Robust PID tuning strategy for uncertain plants based on the Kharitonov theorem
ISA Transactions 39 (2000) 419±431 www.elsevier.com/locate/isatransRobust PID tuning strategy for uncertain plants based on the Kharitonov theorem Ying J. Huang *, Yuan-Jay Wang Institute of Electrical Engineering, Yuan Ze University, 135 Far-East RD., Chungli, 320 TaiwanAbstract In this paper, the Kharitonov theorem for interval plants is exploited for the purpose of synthesizing a stabilizingcontroller. The aim here is to develop a controller to simultaneously stabilize the four Kharitonov-de®ned vortexpolynomials. Dierent from the prevailing works, the controller is designed systematically and graphically through thesearch of a non-conservative Kharitonov region in the controller coecient parameter plane. The region characterizesall stabilizing PID controllers that stabilize an uncertain plant. Thus the relationship between the Kharitonov regionand the stabilizing controller parameters is manifest. Extensively, to further guarantee the system with certain robustsafety margins, a virtual gain phase margin tester compensator is added. Stability analysis is carried out. The controlsystem is proved to maintain robustness at least to the pre-speci®ed margins. The synthesized controller with coe-cients selected from the obtained non-conservative Kharitonov region can stabilize the concerned uncertain plants andful®ll system speci®cations in terms of gain margins and phase margins. # 2000 Elsevier Science Ltd. All rightsreserved.Keywords: Kharitonov theorem; Parameter plane; PID controller; Interval plants1. Introduction theorem  and box theorem  then suggested that the set of transfer functions generated by In the past years, many important results in the changing its perturbed coecients in the pre-area of robust control for uncertain plants have scribed ranges corresponds to a box in the para-been based on the Kharitonovs celebrated theorem meter space and is referred to ``interval plants.[1±6]. The theorem investigates the stability char- Further, the Kharitonov theorem is generalized toacteristics of the interval systems via four vortex obtain a ``Kharitonov region [5,6] in the complexpolynomials with real coecients varying in a plane for the robust stability of linear uncertainbounded range. Kharitonov extended his results systems. In order to guarantee the uncertain sys-into interval polynomials with complex coecients tems a stronger stability characteristic, a virtuallater. Based on the Kharitonov theorem, the edge compensator was introduced such that the closed loop systems can maintain a suitable gain margin (Gm) and phase margin (Pm) . Recently, atten- * Corresponding author. Tel.: +886-3-4638800 ext. 410; fax: tion has been given to the formulation of P, PI+886-3-4633326. and PID controllers to stabilize an interval plant E-mail address: firstname.lastname@example.org (Y.J. Huang). family .0019-0578/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.PII: S0019-0578(00)00026-4
420 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 Despite the various results concerning the synthesizing ®xed and low order controllers, thatrobust stability for uncertain plants, existing simultaneously stabilize a given interval plantmethods in the area of parametric robust control family in the parametric robust control area, isare of analysis type which are passively concerned bene®cial. To achieve this goal, a systematic andonly if a family of Kharitonov polynomials are graphical robust controller design procedure isHurwitz or not. It turns out that a technique for established based on the Kharitonov-de®ned vortex Fig. 1. Feedback control system with a gain phase margin tester. Fig. 2. Stability boundary for GI s.
Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 421polynomials. A speci®c Kharitonov region can be also ful®lls several speci®cations simultaneously.obtained in the parameter plane using the para- For every one of the four vortex polynomials, itmeters of the controller as the axes. Therefore, the suces to plot the constant gain margin and con-stability characteristic for the uncertain system in stant phase margin boundaries in the parameterquestion, with respect to the adjustment of the plane. The overlapped region of speci®cation-controller parameters, is obvious. The aforemen- satis®ed area for each Kharitonov polynomial istioned region constitutes the whole admissible the useful parameter area for the selection of con-stabilizing PID controllers. troller parameters, where the whole uncertain Further, for the purpose of endowing the system, plant can be stabilized.a robust performance in the case of parameter The advantage of the developed method is thatvariation, it is of interest to insert speci®ed gain the procedure to design a robust stabilizing con-margins and phase margins into the characteristic troller is systematic and straightforward. A trialequations, such that the resulted system will and error process is unnecessary. All exploitablemaintain at least the pre-speci®ed robust margins. controller parameters can be obtained from theThe proposed method in this paper not only pro- non-conservative Kharitonov region. Controllersvides a necessary and sucient condition for the selected from the region guarantee the uncertainHurwitz stability of an interval polynomial set but systems a pre-speci®ed safety margin. Moreover, Fig. 3. Stability boundary for G1 s with dierent value of KD .
