AC : A GRAPHICAL USER INTERFACE (GUI) FOR A UNIFIED APPROACH FOR CONTINUOUS-TIME COMPENSATOR DESIGN

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1 AC : A GRAPHICAL USER INTERFACE (GUI) FOR A UNIFIED APPROACH FOR CONTINUOUS-TIME COMPENSATOR DESIGN Minh Cao, Wihita State University Minh Cao ompleted his Bahelor s of Siene degree at Wihita State University in 24. He ompleted his Master s of Siene degree at Wihita State University in 27. He is urrently working as a Senior Eletrial Engineer at Hawker Beehraft Corporation. John Watkins, Wihita State University John M. Watkins is an Assoiate Professor in the Department of Eletrial and Computer Engineering at Wihita State University. He reeived his B.S. degree in eletrial engineering from the University of Nebraska-Linoln in 1989 and his M.S. and Ph.D. degrees in eletrial engineering from The Ohio State University in 1991 and 1995, respetively. His researh interests inlude feedbak ontrol and time-delay systems. Rihard O'Brien, U.S. Naval Aademy Rihard T. O Brien, Jr. (M'97) was born in Jersey City, NJ in August He reeived his Bahelor of Siene in Engineering with Honors in 1991 from Brown University, Providene, RI, Masters of Siene in Engineering and Dotor of Philosophy in Eletrial and Computer Engineering from The Johns Hopkins University, Baltimore, MD, in 1994 and 1998, respetively. Sine 1997, he has been on the faulty of Systems Engineering Department of The United States Naval Aademy where he has been an Assoiate Professor sine 22. Dr. O'Brien's urrent interests inlude H-infinity ontrol and estimation. Amerian Soiety for Engineering Eduation, 28 Page

2 A Graphial User Interfae (GUI) for a Unified Approah for Continuous-Time Compensator Design Abstrat As an alternative to the numerous distint ontroller design algorithms in lassial ontrol textbooks, a simple unified design approah, whih is independent of the form of the linear system information, was developed in previous work for all standard lassial ompensators. This approah is based on a simple root lous design proedure for a proportional-derivative (PD) ompensator. From this proedure, design proedures for unified notation lead, proportional-integral (PI), proportional-integral derivative (PID), and PI-lead ompensator were developed. With this proposed approah, students an onentrate on the larger ontrol system design issues, suh as ompensator seletion and losed-loop performane, rather than the intriaies of a partiular design proedure. One students learn the unified design proess disussed above, it is important that they get an opportunity to apply it to design and laboratory projets. Most real life examples require design iterations. The Graphial User Interfae (GUI) developed in this paper not only makes this feasible, but also makes this an exellent learning opportunity. The authors have implemented the unified ompensator design proedure as a GUI in MATLAB. The GUI presents the user with both root lous and Bode information. Either domain an be used for design. The effet of the design on both domains an be seen instantaneously. The GUI also provides the user with the losed-loop step and Bode response as well. Design speifiations in the time and frequeny domain are easily verified. Introdution In ontrols eduation today, there seems to be gap between the theory taught in the typial undergraduate lassroom and the skills required for pratial appliation of ontrol systems. One obvious reason for this is the lak of undergraduate ontrol system laboratories. The ontrol systems ommunity has reognized this need. 1,2 In many departments around the world, undergraduate ontrol system laboratories are being developed. 3,4 A less obvious reason for this gap is the ookbook approah to ompensator design found in typial lassial ontrol textbooks. 5,6,7 For example, a quik omparison reveals signifiant differenes in the proedures for root lous lead design and root lous Proportional-Integral (PI) design. Even more importantly, there are signifiant differenes in the proedures for lead ompensator design using root lous tehniques and Bode tehniques. Furthermore, for even simple systems, these design proedures may yield poor results. 8 Therefore, students end up onentrating on the different reipes that may or may not yield satisfatory results, and, onsequently, tend to miss the big piture. In previous work, design methods were developed that permit students to apply a simple, unified design approah for six standard ompensators (Proportional-Derivative (PD), lead, Proportional-Integral (PI), lag, Proportional-Integral-Derivative (PID), and PI-lead independent of the form of the system information. 9,1 In root lous design, the omputational proedures are Page

