Unstable Systems - Tracking and Disturbance Attenuation

Transcription

1 Roan rokop, Natalia Volkova, Zdenka rokopova Unstale Systes - Tracking and Disturance Attenuation ROMAN ROKO, NATALIA VOLKOVA, ZDENKA ROKOOVÁ Toas Bata University in Zlin ná. TGM 5555, 76 1 Zlin, Czech Repulic e-ail: prokop@fai.ut.cz Astract: The paper is aied to the design of linear continuous controllers for unstale single input output systes. The controller design is studied in the ring of (Hurwitz) stale and proper rational functions R S. All stailizing feedack controllers are given y a general solution of a Diophantine equation in R S. Then asyptotic tracking and disturance attenuation is otained through the divisiility conditions in this ring. The attention of the paper is focused on a class of unstale systes. Both, one and two degree of freedo (1DOF, DOF) control s are considered. Control loops with a feedforward part significantly reduce overshoots in control responses. The ethodology rings a scalar paraeter for tuning and influencing of controller paraeters. As a result, a class of I, ID controllers are developed ut the approach generates also coplex controllers. Siulations and verification are perfored in the Matla+Siulink environent. Keywords: Unstale systes, Diophantine equation, asyptotic tracking, Disturance attenuation. 1 Introduction The dynaics of any technological plants exhiit unstale ehavior. roaly, the reason can e seen in nonlinearity of any industrial processes and plants. Such nonlinear systes exhiit ultiple steady states and soe of the ay e unstale. The situation where linear systes have unstale poles ay occur e.g. in a continuous-tie stirred exotheric tank reactor, in distillation coluns, in polyerization processes or in a class of iocheical processes where the processes ust operate at an unstale steady state. Moreover, a tie delay can e also an inherent part of any technological plants. The ost frequent tool for feedack industrial control has een still ID controller. It is elieved that ore than 9 % feedack loops are equipped with this controller. Also, a great aount for ID assessing and tuning rules has een developed. The traditional engineering design approach of ID like controllers was perfored either in the frequency doain or in polynoial representation (see e.g. [1], [], [3]). Most of the are scheduled for stale systes without or with tie delay, see e.g. [1], [].The unstale cases are studied e.g. in [4], [5]. In this contriution, a general technique for a class of unstale systes is proposed. The control design is perfored in the ring of proper and Hurwitz stale rational functions R S. All stailizing controllers are given y all solutions of Diophantine equation in this ring and asyptotic tracking and disturance attenuation is then forulated y additional conditions of divisiility. This fractional approach proposed in [6], [1], [17] enales a deeper insight into control tuning and a ore elegant derivation of all suitale controllers. The situation and details for stale and tie-delay free systes can e found in [1] - [16] for various control proles. This technique introduces a scalar paraeter > which influences a control responses and also roust ehaviour. The R S ring also enales to utilize the H nor as a tool for perturation evaluation. Descriptions over Rings Linear continuous-tie dynaic systes have een traditionally descried y the Laplace transfor. So polynoials ecae a asic tool for the staility analysis and controller design. Since the characteristic feedack polynoial has two known (plant) and two unknown (controller) polynoials, the Diophantine equations egan to penetrate into synthesis ethod, see e.g. [1]. However, the ring of polynoials induces soe drawacks with solutions of Diophantine equations. Alost all fro the infinite nuer of solutions cannot e used for controller transfer functions ecause they are not proper, see e.g. [11], [15]. These proles were overcoe y introducing of the different ring of proper and stale rational functions. The pioneering work in the so called fractional approach is the work [18], further extension can e found in [11], [15]. Siply speaking, a ratio of polynoials is replaced y a ratio of two Hurwitz stale and proper rational functions. In this paper, the following ring R S () is utilized. Transfer functions of linear single input single output dynaic systes have een traditionally expressed as a ratio of two polynoials. The set of polynoials (for continuous-tie systes in the ISSN: Issue 3, Volue 6, March 11

