Fast Sketching of Nyquist Plot in Control Systems

Journal of cience & Technology Vol ( No( 006 JT Fast ketching of Nyquist lot in Control ystems Muhammad A Eissa Abstract The sketching rules of Nyquist plots were laid-out a long time ago, but have never been modified This paper examines simple rules for faster sketching of the most famous Nyquist plots These rules allow the estimation of some systems transfer functions directly from the plot, leading to a better conception of system design and the compensation procedure Keywords: Nyquist plot starting-point, Diving angle into the origin, imilar sketch transfer functions, Reduced transfer functions, tability centre (-, 0, ositive pole, ositive zero Introduction The Nyquist criterion is a semi-graphical method that determines the stability of a closed loop system by investigating the properties of the frequency-domain plot The Nyquist plot of the open loop transfer function GH ( is a plot of GH( in the polar coordinate of Imag[GH( ] versus Real[GH( ] as the input frequency 0 It is a utilization of the properties of open loop transfer function to find the performance of a closed-loop system [],[] To show this, we will sketch the Nyquist contour of the following example: GH ( ( 5 ( ( 4 Transform the transfer function TF = GH(s into the frequency domain GH( its modulus R and argument, where is the Input frequency (r/s to obtain R ( GH ( i 4 6 5 6 ( arg[ GH ( i ] 5 Calculate the modulus and argument of GH(i forming the following table: ( 4 for various values of frequencies, Associate professor at anaa University, Electrical Engineering Department - Email: M-A-Eissa@Maktoobcom

Journal of cience & Technology Vol ( No( 006 JT Table (: hows the magnitude and argument of the Nyquist plot R ( GH ( i ( arg[ GH( i ](deg 0-90 0 645-9 05 7-9985 0584-088 0 0088-609 0-80 ketch the previous modulus against its argument R on polar coordinates (real and imaginary coordinates This is the most difficult and tedious part of the Nyquist plot which we need to simplify ee Figure ( Fig ( The most difficult and tedious part of Nyquist plot as 0 Obtain the conjugate plot when 0 Figure ( - A Close the polar contour in the clockwise [CW] direction using infinite semi-circles equal to the type L of the transfer function Figure ( B Fig ( The processes of the Nyquist lot The old Nyquist stability criterion states that[],[]: The closed-loop system, which has a stable open-loop transfer function TF = GH(i stable if and only if : is

Journal of cience & Technology Vol ( No( 006 JT Where: N N + 0 is the total number of clockwise encirclements of the contour to the stability center is the number of positive poles in the TF Of course, when the system has a positive pole, then it is unstable (at least for small values of amplifier gain In this case, the Nyquist contour must enclose the stability center ( as will be clarified later in equation 0, and the stability criterion can be reduced to: N 0 It is clear that the contour does not enclose the stability center (-, 0 in the CW direction, ie the system is stable, if that was happened then it is unstable The sketching procedure of a Nyquist plot is very tedious, requiring various difficult calculations These complicated calculations are the main source of error when performing a Nyquist plot (table, usually resulting in an inaccurate plot ome designers still insist on using the classical method of sketching the Nyquist plot [][6][7], wheres others have mentioned inadequate ideas to initiate this research [][4][5], especially[], though he did not properly utilize his idea A new research work offered in [8] and [9], has introduced purely graphical methods to analyze the control systems with the help of isoclines The use of a computer to accelerate the sketch has solved the roblem [], but it still difficult to use In this paper, simple bases for Nyquist-plot will be presented and casted in robust rules These rules facilitate the sketching procedure, and maheit much easier and faster for the designer to deal with the Nyquist plot in control systems This plot can be done immediately, and without any calculation Therefore, this method leads to a better undersding of the system transfer function, allows for greater conception of the design procedure, and also it introduces a faster way of compensation Finally, it allows the estimation of some system transfer functions directly and immediately from the Nyquist plot when the number of zeroes are given If it is not given, then this method helps to obtain only the reduced transfer function which has no zeroes,(ie all zeroes are eliminated with a corresponding numbers of poles imple Rules for the Famous Nyquist lots In the following discussion we introduce simple rules and classifications for the famous Nyquist plots These rules only help in sketching the plot when the frequency ( = 0 The conjugate part of the plot can then be easily obtained, using the old procedure to close and complete the Nyquist plot Rule (: The start of the Nyquist plot occurs only from one of the following infinitesimal directions (or positions on the X Y plane, North, outh, East and West depending upon the type of the control system see Figure ( Fig ( hows the starting positions of Nyquist plot