3 based on the open-loop transfer funtion whereas, in Bode design, the omputational proedures are based on the magnitude and phase of the open-loop frequeny response. With this approah, students an onentrate on the larger ontrol system design issues, suh as ompensator seletion and losed-loop performane, rather than the intriaies of a partiular design proedure. The unified design approah in this paper is based on a simple proedure for root lous Proportional-Derivative (PD) design. Design proedures for the other five ompensators are based on this simple PD design proedure. The unified design approah has been applied suessfully in lassial ontrol lasses at the U.S. Naval Aademy. One students learn the unified design proess disussed above, it is important that they get an opportunity to apply it to design and laboratory projets. Most real life examples require design iterations. The Graphial User Interfae (GUI) developed in this paper not only makes this feasible, but also makes this an exellent learning opportunity. The authors have implemented the unified ompensator design proedure as a GUI in MATLAB. The GUI presents the user with both root lous and Bode information. Either domain an be used for design. The effet of the design on both domains an be seen instantaneously. The GUI also provides the user with the losed-loop step and Bode response as well. Design speifiations in the time and frequeny domain are easily verified. A lassial ontrol design GUI is inluded in the MATLAB Control System Toolbox. It is alled the SISO Design Tool, where SISO refers to Single-Input Single-Output transfer funtions. 11 Like the GUI in this paper, the SISO Design Tool allows the student to view both root lous and Bode information. Unlike the GUI in this paper, the built in design tools are not based on the unified design proedures of the author s previous work. 9,1 Consequently, it does not provide the same seamless integration for the students as they transition from simple lassroom exerises to more ompliated lab and design projets. Compensator Design The integrated design proedure using time or frequeny domain plant data requires a generalization of the angle riterion from root lous design. The standard losed-loop system is shown in Figure 1 where K is the ontrol gain, Gs ( ) is the ompensator and Gp ( s ) represents the plant dynamis. Rs () - K G () s Gp () s Ys () Fig. 1. Closed-loop blok diagram. In root lous design, the ompensator must satisfy the well-known angle and magnitude riteria G( sd) Gp( sd) 18 (1) K G ( s ) G ( s ) 1 d p d Page

4 2 where the design point an be written as s j 1 in terms of the desired d n n damping ratio and the desired natural frequeny. In Bode design methods, the speifiations are inorporated through the desired phase margin PM and the gain rossover frequeny and result in another set of angle and magnitude onstraints (2) g G ( j ) G ( j ) 18PM g p g K G ( j ) G ( j ) 1 g p g Using standard 2 nd order assumptions, 5 the PM and g an also be determined from the ontinuous-time design point as 1 2 (3) PMtan n (4) g tan(pm) Equations (1) and (2) an be ombined to get the generalized angle and magnitude onstraints G( s) Gp( s) 18 (5) K G ( s ) G ( s ) 1 p where the generalized design point is sd, root lous (6) s jg, Bode and the desired angle in the angle onstraint is, root lous (7) PM, Bode Using the angle onstraint in (5), the desired ompensator angle an be omputed from the plant information and the design point without knowledge of the ompensator type. : G ( s ) 18 G ( s ) (8) p In root lous methods, determines a geometri relationship between the design point and the ompensator poles and zeros. In Bode methods, is the phase that must be added at the gain rossover frequeny. PD Compensator n In this unified method, the design proedures for all ompensators are based on the PD design proedure. 9,1 The PD ompensator has a transfer funtion G ( s) s z where the zero is hosen to meet the design speifiations. The angle of the PD ompensator at the design point is Page

