Autotuning of anisochronic controllers for delay systems

Autotuning of anisochronic controllers for delay systes ROMAN PROOP, LIBOR PEAŘ, RADE MATUŠŮ, JIŘÍ ORBEL Faculty of Applied Inforatics, Toas Bata University in Zlín Ná. T.G.Masaryka 5555, 76 Zlín, CZECH REPUBLIC prokop@fai.utb.cz Abstract: The paper brings a cobination of a biased-relay feedback experient and an algebraic control design ethod for tie-delay systes. The cobination results in a new principle of autotuning for a wide class of single input-output dynaic systes. The estiation of the controlled process is based on asyetrical liit cycle data. Then, a stable transfer function with a dead-tie ter is identified. The controller is coputed through solutions of Diophantine equations in the ring of stable and proper retarded quasipolynoial eroorphic functions (R MS ). Controller paraeters are tuned through a pole-placeent proble as a desired ultiple root of the characteristic closed loop equation. The controller design in this ring yields a Sith-like feedback controller with the realistic PID structure. The ethodology offers a scalar tuning paraeter > which can be adjusted by a suitable principle or further optiization. The first and second order tie-delay transfer functions can sufficiently estiate systes of quite high orders. The developed principles are illustrated by exaples in the Matlab + Siulink environent. ey-words: Autotuning, Relay experient, Algebraic control design, Pole-placeent proble. Introduction The developent of various autotuning principles was started by a siple syetrical relay feedback experient proposed by Åströ and Hägglund [] in the year 984. The ultiate gain and ultiate frequency are then used for adjusting of paraeters by original Ziegler-Nichols rules. During the period of ore than two decades, any studies have been reported to extend and iprove autotuners principles; see e.g. [], [3], [4], [8], [9]. The extension in relay utilization was perfored in [], [5], [7], [4] by an asyetry and hysteresis of a relay. Over tie, the direct estiation of transfer function paraeters instead of critical values began to appear. Experients with asyetrical and dead-zone relay feedback are reported in []. Also, various control design principles and rules can be investigated in entioned references. Nowadays, alost all coercial industrial PID controllers provide the feature of autotuning. Tie delay systes constitute an indispensable faily of industrial processes. A feedback loop is the ost efficient anner how to change syste properties. However, thanks to the feedback loop, tie delay notably affects whole syste dynaics. During recent decades various approaches and algoriths have been researched for copensating the influence of tie delay in a feedback loop. In addition to that, any control design principles to obtain satisfactory loop behavior have been presented. There surely exist several classifications of control design ethods for tie delay systes. Nowadays, three ain groups doinate. The first group contains approaches based on Sith predictor structure, or ore precisely its odifications [8], [4]. These ethods assue odel of the controlled syste in feedback loop, thus, it pertains into IMC (Internal Model Controllers). Second group consists of predictive based approaches, ainly using statespace description [5]. Last but not least, third group of algebraic approaches is assued [], [6] [8]. Extension to retarded quasipolynoials utilized in this paper are studied in e.g. [9], []. PID w e u y Process _ Relay Fig. : Block diagra of an autotuner This contribution brings a novel cobination of identification test, ade with help of biased relay with hysteresis, and algebraic controller design approach, based on solution of Diophantine equation in a special R MS ring. A transfer function of the first and second order with tie constant and tie delay is assued as an exaple for control applications giving a class of a PID like ISBN: 978--684-46-6

