Interactive tools can be used to complement books and

Transcription

1 » LECTURE NOTES Interactive Learning Modles for PID Control Using Interactive Graphics to Learn PID Control and Develop Intition JOSÉ LUIS GUZMAN, KARL JOHAN ÅSTRÖM, SEBASTIAN DORMIDO, TORE HÄGGLUND, MANUEL BERENGUEL, and YVES PIGUET Interactive tools can be sed to complement books and lectres [] [4]. This article describes three interactive learning modles that are designed to develop intition as well as a working knowledge of proportional-integralderivative (PID) control. These three modles comprise a package called interactive learning modles for PID (ILM- PID). B illstrating concepts sch as tning, robstness, loop shaping, and antiwindp, ILM-PID can be sed for demonstrations, exercises, and self-std. The main objective of the interactive modles is to explain basic concepts of PID control withot considering implementation aspects. Althogh most PID controllers are implemented as sampled-data control sstems, analsis and design are traditionall performed in continos time assming that the sampling rate for sbseqent digital implementation is sfficientl fast. Implementation isses, sch as aliasing, selection of the sampling time, signal prefiltering, inflence of the discretization algorithms, and bmpless parameter changes, ma be the aim of a ftre interactive modles focsed on implementation aspects for PID control. The modles of ILM-PID have mens for selecting process transfer fnctions and controller strctres. In addition, parameters can be set, and reslts can be stored and loaded. A graphic displa of time and freqenc responses is a central part. The plots can be maniplated directl b dragging points and lines and b sing sliders. Parameters that characterize performance and robstness are displaed. Each modle has two icons called Instrctions and Theor. Instrctions provides access to a docment that contains sggestions for exercises, while Theor provides access to relevant theor b means of the Internet. The modles are implemented in Ssqake [5], a Matlab-like langage with fast exection and capabilities for interactive graphics. The following sections describe three modles that illstrate closed-loop fndamentals (PID Basics), loopshaping design (PID Loop Shaping), and integrator windp (PID ). Readers are encoraged to visit the Web site [6] to experience the interactive featres of ILM-PID. The modles are available for Windows, Mac, Digital Object Identifier.9/MCS and Linx operating sstems and can be freel downloaded from the Ssqake Web site [7] as described in Downloading and Using ILM-PID. PID BASICS The modle PID Basics is designed to explore the properties of a simple feedback loop b showing the time and freqenc responses of a closed-loop sstem and demonstrating how these responses are inflenced b the choice of controller parameters. A block diagram of a basic feedback loop is shown in Figre, where P and C are the process and controller transfer fnctions, respectivel, and F is the filter transfer fnction for the setpoint. The sstem has three inpts representing the setpoint sp, the load distrbance d, and the measrement noise n. It is assmed that the load distrbance acts at the process inpt and that the measrement noise acts at the process otpt. The controller mst redce the effect of the load distrbance and make the process variable x follow the setpoint sp, while not injecting too mch measrement noise. In addition, the closedloop sstem mst be insensitive to variations in the process dnamics. At least three signals are of interest, namel, the process otpt signal x, the measred otpt signal, and the control signal. Tracing signals in the block diagram in Figre gives the relations X = PCF + PC Y P sp + + PC D PC + PC N = FTY sp + PSD TN, () Y = PCF + PC Y P sp + + PC D + + PC N = PCFY sp + PSD + SN, () U = CF + PC Y sp PC + PC D C + PC N = CFSY sp TD CSN, (3) where capital letters denote Laplace transforms of the corresponding time fnctions, S = /( + PC) is the sensitivit fnction, and T = PC/( + PC ) is the complementar sensitivit fnction. Notice that the inpt-otpt relations are completel characterized b the six distinct transfer 8 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER X/8/$5. 8IEEE Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

2 Downloading and Using ILM-PID Interactivit, which is the main featre of the tools described in this work, is difficlt to explain in written text. The best wa to appreciate the tools is to se them. We strongl recommend that the reader download and se them in parallel with reading this article. Exectable versions for PC, Mac, and Linx are freel available for download from the Calerga Web site [7]. No licenses are reqired, and the exectable modles can be freel distribted to stdents and colleages. fnctions in () (3). These transfer fnctions are called the gang of six in [8]. To analze the closed-loop sstem it is necessar to consider all six transfer fnctions. The time responses of the six transfer fnctions are illstrated b showing the response of the process otpt and control signals to a step in the setpoint, a step in the load distrbance, and wideband measrement noise, as illstrated in Figre. A mix of time and freqenc responses can also be displaed. Process models in the form of rational transfer fnctions with a time dela can be chosen from a men that provides a collection of transfer fnctions. An arbitrar transfer fnction can also be entered sing the standard Matlab format. The process gain and time dela can be changed interactivel sing sliders. The PID controller has the strctre ( U = K by sp Y + ) st d (Y sp Y) Y, st i + st d /N d where K is the proportional gain, T i is the integral time, T d is the derivative time, N d is a parameter of the derivative term, and b is the setpoint weight. sp F Σ Controller e C FIGURE Basic feedback loop having two degrees of freedom. P and C are the process and controller transfer fnctions, respectivel, and F is the filter transfer fnction on the setpoint. The variable sp is the setpoint, e is the tracking error, is the controller otpt, d is the load distrbance, x is the process variable, n is the measrement noise, is the measred otpt signal, and v is the controller otpt corrpted b the load distrbance d. Σ d ν P Process x Σ n OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 9 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

