PM diagram of the Transfer Function and its use in the Design of Controllers


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1 PM diagram of the Transfer Function and its use in the Design of Controllers Santiago Garrido, Luis Moreno Abstract This paper presents the graphical chromatic representation of the phase and the magnitude of the transfer function G(s) of a system and its educational use in the design of the most typical controllers. The magnitude is represented in decibels and the phase is represented by colours. This representation permits to put a face to the transfer functions, deepening our intuitive understanding of transfer functions. An important characteristic of this diagram is that permits to read the phase and the gain margins directly and the imaginary axis cut represents the Bode diagram. It permits to see intuitively the connexions among the different diagrams. It is also possible to put the grid with damping ratio ζ and frequency ω n. In summary, this PM diagram can be useful, especially in Control Education because improves the intuition about transfer functions, and it can be used in a Classical Control Course to complement the design of controllers using the Root Locus diagram. Index Terms Root Locus, PID controller, Lead network, Bode plot, Phase margin, Gain margin. I. INTRODUCTION A picture is worth a thousand words. There are many possible representations of the transfer function G(s) of a system such as the Root Locus, the Bode diagram, the Nyquist diagram and the Nichols diagram. In all of them, it is possible to see some characteristics of the system and to create different kinds of controllers to improve the behaviour of the system. Nowadays, with tools such as SISOTOOL of Matlab, it is possible to see many of these diagrams at the same time and the time response of the system to tune the controller. The graphical representation of functions is one of the most important mathematical tools because it allows us to understand the behaviour of the functions. While it is easy to represent the graph of a real function in the plane, the graph of a single variable complex function is more problematic. The reason adduced about why we can not represent the Transfer Function is because G(s) : C C and as C is equivalent to represent two real variables, it is necessary to have 4 axis to represent the Transfer Function G(s). Our brain is trained to visualize objects in three spatial dimensions, while the graphs of complex functions live in a fourdimensional space. Hence most of us are unable to imagine such an object. Complex functions have the reputation of being mysterious entities; seeing these strange objects may help to overcome the fear one might feel while dealing with them. Santiago Garrido and Luis Moreno are with the Robotics Lab., Carlos III University, Madrid, Spain. The Phase Magnitude (PM) diagram makes it possible to put a face to the functions, deepening our intuitive understanding of basic and advanced concepts in complex analysis. They reveal intrinsic structures behind the formulas, literally open our eyes to the wonderful realm of complex functions, and may serve students, teachers, scientists, and engineers as simple and efficient tools in their work. A possibility is to represent the magnitude and the phase of the function in each point with level curves. Cavicchi[6],[7] tried this solution in 996 and 23, but the problem was the bad resolution and the difficulty of reading his diagrams. But the situation has changed with the modern mathematical program computers that permit us to represent the fourth spatial dimension using the colourcoded values of the phase on the domain of the function. This is the solution adopted in this paper. Another problem with the representation of transfer functions is that they are rational and they have poles. In these points the function goes to infinity and far from the poles and zeros, the function is quite plane. In these conditions, the level curves are too close near the poles and zeros, and there are not level curves in the almost plane zones. A possible solution to solve this problem is the technique adopted in the Complex Variable books [3], [4], [5]: to represent the arctan of the function G(s), but in this case the magnitude level curves are not equally spaced and the diagram can t be used numerically. The best solution for Control applications is the solution adopted by Bode in its diagram: to use the vertical scale in decibels y = 2 log (x), because in this way the zones near the poles have less level curves and the level curves are equally spaced: the space between two consecutive magnitude level curves is the same in decibels. The solution adopted in this paper is inspired in [3], but represents the complex function by its module in decibels: 2 log G(s) and its phase in rescaled hsv coded colours. In the figures of the examples of controllers each phase band represents and each magnitude band 2 db. In this case the module level curves are equally spaced, and it is possible to read in the representation almost directly the gain and the phase that has to add the controller and use the values directly in the solution of problems and exercises. Different extensions of Root Locus and related diagrams have been used by many researchers. Ogata[] treats the Constant Gain Locus diagram and the Contour Roots in his book. Also, Kuo[2] refers to the Contour Root Locus. Cavicchi[6], [7] uses the PhaseRoot diagram that represent the level curves of gain and phase. More recently, Cerone[8] proposes the Constant Magnitude Loci for Control Education.
