Interactive Learning Modules for PID Control
Contents
1. gt LECTURE NOTES Interactive Learning Modules for PID Control Using Interactive Graphics to Learn PID Control and Develop Intuition JOS LUIS GUZMAN KARL JOHAN STR M SEBASTIAN DORMIDO TORE H GGLUND MANUEL BERENGUEL and YVES PIGUET nteractive tools can be used to complement books and 1 4 This article describes three interactive learning modules that are designed to develop intuition as well as a working knowledge of proportional integral derivative PID control These three modules comprise a package called interactive learning modules for PID ILM PID By illustrating concepts such as tuning robustness loop shaping and antiwindup ILM PID can be used for demonstrations exercises and self study The main objective of the interactive modules is to explain basic concepts of PID control without considering implementation aspects Although most PID controllers are implemented as sampled data control systems analy sis and design are traditionally performed in continuous time assuming that the sampling rate for subsequent digi tal implementation is sufficiently fast Implementation issues such as aliasing selection of the sampling time sig nal prefiltering influence of the discretization algorithms and bumpless parameter changes may be the aim of a future interactive modules focused on implementation aspects for PID control The modules of ILM PID have menus for selecting process transfer functions and c2. Simple exercises can be used to provide training in loop shaping For instance with the above process it is instruc tive to calculate the gain for a proportional controller for which the closed loop system changes from stable to unstable Before using PID Loop Shaping the result can be calculated analytically which yields LL iw LC iw P iw 180 a 180 w 1 iw 1 4 IL i C i P iw 1 Oil 1 k 1 k 4 iw 1 4 PID Loop Shaping can be used to verify the result inter actively as shown in Figure 11 This exercise challenges students and encourages them to make observations while relating theory to images to develop a broader and deeper understanding On the other hand free interactive designs can also be performed to compare the results with other design methods For instance PID Loop Shaping can be used to design a PID controller interactively for the process P s 1 s 1 where the maximal sensitivity value M must be less than 1 5 A PID controller that satisfies this constraint is obtained when k 0 92 T 1 8 k 0 5 T4 1 03 and kg 0 95 The AMIGO frequency method can also be used for design and the results can be com pared The resulting controller is given by k 1 2 T 2 48 kj 0 48 Tg 0 93 and kg 1 12 Figure 12 shows the Nyquist plots and time responses using PID Basics for both designs in blue for the free PID con troller and in red for the AMIGO method
3. The resulting values of Ms are 1 49 for free PID and 1 46 for the AMIGO method c d FIGURE 10 Nyquist plot modifications depending on the controller type a P controller b controller c PI controller and d PD con troller a The modification of L i w in the direction of P i using a P controller with gain k 2 blue curve and k 2 6 red curve The same study for an l controller is shown in b with k 1 red curve and k 0 6 blue curve where L w is modified in the direction i P iw c d Pl or PD controllers are used respectively In these cases the compensated point at the frequency w is calculated as the sum of two vectors namely the proportional vector and the integral or derivative vector OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 127 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply Effect of the Target Point The target point on the Nyquist plot can be reached using an unconstrained design by selecting the Free option The controller gains are interactively adjusted as shown in the free tuning example Another approach is to use 5 12 where the controller gains are calculated after the target point is defined As discussed above the target point can be fixed or constrained in various ways either at any point to specific values for phase margin and gain margin or to maximal values of
4. in black and the loop transfer functions L s P s C s in red Three different views can be shown depending on the tuning options Figure 9 shows two views the left one for Free tuning and the right one for Constrained PID tun ing A third view is shown in Figure 8 where two designs are shown simultaneously The design and target points can be modified interactively on this graphic The design point is shown in green on the Nyquist curve of the process The target point is represented in light green in the case of Free tuning and in black for constrained tuning as shown in Fig ure 8 The slope of the target point can also be changed TABLE 1 Sensitivity circles This table describes the center and radius of circles that define the loci for constant sensitivity M constant complementary sensitivity M constant mixed sensitivity and equal sensitivities M M M 8 Radius 1 M s E M 2 M XTE 2 Contour Center M Circle 1 M circle 2 Me X1 Xo M circle mM Ms 1 X2 max M Ms 1 _M xi max Me as OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 125 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply interactively For Free tuning the controller gains are shown as arrows in the Nyquist plot The controller gains can be modified interactively by dragging the ends of the arrows Figures 8 and 9 sho
5. loop to be shaped by dragging on the control parameters Constrained PI and Constrained PID permit the calcula tion of the controller parameters based on some con straints on the target point That is the focus can be placed on how the loop transfer function changes when controller parameters are modified which reveals the parameter values required to obtain a given shape of the loop transfer function For PI and PD control the mapping is uniquely given by one point For PID control it is also possible to obtain an arbitrary slope 3 of the loop transfer function at the target point When the Free tuning option is selected sliders are used to modify the controller gains k ki and kg as shown in Figure 8 The controller gains can also be changed by dragging arrows as illustrated in the same figure From 4 the proportional gain changes L iw in the direction of P iw the integral gain k changes L iw in the direction of iP iw and the derivative gain kg changes L iw in the direction of iP iw For the Constrained PI and Constrained PID tuning options the target point can be limited to move on the unit circle the sensitivity circles or the real axis In this Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply way loop shaping is enabled with specifications on gain and phase margins or on the sensitivities In the case of Constrained PI it
6. 4 2008 at 08 27 from IEEE Xplore Restrictions apply Process Output Graphics o Save wDelete y d 0 5 0 0 20 40 Time a Controller Output oP al u 0 1 0 20 40 Time b FIGURE 7 Load disturbance response for a PI controller using the approximate M constrained integral gain optimization AMIGO step method This method enables the compromise described in Figure 6 focusing on load disturbances by maximizing integral gain and adding a robustness constraint The a process output and b control signal to load disturbances respectively for a PI controller designed using AMIGO and with K 0 414 and T 2 66 The slow response compared with Figure 6 corresponds to increased stability margins l8 x CEEE gee l8 x Sam tae l Rao ILM PID LoopShaping Instructions Process 3 ry 5 Controller Tuning Constrained PI O Constrained PID Free Constraints ONo OPm OGm Ms OM OM wdesign 0 89 slope 47 1 Controller type Op OI Om Om pp 148 Ti 1 80 ket 3 40 To 1 28 Robustness and Performance Ms 2 00 Mt 140 Ws 063 Wt 140 Gm 299 Pm 42 66 so a e Wee 1 79 Wt 083 2 0 2 FIGURE 8 The user interface of the module PID Loop Shaping showing both Free and Constrained PID tuning The loop transfer function is shown for two designs under the Free design option Pro portional integral and derivative actio