3 The Interactive Tool The main screen of the tool is shown in Figre 3. The process is characterized b the parameter grop located on the left-hand side of the screen, jst below the icons (see Figre 3). The process is shown smbolicall together with several interactive elements for changing the representative parameters of the process. The transfer fnction in Figre 3 is G(s) = K p (s + ) n, where the gain K p and order n are the interactive elements, with nmerical vales K p = and n = 4. When the ser modifies an plant parameter, the smbolic representation of the process transfer fnction is immediatel pdated, and its effect is reflected on the remaining graphic elements. Five bttons are available for selecting the desired controller. The bttons correspond to proportional (P), integral (I), proportional-integral (PI), proportional-derivative (PD), and proportional-integral-derivative (PID). Several sliders are available below the radio bttons for modifing the controller parameters. The nmber of sliders shown depends on the chosen controller. For instance, Figre 3 shows five sliders since the PID controller is selected. Performance and Robstness Information Parameters that characterize performance and robstness are also displaed on the screen. The performance criteria are based on the setpoint response, the load distrbance response, and the noise response. The setpoint response is characterized b the integral absolte error (IAE) and the overshoot (overshoot). The load distrbance response is characterized b the integral absolte error (IAE), the integral gain k i = K/T i (ki), the maximal error (emax), and the time to reach the maximm (tmax). The integral absolte errors and the maximal error vales are normalized to nit step changes in setpoint and load distrbances. The response to measrement noise is characterized b the standard deviations of the process variable x (sigma_x), measred otpt (sigma_), and control signal (sigma_). The robstness measres are maximal sensitivit (Ms), maximal complementar sensitivit (Mt), gain margin (Gm), and phase margin (Pm). This information can be dplicated to compare two designs, as shown below. A more detailed description of these measres can be fond in [8]. Graphics Two graphics are shown on the right-hand side of the tool (Figre 3). Three representation modes can be selected from the Settings men. These modes are time domain, freqenc domain, and freqenc/time domain. The time domain mode is shown in Figre 3, where the time responses for the sstem otpt (Process Otpt) and inpt (Controller Otpt) are displaed. The initial part of the plots ( < t < 3) shows the response to a step change in the setpoint represented b the transfer fnctions FT and CFS in () (3). The middle portions of the plots (3 < t < 6) show the response to a step in the load distrbance represented b the transfer fnctions PS and T in () (3). The last portions of the plots (t > 6) show the response to wideband measrement noise, which is represented b the transfer fnctions S and CS in () (3). Several elements on the graphics are available for interacting with the application. The vertical green line at time FIGURE Control sstem responses illstrating basic feedback sstem properties. To analze the feedback loop, it is essential to consider six responses. These responses, which are referred to as the gang of six [8], are described b transfer fnctions in () (3). One wa to present this information is to show the process otpt and the controller otpt for step commands in setpoint and load distrbances, as well as the response to sensor noise, as shown here. FIGURE 3 The ser interface of the modle PID Basics. The plots show the time response of the transfer fnctions in () (3) [8]. Several graphical elements, shown on the same screen, are sed to interactivel analze feedback fndamentals sing PID control. This example provides a comparison between PI (ble) and PID (red) controllers. IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

4 t = allows the setpoint amplitde to be modified. The green and black vertical lines located in the middle of the graphics allow setting the vale and time instant for load distrbances and measrement noise, respectivel. The vertical and horizontal scales can be changed sing the black triangles (, ) available in the graphics. For instance, in Figre 3, the setpoint is set to one, the load distrbance is set to.9 at t = 3, and the measrement noise is set to. at t = 6. It is also possible to find the vale for the inpt or otpt signal at a specific time b placing the mose over the crve. Figre 3 shows an example in which, at the time instant t = 37.78, the otpt and inpt signals are.6 and.38, respectivel. All of these options are available in both graphics, that is, Process Otpt and Controller Otpt. The checkboxes save and delete above the Process Otpt graphic provide the abilit to store a simlation for comparison. When the save btton is selected, the crrent design is frozen and displaed in ble, and a new design in red appears, allowing the two designs to be compared. Performance and robstness parameters are dplicated, displaing the vales in ble and red colors associated with each design. The Process Otpt and Controller Otpt graphics indicate the vales of the controller parameters for both designs. Figre 3 presents an example that compares the response of PI (K =.43, T i =.7, b = ) and PID (K =.3, T i = 3.36, T d =., b =.54, N d = ) controllers. Althogh the PID controller provides a better response to load distrbances b reacting faster, the noise also generates more control action. The delete option can be selected to remove a design. If the transfer fnction of the process or an inpt signal sch as a setpoint, load distrbance, and measrement noise are altered, both sets of reslts are affected simltaneosl. Onl two designs are stored to keep the ser interface simple. Additional options for the time-domain mode are shown above the Controller Otpt graphic. These options show the proportional (P), integral (I), and derivative (D) signals of the controller. The freqenc domain mode is shown in Figre 4. When this mode is selected from the Settings men, the left side of the tool remains nchanged. However, in this case the time responses are replaced b the magnitde and phase plots Transfer Fnction Magnitde and Transfer Fnction Phase. The vertical and horizontal scales can be interactivel modified in the same wa as in the time domain. The magnitde and phase for a specific freqenc can be fond b placing the mose over the signals as shown in Figre 4.. Transfer Fnction Magnitde S T PS CS FT FCS L Sn(w) = 3.46 S T Sd Sn FT FSn L Process Otpt Process Otpt Process Inpt. Freqenc 4 6 Time Transfer Fnction Phase S T Sd Sn FT FSn L. Transfer Fnction Magnitde Mag Phase S T PS CS FT FCS L S T Sd Sn FT FSn L. Freqenc. Freqenc FIGURE 4 Time- and freqenc-domain analsis sing the interactive tool. Freqenc domain. The graphical part of PID Basics is shown for the freqenc-domain mode, where the Transfer Fnction Magnitde and the Transfer Fnction Phase graphics are displaed. In this mode the ser can std the transfer fnctions in () (3) in the freqenc domain sing checkboxes placed above the Transfer Fnction Magnitde graphic. Time and freqenc responses, simltaneosl. Above the graphics, the two bttons let the ser choose between the otpt or inpt for the time domain, and magnitde or phase for the freqenc domain. OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