2 Roan rokop, Natalia Volkova, Zdenka rokopova Laplace transfor with the indeterinate s) is a ring [11], [15], [18]. However, there are various rings for studying specific features [11], [18]. The ring R S () denotes the set of rational functions having no poles in the plane Re(s) -. It eans that this set includes all Hurwitz stale and proper rational functions in this region. The transcription of a transfer function is very siple, nuerator and denoinator is divided y the sae stale polynoial of the appropriate degree. Generally, polynoial transfer functions in the ring R S () take the for: ( s) n ( s) ( ) G( s) ; a( s) a( s) (1) n ( ) n ax(deg a, deg ) where >. Also, signals in control systes can e expressed siilarly. The stepwise reference signal w and haronic disturance v are in the rational description given y ratios: 1 1 Gw w s F s w v G F v v 1 ( ) ω ( ) () (3) The load disturance n is supposed also in the for of (), (3). The divisiility of eleents in R ps is defined through the all unstale zeros (including infinity) of the rational functions, see [18] for details. is asyptotic stale and the tracking error e( t) w( t) y( t) tends to zero. Moreover, a stepwise disturance n(t) has to e eliinated without a non-zero steady-state error (disturance attenuation). Naturally, any other syntheses can e found in literature. A handook for general utilization of I, ID controllers is []. Integrating tie delay systes are studied in [19], [1], [3]. Roust control is referred in [] while general stailization is analyzed in [4]. Experiental verification of I/ID controllers is analyzed in [5] and control with tie-varying delay is studied in [7]. 3 Control and Disturance Rejection Design in R S Suppose a general closed loop control syste depicted in Fig.1. The controller C(s) generates the control variale u according the equation: u R w Q y+ n (4) where n is a load disturance. Note that a traditional one degree-of-freedo (1DOF) feedack controller operating on the tracking error is otained for Q R. This is shown in Fig. while the DOF (without load disturance) is depicted in Fig.3. Basic relations following fro the DOF configuration are B R Q y u+ v u w y+ n (5) A and w, v, n are independent external inputs into the closed loop syste. Further, the following equations hold: BR G A G B G y + + A+ BQ F A+ BQ F A+ BQ F w v n w v n (6) The 1DOF (FB) is otained for RQ (depicted in Fig.) and the last relation gives the controlled error e w-y: Fig.1. General closed loop syste The asic control prole is then forulated as follows within the context of Fig.1: Consider the known transfer function (1), the reference and disturance (), (3). The task is to design a proper transfer function C(s) so that the closed loop syste A G A G B G e + + A+ BQ F A+ BQ F A+ BQ F w v n w v n (7) ISSN: Issue 3, Volue 6, March 11

3 Roan rokop, Natalia Volkova, Zdenka rokopova Fig.. Structure 1DOF (FB) of the close loop syste The first step of the control design is to stailize the syste y a proper feedack loop. It can e forulated in an elegant way in R S y the Diophantine equation: A+ BQ 1 (8) with a general solution for SISO systes +BT, QQ - AT; where T is free in R S and, Q is a pair of particular solutions (Youla Kučera paraeterization of all stailizing controllers). Details and proofs can e found e.g.[1], [15], [16], [18]. Then control error for the DOF : Gw Gv Gn e ( 1 BR) + A + B (9) Fw Fv Fn Now, it is necessary to solve oth s 1DOF and DOF separately. For asyptotic tracking and the DOF (FBFW), the second Diophantine equation gets the for: Fw Z+ BR 1 (1) where Z R S is not used in the control law. Fig.3. Structure DOF (FBFW) of the close loop syste The tracking error e tends to zero if a) F w divides A for 1DOF (11) ) F w divides 1-BR for DOF (1) Another control prole of practical iportance is disturance rejection and disturance attenuation. In oth cases, the effect of disturances v and n should e asyptotically eliinated fro the plant output. Since the oth disturances are external inputs into the feedack part of the syste, the effect ust e processed y a feedack controller. It eans that the second and third parts in (11) and (1) are A G A+ BQ F B G A+ BQ F v v n n (13) (14) They ust elong to R S (s), i.e. all A+BQ, F v, F n should cancel. In other words, a ultiple F v, F n ust divide. More precisely F v, ust divide the ultiple A and F n the ultiple B. When define relatively prie eleents A, F v and B, F n in R S (s) A A B B, (15) Fv Fv Fn Fn Then the prole of disturance rejection and attenuation is solvale if and only if the pairs F v, B and F n, B are relatively prie and the feedack controller is given y Q Q C F 1 vf (16) n where 1, Q is any solution of the equation AF F + BQ (17) v n Siple controllers for unstale systes The fractional approach perfored in the ring R ps enales a control design in a very elegant way. roaly, the siplest unstale syste is an integrator with the transfer function: G( s) s The asic stailizing equation (8) takes the for s p q, (18) + 1 and all solutions can e expressed y s 1 + T; Q T (19) () ISSN: Issue 3, Volue 6, March 11