7. Tialx salm e ol Pe Aae Process Output C Linear E Windup E Antiwindup CIPaWindup CPB AntMindup FIGURE 17 The user interface of the module PID Windup showing the windup phenomenon and application of the antiwindup tech nique Several graphical elements are used to interactively analyze typical problems and solutions associated with windup The exam ple shown in the figure illustrates the windup phenomenon in blue and the result of applying the antiwindup technique in green Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply controllers with integral action can be selected l Pl PID where several sliders are available to change the controller parameters including the tracking time constant Tt A sat uration graphic is also available in this zone The Actuator Saturation graphic allows the saturation limits to be deter mined by dragging the small red circle located on the upper saturation value In this graphic a symmetric satu ration is selected for pedagogical purposes Graphics Time responses for process output control signal and integral action are available in three graphics namely Process Output Controller Output and Integral Term In the same way as in PID Basics multiple interactive graph ical elements can be used to change the setpoint load dis turbance measurement noise or horizontal a
8. called PB Windup and PB Antiwindup appear near the top of plot Process Output The activation of these options shows the proportional bands for the windup and antiwindup cases in the Process Output graphic The proportional bands are shown as dotted green and blue curves respectively as shown in Figure 18 Process Output O Linear X Windup O Antiwindup amp PB Windup OPB Antiwindup 0 20 40 60 0 20 40 60 FIGURE 18 Example of the windup phenomenon with proportional band for a K 1 and b K 0 4 In 8 the notion of proportional band is described as being a useful tool for understanding the effects of windup The proportional band is an interval such that the actuator does not saturate when the instantaneous value of the process output or its predicted value is inside this band These plots show two examples demonstrating how the control signal is saturated when the process output is inside the band shown in blue The interactive pink line of the graphics can be used to test this idea OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 131 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply Settings Menu The Settings menu has the same structure as in PID Basics and PID Loop Shaping The process transfer function can be chosen from the entry Process Transfer Functions and numerical values of the parameters can be introduced using Controller P
9. noise and b tuning using rules a illustrates the disadvantage of using a short tracking time constant The short pulse disturbance at time t 10 results in excessive reduction of the integral term and a large distur bance in the process output In b the choice is Tte T Ta 2 OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 133 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply engineers The interactive learning modules developed in this work are freely available from the authors 7 to test these interactive features in control education and professional training REFERENCES 1 M Johansson M Gafvert and K J str m Interactive tools for educa tion in automatic control IEEE Contr Syst Mag vol 18 no 3 pp 33 40 1998 2 J L Guzman M Berenguel and S Dormido Interactive teaching of con strained generalized predictive control IEEE Contr Syst Mag vol 25 no 2 pp 79 85 2005 3 J Sanchez S Dormido and F Esquembre The learning of control concepts using interactive tools Comput Appl Eng Educ vol 13 no 1 pp 84 98 2005 4 J L Guzman Interactive control system design Ph D dissertation Uni versity of Almeria Spain Online Available http www ual es joguz man PhdThesisGuzman pdf 5 Y Piguet SysQuake 3 User Manual Lausanne Switzerland Calerga Sa
10. represented by the transfer functions FT and CFS in 1 3 The middle portions of the plots 30 lt t lt 60 show the response to a step in the load dis turbance represented by the transfer functions PS and T in 1 3 The last portions of the plots t gt 60 show the response to wideband measurement noise which is repre sented by the transfer functions S and CS in 1 3 Several elements on the graphics are available for inter acting with the application The vertical green line at time e T 18 File Edt Settings Plots Figure Layout View Window Help 18 x slam Lele of REAA t 37 78 y 1 62 K 1 13 Ti 3 36 Td 1 21 b 0 54 N 10 K 0 43 Ti 2 27 b 0 ILM PID Basics XN Instructions Theory Process 1 4 eit k f G s o u 100s 1 100s 1 100s 1 100s 1 f f Kp 1 l 2 0 Controller kii Teed he Om Or 0 20 40 60 Time Tt 3 36 li Controller Output Op or Op Te 1 21 li oT y N10 I K 1 13 Ti 3 36 Td 1 21 b 0 54 N 10 t 0 54 1 i K 0 43 Ti 2 27 b 0 j f f f Performance Set point response Load disturbances A IAE WE 29 525 mex 7 590 overshoot 000 k wm 019 emex 046 069 Noise response sigma_x 7000 0 000 sigma_y 092 0 021 sigma_u ioe 0010 Robustness Ms i140 MIM 100 Gm 10 509 Pm ED 69 99 Time FIGURE 3 The user interface of the module PID Basics The plots show
11. the interactive elements with numerical values Ky 1 and n 4 When the user modifies any plant parameter the sym bolic representation of the process transfer function is immediately updated and its effect is reflected on the remaining graphic elements Five buttons are available for selecting the desired con troller The buttons correspond to proportional P inte eral I proportional integral PI proportional derivative PD and proportional integral derivative PID Several sliders are available below the radio buttons for modifying the controller parameters The number of sliders shown depends on the chosen controller For instance Figure 3 shows five sliders since the PID controller is selected Performance and Robustness Information Parameters that characterize performance and robustness are also displayed on the screen The performance criteria are based on the setpoint response the load disturbance response and the noise response The setpoint response is characterized by the integral absolute error IAE and the 0 10 20 30 40 50 O 0 0 5 0 5 O 10 20 30 40 50 60 FIGURE 2 Control system responses illustrating basic feedback sys tem properties To analyze the feedback loop it is essential to con sider six responses These responses which are referred to as the gang of six 8 are described by transfer functions in 1 3 One way to present this information is to show the process output
12. the sensitivity functions Figure 13 shows an example in which the target point is set to the point 0 5 0 57 Two constrained designs are shown for the design frequency w 0 6 rad s The red curve repre sents a system compensated by a constrained PID with k 1 32 k 1 02 and kg 2 15 while the blue curve rep resents a constrained PI with k 1 32 and k 0 15 Although both controllers reach the target point better results are obtained for the PID controller because the slope can be freely adjusted the value for this example is J 22 The PID controller provides better robustness properties with M 1 45 kj 1 02 Gm 5 32 and Pm 40 15 versus a PI controller with M 1 83 Kp 015 Gy 2 69 and Pr 75 77 Similar examples can be used to restrict the target point for phase margin gain margin or maximal values of the sensitivity functions Figure 14 a shows an example where a combined sensitivity constraint is required for Ms lt 2 and M lt 2 This constraint is fulfilled in two different ways namely by using Constrained PID red curve and Constrained PI blue curve Another example combining sensitivity function and gain margin constraints is shown in Figure 14 b with the specification that the gain margin be equal to three and Ms lt 2 These specifications are FIGURE 11 Stability limit on the critical point 1 Oi A typical example for presenting loop shaping is to search for the lowest gain th
