INTERNET PID CONTROLLER DESIGN: M. Schlegel, M. Čech

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1 INTERNET PID CONTROLLER DESIGN: M. Schlegel, M. Čech Deartment of Cybernetics, University of West Bohemia in Pilsen fax : , schlegel@kky.zcu.cz, mcech@kky.zcu.cz Abstract: The urose of this aer is to introduce a simle Internet tool (Java alet) for PID controller design free accessible on The alet imlements the Nyquist lot shaing according to gain and hase margins and/or sensitivity functions shaing. It is based on the robustness regions method. This aroach is suitable esecially for any non-oscillatory or slightly oscillatory linear system with dead time. The authors believe, that the Java alet will be worthwhile for industrial ractitioners. Coyright 004 IFAC Keywords: robust control, PID, Nyquist lot shaing, Java alet. INTRODUCTION PID controllers are widely used in industrial ractice more than 60 years. The develoment went from neumatic through analogue to digital controllers, but the control algorithm is in fact the same. The PID controller is a standard and roved solution for the most of industrial control alications. In site of this fact, there is not some standard and generally acceted method for PID controller design based on known rocess model. Over the years, there are many formulas derived to tune the PID controller. But there exist only a few universal rocedures, which can be used for arbitrary order irrational or non-minimum hase transfer functions. Some of these are described in (Shafei and Shenton, 994, 997; Ackerman and Kaesbauer, 99, 003). The classical D-artition method (Neimark, 948, 998; Bimbirekov 994) is used for the ideal PID controller design where the gain and hase margins are secified. For the real PID controller (filtered derivative art) and for more general design secifications, this method must be modified as it is shown in Sections and 3. The arising algorithm seems to be universal and still comrehensible for eole in industrial ractice. That is why we decided to create a Java alet accessible on Internet. The alet is described in Section 4. Illustrating examle is given in Section 5.. DESIGN METHOD PRINCIPLE Consider the control loo shown in Figure with the PI(D) controller C(s) and the lant described by a stable transfer function P(s). w - e C u l Fig.. Control loo The variables w, e and y denote the setoint, control error and rocess variable, resectively. The variables l and n denote disturbances affecting the control loo. It is well known, that required loo erformance (e. g. gain and hase margins) could be reached by shaing of the Nyquist lot L( j) = C( j) P( j). () For examle, the minimum gain margin (Gm>) and minimum hase margin 60 (Pm 60 ) are equivalent with requirement that oints X = -/ and X = /(+ j 3) in Figure lie on the left side of P n y

2 L( j) or just on this curve. If the secifications Gm < 3 and Pm < 90 are required, the oints X 3 = -/3 and X 4 = -j should lie on the right side of L( j ). In such way the oints X, X,, X 3, X 4 define required Nyquist lot shae. Similarly, one can roceed with, if the limitation of the sensitivity function S( j) = + L( j) is required in the form su S( j) Ms, () or if the limitation of the comlementary sensitivity function L( j) T( j) = + L( j) is required in the form su T( j) M. (3) It can be simly roved, that condition () (res. (3)) is equivalent with the requirement that the Nyquist lot L( j) does not have any intersection with a circle having center M c = - (res. c = ) (4) M and radius M R = (res. R = ). (5) M s M If we want to rovide the disturbance rejection at a given frequency range and the roer bandwidth of the closed loo, the following conditions are suitable su S( j) ε, (6) < 0, s > su < T, + ) T( j) ε. s T (7) The general roblem of (comlementary) sensitivity function shaing (Schlegel and Mertl, 004) is shown in Figure 4. Now, consider the case, when only one shaing oint X = u+jv is secified in the Nyquist lot lane. Our aim is to find all ossible airs of arameters k, k i of the PI controller with transfer function ki Cs () = k +, s for which the oint X lies on the left