5 (9) G ( ) ( ) s s z Therefore the ompensator zero is given by (1) z tan ( ) where s j is the design point in the unified notation. For this ompensator, and eah ompensator to follow, the gain K is omputed using the magnitude onstraint in (5). There is a limit to the improvement that the PD ompensator an ahieve. In general, the ompensator zero should be minimum phase beause a non-minimum phase ompensator an lead to poor performane and/or instability in the losed-loop system. Under this assumption, the maximum value for is ( s ) and is ahieved by the derivative ompensator G ( s) s. As z approahes infinity, approahes. It follows that the design point an be ahieved or, equivalently, that the PD ompensator design problem is feasible if and only if where,max s,,max. Feasibility relationships are shown in Table 1 for other ompensators (lead, PI, PID, PI-lead) using relationships between these ompensators and the PD ompensator. Lead Compensator Table 1. Feasibility of Compensator Types Compensator Feasibility relationship PD, lead,max PI, lag -,max PID, PI-lead -, max, max s z The lead (pratial PD) ompensator has a transfer funtion G () s where z p. The s p angle of lead ompensator at the design point s is G( s) ( s z) ( s p) z p and the lead pole and zero must be seleted to satisfy the angle onstraint (9), or equivalently,. z p For a given desired ompensator angle, the angle ontributed by the lead ompensator zero z must be greater than that for a PD ompensator due to the angle ontributed by the lead ompensator pole p. Figure 2 provides a graphial omparison of the pole and zeros loations of the PD ompensator and those of the lead ompensator. From (16), it follows that the PD ompensator angle is obtained if p. In this sense, the PD ompensator design establishes limits on the lead ompensator design. Speifially, the lead zero must be smaller that the PD zero, i.e., zlead zpd. Page

6 s Im p z - p -zpd -zlead Re Fig. 2. Relationship between PD and lead ompensators. After the lead zero is hosen the lead pole is omputed from p where tan( ). p z Proedures for other ompensators an be found in previous work 9,1. A omplete list of the ompensators inluded in the GUI is provided in Table 2. p GUI Design In order to make the disussion of the GUI simpler, the GUI an be divided into 3 regions. The region on the left side of the GUI displays the unompensated and ompensated root loi. The right side of the GUI displays the unompensated and ompensated Bode magnitude and Bode phase plots. The entral region of the GUI is where the user enters the system speifiations. The user an selet the ompensator type, system plant, design point,, and ompensator zeros (if appliable based on the ompensator hosen). When the Compensator drop-down box is seleted, the GUI gives the user the hoie of seleting between the ompensators listed in Table 2. For the PID and PI-Lead ompensators, the user has the ability to selet different ompensator zeros beause the unknown parameters in the ompensator outnumber the onstraints. The system speifiations, whih may be determined from the rise time (T r ), settling time (T s ), peak time (T p ), and perent overshoot (P.O.), are entered in terms of the Design Point and (labeled Phi_d on the GUI) input fields. The Update button is pressed to exeute the GUI. The GUI then plots the root lous and Bode plots in their respetive areas. The Analysis option, in the toolbar, allows the user to plot the open-loop Bode response, losed-loop step response, and losed-loop Bode response. Page

7 Table 2. Compensator Types Compensator Desription Transfer Funtion PD Lead PI Lag Proportional- Derivative Lead Proportional- Integral Lag G G G G Transient Response Steady-State Error (e ss ) (s z) Improve - (s z) Improve - (s p) (s z) Degrade Improve s (s z) Degrade Improve (s p) PID* PID** Proportional- Integral-Derivative Proportional- Integral-Derivative G 2 (s z) G s Improve Improve (s z1)(s z 2 ) s Improve Improve PI-Lead* PI-Lead** 2 Proportional- (s z) G Degrade Improve Integral-Lead s(s p) Proportional- (s z1)(s z 2 ) G Degrade Improve Integral-Lead s(s p) *oloated zeros **nonoloated zeros z < p for lead ompensator z > p for lag ompensator Example using Root Lous Method In this example, the ompensator for an antenna-angle traker will be designed. The G p (s) in 1 Figure 1 is represented by the transfer funtion G p ( s). 12 The system input U (s) is s(1s 1) the voltage applied to the servo motor. The system output Y (s) is the angular position of the antenna. The design speifiations for a step input are the following: perent overshoot less that 16%, settling time less than 1.66 seonds, and a steady-state error of zero. From basi seond order approximations, one design point that satisfies these onstraints is s j 2.41 j4.13. Page