5 The freqenc response for the gang of six transfer fnctions and the open-loop transfer fnction L(iω) = P(iω)C(iω) can be shown in the graphics sing checkboxes placed above the Transfer Fnction Magnitde graphic. In Figre 4, all transfer fnctions are displaed. Time and freqenc responses can be shown simltaneosl, as illstrated in Figre 4. The pper part represents the time responses, while the lower part shows the freqenc responses. The defalt screen shows the otpt and the magnitde for the time and freqenc domains, respectivel. Above the graphics, the two bttons let the ser choose between the otpt or inpt for the time domain and magnitde or phase for the freqenc domain. This mode is sefl since it is possible to view the effect of parameter modifications on both domains simltaneosl. Settings Men The Settings men of PID Basics is divided into six grops. Arbitrar transfer fnctions can be selected sing the first entr, Process Transfer Fnction. The nmerator and denominator are introdced sing a Matlab form. Process Otpt Graphics: Save Delete K =.83 T i =.9 b= K =.6 T i = b=.5 Specific vales for controller parameters can be entered sing the Controller Parameters men. Time and freqenc responses can be selected from the third entr, Time/Freqenc Domain, which has the options Time Domain, Freqenc Domain, and Both Domains. The reslts can be stored and recalled sing the Load/Save men, which has the options Save Design and Load Design. All data on the screen can be saved sing the option Save Report. From the men selection Simlation the ser can modif the simlation time, change the maximal time dela to avoid slow simlations, and activate the Sweep option to show the reslts for several controller parameters simltaneosl. Parameters are swept between specified limits. This option is available onl in the timedomain mode. When active, new radio bttons appear in the controller-parameters zone to permit the selection of the desired parameter to be swept. The last men option, Examples Advanced PID Book, loads examples from [8], which the ser can explore b modifing parameters. Analsis and Control Design for Load Distrbances Load distrbances are tpicall low-freqenc signals that drive the sstem awa from its desired behavior. The Transfer Fnction Magnitde S T PS CS FT FCS L S Sd 4 Time Controller Otpt P I K =.83 T i =.9 b= K =.6 T i = b=.. Freqenc Transfer Fnction Magnitde S T PS CS FT FCS L S Sd. 4 Time. Freqenc FIGURE 5 Load-distrbance response and inflence of the integral gain k i. For a sstem with P() and a controller with integral action, the low-freqenc approximation is G d sp()/k i, where k i = K/T i is the integral gain. For load distrbances with low-freqenc content, the integral gain k i is a measre of load-distrbance attenation. The process otpts and control signals to load distrbances, respectivel, are shown for two PI controllers with k i vales of.36 (in red) and.3 (in ble). The controller with larger integral gain provides a faster response to load distrbances. FIGURE 6 Freqenc-domain interpretation of the load-distrbance response. Figre 5 shows that high vales of the integral gain k i provide better response to load distrbances. Althogh this rle is tre, it mst be sed carefll. The freqenc-domain responses of G d and S for two PI controllers with k i =.85 and k i =.3, respectivel. As can be seen, large vales of k i impl large peaks of the sensitivit fnction S = /( + PC). Therefore, a tradeoff occrs between load distrbance rejection and robstness. IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

6 response to load distrbances is a ke isse in process control, since most controllers attempt to keep process variables close to desired setpoints [9]. The following example shows the effects of load distrbances and the inflence of the controller parameters. The setpoint and noise amplitdes are set to zero, and the load distrbance is set to.9 at t =. The process transfer fnction is given b G(s) = (s + ) 4. The response of the process variable to load distrbances is given b the transfer fnction G d = P + PC = PS = T C. If P() and the controller has integral action, then the low-freqenc approximation is G d sp()/k i, where k i = K/T i is the integral gain. For load distrbances with low-freqenc content, the integral gain k i is a measre of load-distrbance attenation. Figre 5 shows the loaddistrbance responses for two PI controllers with k i given b.36 (in red) and.3 (in ble). Althogh the controller with larger integral gain provides faster response and smaller vales for IAE and emax to load distrbances, the stabilit margins are redced. Figre 6 shows the freqenc responses of G d and S for two PI controllers with large and small vales of k i (.85 and.3, respectivel). This figre reflects that large vales of k i impl large peaks of the sensitivit fnction. Therefore, a tradeoff becomes necessar between load-distrbance rejection and robstness. Some tning methods allow a tradeoff between robstness and load distrbance response. The approximate M-constrained integralgain optimization (AMIGO) method [8], [] [] maximizes integral gain nder a robstness constraint; see AMIGO Design Method. The reslt of appling AMIGO to this example is shown in Figre 7. The AMIGO-step method is sed to design a PI controller with K =.44 and T i =.66. The response to load distrbances is slower than the reslts presented in Figre 5, bt stabilit margins reslt are improved, with M s =.3 and M t =. AMIGO Design Method PID LOOP SHAPING This section briefl describes the main aspects of PID Loop Shaping. The main screen of the tool is shown in Figre 8. Process The process transfer fnction can be selected and modified depending on the option selected from the Settings men. Althogh load distrbances are often the major consideration in process control, robstness and measrement noise mst also be considered. Reqirements on setpoint response can be dealt with separatel b sing a controller with two degrees of freedom. The Ziegler-Nichols rles for tning PID controllers are especiall inflential. These rles, however, have severe drawbacks, since the se insfficient process information and can ield closed-loop sstems with poor robstness []. Loop shaping [3] can also be sed for PID control, which gives a flexible design method that allows a tradeoff between performance and robstness. The design approach maximizes the integral gain sbject to constraints on the maximm sensitivit. This method is called M-constrained integral gain optimization (MIGO) [8], []. AMIGO (approximate MIGO) design, which is a tning method in the spirit of Ziegler and Nichols, is the reslt of finding simple tning rles for the MIGO method. A large batch of representative processes is selected, inclding a wide variet of sstems with essentiall monotone step responses that are tpicall encontered in process control. Controllers for each process in the batch are then obtained b appling the MIGO design. Having obtained the controller parameters, correlations with normalized process parameters are fond b deriving the AMIGO tning rles. Tables S and S show these tning rles for PI and PID controllers in the time and freqenc domains. Analsis of these rles can be fond in [8]. The main featre of this design method is that it facilitates tradeoffs between robstness and performance. The method ths focses on load distrbances b maximizing the integral gain and adding a robstness constraint. TABLE S Time-domain AMIGO tning rles for first-order time dela (FOTD) models. L represents time dela, T is the time constant, and K p is the static gain of the process. K, T i, and T d are proportional gain, integral time, and derivative time parameters of PID controllers. Controller K T i T d ( ) 5 PI K +.35 LT T 3LT.35L + p (L+T) K pl T +LT+7L ( ) PID T.4L+.8T L L+.T L.5LT.3L+T K p TABLE S Freqenc-domain tning rles. K 8 is the process gain vale at freqenc ω 8, T 8 = (π)/ω 8 is the corresponding period, and κ = K 8 /K p is the gain ratio. K, T i, and T d are proportional gain, integral time, and derivative time parameters of PID controllers. Controller K T i T d PI.6 K8 PID (.3.κ 4 )/K 8.6 +κ T 8 T8 +4.5κ.5( κ).95κ T 8 OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 3 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