2 (a) A single zero (b) A double zero (c) A triple zero Fig. : PM diagram of a zero, a double zero and a triple zero In Fig. are shown the PM diagram of a single zero, G(s) = s, a double zero, G(s) = s 2, and a triple zero G(s) = s 3. As it can be seen the order of the colours is the same as that the original complex plane C. In the case of a double zero, the different colours appear twice and in the case of a triple zero, the different colours appear three times. In Fig. 2 are shown the PM diagram of a single pole, G(s) = /s, a double pole, G(s) = /s 2, and a triple pole G(s) = /s 3. The different colours appear once, twice and three times but in the opposite orientation. In all the PM diagrams presented in this work, the magnitude or modulus of the transfer function G(s) is represented in decibels, i.e., it is represented 2 log G(s). In this way, the difference between two consecutive lines is the same in decibels. The other set of level curves, the coloured ones, represent the phase arg(g(s)). In this case, the scale is linear with red representing and cyan representing 8. That means that the cyan line is the Root Locus and the red line is the Inverse Root Locus. An important characteristic of this diagram is that it permits to read the phase margin directly because it is the phase distance from the actual closed poles position following the same magnitude line until the intersection with the imaginary axis and the Gain Margin that it is the magnitude distance following the cyan line that represents 8 until its intersection with the imaginary axis. The imaginary axis cut represent the Bode diagram. It is also possible to put the grid with damping ratio ζ and the frequency ω n. In the drawings, we have used linear scales for the real and imaginary axis, but if there are poles and zeros placed in different decades, it would be better to put log scales. In order to use these diagrams to calculate controllers C(s) for a system G(s), it is possible to read the phase, calculate the controller C (s) with gain that adds that phase, represent the PM diagram of C (s)g(s), and then multiply by the gain needed to have C(s) = K c C (s). The use of the SISOTOOL has a strange characteristic. You can add zeros and poles and change its positions, and change the gain until by trial and error you have the closed loop poles in the desired position. However, in the educational books about Control the process is different, you know the desired position of the closed loop poles and you have to calculate the controller in order to have the closed loop poles in the required positions. II. EXPERIMENTAL RESULTS The PM diagram permits to read directly the magnitude and the phase of the transfer function in each point. Its difference between the present closed loop poles and the desired ones gives the phase and the gain that has to be added by the controller. The phase and magnitudes can be read directly (approximately) or by clicking in the position to be given by Matlab (more precise). A. Ideal PD design. Suppose you have the system G(s) = (s + )(s + 2) This system has the closed loop poles placed in s,2 =.5 ±.86j as shown in Fig. 4 (point on the right). Its magnitude is G( j) =.5 and its phase is arg(g( j)) = 8 The desired specifications give us that the desired closed loop poles have to be placed in s,2 = 3.4 ± 3.4j, as shown in Fig. 4 (point on the left). The magnitude and phase of G(s) in these points are G( 3.4±3.4j) =.788 and arg(g( 3.4 ± 3.4j)) = If the desired controller is an ideal P D controller C(s) = K c (s + b), then the phase can be calculated using Fig. 5 as [ arctan 3.4 ] b 3.4 [ 8 arctan 3.4 ] arctan =
3 (a) A single pole (b) A double pole (c) A triple pole Fig. 2: PM diagram of a pole, a double pole and a triple pole (a) G(s) = (s+3)(s+4) (s+)(s+2) (b) G(s) = (s+)(s+2)(s+3) (c) G(s) = Fig. 3: PM diagram of different transfer functions (s+)(s+2)(s+3)(s+4) i.e. arctan 3.4 b 3.4 = 54.2 and that means that b = In order to calculate the gain, the magnitude condition can be applied: K = d d 2 d = The desired controller is ( 3.4) 2 = 3.28 C(s) = K(s + b) = 3.28(s + ) = s + b = K c = 7.8 s + = 7.8( +.9s) b The operations can be simplified using the PM diagram of the Fig. 4: The controller will be designed in two steps: The first step is the design of a controller of gain that aport the desired phase (Fig. 4) and the second step is to calculate the necessary gain. Between the cyan square and the green square in (Fig. 4) there are five and a half lines of phase (each one of ) and that means that the required phase of the controller has to be approximately φ = 54. Obviously, the best way of measuring the angle is that it appears in the data tip, as shown in Fig. 4. Applying arctan 3.4 b 3.4 = 54 the zero has to be in b =. The controller of gain that aport the necessary phase is C (s) = s + Now, we represent the function C (s)g(s) = s + = ( +.9s) (s + )(s + 2) as shown in Fig. 6. Now, we are going to design the second step: to calculate the necessary gain to pass from the point