13. 33X 08 25 00 2008IEEE Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply Downloading and Using ILM PID i nteractivity which is the main feature of the tools described in this work is difficult to explain in written text The best way to appreciate the tools is to use them We strongly recommend that the reader download and use them in parallel with reading this article Executable ver sions for PC Mac and Linux are freely available for download from the Calerga Web site 7 No licenses are required and the executable modules can be freely distributed to students and colleagues Process Outpt Process put 15 20 Tranter Sanction Magnitude us Ofer As Ai functions in 1 3 These transfer functions are called the gang of six in 8 To analyze the closed loop system it is necessary to consider all six transfer functions The time responses of the six transfer functions are illustrated by showing the response of the process output and control signals to a step in the setpoint a step in the load disturbance and wideband measurement noise as illustrated in Figure 2 A mix of time and frequency responses can also be displayed Process models in the form of rational transfer func tions with a time delay can be chosen from a menu that provides a collection of transfer functions An arbitrary transfer function can also be entered us
14. a new design in red appears allowing the two designs to be compared Performance and robustness parameters are duplicated displaying the values in blue and red colors associated with each design The Process Output and Controller Output graphics indicate the values of the controller para meters for both designs Figure 3 presents an example that Transfer Function Magnitude KIS RT MPS MCS MFT RFCS ML 0 01 0 1 10 Frequency Transfer Function Phase 200 compares the response of PI K 0 43 T 2 27 b 0 and PID CK 1 13 1 3 36 Ta 1 21 b 0 54 Ny 10 controllers Although the PID controller provides a better response to load disturbances by reacting faster the noise also generates more control action The delete option can be selected to remove a design If the transfer function of the process or an input signal such as a setpoint load dis turbance and measurement noise are altered both sets of results are affected simultaneously Only two designs are stored to keep the user interface simple Additional options for the time domain mode are shown above the Controller Output graphic These options show the proportional P integral I and derivative D signals of the controller The frequency domain mode is shown in Figure 4 When this mode is selected from the Settings menu the left side of the tool remains unchanged However in this case the time responses are replaced by the magnitude and pha
15. and b control signals to load disturbances respectively are shown for two PI controllers with k values of 0 36 in red and 0 30 in blue The controller with larger integral gain provides a faster response to load disturbances 122 IEEE CONTROL SYSTEMS MAGAZINE gt OCTOBER 2008 Specific values for controller parameters can be entered using the Controller Parameters menu Time and frequen cy responses can be selected from the third entry Time Frequency Domain which has the options Time Domain Frequency Domain and Both Domains The results can be stored and recalled using the Load Save menu which has the options Save Design and Load Design All data on the screen can be saved using the option Save Report From the menu selection Simulation the user can modify the simulation time change the maxi mal time delay to avoid slow simulations and activate the Sweep option to show the results for several controller parameters simultaneously Parameters are swept between specified limits This option is available only in the time domain mode When active new radio buttons appear in the controller parameters zone to permit the selection of the desired parameter to be swept The last menu option Examples Advanced PID Book loads examples from 8 which the user can explore by modifying parameters Analysis and Control Design for Load Disturbances Load disturbances are typically low frequency signals that drive the system away from its
16. arameters Essential data and results can be saved and recalled using the Load Save menu options The menu selection Simulation makes it possible to choose the simulation time and activate the Sweep option which can be used to show the results for several values of the tracking time constant Several examples from 8 can be loaded from the Examples entry Examples The following examples illustrate properties of the PID Windup module Understanding the Windup Phenomenon Windup can be studied using the first entry from the Examples option menu This example from 8 uses the pure integrator process P s 1 s controlled by a PI con troller with parameters K 1 T 1 2 and b 1 and with the control signal limited to 0 2 Figure 17 shows the time responses for this example The control signal is satu rated from t 0 The process output and the integral term increase while the control error is positive Once the Process Output O Linear X Windup amp Antiwindup O PB Windup amp PB Antiwindup 40 60 process output exceeds the setpoint the control error becomes negative however the control signal remains sat urated due to the large value of the integral term The time responses are shown in Figure 17 The proportional band can be drawn in this example using the PB Windup checkbox shown in Figure 18 a Using the vertical line the user can see that the process output remains inside the band while the control signal is working i
17. at makes the system unstable This task can be interactively per formed with PID Loop Shaping as shown in this figure 128 IEEE CONTROL SYSTEMS MAGAZINE gt gt OCTOBER 2008 established by maximizing the integral gain k Hence the constraint gain margin is chosen and the target point is located in such a way that Gm 3 Then a Constrained PID controller is selected where the design point and the slope are modified until Ms lt 2 and the integral gain is maximized The final controller is given by k 1 38 k 0 52 and kg 0 54 for 1 02 and 3 32 L Plane Design1 Design2 Graphics XSave ODelete 2 Process Output Graphics X Save O Delete K 2 eZ T093 l NSN K 0 92 T 1 79 Ta 1 039 b N 10 Time Controller Output ar ol oo u kK 1 2 T 251 T4 0 93 b 1 N 10 K 0 92 ilS EOE a N 10 Time FIGURE 12 Example of loop shaping with M lt 1 5 PID Loop Shap ing can be used to compare various designs In this figure a Nyquist plots and b time domain responses generated with PID Basics are shown to compare an unconstrained design k 0 92 Ti 1 8 k 0 5 Ta 1 03 and ka 0 95 with an alternative design developed using the AMIGO frequency method k 1 2 T 2 48 k 0 48 Ty 0 93 and ky 1 12 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply The Deriva