7 Process Otpt Graphics: Save Delete.5 4 Time Controller Otpt P I 4 Time FIGURE 7 Load distrbance response for a PI controller sing the approximate M-constrained integral gain optimization (AMIGO) step method. This method enables the compromise described in Figre 6, focsing on load distrbances b maximizing integral gain and adding a robstness constraint. The process otpt and control signal to load distrbances, respectivel, for a PI controller designed sing AMIGO and with K =.44 and T i =.66. The slow response compared with Figre 6 corresponds to increased stabilit margins. FIGURE 8 The ser interface of the modle PID Loop Shaping, showing both Free and Constrained PID tning. The loop transfer fnction is shown for two designs nder the Free design option. Proportional, integral, and derivative action are maniplated directl b drawing the arrows. In the Constrained PID, tning the target point is constrained to lie on the sensitivit circle. Several process models are available, and their parameters can be modified sing sliders as described in PID Basics. In addition, a free transfer fnction can be selected (men option Interactive TF), where poles and zeros can be defined graphicall as shown in Figre 8. Controller The Controller part of the tool shows the varios parameters and properties of PID Loop Shaping to perform loop shaping. The design point of the process transfer fnction is determined at a specified freqenc ω. The design point is shown b a green circle on the L-plane graphic. The corresponding point of the loop transfer fnction at the freqenc ω is called the target point. The controller sed in PID Loop Shaping is parameterized as C(s) = k + k i s + k ds, which ields the loop transfer fnction ( ) ki L(s) = C(s)P(s) = kp(s) + s + k ds P(s). The point on the Nqist crve of the loop transfer fnction corresponding to the freqenc ω is given b ( L(iω) = kp(iω) + i k ) i ω + k dω P(iω). (4) PID Loop Shaping provides three methods for tning the parameters to move the process transfer fnction from the design point to the target point. These methods are listed in the Tning zone as Free, Constrained PI, and Constrained PID. Free tning allows an nconstrained loop to be shaped b dragging on the control parameters. Constrained PI and Constrained PID permit the calclation of the controller parameters based on some constraints on the target point. That is, the focs can be placed on how the loop transfer fnction changes when controller parameters are modified, which reveals the parameter vales reqired to obtain a given shape of the loop transfer fnction. For PI and PD control the mapping is niqel given b one point. For PID control it is also possible to obtain an arbitrar slope ϑ of the loop transfer fnction at the target point. When the Free tning option is selected, sliders are sed to modif the controller gains k, k i, and k d, as shown in Figre 8. The controller gains can also be changed b dragging arrows, as illstrated in the same figre. From (4), the proportional gain changes L(iω) in the direction of P(iω), the integral gain k i changes L(iω) in the direction of ip(iω), and the derivative gain k d changes L(iω) in the direction of ip(iω). For the Constrained PI and Constrained PID tning options, the target point can be limited to move on the nit circle, the sensitivit circles, or the real axis. In this 4 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