controllers with a Sith predictor structure. The pole placeent proble in R MS ring is forulated through a Diophantine equation and the pole is analytically tuned according to aperiodic response of the closed loop. A general basic schee of the autotuning principle can be seen in Fig.. Relay Feedback Tests An auto-tuning procedure consists of a process identification experient plus a controller design ethod. The traditional ethod was proposed by Åströ and Hägglund [], based on a syetrical relay feedback test when a relay of agnitude h r is inserted in the feedback loop. The result of the original test was the critical point of the open loop Nyquist curve, see e.g. [], [3], [4] and naturally the ultiate period and the liit cycle aplitude generated by process output. However, there are another relays used in identification experients. where a r and T u are depicted in Fig.3 and ε is the hysteresis. Siilarly, the second order odel plus dead tie (SOPDT) is assued in the for: G e ( ) Θs (4) The gain is given by (), the tie constant and tie delay ter can be estiated according to [] by the relation: Tu 4 hr T π π a r T u π T Θ π arctg arctg π Tu ε ar ε (5) w e u Process y - Fig. : Block diagra of an autotuning principle A biased (asyetrical) relay experient according to Fig. used for identification can give the final odel transfer functions with a tie delay ters. It is well known that any stable industrial processes can be adequately approxiated by the odel for first order (stable) systes plus dead tie (FOPDT). It is supposed in the for: G e Θs () and the process gain can be coputed by the relation [4]: ity ity y( t) dt ; i,,3,... u( t) dt () The tie constant and tie delay ters are given by []: Tu 6 u T π π ar T u π T Θ π arctg arctg π Tu ε ar ε (3) Fig. 3. Biased relay oscillation of stable processes. 3 Algebraic Control Design Rings and linear (Diophantine) equations have becoe coon tools in odern control theory before decades. There are several rings, the ring of polynoials R P, the ring of stable and proper rational function R PS etc., see e.g. [], [3], [7] which can be used for control syntheses. Different rings require various approxiations of delay ters which reduce quality of a odel. The ost known is the Pade approxiation, respecting the relative degree of the original transfer function. As a negative consequence, the final controllers have usually higher degrees. This paper utilizes a ring of stable and proper eroorphic functions R MS oitting any approxiation which was developed especially for delay systes by Zítek and učera in [3]. An eleent of this ring is a ratio of two retarded quasipolynoials y(s)/x(s). ISBN: 978--684-46-6

A retarded quasipolynoial x(s) of degree n eans n h n i ( ϑ ) x s s + xij s exp ijs, ϑij (6) i j where retarded refers to the fact that the highest s- power is not affected by exponentials. A ore general notion called neutral quasipolynoials also can be used in this sense, see [8]. A quasipolynoial in the for of (6) is stable when it owns no finite zero s such that Re{s }. For stability tests, see e.g. in [3], [4]. The denoinator of the ratio in R MS is supposed to be stable, while the nuerator y(s) of an eleent in R MS can be factorized in the for y y% exp( Θs), where the ter Θ and y% is any retarded quasipolynoial. The ratio y(s)/x(s) is called proper when the degree of the nuerator is less or equal to the degree of the denoinator. A linear tie-invariant delay syste can be expressed as a ratio of two eleents of the R MS ring. The first order syste with input-output tie delay can be expressed by G ( Θ ) ( ϑ ) exp s b s s+ exp s s B s a s A s s+ exp( ϑ s) s A s, B s R, > MS (7) The traditional feedback loop for the control design is displayed in Fig. 4. Generally, let a odel transfer function be expressed as B s G, A, B RMS (8) A s and a controller be given by a ratio Q s GR, Q, P RMS (9) P s Siilarly, reference and load disturbance signals can be expressed by HW W, HW, FW RMS F () W H D D, H D, FD RMS F () D The ai of the control synthesis is to (internally) stabilize the feedback control syste with asyptotic tracking and load disturbance attenuation. Fig.4: Feedback (DOF) control loop The first step of the stabilization can be forulated in an elegant way in R MS by the Diophantine equation A s P s + B s Q s () where P (s) a Q (s) is a particular solution fro R MS. Since for stable systes, the R MS ring constitutes the Bézout doain (see [3]), the solution of (7) always exists. All stabilizing controllers can be expressed in a paraetric for by + B Z Q s Q s A s Z s P s P s B s Z s P s (3) where Z(s) is an arbitrary eleent of R MS. The special choice of this eleent can ensure additional control conditions. Details and proofs can be found e.g. in [], [3], [7], [9]. Asyptotic tracking and disturbance attenuation result fro expression for E(s) which reads A P A P + B Q B P D A P + B Q E s W s (4) and they lead to the condition that both F w (s) and F D (s) divide P(s). Details about divisibility in R MS can be found, e.g. in [3], [3]. 4 Basic anisochronic autotuners 4. First order odel Let a controlled process be described by a first order delayed odel () where paraeters, T and Θ are estiated via relay identification test () - (3). ISBN: 978--684-46-6