18. d curve This behavior is a consequence of the presence of complex zeros due to T lt 47g The same example is shown in blue for T 4 Ta where this problem is avoided 8 130 IEEE CONTROL SYSTEMS MAGAZINE gt OCTOBER 2008 The Interactive Tool We now describe the main aspects of PID Windup Process The Process area is similar to that described in PID Basics and PID Loop Shaping The time delay is modified using a slider instead of a text edit so that the time delay effect on the antiwindup mechanism can be analyzed Controller The Controller area contains information about the con troller parameters and actuator saturation Three kinds of Actuator Model Actuator FIGURE 16 PID controller with antiwindup scheme where K is the controller proportional gain T is the controller integral time Ta is the controller derivative time Ysp is the setpoint y is the process output eis the tracking error v is the controller output u is the sat urated controller output and amp is the difference between the con troller output v and the saturated controller output u In this scheme the control signal remains unconstrained when the saturation is not active When saturation occurs the integral control action is modi fied until the control signal is out of the saturation limit The modifi cation of the integral element is performed dynamically by adjusting the tracking time constant 7 8 Boece sae oe hee bee er weer oe
19. de the band most of the time 132 IEEE CONTROL SYSTEMS MAGAZINE gt OCTOBER 2008 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply band for the PI controller with antiwindup is shown in the same figure It can be seen that the proportional band is wider than for PI without antiwindup Figure 19 a where the process output remains most of the time The effect of the tracking time constant is illustrated in Figure 19 b for T 0 1 10 50 In this scenario the Sweep menu option is used High values of T make the anti windup too slow to be effective while low values reset the integral term quickly with improved results It may thus seem advantageous to always have small values of T However the next example shows some situations where this choice is not advisable The Tracking Time Constant The tracking time constant is an essential parameter because it determines the reset rate for the integral term of the con troller It may seem advantageous to have a small value for this constant However measurement errors may acciden tally reset the integral term when the tracking time constant is too small The following example illustrates this phenom enon when a measurement error occurs in the form of a short pulse The transfer function of the process is Po Bsa Process Output O Linear X Windup X Antiwindup O PB Windup O PB Antiwindu
20. desired behavior The Transfer Function Magnitude m s or mes ocs OFT OFCs OL 0 1 a 10 Transfer Function Magnitude m s or mes ocs OFT OFCs OL Frequency be eee ee ee eee 0 1 0 1 10 Frequency b FIGURE 6 Frequency domain interpretation of the load disturbance response Figure 5 shows that high values of the integral gain k provide better response to load disturbances Although this rule is true it must be used carefully The frequency domain responses of Gya and S for two PI controllers with a k 0 85 and b k 0 30 respectively As can be seen large values of k imply large peaks of the sensitivity function S 1 1 PC Therefore a tradeoff occurs between load disturbance rejection and robustness Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply response to load disturbances is a key issue in process con trol since most controllers attempt to keep process vari ables close to desired setpoints 9 The following example PID LOOP SHAPING This section briefly describes the main aspects of PID Loop Shaping The main screen of the tool is shown in Figure 8 shows the effects of load disturbances and the influence of the controller parameters The setpoint and noise ampli tudes are set to zero and the load disturbance is set to 0 9 at t 0 The process transfer function is given by G s s 1 Th
21. e response of the process variable to load disturbances is given by the transfer function P T EE 3 Ye 7 eC C If P 0 40 and the controller has integral action then the low frequen cy approximation is Gya sP 0 kj where k K T is the integral gain For load disturbances with low fre quency content the integral gain k is a measure of load disturbance atten uation Figure 5 shows the load disturbance responses for two PI controllers with k given by 0 36 in red and 0 30 in blue Although the controller with larger integral gain provides faster response and smaller values for IAE and emax to load dis turbances the stability margins are reduced Figure 6 shows the frequen cy responses of Gyq and S for two PI controllers with large and small val ues of k 0 85 and 0 30 respectively This figure reflects that large values of k imply large peaks of the sensi tivity function Therefore a tradeoff becomes necessary between load dis turbance rejection and robustness Some tuning methods allow a tradeoff between robustness and load disturbance response The approximate M constrained integral gain optimization AMIGO method 8 10 12 maximizes integral gain under a robustness constraint see AMIGO Design Method The result of applying AMIGO to this example is shown in Figure 7 The AMIGO step method is used to design a PI controller with K 0 414 and T 2 66 The response to load disturba
22. er 1997 Yves Piguet is CEO of Calerga S rl a company he co founded in 2001 to develop and commercialize scientif ic software such as Sysquake of which he is the main developer He received a diploma in microtechnics in 1991 and a Ph D degree in control systems in 1997 both from EPFL Lausanne Switzerland His research interests include robust control real time controllers and their use in mobile robotics and interactive soft S ware and its applications Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply
23. fluenced by the choice of controller parameters A block diagram of a basic feedback loop is shown in Figure 1 where P and C are the process and controller transfer functions respectively and F is the filter transfer function for the setpoint The system has three inputs rep resenting the setpoint ys the load disturbance d and the measurement noise n It is assumed that the load distur bance acts at the process input and that the measurement noise acts at the process output The controller must reduce the effect of the load disturbance and make the process variable x follow the setpoint ys while not inject ing too much measurement noise In addition the closed loop system must be insensitive to variations in the process dynamics At least three signals are of interest namely the process output signal x the measured output signal y and the con trol signal u Tracing signals in the block diagram in Figure 1 gives the relations n PCE F P PC te pc TPC 1 PC FTY PSD TN 1 PCE P 1 Y OO N 1 Ppc Y 14 PC 1 PC PCFYs PSD SN 2 Cr PC Use ee e N 14 PC Y 14PC 1 PC CFSY TD CSN 3 where capital letters denote Laplace transforms of the cor responding time functions S 1 1 PC is the sensitivi ty function and T PC 1 PC is the complementary sensitivity function Notice that the input output relations are completely characterized by the six distinct transfer 1066 0
24. he batch are then obtained by applying the MIGO design Having obtained the controller parameters correlations with normalized process parameters are found by deriving the AMIGO tuning rules Tables S1 and S2 show these tuning rules for PI and PID controllers in the time and frequency domains Analysis of these rules can be found in 8 The main feature of this design method is that it facili tates tradeoffs between robustness and performance The method thus focuses on load disturbances by maximizing the integral gain and adding a robustness constraint TABLE S1 Time domain AMIGO tuning rules for first order time delay FOTD models L represents time delay 7 is the time constant and K is the static gain of the process K 7 and 7 are proportional gain integral time and derivative time parameters of PID controllers Controller K T 15 Lap Ir VLE PI E 0 35 atte KT Doly ae 12LT 7L PID K 0 2 0 457 ole L 0 17 TABLE 2 Frequency domain tuning rules Kiso is the process gain value at frequency wiso hso 27 wigo is the corresponding period and k Kigo Kp is the gain ratio K 7 and 7y are proportional gain integral time and derivative time parameters of PID controllers Controller K Ti Ta 0 16 Tso PI K 144 5 0 3 0 1x4 Kig0 T 142K 0 15 1 1 0 95 Tso OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 123 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November
