Stability Analysis On the Nichols Chart and Its Application in QFT Wenhua Chen and Donald J. Ballance Centre for Systems & Control Department of Mechanical Engineering University of Glasgow Glasgow G12 8QQ UK August, 1997 Abstract A new stability criterion on Nichols chart is presented. It provides an alternative graphical method for stability analysis on Nichols chart. Furthermore this result can be used to test the stability of the new nominal closed-loop system arising in QFT controller design method for nonminimum phase and unstable plants, which is dicult to test with the existing stability criterion. keywords: Stability analysis, Quantitative Feedback Theory, stability criterion, Nichols chart 1 Introduction In QFT and other frequency domain design methods, the robust stability is guaranteed by ensuring that the nominal case is stable and the following equality holds 1 + L(j!) 6= for any allowable uncertainties, where L(j!) is open-loop transmissions, or open-loop plants. This is so-called Zero Exclusion Theorem. In QFT the later is achieved by imposing the robust stability margin condition, for instance, the high-frequency U courtor. It is therefore essential to guarantee the stability of the nominal case in the QFT design procedure and if the nominal case is stable and the robust margin condition is satised, the robust stability follows. Since the loop shaping is performed on the Nichols chart and a graphic CAD environment for QFT controller design was developed as a commercial QFT Toolbox in MATLAB [2], it is very interesting to obtain the graphical stability criterion. In addition, to develop a new stability criterion has itself meaning. The well-known graphical stability criterion is the celebrated Nyquist stability criterion (for example, see [9]). This criterion was extended to the Nichols chart for QFT design recently [7]. When designing QFT controller for non-minimum phase and unstable plants with Horowitz and Sidi method [6, 4], Chen and Ballance [3] show not only the robust bounds but also the stability line R n must be shifted with frequency. 1
j Rn Y db r 6 + Rc + -1 - X -36 - -18 φ - -6 Figure 1: Denitions of positive and negative crossing When the stability line R n moves with frequency in QFT design, it is dicult to use the existing stability criterion directly. That is, it is dic Lemma[7]: For the Nichols and Nyquist plot with the positive and negative crossing denitions in Fig. 1, the followings are equivalent: 1. The feedback system is stable. 2. The one-sheeted full Nichols plot of L(s) does not intersect the point (?18 ; db), and the net sum of its crossings of the ray R n := f(; r ) : =?18, r > dbg is equal to n. 3. The multiple sheeted full Nichols plot of L(s) does not intersect any of the points (2k + 1)q,k = ; 1; 2;, and the net sum of its crossings of the rays R n + 2kq is equal to n. ult to test whether or not the plot L o (s) intersects the stability line graphically in the loop shaping procedure. It is interesting to develop a new graphical stability criterion. In this paper the plot in Nichols chart and the plot in Nyquist chart of L(j!) for! 2 (?1; +1) are called the full Nichols plot and the full Nyquist plot of L(j!) or of the transfer function L(s), respectively. The half parts of the full Nichols plot and the full Nyquist plot of L(s) are called the half Nichols plot and the half Nyquist plot of L(s) respectively, and, for simplicity, called the Nichols plot and the Nyquist plot of L(s) or L(j!) in this paper. Moreover the steady gain of a plant L(s) with n integrals is dened as 2 Stability Criterion k o = lim s! L(s)sn In general the magnitude of the transfer function decreases with frequency and it only increases in a narrow frequency range. In This paper, it is assumed the plot L(s) only intersect the db line for at most one time. A stable plant is considered rst, and then a plant which has one unstable pole. Finally, we show the stability analysis of an open-loop plant with multiple unstable poles can be converted to that of a plant with one unstable pole or a stable plant. 2.1 Stable plants First consider the stable plant with positive steady gain k o. The discuss is based on Figure 2. The Nichols plot of L(s) starts either from the point A 2