8 wa loop shaping is enabled with specifications on gain and phase margins or on the sensitivities. In the case of Constrained PI it is necessar to find controller gains providing the desired target point. Dividing (4) b P(iω) and separating the real and imaginar parts gives k i ω + k dω =I ( L(iω) k =R P(iω) ( L(iω) P(iω) ), (5) ) = A(ω). (6) With k d =, (5) and (6) ield the two parameters of the PI controller. An additional condition is reqired for the Constrained PID tning option. Hence, it is observed that L (s) = C (s)p(s) + C(s)P (s) = C (s)p(s) + L(s)P (s) P(s) = ( k ) i s + k d P(s) + L(s)P (s) P(s) The slope of the Nqist crve is then given b. (7) ( ) il ki (iω) = i ω + k d P(iω) + ic(iω)p (iω). (8) The complex nmber represented b (8) has the phase angle ϑ if Reslts (7) (9) impl that I(iL (iω)e iϑ ) =. (9) ( R L(iω) P ) (iω) k i ω + k P(iω) e iϑ d = R(P(iω)e iϑ = B(ω). () ) Combining () with (5) (6) gives the controller parameters k i = ωa(ω) + ω B(ω), () k d = A(ω) + B(ω), ω () where A(ω) and B(ω) are given b (6) and (), respectivel. The design freqenc ω can be chosen sing the slider wdesign or graphicall b dragging the green circle on the process Nqist crve (black crve in Figre 8). The target point on the Nqist plot and its slope can be dragged graphicall. The slope can also be changed sing the slider slope. Frthermore, it is possible to constrain the target point sing the Constraints radio bttons to the nit circle (Pm), the negative real axis (Gm), circles representing constant sensitivit (Ms), constant complementar sensitivit (Mt), or constant sensitivit combinations (M). When sensitivit constraints are active, the associated circles are drawn in the L-plane plot, and sliders can be sed to modif their vales. The circles are defined in Table. Figre 8 illstrates designs for two PID controllers and a given sensitivit. The target point is moved to the sensitivit circle, and the slope is adjsted so that the Nqist crve is otside the sensitivit circle. The red design shows a PID controller sing Free tning, while the ble design shows a Constrained PID tning. Specifications that cannot be reached are indicated in the tool b giving the integral or derivative gain negative vales in these cases. Robstness and Performance Parameters Robstness and Performance parameters are displaed on the screen below the controller parameters (Figre 8), and these parameters characterize robstness and performance in the same manner as in PID Basics. The vales are maximal sensitivit (Ms), sensitivit-crossover freqenc (Ws), maximal complementar sensitivit (Mt), complementar sensitivit-crossover freqenc (Wt), gain margin (Gm), gain-crossover freqenc (Wgc), phase margin (Pm), and phase-crossover freqenc (Wpc). L-Plane Graphic The L-plane graphic is given in the right-hand side of the PID Loop Shaping men, as shown in Figre 8. This graphic contains the Nqist plots of the process transfer fnction P(s) in black and the loop transfer fnctions L(s) = P(s)C(s) in red. Three different views can be shown depending on the tning options. Figre 9 shows two views, the left one for Free tning and the right one for Constrained PID tning. A third view is shown in Figre 8, where two designs are shown simltaneosl. The design and target points can be modified interactivel on this graphic. The design point is shown in green on the Nqist crve of the process. The target point is represented in light green in the case of Free tning and in black for constrained tning, as shown in Figre 8. The slope of the target point can also be changed TABLE Sensitivit circles. This table describes the center and radis of circles that define the loci for constant sensitivit M s, constant complementar sensitivit M t, constant mixed sensitivit, and eqal sensitivities M = M s = M t [8]. Contor Center Radis M s -circle /M s M t -circle M-circle x = max M t M t x+x ( Ms+ Ms, ) Mt Mt x = max Mt M t x x ( Ms Ms, ) Mt Mt + OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 5 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

9 interactivel. For Free tning, the controller gains are shown as arrows in the Nqist plot. The controller gains can be modified interactivel b dragging the ends of the arrows. Figres 8 and 9 show examples of these arrows. The scale of the graphic can be changed sing 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 3 3 L-Plane Graphics: Save Delete L-Plane Graphics: Save Delete FIGURE 9 The L-plane graphic. The Nqist plots of the process transfer fnction P(s) (black line) and the loop transfer fnction L(s) = P(s)C(s) (red line) are shown. An example of the Free tning design. The controller gains can be changed b dragging arrows, the proportional gain changes L(i ω) in the direction of P(i ω) (ble arrow), the integral gain k i changes L(i ω) in the direction of ip(i ω) (can arrow), and the derivative gain k d changes L(i ω) in the direction of ip(i ω) (magenta arrow). An example of the Constrained PID tning design. In this case, once the ser moves the target point (black circle), the controller parameters are calclated sing (5) (). its vale based on its meaning. Options save and delete can be fond 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 pictres appear, one of which shows the crrent design in red while the other shows the crrent design in ble (see Figre 8). Modifications of the controller parameters affect the crrent (active) design, which can be changed sing the options Design and Design, which appear on the top of the L-plane graphic. Once a design is chosen, the associated crve is switched to red, and the controller zone is modified based on that design. The controller gain vales can be seen b moving the crsor on the crves. Settings Men The Settings men, which is available in the main men of PID Loop Shaping, is divided into for grops, following the same strctre as in PID Basics. The first entr, called Process Transfer Fnction, is sed to choose between several predefined transfer fnctions or to inclde an serspecified transfer fnction throgh two options. The String TF option allows a transfer fnction to be entered smbolicall. For instance, P(s) = / cosh s can be represented as P= /cosh(sqrt(s)). Reslts can be stored and recalled sing the Load/Save men. The option Save Report can be sed to save all essential data in text format, which is sefl for docmenting reslts. Specific vales for control parameters can be entered with Parameters men option. As in PID Basics, the last men option (Examples Advanced PID Book) allows loading examples from [8]. Examples Some of the capabilities of PID Loop Shaping are illstrated b the following examples. Effect of Controller Parameters The prpose of this example is to illstrate how the Nqist plot of the loop transfer fnction changes when the controller parameters are modified. Consider the process P(s) = /(s + ) 4. When a P controller is sed, the proportional gain changes the loop transfer fnction L(iω) = kp(iω) in the direction of P(iω). Figre shows the effect of modifing L(iω) sing a P controller with gain k = (ble crve) and k =.6 (red crve). These crves show how the proportional gain modifies the Nqist plot of the process (black crve) at the freqenc ω (green circle on the black crve) in the direction of P(iω). Figre shows the same std for an I-controller with k i = (red crve) and k i =.6 (ble crve). It can be seen that the integral gain k i changes L(iω) in the direction ip(iω). The derivative gain has the same effect in the direction of ip(iω). When a PI or PD controller is sed, the compensated point at the freqenc ω is calclated as the sm of two vectors, namel, the proportional vector and the integral or derivative vector. Examples of this capabilit are 6 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