25. ing the standard Matlab format The process gain and time delay can be changed interactively using sliders The PID controller has the structure 1 sT4 i ie ee a or p71 1 sT4 N4 Dean esin 2 where K is the proportional gain T is the integral time T4 is the derivative time N4 is a parameter of the derivative term and b is the setpoint weight FIGURE 1 Basic feedback loop having two degrees of freedom P and C are the process and controller transfer functions respective ly and F is the filter transfer function on the setpoint The variable Ysp is the setpoint eis the tracking error u is the controller output d is the load disturbance x is the process variable n is the measure ment noise y is the measured output signal and v is the controller output corrupted by the load disturbance ad OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 119 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply The Interactive Tool The main screen of the tool is shown in Figure 3 The process is characterized by the parameter group located on the left hand side of the screen just below the icons see Figure 3 The process is shown symbolically together with several interactive elements for changing the repre sentative parameters of the process The transfer function in Figure 3 is G o gt EDF where the gain Kp and order n are
26. is necessary to find controller gains pro viding the desired target point Dividing 4 by P iw and separating the real and imaginary parts gives B L iw 22 a ki _ yf E02 _ 2 kaw 3 5 A a 6 With k4 0 5 and 6 yield the two parameters of the PI controller An additional condition is required for the Constrained PID tuning option Hence it is observed that L s C s P s C s P s L s P s P s _ k L s P s 3 ka POS ra E 7 C s P s The slope of the Nyquist curve is then given by ye k Ire iL iw 5 ka Piw 1C i P iw 8 w The complex number represented by 8 has the phase angle if SGL iw e 0 9 Results 7 9 imply that R tio oe et ae t Poe B w 10 Combining 10 with 5 6 gives the controller parameters k wA w B w 11 kj Ao B 12 where A w and B w are given by 6 and 10 respectively The design frequency w can be chosen using the slider wdesign or graphically by dragging the green circle on the process Nyquist curve black curve in Figure 8 The target point on the Nyquist plot and its slope can be dragged graphically The slope can also be changed using the slider slope Furthermore it is possible to constrain the target point using the Constraints radio buttons to the unit circle Pm the negative real axis Gm circles representing con stant sensit
27. ivity Ms constant complementary sensitivity Mt or constant sensitivity combinations M When sensi tivity constraints are active the associated circles are drawn in the L plane plot and sliders can be used to modify their values The circles are defined in Table 1 Figure 8 illustrates designs for two PID controllers and a given sensitivity The target point is moved to the sensi tivity circle and the slope is adjusted so that the Nyquist curve is outside the sensitivity circle The red design shows a PID controller using Free tuning while the blue design shows a Constrained PID tuning Specifications that cannot be reached are indicated in the tool by giving the integral or derivative gain negative values in these cases Robustness and Performance Parameters Robustness and Performance parameters are displayed on the screen below the controller parameters Figure 8 and these parameters characterize robustness and performance in the same manner as in PID Basics The values are maxi mal sensitivity MS sensitivity crossover frequency Ws maximal complementary sensitivity Mt complementary sensitivity crossover frequency Wt gain margin Gm gain crossover frequency Wgc phase margin Pm and phase crossover frequency Wpc L Plane Graphic The L plane graphic is given in the right hand side of the PID Loop Shaping menu as shown in Figure 8 This graphic contains the Nyquist plots of the process transfer function P s
28. llowing examples Effect of Controller Parameters The purpose of this example is to illustrate how the Nyquist plot of the loop transfer function changes when the controller parameters are modified Consider the process P s 1 s 1 When a P con troller is used the proportional gain changes the loop trans fer function L iw kP iw in the direction of P iw Figure 10 a shows the effect of modifying L iw using a P con troller with gain k 2 blue curve and k 2 6 red curve These curves show how the proportional gain modifies the Nyquist plot of the process black curve at the frequency w green circle on the black curve in the direction of P iw Figure 10 b shows the same study for an I controller with k 1 red curve and k 0 6 blue curve It can be seen that the integral gain k changes L iw in the direction iP iw The derivative gain has the same effect in the direc tion of 1P iw When a PI or PD controller is used the com pensated point at the frequency w is calculated as the sum of two vectors namely the proportional vector and the inte gral or derivative vector Examples of this capability are Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply shown in Figure 10 c and d where the process is con trolled by a PI controller k 2 3 and k 0 7 and a PD controller k 2 1 and kg 3 35 respectively
29. n are manipulated directly by drawing the arrows In the Constrained PID tuning the target point is constrained to lie on the sensitivity circle 124 IEEE CONTROL SYSTEMS MAGAZINE gt OCTOBER 2008 Several process models are available and their parameters can be modified using sliders as described in PID Basics In addition a free transfer function can be selected menu option Interactive TF where poles and zeros can be defined graphically as shown in Figure 8 Controller The Controller part of the tool shows the various parameters and properties of PID Loop Shaping to perform loop shap ing The design point of the process transfer function is determined at a specified frequency w The design point is shown by a green circle on the L plane graphic The corre sponding point of the loop transfer function at the frequency w is called the target point The controller used in PID Loop Shaping is parame terized as C s k kgs which yields the loop transfer function kj L s C s P s kP s a tas P s The point on the Nyquist curve of the loop transfer func tion corresponding to the frequency is given by L iw kP iw 5 kw P iw 4 PID Loop Shaping provides three methods for tuning the parameters to move the process transfer function from the design point to the target point These methods are listed in the Tuning zone as Free Constrained PI and Constrained PID Free tuning allows an unconstrained