side of the Nyquist curve. For this urose, let us solve the equation k i L( j) = ( k j )( a( ) + jb( )) = u+ jv (8) for unknown k and k i, where a( ) = Re( P( j)) and b( ) = Im( P( j)). The exressions obtained a( ) u+ b( ) v k = a ( ) + b ( ) [ a( ) v b( ) u] ki = a ( ) + b ( ) (9) define arametric curve in the k, k i lane. This curve together with k, k i axis slits the arametric lane into several regions as it is shown in Figure 3. We are interested only in ositive values of k, k i in the first quadrant. It follows from (8) that the boundary of regions corresonds with PI arameters leading to the Nyquist lot assing through the oint X. All oints inside any region lead to the Nyquist lot having oint X at the same side. More recisely, the number of encirclements of the oint X by the Nyquist lot is the same. Usually, just one region contains suitable oints corresonding to the required relative location of the Nyquist lot and the oint X. Any Nyquist lot shaing roblem with a finite number of shaing oints could be converted to the several one-oint cases described above. If we find corresonding regions R i for all oints X i, i=,,,n, than their intersection - n R = I R contains all oints i= solving our roblem. It is suitable to choose a oint with the maximum k i coordinate from all ossible solutions. Then, the PI controller has the maximum gain in low frequencies and also minimizes the criterion IE = etdt (), when the ste is alied to the setoint. + 0 i X X 3 60 o L(j ) Im X X 4 Fig.. Nyquist lot shaing oint X at the left side from L(j) k oint X encircled twice oint X encircled once Re Fig. 3. Curve (9) and corresonding regions 3. PID CONTROLLER DESIGN The method resented above can be modified for a real DOF PID controller design. The PID controller is in according to ISA norm described by the relation k i

3 T(j) M T ε T S(j) M S T It follows from (0), that the closed loo stability and disturbance resonse are deendent only on arameters k, T i, T d and N, while the closed loo resonse can be indeendently influenced by the arameters b and c. Therefore, the controller design should be divided into two stes. Firstly, we design the DOF PID controller (b = c = ) described by the transfer function Ts d Cs () = k+ + () Ts T i d s+ N and then we tune arameters b and c manually ( b 0, and c=0 are recommended). Since the ε s s Fig. 4. The constrains on sensitivity and comlementary sensitivity functions Ts d U() s = kbw() s Ys () + Es () + [ cw() s Ys ()], Ts T i d s+ N (0) where Y(s), W(s) and U(s) denote the Lalace transforms of the rocess variable, setoint and maniulated variable, and E(s)=W(s)-Y(s). Further, k denotes gain, T i and T d are integral and derivative time constants, b and c are weight coefficients of the set oint in roortional and derivative art and arameter N secifies the degree of derivative art filtering. controller () has four design arameters k, T i, T d and N, we must relate them by additional conditions, in order to the arameter lane method could be used. Fortunataly, arameter N has the clear hysical interretation. When N, we obtain an ideal PID controller, while the derivative art is switched off if N 0. The value of N is usually chosen in the interval,0 according to the signal/noise ratio. Moreover, the ratio f =T d / T i is used to be equal to ¼ (Ziegler and Nichols, 94). Newer studies (Aström and Hägglund, 995) acknowledged correctness of this value esecially for lants with a monotone ste resonse. Note, that f = 0 leads to PI controller and f > /4 enhances the derivative art. When N and f are constant, we only need to determine two arameters k and k i = k /T i like in Section. Comutation of region boundaries is much more comlicated (Schlegel, et al., 003a) but the design technique can be used without changes. Fig. 5. General view of the alet

