof is given that the space of Hurwitz polynomials of degree
Of great interest are the problems of ensuring system stability and quality being solved in the modern statements of the problem [2] as tasks of guaranteeing system robustness, which could be solved by application of the root locus approach. The basic benefit of this approach is that its application itself, by its nature, implies parametric variations (i.e., uncertainty). The root locus approach is a powerful method used for the system synthesis [2] and is notable for its descriptiveness ensuring both calculation of the system robust parameters’ values and possibility of detailed overview of the dynamic properties variation changes, the system response to uncertainties that is particularly important when investigating systems with uncertain and in particular interval parameters.
\nRoot locus approach to the problem is considered in [17, 18, 19, 20, 21, 22, 23]. Paper [17] gives a solution for a compensator synthesis on the basis of the root locus method application. The task of a stable characteristic polynomial synthesis for the interval dynamic system (IDS) by setting up coefficients of the given (initial) unstable one for the case of location of its root locus initial point (where the variable parameter is equal to zero) family within the left half-plane is solved in [21], where the stability is attained via simple setting up the interval of the free term variation.
\nThe above analyzed literature covers various approaches to the uncertainty treatment. However, most of the theoretical works are focused on the tasks of robust stability analysis. The methods for synthesis are not that widely represented, often suffer from complexity and in most cases are enough narrow, which means that they certainly provide instruments for system synthesis, but they are mostly “closed on themselves,” which means that they do not provide the complete picture in the sense of showing up what is happening “under cover,” which is especially important for the qualitative robust system (polynomials) synthesis. The root locus approach is rarely applied even though it represents the dynamic picture of the system response to uncertainties in the most comprehensive way and thus seems to be the most suitable one to deal with uncertainties.
\nAs for polynomial families, the root locus approach gives us the transparent picture of root dynamics making it possible to see as if from the inside, for example, what subfamilies constitute the whole family of uncertain polynomials in terms of their configuration and stability or some other dynamic indicators bearing significant information about the system behavior and thus leading the way for its investigation and synthesis.
\nIn this work, the root locus methods are described for calculating intervals of uncertainty for coefficients of the given (initial) stable or unstable polynomial with coefficients subject to perturbations, which ensure its robust stability. The proposed methods are based on introduction and application of the notions “extended root locus,” “diagram of the root locus parameter function values distribution along the stability bound” and can be used for both synthesis of interval stable polynomials by setting up (adjusting) the unstable ones and analysis of the polynomial behavior under coefficient perturbations. The influence of every coefficient upon the polynomial behavior could be observed.
\nThe work further develops results represented in the papers of Anderson [22] and Kharitonov [4] where they consider the issues of analysis and synthesis of robust interval polynomial families.
\nDefine a polynomial like
where
In the event of coefficient perturbations, a vector of coefficients of (1),
Suppose that coefficients of (1) vary within the following intervals:
where \n
After substituting
where
The root locus method represents a powerful and effective tool for stable and qualitative polynomial synthesis and analysis. However, as it is known, this method allows to consider polynomials with only a single variable coefficient (parameter) and cannot be applied in the cases when all coefficients are uncertain. Therefore, the task is to generalize the root locus method for the cases when the number of variable coefficients is arbitrary and thus to solve the problem of investigation of the uncertain polynomial dynamics and working out methods for synthesis of the robustly stable uncertain (interval) polynomial by setting up the given polynomial (non-Hurwitz or Hurwitz) with constant/variable coefficients and determining intervals of all its coefficients (stability intervals) assuring its robust stability.
\nIf the root locus parameter is
Let us along with the parameter
It is evident that the root locus Eq. (3) represents also the equation of level lines of the
that represents the infinite set of root locus fields and therefore possesses their properties, and from the mathematical point of view, all root locus fields of
Hereinafter the term “root locus” is used in the sense of “Teodorchik – Ewans free root locus” [18].