30. n linear mode and outside the proportional band when the control signal is saturated Large controller gains provide narrow proportional bands with more energetic control signals and therefore longer saturation times while small controller gains give wider proportional bands Figure 18 b illustrates this effect where the proportional controller gain is reduced to 0 4 producing a wider proportional band Antiwindup The process P s 1 s is also useful for visualizing the antiwindup technique The same controller parameters namely K 1 T 1 2 b 1 are used and the tracking time constant is set to T 1 Figure 19 a shows the responses for both cases control with and without anti windup The system with antiwindup remains in satura tion for only a short period of time with the magnitude of the integral term considerably reduced The proportional Process Output O Linear X Windup X AntiwindupO PB Windup O PB Antiwindup 40 60 Time 40 60 Time 0 20 40 60 Time FIGURE 19 Example of the effect of the tracking time constant 7 on in the antiwindup technique a Antiwindup and b effect of T These plots show the results of applying the antiwindup technique to the example shown in Figure 17 The integral signal is considerably reduced allowing the control signal to remain in saturation during a shorter period of time The proportional band for the antiwindup technique is shown in green The process output remains insi
31. nces is slower than the results presented in Figure 5 but sta bility margins result are improved with M 1 32 and M 1 Process The process transfer function can be selected and modified depending on the option selected from the Settings menu AMIGO Design Method Aw load disturbances are often the major consideration in process control robust ness and measurement noise must also be considered Requirements on setpoint response can be dealt with separately by using a controller with two degrees of freedom The Ziegler Nichols rules for tuning PID controllers are especially influential These rules however have severe drawbacks since they use insufficient process information and can yield closed loop systems with poor robustness 11 Loop shaping 13 can also be used for PID control which gives a flexible design method that allows a tradeoff between perfor mance and robustness The design approach maximizes the integral gain subject to con straints on the maximum sensitivity This method is called M constrained integral gain optimization MIGO 8 11 AMIGO approximate MIGO design which is a tuning method in the spirit of Ziegler and Nichols is the result of finding simple tuning rules for the MIGO method A large batch of representative processes is selected including a wide variety of systems with essentially monotone step responses that are typically encountered in process control Controllers for each process in t
32. nd vertical scales see Figure 17 These graphics can simultaneously represent the controlled system in linear nonlinear with windup and nonlinear with antiwindup modes These representations can be configured using the checkboxes located above the Process Output graphic For instance Figure 17 shows an example containing the nonlinear with windup and nonlinear with antiwindup modes The dotted pink vertical line in Figure 18 is helpful for comparing the outputs of the different plots at the same time instant The saturation limits can be altered using the dotted blue horizontal lines available in the Controller Output graphic see Figure 17 The notion of proportional band is useful for under standing the windup effect and is included in PID Windup The proportional band is defined as the range of process outputs such that the controller output is in the linear range Ymin Ymax For a PI controller the propor tional band is limited by I Umax min m 1 y bysp K 13 I umin Ymax bY sp z 14 where I is the integral term of the controller and Umax and Umin are the control signal limits Expressions 13 and 14 hold for PID control when the pro portional band is defined as the band where the predicted output Yp y Ta dy dt is in the proportional band ymin Ymax The proportional band has the width Umax Umin K and is centered around bys I K Umax Umin 2K Two additional checkboxes
33. ntegral may reach large val L Plane e Design10Design2 Graphics x Save o Delete lt 2 0 2 FIGURE 13 Example of a constrained design with a target point of 0 5 0 5 The target point can be constrained to reach arbitrary specifications Once a point is constrained the controller parame ters are automatically calculated This plot shows PI k 1 32 k 0 15 and PID k 1 32 k 1 02 ky 2 15 controllers both reaching the target point The PID controller provides better results due to the use of the slope as an allowable third degree of freedom as described in 11 and 12 where the slope V takes the value 22 ues maintaining the control signal saturated for a long time resulting in large overshoot and undesirable tran sients This problem is known as windup phenomenon 8 Windup can be avoided in different ways Back calcula tion and tracking 8 is illustrated in the block diagram in Figure 16 The system remains unchanged when the satu ration is not active However when saturation occurs the integral term in the controller is modified until the control signal is out of the saturation limit This modification is not performed instantaneously but dynamically with a time constant T called the tracking time constant 8 The module PID Windup shows process outputs and control signals for unlimited control signals limited L Plane Graphics XSave O Delete L Plane Graphics OSave amp Delete i 0
34. om the University of Almer a Spain where he is an assistant professor and a researcher in the Automatic Con trol Electronics and Robotics Group Currently his inter ests center on control education and robust model predictive control techniques with applications to interac tive tools virtual and remote labs and agricultural processes He can be contacted at the Universidad de Almer a Dpto de Lenguajes y Computaci n Ctra Sacra mento s n 04120 Almer a Spain 134 IEEE CONTROL SYSTEMS MAGAZINE gt gt OCTOBER 2008 Karl Johan str m received his M Sc in engineering physics in 1957 and his Ph D in control and mathematics in 1960 from the Royal Institute of Technology in Stock holm Sweden After graduating he worked for IBM Research for five years In 1965 he became professor at Lund Institute of Technology Lund University where he founded the Department of Automatic Control He is now emeritus at Lund University and a part time visiting pro fessor at the University of California at Santa Barbara He is a Life Fellow of the IEEE He has broad interests in con trol and its applications He has received many awards including the IEEE Medal of Honor and the Quazza medal from IFAC Sebasti n Dormido holds a degree in physics from the Complutense University in Madrid Spain 1968 and a Ph D from the University of the Basque Country Spain 1971 In 1981 he was appointed professor of control engi neering at the Unive
35. on Magnitude graphic In Figure 4 a all transfer functions are displayed Time and frequency responses can be shown simulta neously as illustrated in Figure 4 b The upper part rep resents the time responses while the lower part shows the frequency responses The default screen shows the output and the magnitude for the time and frequency domains respectively Above the graphics the two but tons let the user choose between the output or input for the time domain and magnitude or phase for the frequen cy domain This mode is useful since it is possible to view the effect of parameter modifications on both domains simultaneously Settings Menu The Settings menu of PID Basics is divided into six groups Arbitrary transfer functions can be selected using the first entry Process Transfer Function The numerator and denominator are introduced using a Matlab form Process Output Graphics x Save oDelete Koea ee opal K 0 6 T 2 b 1 0 20 40 Time a Controller Output oP ml u K 089 T 22 29 p a OO T 2 D5 0 6 A oe 0 20 40 Time b FIGURE 5 Load disturbance response and influence of the integral gain kj For a system with P O 40 and a controller with integral action the low frequency approximation is Gya SP 0 k where ki K T is the integral gain For load disturbances with low fre quency content the integral gain k is a measure of load distur bance attenuation The a process outputs