4 The rule of otimal arameters choice is also the same: the otimal oint has the maximum k i coordinate. 4. USER DESCRIPTION OF THE APPLET Now, we shortly describe the grahical user interface of the alet, which is free accessible on The alet area is divided into five basic windows (Figure 5) ) Process model (PM). In this window, one can secify a rocess model (transfer function) using one of four ways selected in the right middle. The otions are: coefficients of the numerator and denominator of the transfer function, time constants (Bode), zeroes and oles and the second order lus dead time (SOPDT) form. All forms can be comlemented by the gain and the transort delay. The introduced rocess model should always be confirmed by the OK button. ) Controller (C). The controller design is divided into following stes. a) Define new rocess model b) Choose tye of the controller (PI/PID). Choose the fix arameters N and f. c) Define design secifications in the Nyquist lot lane (DS window). d) By clicking on any oint in satisfactory region in the RP window (the intersection of all regions meeting the design secifications) we obtain the arameters k and k i and related T i, T d. e) Finally, we can change some arameters by hand. Esecially decreasing of the arameter b could cause a lower overshot in closed loo. Moreover, it is also ossible to set all arameters of the DOF ISA PID controller by hand. 3) Design secifications (DS). In this window, we secify the Nyquist lot shae using shaing oints. We can easily secify gain and hase margins or the limitations of the (comlementary) sensitivity function by secial choice of these oints. Points can be added by left mouse button click. All oints are listed in the design secifications list. and can be edited manually there. If the Gain and hase margins checkbox is checked, then it is easy to secify oints on the unit circle or the negative real axes. 4) Robustness regions in PID arameter lane (RP) This window is active only if at least one design secification is defined. The regions corresonding to design secifications are loted in this window. We need to click here to choose or change controller arameters k and k i. 5) Loo erformance (LP). One of four grahs can be chosen in this window. The rocess ste resonse is loted when new rocess is defined. Other grahs are available only if all controller arameters are secified. We can choose the closed loo setoint and load disturbance ste resonse, the sensitivity function or the comlementary sensitivity function. Grah Axes We can change the axes ranges with buttons under each grah. When the button 'auto' is ressed, the ranges of the current lot are set automatically. The best way to zoom is to define the zoom rectangle by mouse dragging. Settings and status line In the lower art of the alet is the settings anel. Here, the ranges for all frequency characteristics can be changed. We can also set the simulation time and the samling eriod of discretization of the rocess and the controller. The status line can be very useful. Actual alet state information as so as the short hel about imortant comonents are rinted here. 5. PRACTICAL EXAMPLE Let us use the alet for a brick ress controller design. Assume, that the brick ress is described by the transfer function 00s e Fs () =. (40s+ )(0s+ ) The controller design is divided into following stes. ) Process model definition We secify the time constants, gain and transort delay in BODE form. After that, we need to confirm the introduced model by the OK button (Figure 6). Fig. 6. Process model definition ) Design secifications We use default values of controller tye (PID), N and f. Then we secify gain and hase margins (P m =60, G m =) by mouse clicking in DS window. We can set the P m and G m values more recisely manually after selection in the design secifications list (Figure 7). If we want to see the standard sensitivity and comlementary sensitivity functions limitations (so called m-circles), we must check the m-circles checkbox. The required Nyquist lot shaing can be reached by choosing oints on these circles. 3) Choice of PID controller arameters K, K i in arameter lane We will see two regions corresonding to the gain and hase margins shaing oints. The region actually selected in design secification list is loted by red color. If we want to fulfill both conditions, we must choose the arameters at the intersection of the regions boundaries by mouse clicking (Figure 0).