Journal of cience & Technology Vol ( No( 006 JT Type zero-control system tarts from the positive x-axis Type one-control system tarts from the negative y- axis Type two-control system tarts from the negative x- axis Type three-control system tarts from the positive y- axis Type four-control system tarts from the positive x- axis again And so on Ie In type zero, four, eight etc the plot starts from a position on the positive x-axis To prove that, let us consider the following transfer function: ( ( GH ( L ( ( ( Where L: is the type of the system transfer function The modulus of that transfer function TF when i is R( GH ( i L ( And the argument is: arg GH ( i ( L( ( When = 0, the radius of the polar plot can take any real value, (Usually =, and the argument of the polar plot becomes: ( arg GH ( i L Tail (4 Where the parameter (Tail is a negligible angle (either to be added to or subtracted from the diving angle, with its sign being very import as it determines on which side of the axis (left or right, over or under the plot will start It is clear that at = 0 the start of the plot depends only on the type of system transfer function and sign of the parameter (Tail This sign can be determined from to the following equation: Where: ( and i j Tail i i j J (5 are the zeroes and poles in TF which have real non-zero values Note: For special case systems with L= -, -, the plot starts from the origin, flies around, and returns the origin or somewhere else on the positive x-axis These cases are thus not part of the famous group Rule (: Before mentioning this rule, we must define the diving angle into the origin dv, which is the limiting angle that the plot makes when it arrives to the origin, iewhen This angle depends on the number of asymptotes in the system transfer function N, where: N = (the number of poles the number of zeroes

Journal of cience & Technology Vol ( No( 006 JT The radius in polar coordinates usually tends to zero when the plot is given by equation ( arg GH ( i ( L( The sum of the arguments when is where, dv, while the argument of dv ( number of zeroes - number of poles* Tail (6 For a realistic control system the number of poles should be the number of zeroes, ie dv ( N Tail (7 This equation shows that the diving angle is dependent upon the number of asymptotes N in the system transfer function ee Figure (4 Fig (4 Various diving angles in relation to the origin As we have four initial sketching positions according to Rule, and four final sketching shapes according to Rule, we can therefore say that in normal circumsces we have 4 * 4 = 6 general shapes of Nyquist plots Note: In case of type zero and when N 0, the diving angle doesn't fall into the origin Therefore, this problem is a special case and needs to be solved classically It might also be considered out of the famous group because it is a lead or lag compensation circuit Let us now analyze the case a positive zero like ( Its argument in the frequency domain is equal to: arg[ i ], This is because of the negative real value of, which means that the angle lies in the second quarter of the complex plane ee Figure (5

Journal of cience & Technology Vol ( No( 006 JT Fig (5 The argument of a positive zero In this case (8 Notice that in the previous equation when = 0 ie the start of the Nyquist plot is shifted by But when = any change to the diving angle when =, meaning that the positive zero doesn't cause Also the effect of a positive pole like ( is opposite to the effect of a positive zero, since its argument is equal to: arg[ ( i ] In the previous equation when = 0 This means that the start of Nyquist plot is shifted by p (9 But when = unaffected by a positive pole? Which means that the diving angle is It is clear that the positive zero rotates the starting point of the Nyquist plot by means of, and the positive pole rotates back this point by means of Thus the effect of a positive pole and zero together is to rotate that starting point by means of one complete infinite circle which means that the contour is enclosing the stability center and the system is unstable Equal to one infinite circle equation (0 We can conclude that the change of sign of pole or zero (positive pole or zero does not change the value of the diving angle, but it shifts the starting point of the plot by means of Therefore, the MATLAB should be modified to sketch a complete infinite circle to show system instability