The odel coprie factorization in the R MS ring can be then expressed by ( Θs) exp G% s+ B A (5) s+ where > is a free (selectable) scalar paraeter. The control loop is considered as a siple feedback syste (Fig. ) with plant and controller transfer functions (8), (9), respectively. Both external inputs (), () are supposed as step functions. The stabilizing Diophantine equation () reads ( Θs) exp P + Q s+ s+ Choose Q which yields (6) according to (8). Thus, the final anisochronic controller structure reads ( exp) GR s GR s s+ Θs () where serves as a tuning paraeter. The denoinator in () has infinite nuber of poles. The construction of this controller is ore coplex than usual PI or PID controllers. 4. Second order odel Now let a plant odel be of the for (4) which can be forulated in R MS as a ratio exp( Θs) ( + ) ( ) ( s+ ) s B s G% () A s P s+ exp Θs (7) Siilarly as in (6) and (7) for a first order odel, a stabilizing (non unique) particular solution of () can obtained as Obviously, this solution does not satisfy the requireents of asyptotical reference tracking and disturbance attenuation, since P ( ), thus, the particular solution ought to be paraeterized as s+ exp( Θs) P s P s B s Z s exp( Θs) Z s s+ (8) In order to have P( s ) in a siple for satisfying P, choose s+ Z s (9) which gives the controller denoinator and nuerator by ( exp) s+ Θs P Q (), Q s P s ( s+ ) exp( Θs) ( ) (3) and the paraeterization (3) enables to satisfy the reference tracking and disturbance attenuation; hence the option results in P s Q s Z s ( Ts ) s+ + ( ) ( ) s + s+ exp Θs The controller structure is then G R ( ) ( ) s + s+ exp Θs (4) (5) (6) Fig. 5 deonstrates the Siulink schee of the anisochronic structure of proposed controller (6). ISBN: 978--684-46-6 3

where paraeter.85 was tuned by the equalization principle. Fig. 5: Matlab-Siulink schee of controller (6) 5 Exaples and Siulations As an exaple, a stable syste with tie delay governed by the transfer function of the third order was chosen G exp( s) ( s+ )( 5s+ )( 3s+ ) (7) The estiation was perfored by the relay feedback experient where asyetric relay with hysteresis was used with adjusted paraeters: h r.5 (. when on, -.5 when off), ε.5. Liit cycles result in a r., T u 56.8. The first order odel () was obtained by the experient approxiation using () - (3) is ( s) exp 6.9 G % 3.45s+ (8) Siilarly, relations () and (5) yield the second order odel (3) Fig. 6: Step responses of (7)-(9) The second order odel for the tuning paraeter 7.37 takes the transfer function GR ( s ) ( ) 4 5.43.3 + + + 3 s 4.74 s 5.43 exp.4s (3) The tuning paraeter can be adjusted by various principles, one of the is the equalization principle proposed in [4]. In both cases (3), (3), coparing the appropriate paraeters defined the final value of tuning paraeters. Control responses for both odels and controllers are copared in Fig. 7 (control variable) and in Fig. 8 (controlled variable). Reference signal w(t) is changed fro to in tie t and the step load disturbance d(t) -. is injected at t 4 s. G % s exp.4s (9) (.3s+ ) The step responses of (7) - (9) are pictured and copared in Fig. 6. Estiated odels (8), (9) were used for the algebraic controller design in the sense of transfer functions (5), (). The ethodology entioned in 4., 4. results in the first and second order anisochronic controllers, respectively. The final first order controller has the for G R.85 3 3.45s+ s+ ( ).85 exp 6.9 (3) Fig. 7: Control variable first and second order synthesis. Control responses are rather slow; however, without abrupt changes of control signals (except ISBN: 978--684-46-6 4