4 Fig. 4: PM diagram of G(s) = (s+)(s+2). The original closed loop poles are s,2 =.5 ±.86j (on the right), and the desired poles are s,2 3.4 ± 3.4j (on the left) Im Fig. 6: PM diagram of the Transfer Function G(s) = of the ideal PD design s+ (s+)(s+2) 3.4j b Fig. 5: Application of the phase condition for the ideal PD design on right of Fig. 6 that are the actual closed loop poles to the point on the left that represent the desired poles. The actual closed loop poles of this system are in s =.59 ±.68j. In this figure, these closed loop poles are in the Root Locus (cyan line) and have magnitude db =. In order to find the gain, it is necessary to read the magnitude at the desired closed loop poles, C (s)g(s) s= j = 25 db =.56, then K c = /.56 = 7.8. The desired controller is s + b C(s) = K c = 7.8 s + = 7.8( +.9s) b B. Design of a lead network. Consider the system G(s) =. It is desired to find a controller with a static coefficient of velocity error K v = 2s, and phase margin of 5. 4K Re The first step is to find the value of K in order to have the required K v K v = lim s sg(s) = lim s s 4K = 2K = 2 s(s + 2) therefore, K =. Now, we represent the PM diagram of G(s) = 4 as shown in Fig.7. In this diagram, we find the point of the imaginary axis with magnitude line of db, the phase margin is the number of phase bands between the actual position and that cyan band that represents 8. In this case there are a bit less than two bands, approximately 8. The lead network C(s) = s + z α s + p = s + T α s + αt has to add a phase φ = γ desired γ actual + 5 = = 37 and the parameter α of the network can be calculated as α = sin(φ) + sin(φ) =.24 As the lead network adds a magnitude of log(α)db in its middle point ω c = αt, we search the point with magnitude K m = log(α) = 6.2 db
5 Fig. 7: PM diagram of G(s) = 4 This point correspond to frequency ω c = 9 rad/sec. Choosing this frequency as the new frequency of transition ω c = αt, it is possible to find the corner frequencies of the lead network and T = αω c = 4.4 αt = ω c α = 8.4 In summary, the lead network is C(s) = 8.4 s s = 4.723s s Where the multiplicative constant α is to have gain equal to. The closed loop poles of the system with the lead network are placed in 6.9 ± 8j as shown in Fig. 8 and its Bode, that it are the values in the imaginary axis of this figure, is shown in Fig.9. III. PHASE MARGIN, GAIN MARGIN AND BODE DIAGRAM It is possible to use the Matlab command sgrid to put over the PM diagram in order to read the values of the damping ratio ζ and the natural frequency ω n as shown in Fig. that represent the system G(s) = (s + )(s + 2)(s + 3) Fig. 8: PM diagram of C(s)G(s) = s+4.4 Magnitude (db) Phase (deg) Bode Diagram 4 s Frequency (rad/s) Fig. 9: Bode plot of the uncompensated system G(s) = 4 (blue), the compensated system C(s)G(s) = s (green), and the lead network C(s) = s s+4.4 s+8.4 (red)
6 In this case, the closed loop poles are placed in s,2 =.8 ±.8j, because of this G(s,2 ) = = db. The cut of the imaginary axis is the Bode diagram shown in Fig.. The gain margin is the value of the intersection of the Root Locus (cyan line) with the imaginary axis, in this case GM = 4.8 db, and the Phase Margin is the number of colour bands following the same zero gain line (white Line) until the intersection with the imaginary axis, multiplied by the value of each band,. In this case there are 8 bands, approximately P M = 8. Finally, in this figure the bandwidth (BW) is the distance over the imaginary axis from the origin to the point in which G(s) = 3 db. In this example, BW =.2 rad/sec. Imag Axis Root Locus Editor for Open Loop (OL) Real Axis Magnitude (db) Phase (deg) G.M.: 4.6 db Freq: 3.32 rad/s Stable loop Open Loop Bode Editor for Open Loop (OL) P.M.: 8.3 deg Freq:.3 rad/s Frequency (rad/s) Fig. : Root Locus and Bode diagram of G(s) = (s+)(s+2)(s+3) Fig. 2: PM diagram Graphic User Interface of the PM Diagram Matlab Toolbox the Bode diagram. It also can be read the bandwidth. Finally the PM diagram permits unify the time and frequency analysis. Fig. : PM diagram of G(s) = (s+)(s+2)(s+3) The authors have written a PM Diagram Toolbox for Matlab that include an Graphic User Interface shown in Fig. 2. IV. CONCLUSIONS This paper presents a graphical chromatic representation of transfer functions that it is very visual and intuitive. In Control Education, it can be used to design controllers in an easier way and more intuitive than with the Root Locus diagram. It also give us an deeper knowledge of the transfer function and unifies the analysis in time and frequency because the cut of the PM diagram with the imaginary axis is the Bode diagram. The Phase and Gain margins can be read directly in the PM diagram in a very visual and natural way, without using REFERENCES [] Ogata, K., Modern Control Engineering, 5th. ed. Prentice Hall, 29. [2] Kuo, B. C., Automatic Control Systems, 6th ed.. Prentice Hall, 99. [3] Wegert, E., Visual Complex Functions: An Introduction With Phase Portraits. BirkhŁuser, 22. [4] Needham, T. Visual Complex Analysis. Clarendon Press, 998. [5] Mathews, J., Howell, R., Complex Analysis for Mathematics and Engineering. Jones and Bartlett Learning, 2. [6] Cavicchi, T.J., PhaseRoot Locus and Relative Stability. IEEE Contr. Syst. Mag., vol 6, pp , 996. [7] Cavicchi, T.J., Phase Margin Revisited: PhaseRoot Locus, Bode Plots, and Phase Shifters. IEEE Trans on Education. vol 46(). pp , 23. [8] Cerone, v., Canale, M. and Regruto, D., Loopshaping Design with Constant Magnitude Loci in Control Education. Int. J. Engng. Ed. vol 24(), pp , 28
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