8 Fig. 3. Graphial User Interfae (GUI) in Matlab The first step is to plot the unompensated plant information. The unompensated root lous is shown in Figure 4. The seond step is to determine if a ompensator is neessary. The root lous is to the right of the design point s, therefore, a ompensator is needed. Beause the plant is a Type 1 system, the steady state error riterion is automatially met for a step input provided that the losed-loop system is stable. A lead ompensator is hosen beause it pulls the root lous left and improves the transient response of the system. The GUI alulates the ompensator angle. The ompensator poles and zeros must be seleted. Figure 2 shows the relationship between the lead ompensator pole, zero, pole angle ( p ), and zero angle ( z ). For a lead ompensator, the zero ( z lead ) must be seleted suh that it is smaller than the zero of a PD ompensator ( z PD ). Using a PD ompensator in the GUI, we get z PD We will hoose z.1 in order to ensure that the design point is as dominant as possible. The value of the lead 1 z is then alulated by z tan. The value of p an be obtained by z p zlead d and p d. The GUI then alulates the value of the ontrol gain K from the tan( p ) magnitude riterion in (5). Next we evaluate, simulate, and (if neessary) redesign. From Figure 5, the P.O.=16% and T s =1.68 seonds. The design speifiation for P.O. was met. However, the design speifiation for T s was not met. Using our seond order approximations, a new design point was hosen as s 3 j4.13. The real part of s,, was inreased in order to improve T s. Figure 6 shows the losed-loop step response for the new design point. The new design point does meet the Page

9 design speifiations with P.O.=1.2% and T s =1.16 seonds. The new ompensator is ( s.1) G ( s) ( s 4.82) Fig. 4. Example with s 2.41 j4.13, z=.1, and 1.4 Step Response 1.2 Peak amplitude: 1.16 Overshoot (%): 16 At time (se): Final Value: 1.8 Rise Time (se):.345 Settling Time (se): 1.68 Amplitude Fig. 5. Closed-loop step response with s 2.41 j4.13, z=.1, and Time (se) Page

10 1.4 Step Response 1.2 Peak amplitude: 1.1 Overshoot (%): 1.2 At time (se): Settling Time (se): 1.16 Final Value: 1.8 Rise Time (se):.358 Amplitude Time (se) Fig. 6. Closed-loop step response with s 3 j4.13, z=.1, and Example using Bode Method We will repeat the example using the Bode method with the plant and design speifiations the same as before. The relationship between the damping ratio and the PM in (3) an be approximated by 1.6 (11) PM From (11) and (4), one design point that satisfies the satisfies the speifiations is s j jg j2.2 with PM 5. The first step is to plot the unompensated Bode plot. The GUI plots the unompensated plot (in blue) in Figure 7. The unompensated Bode phase response annot satisfy the PM requirement for This leads us to the onlusion that a ompensator is needed. A lead g ompensator is seleted in order to improve the transient response. As in the root lous example, is alulated and a lead ompensator zero is hosen with.1. After alulating the pole loation, the GUI alulates the value of the ontrol gain z lead K from the magnitude riterion in (5). The final step is to evaluate, simulate, and (if neessary) redesign. From Figure 8, the P.O.=18.1% and T s =3.27 seonds. Neither one of the design speifiations were met. Using (11) and (4), was inreased to 3 rad/se and the PM to 6º. Figure 9 shows the losed-loop step response for the new design point. With the new design g Page