422 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431the intrinsic property of system stability corre- ``interval plants . The controller C(s) issponding to the drift of the polynomial coecients designed to simultaneously stabilize the systemcan be analyzed manifestly in the demonstrated and track the command signal. The gain phaseparameter planes. margin tester AeÀj is applied in the forward path, where A and are subject to gain margin and phase margin speci®cations, respectively. Let2. Mathematical description F s 1 AeÀj C sG s P Consider the feedback control system as shownin Fig. 1. The general expression of the plant G s is and a0 a1 s a2 s2 Á Á Á am sm Fi s 1 AeÀj C sGi sY i 1Y Á Á Á Y 4Y QG s Y I b0 b1 s b2 s 2 Á Á Á bn s n where Gi s are the edge interval plants. If the where am T 0, bn T 0, ai P aÀ Y a , bj P bÀ Y b , i i j j controller C s can be designed with imposed spe-and n5m. A set of transfer functions can be gen- ci®cations on gain margins and phase margins,erated by changing the perturbed coecients in then it is of no doubt that the compensated systemthe prescribed ranges, which correspond to boxes will maintain robust performance to some degree.in the parameter plane. This is referred to as To show this, we present the following theorem. Fig. 4. A speci®c Kharitonov region (hatched area) in the KP ÀKI plane.
Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 423Theorem 1. Consider the control system as shown in p1 s dÀ dÀ s d s2 d s3 dÀ s4 Á Á Á Y 0 1 2 3 4 SFig. 1, with the de®nitions (1) and (2), the entirefamily G s is stabilizable by a controller C s if andonly if each interval plant Gi s is stabilizable by p2 s dÀ d s d s2 dÀ s3 dÀ s4 Á Á Á Y 0 1 2 3 4 Tthat same controller C s. p3 s d dÀ s dÀ s2 d s3 d s4 Á Á Á Y 0 1 2 3 4 UProof. Consider the characteristic polynomial ofthe closed-loop system shown in Fig. 1, i.e. thenumerator of F s, p4 s d d s dÀ s2 dÀ s3 d s4 Á Á Á Y V 0 1 2 3 4 np s di s i Y R i0 which are associated with Gi s in (3), respectively. It follows that the entire family is Hurwitz stablewhere di are the characteristic coecients, and if and only if the four vortex polynomials (4)±(7)dÀ 4di 4d . According to the Kharitonov theo- i i are all Hurwitz stable. Hence one can determinerem [1,4], every interval polynomial in the family all possible stabilizing controllers Ci s for eachis Hurwitz if and only if the following four Khar- vortex polynomial, and then take their intersection itonov polynomials are Hurwitz, C s C1 s C2 s C3 s C4 s to obtain the Fig. 5. Output responses for the PID controlled interval system.
424 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431non-conservative robust parametric controller for The problem of characterizing all stabilizing PIDthe uncertain plant G s. controllers for the entire family G s is to deter- mine KP , KI and KD such that all the closed loop In the following section, the controller sets, which characteristic polynomials are Hurwitz.stabilize the uncertain plant, are demonstrated to lie Consider the same uncertain system in ,within a speci®c Kharitonov region in the parameterplane. A strict Kharitonov region, which guarantees 5X2 s 2the system a robust margin, can be obtained. G s X IH s s3 b2 s2 b1 s b0 3. PID controller design for uncertain systems Assume that the coecients b0 Y b1 and b2 of the denominator lie within the following bounds, where For the uncertain plant (1), a PID controller 9X54b0 411X5Y 124b1 415, and 3X54b2 44X8. TheC s is developed with the exploitation of a gain desired speci®cations for the control system are sup-phase margin tester in the forward loop. Let the posed to be 5db4Gm410dbY and 30 4Pm460 . Bycontroller C s be inserting a gain phase margin tester AeÀj in the KD s2 KP s KI forward loop, it is found that the characteristicC s X W polynomial is s Fig. 6. A speci®cation-oriented parameter area for G1 s with pre-speci®ed robust margins in gain and phase.
Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 425p s s5 b2 s4 b1 s3 b0 s2 p3 s s5 3X5s4 15s3 11X5s2 Â Â 5X2KD s3 5X2KP 10X4KD s2 5X2KD s3 5X2KP 10X4KD s2 Ã Ã 5X2KI 10X4KP s 10X4KI A os À j sin X 5X2KI 10X4KP s 10X4KI A os À j sin Y II IR From (11), in four Kharitonov polynomials result p4 s s5 4X8s4 15s3 9X5s2 Âp1 s s5 4X8s4 12s3 9X5s2 5X2KD s3 5X2KP 10X4KD s2 Â Ã 5X2KD s3 5X2KP 10X4KD s2 5X2KI 10X4KP s 10X4KI A os À j sin X Ã IS 5X2KI 10X4KP s 10X4KI A os À j sin Y IP The aim here is to ®nd that all possible sets ofp2 s s5 3X5s4 12s3 11X5s2 KP , KI , and KD , which make the characteristic Â polynomials (12)±(15) to be Hurwitz stable. Take 5X2KD s3 5X2KP 10X4KD s2 Ã p1 s for instance, substituting s j3 and equating 5X2KI 10X4KP s 10X4KI A os À j sin Y the real part and imaginary part of (12) to zero, IQ respectively, one obtainsFig. 7. A non-conservative Kharitonov region in the parameter plane with pre-speci®ed gain margin and phase margin speci®cations.
426 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 C1 D2 À C2 D1 D2 sm q1 3Y KD Y PQKP Y IT B1 C2 À B2 C1 and D1 B2 À D2 B1KI Y IU B1 C2 À B2 C1 q1 3Y KD 4X834 À 5X2KD Asin33 À 11X5 10X4KD Aos32where Â Ã j 35 À 12 5X2KD Aos33 10X4KD Asin32 YB1 e q2 3Aos sm q2 3AsinY IV PRC1 e q3 3Aos sm q3 3AsinY IW q2 3 À5X232 j10X43Y PSD1 e q1 3Y KD Y PH q3 3 10X4 j5X23X PTB2 sm q2 3Aos À e q2 3AsinY PI Referring to (16) and (17), JÁB1 C2 À B2 C1 isC2 sm q3 3Aos À e q3 3AsinY PP the Jacobian. Let KD 0X01Y A 1, and 0, a Fig. 8. Bode diagrams for the PID controlled uncertain plants G1 s and G2 s.
Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 427stability boundary is plotted as shown in Fig. 2. pi s, i 2, 3, 4 are portrayed in Fig. 4, and JiThe stability characteristics of the considered (i 1, 2, 3, 4) are the corresponding Jacobian ofpolynomial are totally dierent to the left and each vortex polynomial. We can, therefore, obtainright of the boundary. Resorting to [10,11], it is a speci®c Kharitonov region in the plotted para-concluded that if J b 0, then to the left of the sta- meter plane by ®nding the overlapped region, thebility boundary facing the direction in which 3 hatched area, as shown in Fig. 4. The overlappedincreases is the stable area. Similarly, to the right area of the boundaries constitutes the entire fea-of the stability boundary facing the direction in sible controller sets that can stabilize each parti-which 3 increases is the stable region while J ` 0. cular selected edge polynomial. Within it, the KP ,Then a graphical stability region in the controller KI , and KD of the PID controller can be arbitrarilycoecient parameter space as seen in Fig. 2 can be selected. To show that the obtained controllerobtained. For various KD , a family of stability parameter area can eciently compensate theboundaries is portrayed in Fig. 3. It is realized that uncertain plant, a representative point P1 (KP 2,with a larger KD , the stability region for the con- KI 1, KD 0X01) is picked for demonstration.sidered plant would be larger. The resulting step responses are shown in Fig. 5. It For example, choose KD 0X01. Following the is obvious that the designed controller stabilizessame procedure, the other stability boundaries for all the four interval plants. Fig. 9. Bode diagrams for the PID controlled uncertain plants G3 s and G4 s.