instants of step changes of the reference signal). This result agrees with the philosophy of the equalization ethod which suggests a coproise between a suitable control response and carefulness to actuators. Generally, higher gives faster but ore oscillating control responses, and vice-versa. Naturally, second order approxiation as well as control responses exhibit better and ore acceptable behavior. Fig. 8: Controlled variable first and second order synthesis. 6 Conclusions The contribution brings a novel principle of autotuners with controller design based on a special ring of eroorphic functions. A transfer function with tie delay is estiated fro asyetric liit cycle data by a biased relay with hysteresis. The control synthesis is then perfored through a solution of the Diophantine equation in the ring of proper and stable RQ-eroorphic functions. For first and second order odels the ethodology generates a class of generalized PI or PID controllers in the sense of the Sith predictor. The design ethod brings a scalar tuning paraeter > that can be adjusted by various strategies. The illustrative exaple shows the application of the proposed (first and second order) to control of a higher order syste with tie delay. 7 Acknowledgeents This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic under the Research Plan No. MSM 78835 and by the Europian Regional Developent Fund under the project CEBIA-Tech No. CZ..5/../ 3.89. References: [].J. Åströ and T. Hägglund, Autoatic tuning of siple regulators with specification on phase and aplitude argins. Autoatica, Vol., 984, pp.645-65. [] Ch.Ch. Yu, Autotuning of PID Controllers. Springer, London, 999. [3] A. O Dwyer, Handbook of PI and PID controller tuning rules. London:Iperial College Press, 3. [4].J. Åströ and T. Hägglund, PID Controllers: Theory, Design and Tuning. Research Triangle Park, NC: Instruental Society of Aerica, 995. [5] R.F. Garcia and F.J.P. Castelo, A copleent to autotuning ethods on PID controllers, Iin: Preprints of IFAC Workshop PID, pp. - 4,. [6] R.R. Pecharroán and F.L. Pagola, Control design for PID controllers auto-tuning based on iproved identification, In: Preprints of IFAC Workshop PID, pp. 89-94,. [7] C.C.Hang,.J. Åströ and Q.C. Wang, Relay feedback auto-tuning of process controllers a tutorial review, Journal of Process Control, Vol., No6,. [8] T. Thyagarajan and Ch.Ch. Yu, Iproved autotuning using shape factor fro relay feedback, In: Preprints of IFAC World Congres,. [9] S. Majhi and D.P. Atherton, Autotuning and controller design for unstable tie delay processes, In: Preprints of UACC Conf an Control, 998, pp. 769-774. [] F. Morilla, A. Gonzáles and N. Duro, Autotuning PID controllers in ters of relative daping, In: Preprints of IFAC Workshop PID,, pp. 6-66. [] M. Vítečková and A. Víteček Experientální identifikace etodou relé (in Czech), In: Autoatizácia a inforatizáci, 4. [] M. Vidyasagar, Control syste synthesis: a factorization approach. MIT Press, Cabridge, M.A. (987). [3] V. učera, Diophantine equations in control - A survey, Autoatica, Vol. 9, No. 6, 993, pp. 36-75. [4] R. Gorez and P. lán, Nonodel-based explicit design relations for PID controllers, In: Preprints of IFAC Workshop PID,, pp. 4-46. [5] I. aya and D.P. Atherton, Paraeter estiation fro relay autotuning with asyetric liit cycle data, Journal of Process Control, Vol., No4,, pp. 49-439. [6] M. Fliess, R. Marques, and H. Mounier, An extension of predictive control, PID regulators ISBN: 978--684-46-6 5

and Sith predictors to soe linear delay systes. International Journal of Control, Vol. 75, No., pp. 78 743,. [7] R. Prokop and J.P. Corriou, Design and analysis of siple robust controllers, Int. J. Control, Vol. 66, 997, pp. 95-9. [8] L. Pekař and R. Prokop, Soe observations about the RMS ring for delayed systes, In: Preprints of the 7 th Int. Conf. on Process Control 9, Slovakia, 9). 8-36. [9] R. Prokop, orbel, J., Prokopová, Z., Relay based autotuning with algebraic control design, In: Preprints of the 3rd European Conf. on odelling and Siulation, Madrid, 9, s. 53-536. [] R. Matušů and R. Prokop, Robust Stabilization of Interval Plants using ronecker Suation Method. In: Last Trends on Systes, 4th WSEAS International Conference on Systes, Corfu Island, Greece,, pp. 6-65. [] L. Pekař and R. Prokop, Non-delay depending stability of a tie-delay syste. In: Last Trends on Systes, 4th WSEAS International Conference on Systes, Corfu Island, Greece,, pp. 7-75. [] L. Pekař and R. Prokop, Control of Delayed Integrating Processes Using Two Feedback Controllers: RMS Approach, In: Proceedings of the 7th WSEAS International Conference on Syste Science and Siulation in Engineering, Venice, 8, pp. 35-4. [3] P. Zítek and V. učera, Algebraic design of anisochronic controllers for tie delay systes, International Journal of Control, Vol. 76, No. 6, pp. 95-9, 3. [4] T.Vyhlídal, Anisochronic first order odel and its application to internal odel control, In: Preprints of ASR Seinar. Prague,, pp. -3. ISBN: 978--684-46-6 6