Journal of cience & Technology Vol ( No( 006 JT The effect of amplifier gain K on system stability The main reasons for unstable open loop control system are: ositive poles and zeroes High values of the amplifier gain K High values of the deterioration index (discussed later in this paper ometimes, when transforming a system from an open-loop into a closed-loop feed back control system, it becomes stable This is because the feedback closure decreases the amplifier gain K, so that the shape of the Nyquist plot is changed from enclosing into notenclosing the stability center see Figure (6 The following system has been sketched for two values of amplifier gain K: GH( ( K ( The inner sketch with small amplifier gain K is stable, while the outer sketch with high amplifier gain K is unstable Fig (6 The effect of amplifier gain on system stability Application Examples on the imple Rules Ex-: Let us analyze the following transfer function: GH ( ( ( ( The system is of type, meaning that it starts from the negative Y-axis The sign of the parameter (Tail can be calculated according to equation 5 Tail Negative This means that the start of the plot will be from the left hand side of the negative Y-axis (Refer to Figure ( The number of asymptotes = ie The diving angle dv (Refer to Figure (4 The plot will then be completed using the classical steps Note: the plot can be sketched directly without writing the previous analysis (Refer to Figure (7A

Journal of cience & Technology Vol ( No( 006 JT Ex-: Another example to further clarify the rules: GH( ( ( ( The system is of type, meaning that it is expected to start from the negative X-axis (refer to Figure ( However due to the positive zero, the start is shifted by means of, ie it starts from the positive real X-axis The index N = asymptotes ie the diving angle dv = (Refer to Figure (4 The completion of the plot will be done using the classical steps, ie we need close the contour Thus the system is unstable Notice that the plot can be sketched directly without writing the previous analysis (ee Figure (7B to Fig (7 The sketches of the last two Nyquist plots Ex-: ketch the Nyquist plot of the transfer function: GH ( s s( s ( s 4( s 5 The system is of type, meaning that the plot must start from the negative Y-axis (also the effect of the positive pole and zero eliminate each others The index N = asymptotes ie the diving angle dv = (Refer to Figure (4 We need half infinite-circle to close the contour ( type The system looks stable (even on MATLAB, but the system has a positive pole and a positive zero, therefore its contour must be closed by an extra infinite circle according to equation (0 This implies that the contour encloses the stability center and the system is unstable These problems happen only when the number of positive poles is equal to the number of the positive zeroes This might be the only reason for considering the number of positive poles in the old Nyquist stability criterion It is common that the system is unstable when having a positive pole (see Figure (8

Journal of cience & Technology Vol ( No( 006 JT Fig (8 ketch of the last Nyquist plot 4 Estimating the Transfer Function from the lot This method shows that the number of poles and zeros are not directly import in determining the shape of the plot For example the following two transfer functions have similar plots GH GH ( ( ( ( ( ( ( ( ( ( That is because they are of the same type and have the same number of asymptotes N The number of zeroes is useful to know when revealing the transfer function from the Nyquist plot since if we know them, we can estimate the general shape of the transfer function from the plot (as will be discussed in Ex 4 The value of the poles and zeroes can be obtained from the value of the modulus R(, (refer to equation ( R( i GH( i L From Table (, we can substitute to obtain a number of equations equal to the number of unknown poles and zeroes (n, R R ( ( R n ( n value value value n Then this system of equations can be solved to obtain the unknown values of poles and zeroes

Journal of cience & Technology Vol ( No( 006 JT Ex-4: Another example: Let us analyze the following Nyquist plot (given that the system has one negative zero, and all the poles in the transfer function are negative Imaginary GH(i Fig (9 Nyquist plot of unknown transfer function The sketch starts from the negative Y-axis, meaning that it is of type (see rule and Figure ( The diving angle into the origin denotes that it has four asymptotes ie 4 N (refer to rule ( It is given that the system has one negative zero, thus: No of poles = 4 + = 5 (refer to equation (6 The transfer function has one zero (given and 5 poles, also of type Therefore: GH ( ( ( ( ( ( 4 This idea contributes to a greater undersding of the compensation procedure It should be noted that in generalm the addition of a lead or lag circuit etc, will not change the general shape of the Nyquist plot, but will affect the stability only, moving the plot closer or further away from the stability centre ( -, 0 It is known that moving the pole to the left or moving the zero to the right helps to improve the system stability This fast method of Nyquist plot is not suitable for transfer functions of type zero and has N 0, because the diving angle dv does not fall into the origin (see Table ( in the appendix