428 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431Table 1 end, the constant gain margin boundaries andGain margins and phase margins for the family of uncertain constant phase margin boundaries are plotted inpolynomials with designed PID controller (KP 1X72, the KP ÀKI plane. It isolates speci®ed gain marginKI 0X05, KD 0X01) and phase margin regions in the parameter planeb2 b1 b0 Gain margin (db) Phase margin (deg) to allow ¯exible choice of the controller para-3.5 12 9.5 5.5293 42.1600 meters. For the edge polynomial p1 s, the hatched 11.5 5.1832 50.6060 area as shown in Fig. 6 is the suitable area for selecting the stabilizing controller coecients. 15 9.5 9.0530 44.9010 11.5 8.7540 54.0440 Concurrently, system speci®cations can be achieved.4.8 12 9.5 6.0765 30.5700 A non-conservative Kharitonov region is 11.5 6.0978 38.0700 obtained, as seen in Fig. 7, by searching for the 15 9.5 10.0000 37.9260 intersectional region of the speci®cation-limited 11.5 9.9410 51.7380 areas from all edge polynomials pi s, i 1, 2, 3, 4. This region constitutes all the possible stabilizing PID controller sets for system with robust safety margins. In other words, the KP , KI , and KD can Successively, we provide the uncertain system be readily chosen from the non-conservativewith a robust performance through specifying the Kharitonov region. A representative point P2gain margins and phase margins. Toward this (KP 1X72, KI 0X05, KD 0X01) is selected.Fig. 10. The variation of gain margins subject to parameteric change of the control system with the selected PID controller(KP 1X72, KI 0X05, KD 0X01).
Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 429Bode diagrams are plotted in Figs. 8 and 9. It is conservative Kharitonov region in the coecientveri®ed that the designed controller can really sta- plane.bilize the interval system with robust margins. Output responses are shown in Fig. 12. For aGain margins and phase margins for the con- standard second system, the speci®cations in termssidered uncertain plant with designed PID con- of gain margin and phase margin are de®nedtroller (KP 1X72, KI 0X05, KD 0X01) are exactly corresponding to time domain perfor-tabulated in Table. 1. The resulting gain margins mance requirement. However, for the consideredand phase margins subject to the uncertain coe- system, to inherit the same design idea, it is ofcients b0 , b1 , and b2 with the selected controller are much interest to de®ne interval gain margin andillustrated in Fig. 10 and Fig. 11, respectively. phase margin speci®cations such that the timeObviously, the speci®cations 5db4Gm410dbY and domain performance can also be guaranteed to30 4Pm460 are satis®ed. Note that for any some degree. From Fig. 12, it is seen that the riseother KD , the whole exploitable PID coecients time is less than 2 s and the maximum overshoot iscan be obtained, similarly, by searching the non- less than 50%. Henceforth, the linkage of the timeFig. 11. The variation of phase margins subject to parameteric change of the control system with the selected PID controller(KP 1X72, KI 0X05, KD 0X01).
430 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 Fig. 12. Output responses for the four interval plants with KP 1X72, KI 0X05, KD 0X01.performance and the gain and phase margin spe- The relationship between the varying coecientsci®cations can also be evinced. of the uncertain characteristic polynomial and its corresponding stable area in the plotted parameter plane is studied. Simulation results reveal the4. Conclusions remarkable performance and the eectiveness of the developed method. In future work, the devel- The so-called interval plants are often used for oped method will be extended to uncertain systemsdealing with uncertain dynamical systems. A with adjustable parameters and time delay, androbustness stability analysis is useful to provide systems with inherent nonlinearities.some ®nite checking of the stability of the closedloop system so that one can design the controller tostabilize the whole uncertain plant. However, muchof the present success in the control of intervalsystems is restricted to the analysis issue. In this Referencespaper, a parameter plane method based on thegain phase margin tester method and Kharitonov  V.L. Kharitonov, Asymptotic stability of an equilibriumtheorem is adopted to compensate the uncertain position of a family systems of linear dierential equa-plants. A non-conservative Kharitonov region can tions, Dierentialnye Uraveniya 14 (1978) 2086±2088.  V.L. Kharitonov, On a generalization of a stability criterion,be obtained systematically and graphically such Izv, Akad. Nauk. Kazach. SSR Ser. Fiz. Mat 1 (1978) 53±57.that a PID controller with coecients selected  A.C. Bartlett, Root locations for an entire polytope offrom the region will stabilize the entire interval polynomials: it suces to check the edges, MCSS 1 (1988)plants. The design procedure is straightforward. 61±71.
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