10 shown in Figre (c) and (d), where the process is controlled b a PI controller (k =.3 and k i =.7) and a PD controller (k =. and k d = 3.35), respectivel. Simple exercises can be sed to provide training in loop shaping. For instance, with the above process, it is instrctive to calclate the gain for a proportional controller for which the closed-loop sstem changes from stable to nstable. Before sing PID Loop Shaping, the reslt can be calclated analticall, which ields L(iω) = C(iω)P(iω) = 8, k = 8, ω =. (iω + ) 4 L(iω) = C(iω)P(iω) = + i, k (iω + ) 4 =, k = 4. PID Loop Shaping can be sed to verif the reslt interactivel, as shown in Figre. This exercise challenges stdents and encorages them to make observations while relating theor to images to develop a broader and deeper nderstanding. On the other hand, free interactive designs can also be performed to compare the reslts with other design methods. For instance, PID Loop Shaping can be sed to design a PID controller interactivel for the process P(s) = /(s + ) 4, where the maximal sensitivit vale M s mst be less than.5. A PID controller that satisfies this constraint is obtained when k =.9, T i =.8, k i =.5, T d =.3, and k d =.95. The AMIGO-freqenc method can also be sed for design, and the reslts can be compared. The reslting controller is given b k =., T i =.48, k i =.48, T d =.93, and k d =.. Figre shows the Nqist plots and time responses sing PID Basics for both designs, in ble for the free PID controller and in red for the AMIGO method. The reslting vales of M s are.49 for free PID and.46 for the AMIGO method. (c) (d) FIGURE Nqist plot modifications depending on the controller tpe. P controller, I controller, (c) PI controller, and (d) PD controller. The modification of L(i ω) in the direction of P(i ω) sing a P controller with gain k = (ble crve) and k =.6 (red crve). The same std for an I-controller is shown in with k i = (red crve) and k i =.6 (ble crve), where L(i ω) is modified in the direction ip(i ω). (c), (d) PI or PD controllers are sed, respectivel. In these cases, the compensated point at the freqenc ω is calclated as the sm of two vectors, namel the proportional vector and the integral or derivative vector. OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 7 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

11 Effect of the Target Point The target point on the Nqist plot can be reached sing an nconstrained design b selecting the Free option. The controller gains are interactivel adjsted as shown in the free tning example. Another approach is to se (5) (), where the controller gains are calclated after the target point is defined. As discssed above, the target point can be fixed or constrained in varios was, either at an point, to specific vales for phase margin and gain margin, or to maximal vales of the sensitivit fnctions. Figre 3 shows an example in which the target point is set to the point.5.5i. Two constrained designs are shown for the design freqenc ω =.6 rad/s. The red crve represents a sstem compensated b a constrained PID with k =.3, k i =., and k d =.5, while the ble crve represents a constrained PI with k =.3 and k i =.5. Althogh both controllers reach the target point, better reslts are obtained for the PID controller becase the slope can be freel adjsted (the vale for this example is ϑ = ). The PID controller provides better robstness properties with M s =.45, k i =., G m = 5.3, and P m = 4.5, verss a PI controller with M s =.83, k i =.5, G m =.69, and P m = Similar examples can be sed to restrict the target point for phase margin, gain margin, or maximal vales of the sensitivit fnctions. Figre 4 shows an example where a combined sensitivit constraint is reqired for M s and M t. This constraint is flfilled in two different was, namel, b sing Constrained PID (red crve) and Constrained PI (ble crve). Another example combining sensitivit fnction and gain margin constraints is shown in Figre 4, with the specification that the gain margin be eqal to three and M s. These specifications are established b maximizing the integral gain k i. Hence, the constraint gain margin is chosen, and the target point is located in sch a wa that G m = 3. Then, a Constrained PID controller is selected, where the design point and the slope are modified ntil M s and the integral gain is maximized. The final controller is given b k =.38, k i =.5, and k d =.54 for ω =. and ϑ = 3. Process Otpt Graphics: Save Delete 3 L-Plane Design Design Graphics: Save Delete K =. T i =.5 T d =.93 b= N= K =.9 T i =.79 T d =.3 b= N= 5 Time Controller Otpt P I D K =. T i =.5 T d =.93 b= N= K =.9 T i =.79 T d =.3 b= N = 5 Time FIGURE Stabilit limit on the critical point + i. A tpical example for presenting loop shaping is to search for the lowest gain that makes the sstem nstable. This task can be interactivel performed with PID Loop Shaping as shown in this figre. FIGURE Example of loop shaping with M s <.5. PID Loop Shaping can be sed to compare varios designs. In this figre, Nqist plots and time-domain responses generated with PID Basics are shown to compare an nconstrained design (k =.9, T i =.8, k i =.5, T d =.3, and k d =.95) with an alternative design developed sing the AMIGO-freqenc method (k =., T i =.48, k i =.48, T d =.93, and k d =.). 8 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

12 The Derivative Cliff We again consider the process transfer fnction P(s) = /(s + ) 4. It is desirable to maximize the integral gain k i sbject to the robstness constraint M s.4. The reslting controller has the parameters k =.95, k i =.9, and k d =.86, where the Nqist plot of the loop transfer fnction is shown in red in Figre 5. It can be seen that the Nqist crve has a loop, called a derivative cliff. As explained in [8], this featre, which is de to excessive controller phase lead, reslts from having a PID controller with complex poles, which occrs when T i < 4T d. In this example the relation is T i =.33T d. Figre 5 shows, in red, the time response of the controller, which ields oscillator otpts. For comparison, the reslts for a controller with T i = 4T d are shown in ble in Figre 5 and with the controller parameters k =., k i =.36, and k d =.9. The responses for this controller are improved, despite larger overshoot in response to load distrbances. This example is available in the Settings men of PID Loop Shaping. PID WINDUP The prpose of the PID modle is to facilitate nderstanding of integral windp and a method for compensating it [8]. For a control sstem with a wide range of operating conditions, it ma happen that the control variable reaches the actator limits. When this sitation occrs in loops sing a controller with integral action, the feedback loop is broken and the integral ma reach large vales, maintaining the control signal satrated for a long time, reslting in large overshoot, and ndesirable transients. This problem is known as windp phenomenon [8]. can be avoided in different was. Back calclation and tracking [8] is illstrated in the block diagram in Figre 6. The sstem remains nchanged when the satration is not active. However, when satration occrs, the integral term in the controller is modified ntil the control signal is ot of the satration limit. This modification is not performed instantaneosl bt dnamicall with a time constant T t called the tracking time constant [8]. The modle PID shows process otpts and control signals for nlimited control signals, limited 3 L-Plane Graphics: Save Delete L-Plane Design Design Graphics: Save Delete L-Plane Graphics: Save Delete 3 FIGURE 3 Example of a constrained design with a target point of.5.5i. The target point can be constrained to reach arbitrar specifications. Once a point is constrained, the controller parameters are atomaticall calclated. This plot shows PI (k =.3, k i =.5) and PID (k =.3, k i =., k d =.5) controllers, both reaching the target point. The PID controller provides better reslts de to the se of the slope as an allowable third degree of freedom as described in () and (), where the slope ϑ takes the vale. 3 FIGURE 4 Example of a constrained design with sensitivit and gain-margin constraints. These plots show an example where the target point is constrained to reach specified vales for the combined sensitivit fnctions with M s and M t, and gain margin with limited sensitivit vales with Gm = 3 and M s. OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 9 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