\nIntroduce the following system of polynomials:
where
Every polynomial (8) of (i−1) degree is generated from the
Extension
Every (i−1)-th polynomial of (6) is the originative one relative to
where \n
Further in the text, polynomial
Statement 1 is illustrated by Figures 1 and 2. Initial points here are designated by signs “x” (crosses) and letters “p” with the lower indexes, designating the point sequential numbers, and upper indexes, designating the sequential numbers of the corresponding root locus. The root locus sequential number is indicated by a digit next to its corresponding branch.
\nPolynomial
Free root locus portrait (field) for polynomial
Consider Eq. (3) in the sense of four following possible cases:
Specify the set \n
where \n
of intervals \n
that proofs the statement being considered.
\nFor the 4-th degree polynomial represented in Figure 2, the interval \n
find among polynomials of extension (6), the stable polynomial of degree
set up sequentially every coefficient
An algorithm for the robustly stable regular or interval polynomial synthesis is given below.
\nIn case of synthesis of the whole interval polynomial, begin the procedure with the 1-st degree polynomial,
Cross points \n
Properties of this domain and behavior of the interval root locus portrait at the stability bound
by solving the corresponding Eq. (3) after substituting preliminarily into this equation, the appropriate combination [18] of the limit values of each coefficient, from
after substituting previously into (18) the corresponding combinations of coefficients (from
where \n
Synthesis of the interval polynomial of the 3-rd degree.
\nConsider polynomial family
where
and parameter function (18) at the stability bound:
Find the 1-st order derivative of (23) and equate it zero:
On the basis of (23) and (24), it can be stated that the character of parameter (23) distribution along the axis σ is steadily increasing and the single extreme point is located at the origin. Thus, there exists the only one extreme point:
where function (23) gets the minimal value of the set \n
\n\n
Thus, coefficient intervals for the resulting robustly stable polynomial \n
Consider a dynamic system described by the family of interval characteristic polynomials [4, 18, 20, 22] like.
Coefficients of Eq. (26) to be real, positive, and variable within the intervals
Substitute \n
and on the base of (28), write the root locus equation [18, 20] at the stability boundary:
and the parameter equation (parameter function) [18, 20] at the stability boundary:
\nFunctions (29) and (30) imply properties of analyticity and continuity and, thus, the points where axis
constitute on the stability boundary, axis
The region \n
Over the symmetry of the portrait hereinafter, the only upper half-plane
Obtain the extremum parameter function values within \n
Take the first-order derivative of (32) and set it to zero:
After solving Eq. (33), obtain three points of extremum for the majorant parameter function for the field when \n
Rewrite (30) for determination of a minorant parameter function (or a minorant):
In the same way obtain three points of extremum for the minorant, when \n
Evidently, for
Figure 3 represents the character (diagram) of the parameter function (30) distribution along the boundary of stability by its majorant (32) and minorant (35). For better understanding and descriptiveness, the diagram in Figure 3 is shown by strait lines, although it constitutes curves. Region \n
Dω− where the parameter function is getting decreased (
Dωс where increase and decrease regions combine (
A diagram for distribution of the interval system root locus portrait parameter function along the asymptotic stability boundary.
Analyze the region
Within interval [0,
But specific pieces of the positive branches are situated within the right half-plane. For this reason, in some cases, the unstable polynomials could have been found within the whole family (26). However, there certainly could always be found the intervals (27) of stability where the whole family is stable. Name the interval [0,
The interval [
If the interval [z”, ∞] completely belongs to the region
only the negative branches cross the stability boundary
No stable polynomial could be found in (26). This region name the
Specify the region \n
where \n
Define below three possible ways of the real crossing region location and the corresponding stability conditions.
\n\n
In this case \n
Then, define the set
where \n
The following statement can be formulated on the basis of expressions (42) and (47).
\nFrom Statement 4 and the previous conclusions, the following stability condition goes.
\nof the family is stable. Polynomial (49) represents the dominating one.
\nStability is verified using the Stability condition 1. The polynomial parameters are calculated with application of the Statement 4.
\n\n
It happens in case if \n
The above made conclusions allow to formulate the following statement.