Journal of cience & Technology Vol ( No( 006 JT 5 ystem Deterioration Index This index indicates the inherent trouble and instability in the system transfer function, In addition, it gives greater insight about the usefulness of the system This index is givin by: = Number of asymptotes + ystem type number = N + L Table (see appendix provides helpful data and a classification of the system deterioration index can also be obtained (although it is not very strict: When : the system is inherently stable and useful = or 4 : the system is useful and might be stable = 5 : the system is around the critical stability and can be compensated 6 : the system is inherently unstable and useless Unfortunately, the MATLAB sketch is unable to show clearly all the details in table ( (see appendix One possible reason for these troubles is that the computer can not obtain the limit of the argument when 0 or 6 Comments and Conclusions Easy rules and parameters were examined to describe and treat the control systems in a modern and simplified menner The start of the Nyquist plot usually occurs on one of the four directions of the X-axis and Y- axis, and the diving angle also falls into the origin in four directions These rules help in sketching most of the Nyquist plots immediately without performing any calculations 4 This method is very useful at the end of examinations when the time is limited, and the student need to give very quick insight to verify the Nyquist plot 5 This method provides a better conception of the compensation procedure, because it is easy for the designer to change the type or the diving angle in order to obtain better system characteristics 6 It is clear that the positive zero or positive pole cause shifts to the starting position of the Nyquist plots by means of The effect of both together is a complete infinite circle surrounding the stability center, which means that the system is unstable This point can not be demonstrated using MATLAB 7 The change of sign of pole or zero does not affect the value of the diving angle 8 The number of general Nyquist plots is 6; however, one of these plots does not satisfy the rules completely

Journal of cience & Technology Vol ( No( 006 JT 9 This method is not suitable for systems of type zero and has N 0, because the diving angle dv does not fall into the origin, but falls into a point on the positive real axis This is also the case when the type L= -, -, where the sketch starts from the origin, flies around and returns the origin 0 Every box in table (see appendix represents a set of transfer functions having the same type and number of asymptotes They thus have a common Nyquist plot, although they might have an extra lead or lag circuit The compensation with a lead or lag circuit does not change the general shape of the Nyquist plot, but it improves the system characteristics rediction of the transfer function is possible when the Nyquist plot is available and the number of zeroes is known The number of asymptotes and the type number of the system help each other to deteriorate its characteristics 4 In a Nyquist plot, the point (-, 0 is called the stability centre (or stability border 5 In case of a system with a positive pole and zero, MATLAB should be modified to sketch a complete infinite circle to show system instability 6 Although these new rules are very helpful to designers, the classical sketching method is still essential when teaching beginners 7 References: MATLAB [] [] Katsuhiko OGATA, "Modern Control Engineering", rentice Hall International 00 [] Benjamin C KUO, "Automatic Control ystems", rentice Hall Inc 00 [4] aulo J G Ferreira, "Concerning the Nyquist lots of Rational Functions of Nonzero Type", IEEE Transactions on Education, Vol 4, NO, August 999 [5] eter H Greg son, "Using Contests to Teach Design to EE Juniors", Member IEEE, and Timothy A Little, IEEE Transactions on Education, Vol 4, NO, August 999 [6] D Chen, D E eborg, "Design of decentralized I control systems based on Nyquist analysis", Journal of rocess control Volume(issue: ( 00 [7] A Cook, "tability of two-dimensional feedback system", Taylor & Francis Group- Article, London, Oxford shire OX4 4RN [8] Kent H Lundberg, "ole-zero phase maps", IEEE Control ystems Magazine, vol 5, no, pp 84 87, Feb005 [9] Kent H Lundberg and achary Malchano, "Three-Dimensional Visualization of Nichols, Hall, and Robust-erformance Diagrams", ACC004

Journal of cience & Technology Vol ( No( 006 JT Appendix- 8 Appendix A program to sketch the Nyquist plot using MATLAB soft-ware is: In case of zero-pole-gain transfer function gh = zpk( [-z ], [ 0 -p -p ], [k] or in case of a polynomial transfer function gh = tf( [ a a ], [ b b b ] Nyquist(gh Appendix-: The No of asymptotes N The Type L and the famous systems transfer- Table (: howing the relation between the deterioration index functions ( s ( s a b 0 = 0 ( s b = = ( s b ( s ( s b ( s b ( s b b = 0 (s a = = = ( b ( s b ( s b = 4 ( s a ( s a = ( s a = = 4 ( s b = 5 ( s a ( s a ( s a = ( s a ( s a = 4 ( s a = 5 = 6 Notice: that every transfer function is plotted as possible as we can in the stable shape to demonstrate the useful field