DB R D H R 8DB A DB E -18 C F 18 -DB B Figure 2: The Nichols Plot for Stable Plant above the db line or the point B below the db line both on the degree vertical line whether or not the open loop plant has integrals. When a plant has n integrals, we must add the line from to n 9 in Nichols chart as in the Nyquist plot. When the plot L(j!) starts from the point A, it, denitely, must go out the region which is dened by the boundaries DE EF and F H. So the total number for the Nichols plot of L(s) crosses the bounds DF, EF and F H is odd. If the plot L(s) crosses EF for one time then it crosses DE and F H for even times. Moreover the numbers for the positive crossing and the negative crossing as dened in Lemma 1 are equal due to the monotone property of the gain jl(s)j. According to Lemma 1, the closed-loop system is stable. If the plot does not intersect the line segment EF, i.e., the line segment C := f(; r) : r = db;?18 < < 18 g, it intersects the line DE and F H for odd times. Remember the lines DE and F H responding to the stability line R n in Lemma 1. So whether or not the net number for crossings is positive or negative, the closed-loop system is unstable since it is impossible that the net crossing is zero. When the plot starts from the point B, that is, the steady gain jk o j is less than db, the plot will not cross with the stability line R n so the system is always stable. For the stable plant with the negative steady gain k o, similar to the discuss in the above, it is easy to show it is always stable when the steady gain jk o j is less than db and the system is always unstable when the steady gain jk o j is equal or larger than db. So we have the rst result about stability in term of the db line. Proposition 1 : For the stable plant L(s) whose Nichols plot intersects db line at least one time, the feedback system is stable if and only if one of the followings holds 1. the steady gain jk o j is less than db, 2. the Nichols plot L(j!) intersects the line segment C := f(; r) : r = db;?18 < < 18 g for the plant with the positive steady gain k o and jk o j larger than db. 2.2 plants with one unstable pole Next we consider the plant with only one unstable pole. If the steady gain jk o j is less than db, then it is impossible to intersect the stability line R n in Lemma 1 and instability follows. If the steady gain k o is positive and jk o j is larger than db, due to the symmetry of the full Nichols plot of L(s), the full Nichols 3
DB R D A 8DB 1 H R 3 2 DB E -18 C F 18 -DB B Figure 3: Nyquist Plots for Plants with one Unstable pole plot of L(s) crosses the stability line for even times (including zero). Hence the feedback system is unstable. When the steady gain k o is negative and jk o j is larger than db, there are two cases. One is the crossing at! = is positive. Then the feedback systems is stable if and only the net number of other crossings is zero. It implies the feedback system is stable if and only if the Nichols plot will intersect the db line within the phase range (?18 ; 18 ) as the plot 1 in Fig. 3. The other is the crossing in! = is negative. The starting point of the plot in! = + has the phase less than?18 as the plots 2 and 3 in Fig. 3. Then the feedback system is stable if and only if the net number of the positive and negative crossings of the full Nichols plot are 2 since the plant has one unstable pole and there is one negative crossing in! =. Due to the symmetry, the net number of the half part of the full Nichols plot, ie., Nichols plot, is 1. It implies the Nichols plot of L(s) will return to the region combined by the line segments DE, EF and F H, and the Nichols plot must also intersect the db line within the phase range (?18 ; 18 ). Hence whether it is positive crossing or negative crossing at! =, the feedback system is stable if and only if it crosses the line segment C. The result for the plant with one unstable pole is summarised as follows. Proposition 2 : For the plant L(s) with one unstable pole, the feedback system is stable if and only if the half part of the Nichols plot crosses the line segment C := f(; r) : r = db;?18 < < 18 g, the steady gain k o is negative and jk o j is larger than db. 2.3 Plants with several unstable poles The stability criterion for the plant with several unstable poles are now considered. If the number of the unstable poles is even, then it can written as 2k where k is a positive integral k = 1; 2; :. In this case, the feedback system is stable if and only if the net number of positive crossings and negative crossings of the full Nyquist plot is 2k times, ie., the number of the positive and negative crossings of the Nichols plot of L(s) is k. The net number of the positive crossings more than negative crossings for one time implies the phase will adding -36. In the Nichols chart we should add a sheet. So the stability of the closed-loop system can be analysed within the phase range (?k 36? 18 ;?k 36 + 18 ). By shifting the Nichols plot of L(s) right on the multiple sheeted Nichols chart for k times, it can be shown the stability of the feedback system can be analysed by investigating the properties of the shifted Nichols plot within the phase range (?18 ; +18 ), ie., on one sheeted Nichols chart. It is reduced to a stability analysis of the plant with no unstable 4