4 Roan rokop, Natalia Volkova, Zdenka rokopova with T free in R ps. For the integrator, the condition of divisiility etween stepwise F w and A is generically fulfilled ecause they are the sae rational function (as well as polynoial) in the for (). Then the siplest 1DOF controller is proportional with the gain /. The influence of tuning paraeter is shown in Fig.4 where responses for three various paraeters are depicted ( 1). Naturally, this controller is not ale to copensate any load disturance. The siulation in Fig.4 represents the reference (stepwise) change in tie t15 and the load disturance (also stepwise) is injected in t. In this case, the DOF does not ring any iproveent. The steady-state error is inversely proportional to the gain of the controller, so the tuning paraeter >.The steady-state error decreases with increasing paraeter >. The feedforward part of the DOF for integrator (18) is given y (1) with the general solution for R: s R + T () s + which for t gives R( s) s +. The final control law (4) for DOF takes the relation t u( t) ( w y( τ )) dτ y( t) + (3) The control law (3) is a generalized I controller propose y Aströ in [1], [3] as a tool against overshooting. Fig. 6 shows the control responses for the sae values of >. Vanishing of overshoots confir this fact. Fig.4. Siple integrator with controller with 1DOF control Now, it is necessary to find such a free paraeter T Q in () so that controller C ensures asyptotic tracking for a stepwise load disturance (). So, the condition (11) is achieved for t and the 1DOF controller takes the for of I one: Fig.5. Siple integrator with I controller (1DOF ) Q s (1) It is clear that tuning paraeter is incorporated into controller paraeters in a nonlinear way. The influence for control ehaviour is then deonstrated in Fig. 5 (also for 1). Typical I (1DOF) response exhiit a sall overshoots. Fig.6. Siple Integrator with I controller (DOF control ) A it ore coplex situation occurs for disturance rejection with haronic signal (3). Then the paraeterization () leads to the expression: ISSN: Issue 3, Volue 6, March 11

5 Roan rokop, Natalia Volkova, Zdenka rokopova t t s +ω ( ) 1 1+ (4) R r ( ) s + ω, (9) It is necessary to find paraeters t, t 1 satisfying the identity in (). Equating of coefficients in (), the following linear equations for t, t 1 are: + t1 (5) + t ω with the solution t1 ; t ω. Q The resulting feedack controller C( s) has no ore of the I or ID ut it takes the for: Q qs + q1 q s + ω (6) where q t ; q 3 t ; q (7) 1 1 where r. The control responses are shown in Fig.8. Here (also in Fig.7), the depicted haronic signal represents an injected disturance v in the sense of Fig. 1, Fig.. The influence of the feedforward part is siilar as in I reference tracking. The feedforward part reduces overshoots while paraeter > influences the dynaics (speed) of control responses. Two rearkale facts can e seen in Fig.5 - Fig.8. The first one is that increasing value of the tuning paraeter lessens overshoot of the control response. The second one is that the divisiility condition enales to copensate the stepwise load disturance which is injected in the tie t. The control responses for three different values of > are depicted in Fig.7. Fig.8. Integrator with haronic disturance copensation with DOF control Fig.7. Siple integrator with haronic disturance copensation with 1DOF control The feedforward part of the DOF control law is derived in siilar way like in (). The second Diophantine equation has the for s Z+ R 1 (8) with the sae solution (). The resulting feedforward transfer function has the for A second set of controllers for unstale systes can e derived for syste governed y the transfer function: G( s) s a (3) with a >. The stailization feedack equation (8) takes the for s a p + q 1 (31) with all paraeterization solutions + a s a 1 + ; T Q T (3) ISSN: Issue 3, Volue 6, March 11

6 Roan rokop, Natalia Volkova, Zdenka rokopova In this case (for the stepwise reference) the divisiility condition F w \ A is not generically p fulfilled and it is achieved for T t. The final feedack part is again in the for of I controllers: Q q1 q (33) s where + q1 a ; q (34) Siulations for three values (.6, 1.,.5) for the particular case, a.5 are shown in Fig.9. contrast to (3) plant paraeter a is iersed into control law: t + a u( t) ( w y( τ )) dτ y( t) + (35) The control responses for three value of (1., 1.5,.) are depicted in Fig.1. Generally, the DOF always reduces overshoots after step changes of input signals (reference, load disturance). Also the haronic disturance attenuation can e overcoe siilarly as in the case of a siple integrator. In the general solution of stailizing controller in (3) the following divisiility condition has to e fulfilled: s s s ( s ) ( +ω ) 1+ T is divisile y (36) Fig.9. Unstale syste (3) with 1DOF control Condition (36) can e achieved y the choice ts + t1 t T ( ) After soe algeraic anipulations a it ore coplex than in (5) (7) the final feedack transfer function is in the for Q q s + q s + q q s( s + ω ) (37) (38) The explicit expressions for controller paraeters are quite coplex, see [16] for details. The control responses for the 1DOF are shown in Fig. 11. The feedforward part of the DOF follows fro the second Diophantine equation (1) which has the identical general solution (). The final Fig.1. Unstale syste (3) with DOF control Another question is a total rejection of overshoot. It can e achieved y utilizing of control DOF and equation (1) which has the sae for and general solution (). The final control law has also the for of a generalized I controller. In R r ( ) s s ω 3 ( + ), (39) where r. The control responses for the sae values and conditions are shown in Fig. 1. The last two figures again confir the fact that the DOF reduces overshoots also in the presence of haronic disturances. The load disturance attenuation and disturance can e achieved siultaneously as it is shown in Fig. 15 Fig. for the next second order syste. ISSN: Issue 3, Volue 6, March 11