\n\n
We have this when the following conditions are not satisfied: \n
For this case
We have already discussed the increase part of (52), when
Evidently, options (54) and (55) take place when
As options (54)–(57) deliberately indicate instability of the system in whole, consider below only option (53) of the system poles location,
where ω(
In this case proceed just as in (44)–(47) but only substituting ωmax instead of ωmin.
\nFrom condition (59) follows that the system asymptotic stability for part
Because in this case, the portrait represents the compound one (52), check the stability by checking both polynomials, (49) and (60).
\nof family (26) are both stable.
\nFrom the results obtained above also goes that in case (51) the system asymptotic stability can be verified by only a single polynomial of (26) having constant coefficients. The equation to choose depends of condition (49) verification results. If the verification shows that \n
To determine the coefficients of (26), ensuring satisfaction of expressions (53) and (58), Eqs. (30) and (31) are applied. Thus, coefficients
To verify the system stability, the stability conditions 1 and 2 are used. For calculation of the system (polynomial) parameters, expressions (48), (49) and (63) are used.
\nPolynomial stability could be estimated graphically directly from the plots (see Figures 3 and 4).
\nDynamics of the interval system root locus portrait at the asymptotic stability boundary.
Coefficients of the given polynomial (26): \n
[
In Figure 4, the above indicated regions are shown. The points, corresponding to the dominating polynomials (61), (62), are designated by
It is evident that the given polynomial family in whole is unstable. Within region
Dominating polynomials of the family are the following:
Polynomials stability check shows that polynomial (6), which root loci crosses the stability boundary at point \n
Extraction of the stable polynomial subfamily of the given unstable family:
\nThe stable root locus family, satisfying conditions (58) and (59), should cross the stability boundary within the region bounded by interval [
To calculate the maximal value of
Based on (66), accept \n
Based on (59), accept: \n
As per stability condition 2, the root locus portrait subfamily having new modified values of
A method has been worked out for synthesis of asymptotically stable regular or interval polynomial from the given Hurwitz or non-Hurwitz source polynomial with constant/interval coefficients by setting up coefficients of the given one. The root locus approach is used. The task is solved by introduction of notions of the “extended polynomial” (“generalized polynomial”) and the polynomial “extended root locus,” which allows to obtain a descriptive picture of the polynomial root dynamics under coefficient variations and to disclose on this basis the cause of instability. The intervals of uncertainty for each coefficient being set up are specified along the root locus branches.
\nThe above described method based on the “extended root locus” notion is new and allows to extend the application sphere of the root locus method, which is traditionally considered to be the method of system synthesis by only a single parameter (coefficient) variation and with only one variable parameter (coefficient), in both directions: system synthesis by many parameter variations and system synthesis with many parameter variations.
\nInvestigation of the fourth power dynamic system behavior in conditions of the interval parameter variations has also been carried out on the basis of root locus portraits and introduction of the notion of the “diagram of the root locus parameter function values distribution along the stability bound.” Behavior regularities for interval system root locus portraits at the stability boundary have been formulated. On this basis, the stability conditions have been derived, and graphic-analytical method has been worked out for calculating intervals of parameter variation ensuring the system robust stability.
\nIn continuation of the results of Anderson [22] and Kharitonov [4] in this work, it is proved that for the 4th power interval system family asymptotic stability analysis, it is enough to use the only one polynomial of this kind. It is also shown, how to find and extract the stable families from the unstable ones.
\nThe above discussed topic is certainly worth further investigation in the light of continuous progress of both theory and technology. When speaking of the practical implementations, it could be noted that most of the control system synthesis tasks, especially those in the area of robust control, are currently still being solved in a somewhat “local domestic” way, when a designer each time tries to invent a solution to be suitable for the specific application experiencing the lack of more generalized methods. Besides this, a great deal of existing robust control methods share and suffer complexity. In this connection, further in-depth investigation of the uncertain polynomials’ root locus portraits seems helpful, especially the analysis of its composition in terms of configurations variety, constituting subfamilies, placement of various root domains within the prescribed regions in the complex plane and, of course, dynamics. They also could be distinguished for their undoubted descriptiveness.