36 27 18 9 db 3.25 db.5 db 1 1 db 3 db Figure 4: Nichols Plot of Example 1 pole but with mirror modication. Proposition 3 : Consider the plant with 2k unstable poles,k = 1; 2;, and shift the Nichols plot of L(s) right on multiple sheeted Nichols chart for k sheets. The feedback system is stable if and only if the plot intersects the line segment C := f(; r) : r = db;?18 < < 18 g, the steady gain k o is positive and jk o j is larger than db. Proposition 3 implies the stability analysis of the plant with 2k unstable poles can be transfered to that of the stable plant by shifting the Nichols plot. It should be noticed that the Nichols plot might not start within the region combined by the lines DE; EF; F H when the steady gain jk o j is above the db line. So Proposition 3 is mirrorly modied from Proposition 1. For the plant with odd, ie, 2k + 1, k = 1; 2;, unstable poles, similar to the discussion in the above, the stability analysis becomes that of the plant with one unstable after shifting the Nichols plot for k sheets. The result in Proposition 2 is employed directly. Proposition 4 : For the plant L(s) with 2k + 1 unstable poles, k = 1; 2;, after shifting the Nichols plot right on multiple sheeted Nichols chart for k times, the stability analysis of the plant with 2k + 1 unstable poles is equivalent to that of the shifting Nichols plot of the plant with one unstable pole on single sheeted Nichols chart. 3 Examples To compare with the stability criterion in [7], the graphical stability criterion developed in this section is applied in analysis of stability of all examples in [7]. Example 1 : The stable minimum phase plant is given by L(s) = k s + 1 The Nichols plot is shown in Fig. 4 with k = 2. Because it crosses the db line within the range(?18 ; 18 ) and the open-loop plant is stable, it follows from the Proposition 1 that the feedback system is stable. It can also be shown that the open-loop steady gain jk o j is larger than db and the Nichols plot crosses the db line within the range (?18 ; 18 ) for any k > 1. When k 1, the open loop steady gain jk o j is equal to or less than db. Hence following the Proposition 1, the closed-loop system is also stable for all k >. 5
36 27 18 9 db 3.25 db.5 db 1 1 db 3 db Figure 5: Nichols Plot of Example 2 db 3.25 db.5 db 1 1 db 3 db 36 27 18 9 Figure 6: Nichols Plot of Example 3 Example 2 : The plant is the same as the plant in Example 1 but the gain is negative. The Nichols plot for the plant with k =?2 is depicted in Fig. 5. Since the Nichols plot intersects the the db line out with the range (?18 ; 18 ) and the gain is negative, the closed loop system is unstable for k =?2 by using Proposition 1. From the Fig. 5, we can nd the Nichols plot crosses the db line less than or equal to?18 for any k?1 and the steady gain jk o j is less than db, and when?1 < k <, the Nichols plot doesn't cross the db line. Hence from Proposition 1, the closed-loop system is unstable for k?1 and stable for?1 < k <. Example 3. This example has three stable poles L(s) = k (s + 1)(s + 5)(s + 1) The Nichols plot for the plant with the gain k = 3 is shown in Fig. 6. Since the open-loop system is stable, Proposition 1 is applicable and we can conclude the closed-loop system is unstable since the Nichols plot intersects the db line less than?18. Example 4. Consider the following fth order plant L(s) = k(s + 5) 2 (s + 1) (s + 1)(s + 2)(s + 5)(s + )(s + 5) The plot on a Nichols chart is given Fig. 7 for k = 1. From Proposition 1, it is easy to determine that the closed-loop system is stable. Similar to [7], we also can determine the stability range of the parameter k by Proposition 1. However (1) (2) 6
36 27 18 9.5.25 db db db db 6 8 1 db 8 db 1 1 16 1 Figure 7: Nichols Plot of Example 4.25 db.5 db db db db 6 8 8 db 36 27 18 9 Figure 8: Nichols Plot of Example 5 we don't need to count the numbers of the negative and positive crossings as in [7] since there is at most one time crossing of the db line for this plant. Example 5. Now we examine a plant with integrals. A stable type 1 system L(s) = k s(s + 1)(s + 1) is considered. Since the plant has a positive gain and has a integral, the line from to?9 is added and we draw the plot from?9 as! = +. The plot for the plant with k = 1 is shown in Fig. 8. According to the sucient and necessary condition for in Proposition 1, we conclude the closed-loop system is stable for k = 1. By increasing the steady gain k, the plot will move up. When k > 1, the plot crosses the db less than -18, So the closed loop system becomes unstable when k 1. Example 6. Consider a type 1 unstable system L(s) = k s(s? 1) The gain for plotting is k = 1. Because the plant has the negative gain, the Nichols plot starts from?18. The Nichols plot is given in Fig. 9. This is a stability analysis problem for the plant with one unstable pole and the negative gain, so Proposition 2 should be used. Because the plot intersects the db line for the phase less?18, the feedback system is unstable according to the sucient and necessary condition in Proposition 2. Moreover it is impossible to increase (3) (4) 7