7 Roan rokop, Natalia Volkova, Zdenka rokopova where Q q1 q p p 1 p 1; p 3 + a ; 1 3 q + a (3 + a ); q 3 1 (4) (43) Siulations for the case 1 and a.5 and three paraeters are shown in Fig.13 with the 1DOF. Fig.11. Unstale syste (3) with haronic disturance copensation with 1DOF control Fig.13. Unstale syste (4) with 1DOF control Fig.1. Unstale syste (3) with haronic disturance copensation with DOF control The third class of controllers is derived for a frequent case of unstale systes with the integrator in the for G( s) (4) s( s a ) The divisiility condition for a step-wise reference s with Fw is fulfilled, so the stailizing equation (8) also ensures asyptotic tracking. This equation in this case takes the for s( s a) p1 p q1 q + 1 (41) ( ) ( ) ( ) ( ) It is easy to express paraeters p i, q i and the particular controller has the transfer function Fig.14. Unstale syste (4) with DOF control The DOF control synthesis is derived in a very siilar way like in (8), (9). The siulation responses for (4) are depicted in Fig. 14. Naturally, it is possile to derive a controller for haronic disturance rejection. ISSN: Issue 3, Volue 6, March 11

8 Roan rokop, Natalia Volkova, Zdenka rokopova Fig.15. Unstale syste (4) with haronic disturance copensation (frequency1, aplitude1) with 1DOF control Fig.18. Unstale syste (4) with haronic disturance copensation (frequency3, aplitude1) with DOF control Fig.16. Unstale syste (4) with haronic disturance copensation (frequency1, aplitude1) with DOF control Fig.19. Unstale syste (4) with haronic disturance copensation (frequency, aplitude1) with 1DOF control Fig.17. Unstale syste (4) with haronic disturance copensation (frequency3, aplitude1) with 1DOF control Fig.. Unstale syste (4) with haronic disturance copensation (frequency, aplitude1) with DOF control ISSN: Issue 3, Volue 6, March 11