5 Fig. 0. Robustness regions in PID arameter lane Fig. 7. Design secifications definition 4) Manual tuning of PID arameters The corresonding PID arameters aear in C window after clicking at some oint of the PID arameter lane in RP window. Parameters b and c are set at default values b =, c = 0. If we set the arameter b = 0.3, the closed loo behavior will be without overshot. We see the closed loo resonse in LP window. The rocess variable is ainted blue and the maniulating variable is ainted green. Let us look now, how the region shae deends on values N and f. Note that the shae is strongly influenced (mainly for rocesses without dead time) by the value of derivative art filter arameter N. It is clear, that most of methods based on ideal PID controller will not work roerly (Schlegel, et al., 003b), because the influence of N is very strong (Figure b). Fig. 8. Controller arameters (a) Finally, we can ensure that the Nyquist lot asses through both oints and the closed loo has required erformance (Figure 9). (b) Fig.. Deendency of the shae of the region on values N and f for the rocess /(s+) 3 with time delay 0 (a) res. 0 (b) and shaing oint Pm=60. Fig. 9. Closed loo erformance The new version of the alet (PID controller designer.5) rovides extended shaing of the (comlementary) sensitivity function according to Figure 4. Consider now, that we want to fulfil in our examle the closed loo bandwidth requirement su T ( j) 0.. () < 0., + ) By clicking in comlementary sensitivity function (CSF) lot, we can define the corner of rotected area (Figure 3). Then, the new secification is added into the list. Now we can edit the values of ε T

6 and M T more recisely there. Consequently, the third region will aear in RP window. Fig.. CSF shaing region The otimal controller (which fulfils all design secifications) is reresented by the oint lying in the intersection of all three regions with maximum K i coordinate and is highlighted by red colour in Figure. Now, the comlementary sensitivity function has the required shae as you can see in Figure 3. Fig. 3. CSF shaing This aroach can also be used for simultaneous robust stabilization of several rocesses. More examles and detailed documentation of the alet can be found at 6. CONCLUSIONS The main urose of this aer is to introduce a new tool for the PID controller design (free accessible Java alet on based on new Nyquist lot shaing method. This method allows to design the real DOF PID controller for ractical requirements, e. g. gain and hase margins. But the method is suitable also for more comlicated Nyquist lot shae requirements including disturbance rejection in a given frequency range and the roer bandwidth secification. The method works for any linear system (unstable, non-minimum hase, with or without dead time, fractional order). This aroach is useful esecially for stable non-oscillatory or slightly oscillatory rocesses, where the Nyquist lot shae requirements are well known. REFERENCES Ackerman, J., D. Kaesbauer and R. Muench (99). Robust gamma-stability analysis in a lant arameter sace. Automatica, 7, Ackerman, J. and D. Kaesbauer (003). Stable olyhedra in arameter sace. Automatica, 39, Åström, K. J. and T. Hägglund (995). PID controllers. Theory design and tuning, NC: Instrument Society of America, Research Triangle Park, 995. Bimbirekov, B. L. (994). Local D-artition of the sace of regulator adjustable arameters for a linear system and choice of arameters by frequency techniques. Automation and Remote Control, 55, Neimark, Y. I. (948). Structure of the D-artition of the sace of olynomials and the diagram of Vishnegradskii and Nyquist, Dokl Akad Nauk SSSR, 59, 853. Neimark, Y. I. (998). D-artition and robust stability. Comutational Mathematics and Modeling. 9, Shafei, Z. and A.T. Shenton (994). Tuning PIDtye controllers for stable and unstable systems with time delay. Automatica. 30, Shafei, Z. and A.T. Shenton (997). Frequency domain Design of PID Controllers for Stable and Unstable Systems with Time Delay, Automatica, Volume 33, Issue, December 997, Pages 3-3. Schlegel, M., J. Mertl and M. Čech (003a). Generalized robustness regions for PID controllers. In: Proceedings of Process Control Conference 003, Štrbské Pleso, June 003, Slovak Reublic. Schlegel, M., M. Čech and J. Mertl (003b). Are heuristic methods for PID design reliable? In: Proceedings of Process Control Conference 003, Štrbské Pleso, June 003, Slovak Reublic. Schlegel, M. and J. Mertl (004). Stablility regions for PI/PID controllers. In: Proceedings of ICCC 004, Zakoane, June 004, Poland. Ziegler J. G. and N.B Nichols (94). Otimum settings for automatic controllers. Trans. ASME, 64, htt:// Alet documentation and examles