36 27 18 9 db 3.25 db.5 db 1 1 db 3 db Figure 9: Nichols Plot of Example 6 db 3.25 db.5 db 1 1 db 3 db 36 27 18 9 Figure 1: Nichols Plot of Example 7 or decrease the gain such the plot crosses the db line within (?18 ; 18 ). The the closed-loop system is unstable for all k >. Example 7. In this example a non-minimum phase plant is concerned. The plant is given by k(1? s) L(s) = (5) s(s + 1) The Nyquist plot is drawn on a Nichols chart for the gain k = 1 as shown in Fig. 1. Since the gain for this plant is positive and the open-loop plant is stable, the stability of the closed loop system is analysed by Proposition 1. The closed-loop system is unstable since the plot intersects the db line at?18. Obviously the stability requirement is satised if the plot is lowered. So the closed-loop system is stable if and only if < k < 1. All examples in [7] are tested in the above by the stability criterion developed in this paper. The same conclusions are obtained and our criterion is more simple and more clear since only whether the plot crosses the db line is concerned and we don't need to determine the sign of every crossing is positive or negative and count the numbers of the positive and negative crossings. Example 8 : The stable plant with the occision component L(s) = k (s 2 =25 + s=25 + 1)(s + 1) is considered. The Nichols plot for the plant with k = 2 is shown in Fig. 11. The Nichols plot does not monotonically decrease with the frequency!. But the Propositions given in the above section also hold for this plant since the plot (6) 8
36 27 18 9.5.25 db db db db 6 8 1 db 8 db 1 1 16 1 Figure 11: The Nichols Plot for Stable Plant only intersects the db line one time. Since the Nichols plot intersects the db line within the range (-18 ; 18 ) and the open plant is stable, following from Proposition 1, this closed-loop system is stable. Hence the result in this section provides an alternative method for graphical stability test in QFT design. Moreover the advantage of this method in QFT design for non-minimum phase and unstable plants is shown in the next section. 4 Application in QFT Design Horowitz and Sidi [6, 4] develop a method for QFT controller design for uncertain non-minimum phase and unstable plants. The key idea is to convert the loop-shaping problem for an unstable and/or nonminimum phase nominal plant to that for a stable nonminimum phase nominal plant. The reason is in numerical design it is more convenient to work with a minimum phase function because the Bode integrals [1] can be used and the optimal loop shaping can be derived [5, 6]. In addition, although the same limitations imposed by right plane zeros and poles exist whatever choice is made and appear in one form or another, this method explicitly reveals the limitations on L(j!) and makes it much easier to see if the assigned specications can be satised. The method suggested by Horowitz and Sidi [6, 4] is to shift the robust bounds for the unstable non-minimum phase nominal plant to that for the stable and minimum phase nominal plant. This method is now recognised as an eective method to deal with non-minimum phase and unstable plants within QFT formulation (for example see [1, 8]). Recently Chen and Ballance [3] point out that not only the robust bounds but also the stability line for the new nominal plant must be shifted. That is, the stability line R n moves with the frequency!. Similarly, it can be shown that the phase range (?18 ; 18 ) on the db line for testing the stability should also be moved left or right. More specically, when P o (s) = Po(s)A(s), the phase range (?18? arg(a(j!)); 18 - arg(a(j!))) of Po(s) at frequency! corresponds to the phase range (?18 ; 18 ) for P o (s). Next a simple example is illustrated to show how to apply the new stability criterion in QFT design for non-minimum phase and unstable plants. It is shown that it is much easier to test the stability of the nominal closed-loop system graphically with the stability criterion developed in this paper than with the stability criterion in [7]. 9