9 Roan rokop, Natalia Volkova, Zdenka rokopova The synthesis is ased on equation (17) for the 1DOF and on equation (1) for the DOF. The final explicit expressions are not siple and can e found in [16], [19]. The siulation responses are shown in Fig. 15 Fig.. All siulations represent siultaneous disturance rejection and load disturance attenuation for various frequency of the haronic disturance. Siulations confir the fact that the DOF lessens the overshoots after step changes of the reference. However, the increasing frequency of the haronic disturance iplicate the increasing overshoots (undershoots) after the load disturance injection. It is ovious fro Fig. 15, 17, 19 for the 1DOF as well as Fig. 16, 18, for the DOF. 5 Conclusions The task of siultaneous regulation and disturance attenuation for a class of unstale systes is considered. A controller design ethodology is ased on the fractional representation in the ring of proper and stale rational functions. Resulting control laws in 1 DOF give a class of I or ID controllers. It is iportant fro application point of view. The a it ore coplex DOF gives ore sophisticated controllers which have no ore the ID ut the enefit is in control responses. The proposed ethodology rings a scalar paraeter > which enales to tune and influence the roustness and control ehaviour. The tuning paraeter can e chosen aritrarily or it can e a result of soe optiization or calculation. Also proles of disturance attenuation are analysed. The ethodology is applied for three ost frequent cases of unstale systes of the first and second order. All siulations were siulated and verified in the Matla + Siulink environent. Acknowledgents This work was supported y the Ministry of Education, Youth and Sports of the Czech Repulic under the Research lan No. MSM and y the Europian Regional Developent Fund under the project CEBIA-Tech No. CZ.11.5./.1./ References: [1] K.J. Aströ, C.C. Hang,. ersson, and W.K. Ho, Towards intelligent ID control. Autoatica, Vol. 8, No.1, 199, pp [] K.J. Aströ, C.C. Hang and B.C. Li, A new Sith predictor for controlling a process with an integrator and long dead-tie. IEEE Trans. on AC, Vol. 39, No., 1994, pp [3] K.J Aströ and T. Hägglund, ID Controllers: Theory, Design and Tuning. Instruent Society of Aerica, USA, [4] A.M. De aor and M.O Malley, Controllers of Ziegler Nichols type for unstale processes. Int. J. Control, Vol. 49, 1989, pp [5] V. Venkatashankar and M. Chidaara, Design of and I controllers for unstale firsorder plus tie delay systes, Int. J. Control, Vol. 6, No. 1, 1994, pp [6] C.D. Doyle, B.A. Francis and A.R. Tannenau, Feedack Control Theory. Macillan, New York,199. [7] A. Filasová and D. Krokavec, Decentralized roust control design using LMI. In: Acta Montanica Slovaca, Vol.13, No. 1, 8, pp [8] M.J. Grile and V. Kučera, olynoial Methods for Control Systes Design. Springer, Berlin, [9] S.H. Hwang, S.J. Shiu and M.L Lin, ID control of unstale systes having tie delay. In: repr. IFAC ECC Bruxelles, No.559, 1997, pp [1] T. Kailath, Linear syste, rentice Hall, Englewood Cliffs, 198. [11] V. Kučera, Diophantine equations in control - A survey, Autoatica, Vol. 9, No.6, 1993, pp [1] R. ada Sree and M.Chidaara, Control of Unstale Systes. Alpha Science Int., Oxford, 6. [13] R. Matušů and R. rokop, Roust Stailization of Interval lants using Kronecker Suation Method. In: Last Trends on Systes, 14th WSEAS International Conference on Systes, Corfu Island, Greece, 1, pp [14] L. ekař and R. rokop,non-delay depending staility of a tie-delay syste. In: Last Trends on Systes, 14th WSEAS International Conference on Systes, Corfu Island, Greece, 1, pp [15] R. rokop, L. ekař and J. Korel, Autotuning for delay systes using ero,orphic functions. In: roceedings of the 9th IFAC Wokshop on Tie Delay Systes, rague, Czech Rep., 1, pp. [CD-ROM]. [16] R. rokop and J.. Corriou Design and analysis of siple roust controllers, Int. J. Control, Vol. 66, No. 6, 1997, pp ISSN: Issue 3, Volue 6, March 11

10 Roan rokop, Natalia Volkova, Zdenka rokopova [17] R. Matušů, K. Vaneková, R. rokop and M. Bakošová, Design of Roust I Controllers and their Application to a Nonlinear Electronic Syste,, Journal of Electrical Engineering, Bratislava, 1, pp [18] M. Vidyasagar, Control syste synthesis: A factorization approach. MIT ress, Caridge, M.A., [19] L. ekař and R. rokop, Control of Delayed Integrating rocesses Using Two Feedack Controllers: RMS Approach, In: roceedings of the 7th WSEAS International Conference on Syste Science and Siulation in Engineering, Venice, 8, pp [] D. Krokavec and A. Filasová, ole assignent in roust state oserver design. In: AT& Journal. Vol. 1, č. (7), s [1] T.Vyhlídal, Anisochronic first order odel and its application to internal odel control, In: reprints of ASR Seinar. rague,, pp [] O Dwyer, Handook of I and ID controller tuning rules. London: Iperial College ress, 3. [3] L. ekař, & R. rokop: Algeraic Control of integrating rocesses with Dead Tie y Two Feedack Controllers in the Ring RMS. Int. J. of Circuits, Systes and Signal rocessing, Volue, Issue 4, 8, pp ISSN: [4] L. ekař & R. rokop. Stailization of a delayed syste y a proportional controller. International Journal of Matheatical Models and Methods in Applied Sciences, Volue 4, Issue 4, 1, pp ISSN: [5] R. Matušů & R. rokop. Experiental verification of design ethods for conventional I/ID controllers. WSEAS Trans. on Systes and Control, Vol. 5, Issue 5, 1, pp [6] R. Matušů & R. rokop. Control of systes with tie-varying delay: A coparison study. 1th WSEAS International Conference on Autoatic Control, Modelling and Siulation, ACMOS '1, 1, pp ISSN: Issue 3, Volue 6, March 11