\nPolynomial equation approach in the design technique [16], and root locus technique in particular, is descriptive, clear, and easy to use and computerize and thus could be helpful in many application areas including the areas of industry, biology, medicine, etc. It can be used for proper parameterization of robust drive controllers, for example, in the area of railway traffic control, in particular for the cases of tackling the problems of breaking and skidding.
\nThe author acknowledges the support of this work by the Belarusian Republican Foundation for Fundamental Research.
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His studies in robotics lead him not only to a PhD degree but also inspired him to co-found and build the International Journal of Advanced Robotic Systems - world's first Open Access journal in the field of robotics.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"441",title:"Ph.D.",name:"Jaekyu",middleName:null,surname:"Park",slug:"jaekyu-park",fullName:"Jaekyu Park",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/441/images/1881_n.jpg",biography:null,institutionString:null,institution:{name:"LG Corporation (South Korea)",country:{name:"Korea, South"}}},{id:"465",title:"Dr.",name:"Christian",middleName:null,surname:"Martens",slug:"christian-martens",fullName:"Christian Martens",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Rheinmetall (Germany)",country:{name:"Germany"}}},{id:"479",title:"Dr.",name:"Valentina",middleName:null,surname:"Colla",slug:"valentina-colla",fullName:"Valentina Colla",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/479/images/358_n.jpg",biography:null,institutionString:null,institution:{name:"Sant'Anna School of Advanced Studies",country:{name:"Italy"}}},{id:"494",title:"PhD",name:"Loris",middleName:null,surname:"Nanni",slug:"loris-nanni",fullName:"Loris Nanni",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/494/images/system/494.jpg",biography:"Loris Nanni received his Master Degree cum laude on June-2002 from the University of Bologna, and the April 26th 2006 he received his Ph.D. in Computer Engineering at DEIS, University of Bologna. On September, 29th 2006 he has won a post PhD fellowship from the university of Bologna (from October 2006 to October 2008), at the competitive examination he was ranked first in the industrial engineering area. He extensively served as referee for several international journals. He is author/coauthor of more than 100 research papers. He has been involved in some projects supported by MURST and European Community. His research interests include pattern recognition, bioinformatics, and biometric systems (fingerprint classification and recognition, signature verification, face recognition).",institutionString:null,institution:null},{id:"496",title:"Dr.",name:"Carlos",middleName:null,surname:"Leon",slug:"carlos-leon",fullName:"Carlos Leon",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Seville",country:{name:"Spain"}}},{id:"512",title:"Dr.",name:"Dayang",middleName:null,surname:"Jawawi",slug:"dayang-jawawi",fullName:"Dayang Jawawi",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Technology Malaysia",country:{name:"Malaysia"}}},{id:"528",title:"Dr.",name:"Kresimir",middleName:null,surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/528/images/system/528.jpg",biography:"K. Delac received his B.Sc.E.E. degree in 2003 and is currentlypursuing a Ph.D. degree at the University of Zagreb, Faculty of Electrical Engineering andComputing. His current research interests are digital image analysis, pattern recognition andbiometrics.",institutionString:null,institution:{name:"University of Zagreb",country:{name:"Croatia"}}},{id:"557",title:"Dr.",name:"Andon",middleName:"Venelinov",surname:"Topalov",slug:"andon-topalov",fullName:"Andon Topalov",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/557/images/1927_n.jpg",biography:"Dr. Andon V. Topalov received the MSc degree in Control Engineering from the Faculty of Information Systems, Technologies, and Automation at Moscow State University of Civil Engineering (MGGU) in 1979. He then received his PhD degree in Control Engineering from the Department of Automation and Remote Control at Moscow State Mining University (MGSU), Moscow, in 1984. From 1985 to 1986, he was a Research Fellow in the Research Institute for Electronic Equipment, ZZU AD, Plovdiv, Bulgaria. In 1986, he joined the Department of Control Systems, Technical University of Sofia at the Plovdiv campus, where he is presently a Full Professor. He has held long-term visiting Professor/Scholar positions at various institutions in South Korea, Turkey, Mexico, Greece, Belgium, UK, and Germany. And he has coauthored one book and authored or coauthored more than 80 research papers in conference proceedings and journals. 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