Open loop: 5 Closed loop: Frequency: 384.97deg,45.41dB.13deg,.4dB n/a rad/sec 3 5 1 6 3 2 1.3.1 1 1 1 3 35 3 25 15 1 5 X: Phase (degrees) Y: Magnitude (db) Figure 12: Shifting Bounds and Phase Range of Stability Example 9 : Consider a simple second order unstable plant P (s) = k(s + a) s(s? 2:5) (7) where the uncertain parameters k and a are within the same ranges in k 2 [1; 1]; a 2 [:1; 1] (8) For simplicity, only the robust stability is considered and robust performance is not considered. The robust stability can be imposed by L(s) 1 + L(s) 2:1 for all a 2 [:1; 1]; k 2 [1; 1]; (9) The plant under the parameters k = 1 and a = 1 is chosen as the nominal case. P o (s) = (s + 1) s(s? 2:5) (1) This plant has one unstable pole. According to Horowitz and Sidi method[6, 4, 3], choose Po(s) (s + 1) = (11) s(s + 2:5) as the new stable minimum phase nominal plant. The shifting bounds and the stability lines at the frequencies! i = :1,.3, 1, 2, 3, 6, 1, 5 are plotted in Fig. 12. A controller G(s) = 6:582: (12) is designed by loop shaping [3]. As shown in Fig. 12, it is possible for the Nyquist plot to intersect the stability line within the frequency band! 2 [1; 2]. Since, similar to the most of the Nichols plots, the phase doesn't monotonely vary with frequency! in this band, it is dicult to determine whether the loop transmission L(s) under the controller intersects the shifted stability lines R(!) and the numbers of the positive and negative crossings is also dicult to be counted. So it is dicult to analyse the stability of the nominal closed-loop system under the controller G(s) (12) by the stability criterion [7]. This is also our motivation to develop the new stability criterion. 1
Now the stability criterion proposed in the above section is used to analyse the stability of the nominal closed-loop system in Fig. 12. Since this nominal plant P o (s) with one unstable pole and the gain is negative, according to Proposition 2, the closed-loop system is stable if and only if the plot should intersect the db line within the phase (?18 ; 18 ). When the robust bounds of the nominal plant P o (s) are shifted to that of the nominal plant P o(s), the phase range also shifts from (?18 ; 18 ) to (?18? arg(a(j!)),18?arga(j!))) as shown in Fig. 12. It is easy to show the plot crosses the db line near frequency! = 6 with the phase about?75. The phase range for stability becomes (?1 ; 2 ) at frequency! = 6. Clearly it is within this phase range. Following the necessary and sucient condition for stability in Proposition 2, the nominal closed-loop system under the controller is stable. In other words, with the new stability criterion in hand, it is very easy to know how to shape the loop transmission L(s) such that the nominal stability of the nominal closedloop system is guaranteed when the QFT design is performed in the graphical CAD environment. This is the main advantage of the graphical stability criterion. For example, the nominal closed-loop system is always stable when the controller gain is increased in this example. The nominal closed-loop system is also stable until decreasing the control gain such that the plot crosses the db line at frequency 2, 5 Conclusion In QFT, when Horowitz and Sidi[6, 4] method is employed to design robust controllers for non-minimum phase and unstable plants, Chen and Ballance [3] point out that the stability of the new nominal plant must be reformulated. That is the stability line varies with frequency. It is dicult to analyse the stability of the closed-loop nominal systems by the existing stability criterion. A new graphical stability criterion on Nichols chart was proposed, which is based on the phase information when the plot gain is db. As shown in the last example, this criterion provides an eective method for stability analysis of the new nominal systems. It utilises the gain property of the transfer function. Moreover although this criterion is applicable for the plot crossing the db line at most one time, as we shown in this paper, it is an eective alternative method for stability analysis of most of physical systems since most of plants have monotone property of the gain. References [1] H. W. Bode. Network Analysis and Feedback Amplier Design. van Nostrand, New York, 1945. [2] C. Borghesani, Y. Chait, and O. Yaniv. Quantitative Feedback Theory Toolbox User Manual. The Math Work Inc., 1995. [3] Wenhua Chen and Donald J. Ballance. QFT design for uncertain nonminimum phase and unstable plants. In Proceedings of the 1998 American Control Conference, Philadelphia, U.S.A., 1998. 11
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