13 control signals withot antiwindp, and limited control signals with antiwindp. The ser interface is shown in Figre 7. Process models and controller parameters can be selected in the same wa as in the other modles. The satration limits of the control signal can be determined either b entering the vales or b dragging the lines in the satration scheme. L-Plane Design Design Graphics: Save Delete The Interactive Tool We now describe the main aspects of PID. Process The Process area is similar to that described in PID Basics and PID Loop Shaping. The time dela is modified sing a slider instead of a text edit, so that the time dela effect on the antiwindp mechanism can be analzed. Controller The Controller area contains information abot the controller parameters and actator satration. Three kinds of e = sp KT d s K Σ v Actator Model Actator K/T i Σ /s Σ + Process Otpt Graphics: Save Delete 3 K =.93 T i =.3 T d =3.9 b=.5 N= K =. T i =3.5 T d =.76 b=.5 N= /T t e s FIGURE 6 PID controller with antiwindp scheme, where K is the controller proportional gain, T i is the controller integral time, T d is the controller derivative time, sp is the setpoint, is the process otpt, e is the tracking error, v is the controller otpt, is the satrated controller otpt, and e s is the difference between the controller otpt v and the satrated controller otpt. In this scheme the control signal remains nconstrained when the satration is not active. When satration occrs, the integral control action is modified ntil the control signal is ot of the satration limit. The modification of the integral element is performed dnamicall b adjsting the tracking time constant T t [8]. 5 Time Controller Otpt P I D K =.93 T i =.3 T d =3.9 b=.5 N= K =. T i =3.5 T d =.76 b=.5 N= 5 Time FIGURE 5 Derivative cliff example. Nqist plot and timedomain responses. This example shows that optimization of k i, which is aimed at flfilling robstness specifications, can provide controllers with excessive phase lead, as represented b the loop in the red crve. This behavior is a conseqence of the presence of complex zeros de to T i < 4T d. The same example is shown in ble for T i = 4T d, where this problem is avoided [8]. FIGURE 7 The ser interface of the modle PID, showing the windp phenomenon and application of the antiwindp techniqe. Several graphical elements are sed to interactivel analze tpical problems and soltions associated with windp. The example shown in the figre illstrates the windp phenomenon (in ble) and the reslt of appling the antiwindp techniqe (in green). 3 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

14 controllers with integral action can be selected (I, PI, PID), where several sliders are available to change the controller parameters, inclding the tracking time constant Tt. A satration graphic is also available in this zone. The Actator Satration graphic allows the satration limits to be determined b dragging the small red circle located on the pper satration vale. In this graphic, a smmetric satration is selected for pedagogical prposes. Graphics Time responses for process otpt, control signal, and integral action are available in three graphics, namel, Process Otpt, Controller Otpt, and Integral Term. In the same wa as in PID Basics, mltiple interactive graphical elements can be sed to change the setpoint, load distrbance, measrement noise, or horizontal and vertical scales (see Figre 7). These graphics can simltaneosl represent the controlled sstem in linear, nonlinear with windp, and nonlinear with antiwindp modes. These representations can be configred sing the checkboxes located above the Process Otpt graphic. For instance, Figre 7 shows an example containing the nonlinear with windp and nonlinear with antiwindp modes. The dotted pink vertical line in Figre 8 is helpfl for comparing the otpts of the different plots at the same time instant. The satration limits can be altered sing the dotted ble horizontal lines available in the Controller Otpt graphic (see Figre 7). The notion of proportional band is sefl for nderstanding the windp effect, and is inclded in PID. The proportional band is defined as the range of process otpts sch that the controller otpt is in the linear range [ min, max ]. For a PI controller, the proportional band is limited b min = b sp + I max, K (3) max = b sp + I min, K (4) where I is the integral term of the controller, and max and min are the control signal limits. Expressions (3) and (4) hold for PID control when the proportional band is defined as the band where the predicted otpt p = + T d (d/dt) is in the proportional band [ min, max ]. The proportional band has the width ( max min )/K, and is centered arond b sp + I/K ( max + min )/(K). Two additional checkboxes called PB and PB Antiwindp, appear near the top of plot Process Otpt. The activation of these options shows the proportional bands for the windp and antiwindp cases in the Process Otpt graphic. The proportional bands are shown as dotted green and ble crves, respectivel, as shown in Figre 8. Process Otpt Linear Antiwindp PB PB Antiwindp Process Otpt Linear Antiwindp PB PB Antiwindp Time 4 6 Time Controller Otpt Controller Otpt Time Integral Term I 4 6 Time Integral Term I 4 6 Time 4 6 Time FIGURE 8 Example of the windp phenomenon with proportional band for K = and K =.4. In [8] the notion of proportional band is described as being a sefl tool for nderstanding the effects of windp. The proportional band is an interval sch that the actator does not satrate when the instantaneos vale of the process otpt or its predicted vale is inside this band. These plots show two examples demonstrating how the control signal is satrated when the process otpt is inside the band shown in ble. The interactive pink line of the graphics can be sed to test this idea. OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 3 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