36. ontroller structures In addition parameters can be set and results can be stored and loaded A graphic display of time and frequency responses is a central part The plots can be manipulated directly by dragging points and lines and by using sliders Parameters that characterize performance and robustness are displayed Each module has two icons called Instruc tions and Theory Instructions provides access to a docu ment that contains suggestions for exercises while Theory provides access to relevant theory by means of the Inter net The modules are implemented in Sysquake 5 a Mat lab like language with fast execution and capabilities for interactive graphics The following sections describe three modules that illustrate closed loop fundamentals PID Basics loop shaping design PID Loop Shaping and integrator windup PID Windup Readers are encouraged to visit the Web site 6 to experience the interactive features of ILM PID The modules are available for Windows Mac Digital Object Identifier 10 1109 MCS 2008 927332 118 IEEE CONTROL SYSTEMS MAGAZINE gt OCTOBER 2008 and Linux operating systems and can be freely down loaded from the Sysquake Web site 7 as described in Downloading and Using ILM PID PID BASICS The module PID Basics is designed to explore the proper ties of a simple feedback loop by showing the time and fre quency responses of a closed loop system and demonstrating how these responses are in
37. p Windup Antiwindup 0 5 10 15 Time and the controller is a PID controller with K 3 5 T 05T 01N E Figure 20 a shows the control results A large transient appears after the pulse and the integral term is excessively reduced Various rules are suggested in 8 for choosing the tracking time constant One choice is T T Tg 2 Fig ure 20 b shows an example with T T Tg 2 0 33 where the response is considerably improved CONCLUSIONS In this work a set of interactive modules that comprise ILM PID is presented to support the teaching and learn ing of basic automatic control concepts These tools are intended mainly to include interactivity in the visual con tent of 8 The modules focus on PID control studying feedback fundamentals from the standpoint of the time and frequency domains including robustness issues mea surement of noise filtering load disturbance rejection and windup phenomenon The importance of interactivity in automatic control education has been shown in the context of teaching and learning In the authors experience interactivity offers excellent support to education and learning by enhancing the motivation and participation of future Process Output O Linear X Windup amp Antiwindup O PB Windup O PB Antiwindup Windup Antiwindup 0 5 10 15 Time 0 5 10 15 Time 0 5 10 15 Time FIGURE 20 Tuning the tracking time a Reset by measurement
38. res appear one of which shows the current design in red while the other shows the current design in blue see Figure 8 Modifications of the controller parame ters affect the current active design which can be changed using the options Design 1 and Design 2 which appear on the top of the L plane graphic Once a design is chosen the associated curve is switched to red and the controller zone is modified based on that design The controller gain values can be seen by moving the cursor on the curves Settings Menu The Settings menu which is available in the main menu of PID Loop Shaping is divided into four groups following the same structure as in PID Basics The first entry called Process Transfer Function is used to choose between sev eral predefined transfer functions or to include an user specified transfer function through two options The String TF option allows a transfer function to be entered symboli cally For instance P s 1 cosh ys can be represented as P 1 cosh sqrt s Results can be stored and recalled using the Load Save menu The option Save Report can be used to save all essential data in text format which is use ful for documenting results Specific values for control parameters can be entered with Parameters menu option As in PID Basics the last menu option Examples Advanced PID Book allows loading examples from 8 Examples Some of the capabilities of PID Loop Shaping are illustrated by the fo
39. rl 2004 Online Available http www calerga com 6 J L Guzman K J str m S Dormido T H gglund and Y Piguet Interactive learning modules for PID control in Proc 7th IFAC Symp Advances in Control Education 2006 Madrid Spain Online Available http aer ual es ilm 7 ILM Web site Interactive Learning Modules 2005 Online Available http www calerga com contrib 1 index html 8 KJ str m and T H gglund Advanced PID Control Research Triangle Park NC Instrum Soc Amer 2005 9 F G Shinskey Process Control Systems Application Design and Tuning New York McGraw Hill 1996 10 T H gglund and KJ str m Revisiting the Ziegler Nichols tuning rules for PI control Asian J Contr vol 4 no 4 pp 364 380 2002 11 T H gglund and K J str m Revisiting the Ziegler Nichols step response method for PID control J Process Contr vol 14 no 6 pp 635 650 2004 12 T H gglund and KJ str m Revisiting the Ziegler Nichols tuning rules for PI control Part II the frequency response method Asian J Contr vol 6 no 4 pp 469 482 2004 13 K J str m H Panagopoulos and T H gglund Design of PI controllers based on non convex optimization Automatica vol 34 no 4 pp 585 601 1998 AUTHOR INFORMATION Jos Luis Guzm n joguzman ual es earned a computer science engineering degree in 2002 and a Ph D in 2006 both fr
40. rsidad Nacional de Educaci n a Dis tancia His activities include computer control of industrial processes model based predictive control robust control and model and simulation of continuous processes He has authored and coauthored more than 150 technical papers in international journals and conferences From 2002 to 2006 he was president of the Spanish Associ ation of Automatic Control CEA IFAC Tore H gglund received his M Sc in engineering physics in 1978 and his Ph D in automatic control in 1984 both from Lund Institute of Technology Sweden where he is currently a professor Between 1985 and 1989 he worked for Alfa Laval Automation now ABB on the development of industrial adaptive controllers He is a coauthor of Advanced PID Control and Process Control in Practice His current research interests are in the areas of process control tuning and adaptation and supervision and detection Manuel Berenguel is a professor at the University of Almer a Spain He earned an industrial engineering degree and Ph D from the University of Seville Spain where he has been a researcher and associate professor for six years His research interests are in predictive adaptive and robust control with applications to solar energy systems agriculture and biotechnology He has authored and coauthored more than 100 technical papers in international journals and conferences and is coauthor of the book Advanced Control of Solar Plants Spring