11 point, the system does meet the design speifiations with the P.O.=8.77% and T s =1.41 seonds. (.1) The new ompensator is ( ) 18 s G s. ( s ) Fig. 7. Example GUI with s j j2.2, z=.1, and 5 g 1.4 Step Response 1.2 Peak amplitude: 1.18 Overshoot (%): 18.1 At time (se): Final Value: 1.8 Time (se):.833 Amplitude:.96 Settling Time (se): 3.27 Amplitude Fig. 8. Closed-loop step response with s j j2.2, z=.1, and 5 Time (se) g Page

12 1.4 Step Response Peak amplitude: 1.9 Overshoot (%): 8.77 At time (se):.935 Settling Time (se): 1.41 Final Value: 1.8 Rise Time (se):.445 Amplitude Time (se) Fig. 9. Closed-loop step response with s j j3, z=.1, and 6 Ball and Beam Laboratory Exerise To demonstrate how the GUI an be used in the laboratory, onsider the following projet involving the ball and beam apparatus shown in Figure 1. A ontrol arhiteture with two loops an be used to ontrol the position of the ball on the beam. First, an inner loop is designed to ontrol the motor position and onsequently the angle of the beam. The outer loop is then designed to ontrol the position of the beam. The outer loop is often designed assuming that the inner loop is arbitrarily fast. In this experiment, the students were asked to study the robustness of their outer loop ontrol design to the unmodeled dynamis of the inner loop. To aomplish this, the students needed to develop several designs for the inner loop ontrol, eah with a different speed of response. Normally, this would have required a fairly lengthy and tedious design proess. However, with the GUI, the students were able to quikly design several inner loop ontrols, implement them on the ball and beam apparatus, and study the robustness properties that were the main goal of the exerise. g Page

13 Fig. 1. Ball and beam apparatus Conlusion The GUI an be a very effetive teahing tool. It enables the user to see how both the root lous and the Bode plots an be used for ontrol system design. It is easy to demonstrate whether a partiular ompensator is needed. It an also be used to verify whether the design speifiations are met. The GUI also allows the user to pik either one of the ompensator design methods in the unified approah. If the root lous method is hosen, s sd and is. If the Bode method is hosen, s j and PM. g Bibliography 1. P. Dorato, Undergraduate Control Eduation in the U.S., IEEE Control Systems Magazine, vol. 19, no. 5, pp , P. Antsaklis, T. Basar, R. DeCarlo, N.H. MClamroh, M. Spong, and S. Yurkovih, Report on the NSF/CSS Workshop on New Diretions in Control Engineering Eduation, IEEE Control Systems Magazine, vol. 19, no. 5, pp , B. Armstrong and R. Perez, Controls Laboratory Program with an Aent on Disovery Learning, IEEE Control Systems Magazine, vol. 21, no. 1, pp.14-21, J. M. Watkins and R. T. O Brien, Jr., A Novel Approah to Control Systems laboratory, Proeeding of the 23 ASME International Mehanial Engineering Congress & Exposition, Washington, DC, November, 24. Page

14 5. N. S. Nise, Control Systems Engineering, 3rd Edition, John Wiley & Sons, New York, K. Ogata, Modern Control Engineering, 4th Edition, Prentie-Hall, New Jersey, R. C. Dorf and R. H. Bishop, Modern Control System, 9th Edition, Prentie-hall, New Jersey, R. C. Garia and B. S. Hek, An Interative Tool for Classial Control Design Eduation, Proeeding of the Amerian Control Conferene, pp , San Diego, CA, June R. T. O Brien, Jr. and J. M. Watkins, A Streamlined Approah for Teahing Root Lous Compensator Design, Proeedings of Conferene on Deision and Control, Orlando, FL, Deember R. T. O Brien, Jr. and J. M. Watkins, A Unified Approah for Teahing Root Lous and Bode Compensator Design, Proeedings of the Amerian Control Conferene, Denver, CO, June Control System Toolbox User s Guide, The MathWorks, In., Massahusetts, G. Franklin, J. Powell and M. Workman, Digital Control of Dynami Systems, 3rd Edition, Addison Wesley Longman, California, Page