15 Settings Men The Settings men has the same strctre as in PID Basics and PID Loop Shaping. The process transfer fnction can be chosen from the entr Process Transfer Fnctions, and nmerical vales of the parameters can be introdced sing Controller Parameters. Essential data and reslts can be saved and recalled sing the Load/Save men options. The men selection Simlation makes it possible to choose the simlation time and activate the Sweep option, which can be sed to show the reslts for several vales of the tracking time constant. Several examples from [8] can be loaded from the Examples entr. Examples The following examples illstrate properties of the PID modle. Understanding the Phenomenon can be stdied sing the first entr from the Examples option men. This example from [8] ses the pre integrator process P(s) = /s controlled b a PI controller with parameters K =, T i =., and b =, and with the control signal limited to ±.. Figre 7 shows the time responses for this example. The control signal is satrated from t =. The process otpt and the integral term increase while the control error is positive. Once the process otpt exceeds the setpoint, the control error becomes negative, however the control signal remains satrated de to the large vale of the integral term. The time responses are shown in Figre 7. The proportional band can be drawn in this example sing the PB checkbox shown in Figre 8. Using the vertical line, the ser can see that the process otpt remains inside the band while the control signal is working in linear mode and otside the proportional band when the control signal is satrated. Large controller gains provide narrow proportional bands, with more energetic control signals and therefore longer satration times, while small controller gains give wider proportional bands. Figre 8 illstrates this effect, where the proportional controller gain is redced to.4, prodcing a wider proportional band. Antiwindp The process P(s) = /s is also sefl for visalizing the antiwindp techniqe. The same controller parameters, namel, K =, T i =., b =, are sed, and the tracking time constant is set to T t =. Figre 9 shows the responses for both cases control with and withot antiwindp. The sstem with antiwindp remains in satration for onl a short period of time, with the magnitde of the integral term considerabl redced. The proportional Process Otpt Linear Antiwindp PB PB Antiwindp Process Otpt Linear Antiwindp PB PB Antiwindp Antiwindp Antiwindp 4 6 Time Controller Otpt. Antiwindp. Integral Term I 4 6 Time Antiwindp 4 6 Time Controller Otpt. Antiwindp. Integral Term I 4 6 Time Antiwindp 4 6 Time 4 6 Time FIGURE 9 Example of the effect of the tracking time constant T t on in the antiwindp techniqe. Antiwindp and effect of T t. These plots show the reslts of appling the antiwindp techniqe to the example shown in Figre 7. The integral signal is considerabl redced, allowing the control signal to remain in satration dring a shorter period of time. The proportional band for the antiwindp techniqe is shown in green. The process otpt remains inside the band most of the time. 3 IEEE CONTROL SYSTEMS MAGAZINE» OCTOBER 8 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.

16 band for the PI controller with antiwindp is shown in the same figre. It can be seen that the proportional band is wider than for PI withot antiwindp [Figre 9], where the process otpt remains most of the time. The effect of the tracking time constant is illstrated in Figre 9 for T t =.,, 5. In this scenario, the Sweep men option is sed. High vales of T t make the antiwindp too slow to be effective, while low vales reset the integral term qickl with improved reslts. It ma ths seem advantageos to alwas have small vales of T t. However, the next example shows some sitations where this choice is not advisable. The Tracking Time Constant The tracking time constant is an essential parameter becase it determines the reset rate for the integral term of the controller. It ma seem advantageos to have a small vale for this constant. However, measrement errors ma accidentall reset the integral term when the tracking time constant is too small. The following example illstrates this phenomenon, when a measrement error occrs in the form of a short plse. The transfer fnction of the process is P(s) = (.5s + ), and the controller is a PID controller with K = 3.5, T i =.5, T d =.4, N d =, b =, and T t =. Figre shows the control reslts. A large transient appears after the plse, and the integral term is excessivel redced. Varios rles are sggested in [8] for choosing the tracking time constant. One choice is T t = (T i + T d )/. Figre shows an example with T t = (T i + T d )/ =.33, where the response is considerabl improved. CONCLUSIONS In this work a set of interactive modles that comprise ILM-PID is presented to spport the teaching and learning of basic atomatic control concepts. These tools are intended mainl to inclde interactivit in the visal content of [8]. The modles focs on PID control, stding feedback fndamentals from the standpoint of the time and freqenc domains, inclding robstness isses, measrement of noise filtering, load-distrbance rejection, and windp phenomenon. The importance of interactivit in atomatic control edcation has been shown in the context of teaching and learning. In the athors experience, interactivit offers excellent spport to edcation and learning b enhancing the motivation and participation of ftre Process Otpt Linear Antiwindp PB PB Antiwindp Process Otpt Linear Antiwindp PB PB Antiwindp Antiwindp Antiwindp 5 5 Time 5 5 Time Controller Otpt Antiwindp Controller Otpt Antiwindp 5 5 Time 5 5 Time Integral Term I 5 Antiwindp Integral Term I 5 Antiwindp Time 5 5 Time FIGURE Tning the tracking time. Reset b measrement noise and tning sing rles. illstrates the disadvantage of sing a short tracking time constant. The short plse distrbance at time t = reslts in excessive redction of the integral term and a large distrbance in the process otpt. In the choice is T t = (T i + T d )/. OCTOBER 8 «IEEE CONTROL SYSTEMS MAGAZINE 33 Athorized licensed se limited to: Lnds Universitetsbibliotek. Downloaded on November 4, 8 at 8:7 from IEEE Xplore. Restrictions appl.