41. se plots Transfer Function Magnitude and Transfer Function Phase The vertical and horizontal scales can be interactively modified in the same way as in the time domain The magnitude and phase for a specific frequency can be found by placing the mouse over the signals as shown in Figure 4 a Process Output Process Output OProcess Input Transfer Function Magnitude Mag OPhase XS RT MPS RCS MFT RFCS ML 0 01 0 1 10 b Frequency FIGURE 4 Time and frequency domain analysis using the interactive tool a Frequency domain The graphical part of PID Basics is shown for the frequency domain mode where the Transfer Function Magnitude and the Transfer Function Phase graphics are displayed In this mode the user can study the transfer functions in 1 3 in the frequency domain using checkboxes placed above the Transfer Function Magnitude graphic b Time and frequency responses simultaneously Above the graphics the two buttons let the user choose between the output or input for the time domain and magnitude or phase for the frequency domain OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 121 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply The frequency response for the gang of six transfer functions and the open loop transfer function L iw P iw C iw can be shown in the graphics using checkboxes placed above the Transfer Functi
42. the time response of the transfer functions in 1 3 8 Sev eral graphical elements shown on the same screen are used to interactively analyze feedback fundamentals using PID control This example provides a comparison between PI blue and PID red controllers Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply t 0 allows the setpoint amplitude to be modified The green and black vertical lines located in the middle of the graphics allow setting the value and time instant for load disturbances and measurement noise respectively The ver tical and horizontal scales can be changed using the black triangles A V available in the graphics For instance in Figure 3 the setpoint is set to one the load disturbance is set to 0 9 at t 32 and the measurement noise is set to 0 02 at t 60 It is also possible to find the value for the input or output signal at a specific time by placing the mouse over the curve Figure 3 shows an example in which at the time instant t 37 78 the output and input signals are 1 62 and 0 38 respectively All of these options are available in both graphics that is Process Output and Controller Output The checkboxes save and delete above the Process Output graphic provide the ability to store a simulation for comparison When the save button is selected the current design is frozen and displayed in blue and
43. tive Cliff We again consider the process transfer function P s 1 s 1 It is desirable to maximize the integral gain k subject to the robustness constraint Ms lt 1 4 The resulting controller has the parameters k 0 925 k 0 9 and k4 2 86 where the Nyquist plot of the loop transfer function is shown in red in Figure 15 a It can be seen that the Nyquist curve has a loop called a derivative cliff As explained in 8 this feature which is due to excessive con troller phase lead results from having a PID controller with complex poles which occurs when T lt 4T4 In this exam ple the relation is T 0 33Tg Figure 15 b shows in red the time response of the controller which yields oscillatory outputs For comparison the results for a controller with T 4Tq are shown in blue in Figure 15 a and b with the controller parameters k 1 1 kj 0 36 and kg 0 9 The responses for this controller are improved despite larger overshoot in response to load disturbances This example is available in the Settings menu of PID Loop Shaping PID WINDUP The purpose of the PID Windup module is to facilitate understanding of integral windup and a method for com pensating it 8 For a control system with a wide range of operating conditions it may happen that the control vari able reaches the actuator limits When this situation occurs in loops using a controller with integral action the feed back loop is broken and the i
44. w examples of these arrows The scale of the graphic can be changed using the red triangle located at the bottom of the vertical axis As noted above it is possible to impose constraints on the target point The graphical representation of the target point is modified depending on the constraint selected restricting L Plane Graphics O Save O Delete r z 7 2 Zel 2 2 a L Plane Graphics O Save O Delete 0 2 3 2 0 2 b FIGURE 9 The L plane graphic The Nyquist plots of the process transfer function P s black line and the loop transfer function L s P s C s red line are shown a An example of the Free tuning design The controller gains can be changed by dragging arrows the proportional gain changes L w in the direction of P iw blue arrow the integral gain k changes L w in the direc tion of P Uiw cyan arrow and the derivative gain ky changes L i in the direction of i P iw magenta arrow b An example of the Constrained PID tuning design In this case once the user moves the target point black circle the controller parameters are calculated using 5 12 126 IEEE CONTROL SYSTEMS MAGAZINE gt gt OCTOBER 2008 its value based on its meaning Options save and delete can be found above the L plane graphic These options have the same meaning as in PID Basics making it possible to save designs to perform comparisons Once the save option is active two pictu
45. y and the controller output u for step commands in setpoint and load distur bances as well as the response to sensor noise as shown here 120 IEEE CONTROL SYSTEMS MAGAZINE gt OCTOBER 2008 overshoot overshoot The load disturbance response is characterized by the integral absolute error IAE the inte gral gain kj K T ki the maximal error emax and the time to reach the maximum tmax The integral absolute errors and the maximal error values are normalized to unit step changes in setpoint and load disturbances The response to measurement noise is characterized by the standard deviations of the process variable x sigma_x measured output y sigma_y and control signal u Sigma_u The robustness measures are maximal sensitivi ty Ms maximal complementary sensitivity Mt gain margin Gm and phase margin Pm This information can be duplicated to compare two designs as shown below A more detailed description of these measures can be found in 8 Graphics Two graphics are shown on the right hand side of the tool Figure 3 Three representation modes can be selected from the Settings menu These modes are time domain frequency domain and frequency time domain The time domain mode is shown in Figure 3 where the time responses for the system output Process Output and input Controller Output are displayed The initial part of the plots 0 lt t lt 30 shows the response to a step change in the setpoint
46. z 29 3 2 0 2 b FIGURE 14 Example of a constrained design with sensitivity and gain margin constraints These plots show an example where the target point is constrained to reach specified values for the a com bined sensitivity functions with M lt 2 and M lt 2 and b gain margin with limited sensitivity values with Gm 3 and M lt 2 OCTOBER 2008 IEEE CONTROL SYSTEMS MAGAZINE 129 Authorized licensed use limited to Lunds Universitetsbibliotek Downloaded on November 4 2008 at 08 27 from IEEE Xplore Restrictions apply control signals without antiwindup and limited control signals with antiwindup The user interface is shown in Figure 17 Process models and controller parameters can be selected in the same way as in the other modules The saturation limits of the control signal can be determined either by entering the values or by dragging the lines in the saturation scheme L Plane ODesign1 Design2 Graphics X Save Delete Process Output Graphics X Save O Delete K 0 93 mE mA b 0 5 N 10 K 1 1 7 3 05 Ty 0 76 b 0 5 N 10 K 0 93 7 1 08 7y 3 09 b 0 5 N 10 K 1 1 7 3 05 Ty 0 76 b 0 5 N 10 FIGURE 15 Derivative cliff example a Nyquist plot and b time domain responses This example shows that optimization of k which is aimed at fulfilling robustness specifications can provide controllers with excessive phase lead as represented by the loop in the re
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